Abstract
This work focuses on the numerical modeling of fracture and its propagation in heterogeneous materials by means of hierarchical multiscale models based on the FEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^2}
method, addressing at the same time, the problem of the excessive computational cost through the development, implementation and validation of a set of computational tools based on reduced order modeling techniques.
For fracture problems, a novel multiscale model for propagating fracture has been developed, implemented and validated. This multiscale model is characterized by the following features:
- At the macroscale level, were adapted the last advances of the Continuum Strong Discontinuity Approach (CSDA), developed for monoscale models, devising a new finite element exhibiting good ability to capture and model strain localization in bands which can be intersect the finite element in random directions; for failure propagation purposes, the adapted Crack-path field technique oliver2014crack, was used.
- At the microscale level, for the sake of simplicity, and thinking on the development of the reduced order model, the use of cohesive-band elements, endowed with a regularized isotropic continuum damage model aiming at representing the material decohesion, is proposed. These cohesive-band elements are distributed within the microscale components, and their boundaries.
The objectivity of the solution with respect to the failure cell size at the microscale, and the finite element size at the macroscale, was checked. In the same way, its consistency with respect to Direct Numerical Simulations (DNS), was also tested and verified.
For model order reduction purposes, the microscale Boundary Value Problem (VBP), is rephrased using Model Order Reduction techniques. The use of two subsequent reduction techniques, known as: Reduced Order Model (ROM) and HyPer Reduced Order Model (HPROM or HROM), respectively, is proposed.
First, the standard microscale finite element model High Fidelity (HF), is projected and solved in a low-dimensional space via Proper Orthogonal Decomposition (POD). Second, two techniques have been developed and studied for multiscale models, namely: a) interpolation methods, and b) Reduced Order Cubature (ROQ) methods An_2009. The reduced bases for the projection of the primal variables, are computed by means of a judiciously training, defining a set of pre-defined training trajectories.
For modeling materials exhibiting hardening behavior, the microscale displacement fluctuations and stresses have been taken as primal variables for the first and second reductions, respectively. In this case, the second reduction was carried out by means of the stress field interpolation. However, it can be shown that the stress projection operator, being computed with numerically converged snapshots, leads to an ill-possed microscale reduced order model. This ill-poseddness is deeply studied and corrected, yielding a robust and consistent solution.
For the model order reduction in fracture problems, the developed multiscale formulation in this work was proposed as point of departure. As in hardening problems, the use of two successive reduced order techniques was preserved.
Taking into account the discontinuous pattern of the strain field in problems exhibiting softening behavior. A domain separation strategy, is proposed. A cohesive domain, which contains the cohesive elements, and the regular domain, composed by the remaining set of finite elements. Each domain has an individual treatment. The microscale Boundary Value Problem (BVP) is rephrased as a saddle-point problem which minimizes the potential of free-energy, subjected to constraints fulfilling the basic hypotheses of multiscale models.
The strain flucuations are proposed as the primal variable for the first reduction, where the high fidelity model is projected and solved into a low-dimensional space via POD. The second reduction is based on integrating the equilibrium equations by means of a Reduced Order Quadrature (ROQ), conformed by a set of integration points considerably smaller than the classical Gauss quadrature used in the high fidelity model.
This methodology had been proven to be more robust and efficient than the interpolation methods, being applicable not only for softening problems, but also for hardening problems.
For the validation of the reduced order models, multiple test have been performed, changing the size of the set of reduced basis functions for both reductions, showing that convergence to the high fidelity model is achieved when the size of reduced basis functions and the set of integration points, are increased. In the same way, it can be concluded that, for admissible errors (lower than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 5%} ), the reduced order model is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sim{110}}
times faster than the high fidelity model, considerably higher than the speedups reported by the literature.
dummy chapter
dummy AcronymsAcronyms
Acronyms
BVPBoundary Value Problem CSDAContinuum Strong Discontinuity Approach DNSDirect Numerical Simulation EBAExpanded Basis Approach EFEMEmbedded Finite Element Methodology EIMEmpirical Interpolation Method FEFinite Element Method FEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^2 FEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \times FE hierarchical multiscale technique HFHigh Fidelity HPROMHigh-Performance Reduced Order Model HROMHyper-Reduced Order Model MORModel Order Reduction PODProper Orthogonal Decomposition ROMReduced Order Model ROQReduced Optimal Quadrature RUCRepeating Unit Cell RVERepresentative Volume Element SVDSingular Value Decomposition
Research Summary
1 Introduction
1.1 State of the Art
In this section, a general insight about multiscale approaches, failure modeling and Model Order Reduction, is given. The aim here is to describe the most important contributions along the history of the continuum mechanics community in these research fields.
1.1.1 Multiscale modeling of heterogeneous materials
During the last decades, a large variety of multiscale strategies focusing on the study and analysis of the mechanical behavior of heterogeneous materials, have been proposed by the computational mechanics community. Based on the work of Bohm_2013, these strategies may be divided into three main groups:
- Mean-Field approaches: based on the seminal contributions developed by Eshelby_1957 and Mori_Tanaka_1973. In these approaches, the microfields within each constituent of an heterogeneous material, are approximated by their phase averages, typically, phase-wise uniform stress and strain fields are employed. Recently, the application of these approaches to nonlinear modeling in composites has become a subject of active research.
- Bounding Methods: Variational principles are used to obtain upper and lower bounds of the overall mechanical properties (elastic tensors, secant moduli, yield thresholds, among others). Bounding Methods are closely related with Mean-Field approaches, because many analytical bounds are obtained on the basis of phase-wise constant fields.
The formal treatments were provided by, , Nemat_Nasser,Bornert_1996,Ponte_Suquet_1998,Markov_2000,Milton_2002,Torquato_2002. Two of the most relevant results of this kind of models are, the upper bounds of Voigt_1889, and the lower bounds of Reuss_1929. Posteriorly, Hill extends those bounds to tensorial entities, particularly, for constitutive tangent tensors Hill_1952.
Generally speaking, these bounds, while universal and very simple to assess, offer poor approximations. The reason relies on the fact that these measures do not contain any information on the geometry of the different phases beyond the phase volume fraction, being too slack for many practical purposes. However, considerably tighter bounds on the macroscopic behavior can be obtained from a variational formulation Hashin_Shtrikman_1963.
- RVE based approaches: these approximations are based on studying discrete microstructures, aiming at evaluating the microscale fields, fully accounting for the interactions between different phases. Homogenization is used as a strategy to upscale the resulting averaged variables. This homogenization strategy uses representative volumes, which copy as much as possible the material heterogeneities. These representative volume elements must be large enough to capture a statistically representative solution of the material behavior, but, also their size must be limited in comparison with the macroscopic characteristic length - (separation of scales).
The hypothesis of these approaches are properly fulfilled if there is a marked scale separation between the phenomena observed at the macroscale, and the ones observed at the microscale. However, nowadays, new approaches have been proposed to overcome this limitation.
Models based on the existence of a RVE can be divided into two main groups:
- Hierarchical multiscale models: in these models, the RVE is used to evaluate the mechanical response associated with a wide range of predetermined parameters (typically, strain histories, pore-pressure loads, thermal process, under static and dynamic scenarios, among others), whose mechanical response are usually assessed in terms of homogenized stresses.
Once these set of solutions is at one's disposal, this data can be used from several manners, for instance, constructing quasi-phenomenological models at the macroscale, where, this database acts as a pre-computed black box which relates bi-univocally, input and output variables. Finally, the user does not need to solve a macro-micro coupled problem, commonly costly from the computational point of view.
In consequence, it can be built a parametric grid based on the strain space whose nodes are related with homogenized stress tensors produced by the microscale mechanical response; in order to obtain a random solution, these nodal values can be interpolated via Neural Networks, or Finite Element approximations whose interpolation level are based on the spatial variation of the homogenized stress tensor. In recent works, , Ferrer_et_Al_2016, this methodology is applied to multiscale shape optimization, in which, the set of precomputed strain histories, forming a parametrized spherical domain, are associated to their corresponding homogenized stress states, and microscale optimal shapes.
However, the effectivity of this methodologies is based on the dimensionality of the input parameters, the sampling becomes complex for high-dimensional input parameters, then, those methodologies are inappropriate for microscales involving complex morphologies and/or complex phenomena. Apart from the previous application in multiscale optimization, the most representative methodologies are presented in Gurson_1975,Tvergaard_1981,Giusti_Blanco_Gurson
- Hierarchical models: the RVE, subjected to consistent boundary conditions1 is used to obtain a detailed microscale response. The link between scales is reached by means of an energetic identity, such as the Hill-Mandel Principle of Macro-Homogeneity Hill_1965,Mandel:1971, or even on more general approaches, like the Principle of Multiscale Virtual Power Blanco2016. In the context of the Finite Element Method (FEM), this methodology is known as FEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^2
Feyel_Chaboche_2000.
In virtue of the potential applications in microstructures with complex morphologies, the FEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^2
technique is suitable to deal with problems facing material instabilities, like fracture processes. In this sense, some approaches have been proposed Belytschko_multiscale1,Belytschko_multiscale2,Nguyen_et_al_2010,Nguyen_et_al_2011,Matous_et_al_2008,coenen2012multi,Toro_al_FOMF_2014, among others. One of the main contributions of this work, is an alternative approach with marked differences with respect to the previous ones.
- Concurrent models: the RVE is embedded into the macroscale geometry, and the corresponding kinematic compatibility is guaranteed via Lagrange multipliers, similarly to Domain Decomposition approaches.
Like the above mentioned models, concurrent models are widely applied. In approaches based on the Finite Element Method, the embedded meshes are not necessarily compatible. However, its computational viability is only for cases with small scale separation, this feature becomes into its main disadvantage. Some concurrent approaches in the field of fracture mechanics have been recently developed, , lloberas2012multiscale.
(1) Consistent in the sense that, all possible boundary conditions have to be compatible with the strains obtained at the macroscale.
1.1.2 Fracture mechanics
1.1.2.1 Monoscale Fracture Approaches
The study and analysis of fracture in solids has been a topic of research since the last century. The seminal works on this topic were focussed on the Elastic Fracture Mechanics. However, its generalization to nonlinear material behavior is a non trivial task.
Procedures based on the concept of free energy anderson, becomes into one of the early works in the field of Fracture Mechanics. The concept of fracture energy appears as a consequence of this approach.
Starting from the concept of fracture energy, which has become a central issue in nonlinear fracture mechanics modeling, several techniques have been developed:
- Cohesive models: Based on the introduction of interfaces embedded into a continuum medium. These interfaces admit the development of displacements discontinuities. Cohesive forces across the interfaces act opposing to the crack opening, diminishing as material degradation takes place. The energy necessary to produce a crack is equal to the fracture energy. Some applications of this kind of models are found in pandolfi1999finite,molinari2007cohesive,Toro_al_2014,rodrigues20162d.
- Continuum regularized models: Characterized by a continuum constitutive law displaying a softening response. These kind of approaches are subjected to material instabilities and bifurcation processes, causing ill-possedness of the problem from the mathematical point of view. As a remedy to this flaw, a constitutive law regularization (localization limiters) is introduced, ensuring mesh objective solutions. Some proposed models can be found in Pijaudier_Cabot_Bazant_1987,Tvergaard_et_al_1995,Pijaudier_Cabot_Bazant_1987,Aifantis_1984b,Borst_Muhlhaus_1992,Peerlings_et_al_1996,Peerlings_et_al_2001,Peerlings_et_al_2002,Steinmmann_et_al_1991,Muhlhaus_et_al_1987, among others.
- CSDA: This approach establishes a link between cohesive models and continuum models. Its fundamentals have been presented in the seminal work Simo_et_al_1993, posteriorly improved and applied to many applications in static and dynamic scenarios Oliver_2000,Oliver_et_al_2002,Oliver_Huespe_2004a,Oliver_Huespe_2004b,oliver2014crack. This approach provides an unified theory, which goes from the continuum description to the degradation and posterior material failure exhibiting displacement discontinuities (cracks). In this context, the continuum constitutive model subjected to a kinematics inducing displacement discontinuities represents also a “projected” cohesive law on the crack surface.
1.1.2.2 Multiscale Fracture Approaches
The study of heterogeneous materials subjected to softening, and, therefore, to degradation and failure, through multiscale approaches brings additional challenges. The fundamental reason lies in two aspects: (a) it becomes imperative the use of regularized constitutive theories at both scales in order to ensure the well-possednes of the multiscale problem. (b) The size effect, intrinsically related to the fracture energy, and extensively studied by Bazant_Planas_1998. As a result of this, the homogenized stress tensor, in the post-citrical regime, becomes extremely sensible to the RVE size.
The second issue is the existence of the RVE Gitman_et_al_2007, and the fulfillment of the basic hypothesis in multiscale modeling.
The necessity to develop specific homogenization techniques, becomes a starting point for obtaining consistent multiscale formulations. Belytschko_multiscale1 has proposed a methodology that excludes the localization domain in the homogenization process. More recently, Belytschko_multiscale2 proposed a predetermined size of RVEs. Matous_et_al_2008 describes a novel methodology, based on the existence of a macroscopic adhesive interface, which links the macroscale jump of displacements with an equivalent jump at the microscale, imposed by consistent boundary conditions.
Recent works Nguyen_et_al_2010,Nguyen_et_al_2011, describe the material failure by means of nonlocal gradient theories. In this kind of approaches, and, in contrast with other alternatives, the homogenization of the stress field during the post-critical regime, is carried out at the localization zone (this zone corresponds to a subdomain of the RVE). However, other authors claimed some inconsitencies related to this kind of approaches, particularly, about the fact that kinematics at the macroscale is not equivalent to the kinematics modeled at the microscale.
Toro_al_FOMF_2014 presents a more consistent theory which some several shortcomings given by the previous approaches, however, this the homogenization procedure evolves along time, producing an algorithm too invasive.
1.1.3 Model Order Reduction
In general, the FEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^2
method involving fine space-time discretization and time-dependent homogenization procedures, involves an enormous computational cost, being even larger when facing fracture mechanics problems.
Although no doubt the most versatile and accurate homogenization technique, with no other limitation in scope than the hypotheses of statistical homogeneity and scale separation, the direct computational homogenization approach violates squarely the modeling precept outlined at the outset –it does not discriminate between essential and irrelevant features in solving the fine-scale BVP–, making the accuracy/parsimony balance to tilt unduly towards the accuracy side and far from the parsimony one. The consequence is its enormous computational cost (in comparison with analytical and semi-analytical homogenization techniques).
The idea of exploiting the combination of dimensionality reduction and multiscale modeling is certainly not new. A survey of the related literature reveals that, over the last decade, researchers from various scientific disciplines dealing with multiscale problems have begun to consider the model reduction as a potential route –complementary to improvements in software and hardware power –to diminish the often unaffordable cost of multiscale simulations. In the specific context of homogenization-based multiscale methods, the application of model reduction techniques has been addressed by several authors, namely, Ganapathysubramanian_2004,yvonnet2007reduced,Boyaval_2007,Monteiro2008,Nguyen_ROM_2008. The strategy adopted in all these works for constructing a cost efficient model of the micro-cell is the standard reduced basis method. The gist of this strategy is to project the governing equations onto a low-order subspace spanned by carefully chosen bases Amsallem_2009.
1.1.3.1 Reduced basis techniques (ROM)
Reduced basis methods, in its standard form, suffer from an important limitation when handling nonlinear problems: they reduce notably the number of degrees of freedom –and thus the pertinent equation solving effort–, yet the computational cost associated to the evaluation of the internal forces and jacobians at quadrature points remains the same. Standard reduction methods prove, be effective only when dealing with micro-cells whose constituents obey simple constitutive laws (linear elasticity). In a general inelastic case, the calculation of the stresses at each gauss point is, on its own, a computationally expensive operation and dominates the total cost of the computation. As a consequence, the speed up provided by standard model reduction methods in nonlinear scenarios is practically negligible, and may not compensate the cost associated to the offline construction of the reduced-order bases.
1.1.3.2 High-performance reduced order modeling techniques (HPROM)
The origin of the first effective proposal on this issue can be traced back to the seminal work of Barrault_2004, who suggested to approximate the nonlinear term in the reduced-order equations by a linear combination of a few, carefully chosen basis functions. In the spirit of a offline/online strategy, in the standard reduced basis approach, these spatial bases are computed offline from full-order snapshots of the non-linear term, whereas the corresponding parameter-dependent modal coefficients are determined online by interpolation at a few (as many as basis functions), judiciously pre-selected spatial points. As in classical reduced bases methods, the efficiency of this second or collateral reduction is predicated on the existence of a moderate number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M \ll N}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
is the original dimension of the problem) of basis functions whose span accurately approximate the manifold induced by the parametric dependence of the nonlinear contribution. The interpolation method developed by Barrault_2004 is known as the EIM; the main ingredients of this method are: a) the use of a greedy algorithm to generate a set of maximally independent bases from the collection of snapshots of the nonlinear term, on the one hand; and b) the recursive selection – also via a greedy algorithm – of spatial locations where the error between the full-order bases and their reconstructed counterparts is maximum1.
In solution methods in which the governing equations are used in its variational form (as in the FE), the reduction of the computational complexity arising from nonlinearities can be, alternatively, achieved by approximating the integrals in which the offending nonlinear function appears, rather than the function itself, as done in the interpolatory and least-square reconstruction techniques discussed above. Based on this observation, An_2009 propose a quadrature scheme devised for fast-run integration of the subspace spanned by a representative set of snapshots of the nonlinear integrand.
In what follows, we shall consider as equivalent the appellations HPROM and HROM to refer to reduced basis methods combined with interpolatory or least-square reconstruction schemes.
1.1.3.3 Reduction Order Modeling in fracture problems
The development of reduced models for non-homogeneous materials has been tackled in numerous previous contributions, such as michel2001computational, where the proposed reduction techniques are based on Fourier's transforms, or yvonnet2007reduced, where a reduced model is applied the homogenization analysis of hyperelastic solids subjected to finite strains. Also, the work in Ryckelynck2009hyper develops a hyper-reduced model of a monoscale analysis which consider nonlinear material behavior. However, the existing literature barely considers reduced order modeling of non-smooth problems, as is the case of fracture, where discontinuous displacements occur. The multiscale case, when fracture takes place at both scales of the problem, makes the task even much harder. Indeed, only very few contributions have been presented in the literature about this topic, see for example: oskay2007eigendeformation, which follows an eigendeformation-based methodology, or zhang2016reduced,kerfriden2013partitioned that resort to global–local approaches.
The previous approaches combine projection techniques and, in some cases, empirical criteria to integrate the equilibrium equations in the domain. However, these are ussually ad-hoc techniques, that had been applied to problems with relatively simple crack propagation schemes. Currently, some researchers consider the effective model order reduction of fracture processes, an insolvable problem. This work will reconsider this statements, by developing a robust HPROM formulation, for multiscale fracture problems resulting in high computational speedups.
(1) Maximum in the sense that, the selected points have to be taken from components in which the error between the high fidelity and the HPROM solutions is greater.
1.2 Adopted Approach
The approach adopted in this work, uses a FE method and multiscale hierarchical models. Particularly the FEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^2
approach is used, which involves two scales (macroscale and micro/meso-scale) both discretized via finite elements. Infinitesimal strain setting, and first-order homogenization are assumed.
For fracture modeling purposes, the CSDA is adapted to the multiscale setting, and used for modeling propagating fracture at the macroscale level. At the microscale level, the use of predefined cohesive bands, distributed within the components and its interfaces, is proposed. These cohesive bands are endowed with regularized continuum damage models, which induce the crack initiation and propagation.
The Model Order Reduction techniques used in this work, are based on the POD, defining the projection of the full order model into a low-dimension small space, and, on the use of novel interpolation and ROQ schemes to diminish the computational cost generated by the multiscale problem.
1.3 Objectives and Scope
The main objectives of this work are:
- To develop a consistent and minimally intrusive multiscale hierarchical approach for propagating fracture with proper transfer of energy across scales.
- To develop, implement and validate a set of computational tools to efficiently reduce the unaffordable computational cost associated to the FEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^2
approach.
to develop, implement and validate a set of computational tools that attempts to reduce, considerably, the unaffordable computational cost induced by complex constitutive models, involving interaction between scale levels. Being as general as possible, to solve problems in solids mechanics, exhibiting either, hardening or softening behavior.
Although, multiscale models exhibiting hardening behavior have been widely studied, attention is here focussed in developing, implementing and validating, a multiscale models for propagating fracture.
In consequence, a number of complementary objectives appears in the scenario of this work:
- Study and analyze the recent developments of the CSDA, in order to develop an objective, robust and efficient fracture propagation scheme, for multiracial modeling.
- Explore the viability to reduce the computational cost via High Performance Model Order Reduction (HPROM) techniques.
- Review the state of the art of the MOR techniques, aiming at contributing to its development for the simulation of multiscale problems.
1.4 Outline
The remainder of this manuscript is organized in four chapters. Chapter 2 is devoted to the derivation of the multiscale model for propagating fracture, including a brief introduction to the fundamentals of the computational homogenization used in the proposed approach. Chapter 3 deals with the derivation of reduced order models for multiscale, smooth and non-smooth (fracture), problems, as well as some numerical results obtained from the developed models. Chapter 4 provides some concluding remarks and identifies areas for future research. In Appendix A, the participations in national and international conferences, and specialized workshops are listed. In Appendix B, a short summary of the supporting papers is presented. Finally, in Appendix C, the scientific publications supporting this work, and co-authored by the author, are annexed.
2 Multiscale modeling approach to fracture problems
As pointed out earlier, many ways of approaching the study of microscale behavior has been proposed. Particularly, two main procedures dominate the scenario of hierarchical multiscale modeling at present: analytical approaches mainly based on the asymptotic expansion analysis, and numerical approaches based on variational formulations supported by physical arguments. The later relies on the concept of Representative Volume Element (RVE), being the Computational Homogenization approach, the most common one. The model reduction techniques developed in this work will be applied to hierarchical multiscale models derived from the latter approach. Their fundamental assumptions are presented in its more conventional form. For a more in-depth description of the underlying axiomatic framework, the readers are referred to Miehe_et_al_1999,Feyel_Chaboche_2000,Kouznetsova_thesis,Feijoo_multi_scale_2006,DragoPindera:2007,WeinanE:2011
2.1 Computational Homogenization
In the context of two scale (macroscale – micro/mesoscale) problems, computational homogenization of materials is generally regarded as a way of obtaining point-wise stress–strain constitutive models at the macroscale, accounting for complex micro/mesoscopic material morphology.
The homogenization approach used in this work –commonly know as first-order homogenization– is only valid for materials that display either statistical homogeneity or spatial periodicity.
In consequence, depending on the morphology and random distribution of constituents at the microscale, the definition and existence of a representative sample RVE plays an important role in the material characterization of heterogeneities at the macroscale.
This representative sample, hereafter denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu }}\in {\mathbb{R}^d}(d=2,3)} , is assumed to exhibit several features. One of those corresponds to the size indifference property Terada2000,Kouznetsova_thesis,DragoPindera:2007, which states that if the size of this sample is increased, the response remains identical regardless the admissible boundary conditions on the RVE. The lower size limit for the RVE satisfying the size indifference property is represented by the characteristic length-scale denoted as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {h_\mu }} , giving rise to the existence of the RVE, whereas in microstructures that display periodicity, is known as RUC, or simply unit cell. Furthermore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu }}}
has to be small enough to be regarded as a point at the macroscale GrossSeelig:2011 (, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {h_\mu }\ll {L}}
, being Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {L}}
the characteristic length of the macroscale Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}}}
, see Fig. 1) this is the so-called scale separation hypothesis.
This section presents a summary of the multiscale variational formulation used in this work. This approach is based on the following fundamental hypotheses:
- The infinitesimal strains setting is used.
- Quasiestatic problems are considered1. A monotonically increasing pseudo-time variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}
is used, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t \in [0,t_f]}
, being Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_f}
the final time of the analysis. In the development of the multiscale model for fracture modeling, the incremental form of the equilibrium equations is used, due to the fact that the kinematic enhancement is modified along time. This issue is fully detailed in Sec. 2.2.
- The multiscale problem is restricted to two scales, although it can be easily extended to additional scales. The macroscale, usually identified as the structural scale, is denoted as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}}}
, and its material points are denoted as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{x}} . The representative sample Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu }}}
is defined as meso/microscale, in which, every material point is denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{y}}
. In addition, for the sake of clarity, entities at the small scale, are identified by the subindex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu } , see Fig. 1.
- The body at the macroscale, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}}}
, is subjected to predefined force or displacement actions, applied along its boundary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{\Gamma }} . This boundary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{\Gamma }}
is supposed to be smooth by parts, and it can be splitted into two parts, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{\Gamma }_{D}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{\Gamma }_{N}}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{\Gamma }_{D} \cap \mathbf{\Gamma }_{N} = \emptyset }
, representing the domains in which Dirichlet and Newman boundary conditions are imposed.
- Every point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{x}}
at the macroscale is related with a corresponding heterogeneous microstructural representative domain (the RVE), assuming the existency of a scale separation, so that the representative length at the microscale Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {h_\mu }}
is considerably smaller than that representing the macroscale Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {L}}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {h_\mu }\ll {L}}
).
- The body at the macroscale, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}}}
, is idealized as a statistically homogeneous media, in which the mechanical state at a generic point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{x}}
is characterized by the strain and stress tensors, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\varepsilon }}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\sigma }}}
, respectively. The macroscale strain tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\varepsilon }}}
is the input variable for the microscale, the corresponding output variables are the homogenized stress Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\sigma }}}
and the homogenized tangent constitutive Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{C}}
tensor (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\dot{\boldsymbol{\sigma }}}=\mathbb{C}:{\dot{\boldsymbol{\varepsilon }}}}
). In this way, the multiscale model can be interpreted as a constitutive model that, given a strain history, returns the stress and tangent constitutive tensors histories, accounting for the morphology and the interaction of the different components at the microscale, see Fig. 1.
- The Hill-Mandel Variational Principle of Macro-Homogeneity Hill_1965,Mandel:1971, which states an equivalence between the virtual power densities between micro- and macroscale, and requires the adoption of specific kinematically admissible displacement fluctuations at the RVE, is adopted.
|
| Figure 1: Macrosctructure with an embedded local microstructure. |
(1) However, in one article supporting this work, dynamic problems are also considered. See Hernandez_HPROM_2017
2.1.1 RVE kinematics and strain tensor
In the context of the adopted first-order homogenization setting, the microscopic velocity field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\dot{\boldsymbol{u}}_{\mu }}}
can be splitted as follows:
|
(2.1) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\dot{\boldsymbol{u}}}}
stands for the velocity at the macroscale, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\dot{\boldsymbol{\varepsilon }}}}
stands for the rate of infinitesimal macroscopic strain tensor, the term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\dot{\boldsymbol{\varepsilon }}}\cdot \boldsymbol{y}}
is a velocity term that varies linearly with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{y}}
, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\dot{{\tilde u}}}_\mu }}
the velocity fluctuations. The decomposition of the rate of microscopic strain tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\dot{\boldsymbol{\varepsilon }}_\mu }}
in the finite element framework yields, from the spatial differentiation of Eq. 2.1:
|
(2.2) |
The starting point of multi-scale constitutive settings, is the assumption that the rate of macroscopic strain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\dot{\boldsymbol{\varepsilon }}}} , at a point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{x}}
of the macro-continuum, is the volume average of the rate of microscopic strain 2.2, over the RVE associated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{x}}
. This assumption is also interpreted as the fact that the microscale deformations only influence the macroscale behavior through its volume average.
|
(2.3) |
In virtue of 2.2 and 2.3, this condition is equivalent to impose the volume average of the symmetric gradient of the velocity fluctuations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {{\boldsymbol{\nabla }^s_{\boldsymbol{y}}}{\boldsymbol{\dot{{\tilde u}}}_\mu }}}
to vanish. This condition can be written using the Gauss theorem as a constraint over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde{u}}_\mu }}
, involving the whole volume of the RVE, as follows:
|
(2.4) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}_{\mu }^{u}}
is defined as the space of admissible microscale velocity fluctuations in the RVE, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _\mu }
stands for the boundary of the domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}}}
, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\nu }_{\mu }}
is the unit normal vector on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _\mu }
. Equation 2.4 is also known as the minimum constraint boundary condition.
Eq. 2.4, can be cast four well-known classes of multiscale constitutive models emphasizing that the different classes of multi-scale models differ from one another only in the definition of the subspace Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}_{\mu }\subset \mathcal{\tilde V}_{\mu }^{u}}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\dot{{\tilde u}}}_\mu }}
belongs Feijoo_multi_scale_2006:
- The Taylor, or homogeneous micro-cell strain field model. This model is also refered to as the rule of mixtures,
- The linear RVE boundary displacement model,
- Periodic RVE boundary displacement fluctuations, and
- The minimum kinematical constraint (or uniform traction model), or uniform RVE boundary traction model.
iffalse The use of different definitions of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}_{\mu }}
for a given RVE produces, in general, different estimates of the corresponding macroscopic constitutive response, all those multi-scale models may be related with the following expression:
|
(2.5) |
The Taylor model is obtained by choosing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}_{\mu }= \mathcal{\tilde V}_{\mu }^{Taylor} \equiv \{ \boldsymbol{0}\} } , and gives the stiffest solution (most kinematically constrained) to the microscopic equilibrium problem, followed by the other multi-scale models, until the uniform traction model which produces the most compliant (least kinematically constrained) solution.
fi
The use of different definitions of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}_{\mu }}
for a given RVE produces, in general, different models of the corresponding macroscopic constitutive response. This response can be related in terms of stiffness, being the Taylor model (obtained by choosing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}_{\mu }= \mathcal{\tilde V}_{\mu }^{Taylor} \equiv \{ \boldsymbol{0}\} }
the stiffest one), to the uniform traction model which produces the most compliant (least kinematically constrained) solution. Throughout this work, different boundary conditions have been used. In problems exhibiting hardening responses, all those alternative models have been studied, however, in problems involving softening behavior, the set was reduced to the minimum kinematical constraint, the reason relies on the fact that this boundary condition is considered as the weakest one to model material fracture, in the sense that the crack activation can take place in a random way. In addition, the crack opening can be acting freely with no restrictions (clearly induced by more constrained boundary conditions). This is the case of periodic boundary conditions, which induces periodic failure mechanisms, being considered a strong limitation.
The actual set of kinematically admissible velocity fields Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}_{\mu }} , together with the associated space of virtual kinematically admissible velocities at the microscale, denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{W}_{\mu }^{u}} , play a fundamental role in the variational formulation of the equilibrium problem of the microscale. This space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{W}_{\mu }^{u}}
can be defined as follows:
|
(2.6) |
In virtue of 2.4, and the fact that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}_{\mu }^{u}}
is itself a vector space, it can be concluded from 2.4 that:
|
(2.7) |
Furthermore, the same arguments can be applied to the total form, and establish that any kinematically admissible displacement fluctuation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde{u}}_\mu }}
belongs also to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}_{\mu }^{u}}
.
2.1.2 Hill-Mandel Principle of Macro-Homogeneity
The scale bridging equations are completed by introducing the Hill-Mandel Principle of Macro-Homogeneity Hill_1965,Mandel:1971. Based on physical arguments, this Principle states that the macroscopic stress power equates the volume average over the RVE of the microscopic stress power, making both, macroscale and microscale, continuum descriptors energetically equivalent. Thus, departing from:
|
(2.8) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{E}} , stands for the space of all second order macroscopic strain tensor functions, Eq. 2.8 is similar to Eq. 2.2, but for admissible strain variations. Therefore, the following identity holds:
|
(2.9) |
In particular, taking Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\delta \boldsymbol{\tilde{u}}_\mu }=\boldsymbol{0}} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \forall {\delta \boldsymbol{\varepsilon }}\in \mathcal{E}} , yields:
|
(2.10) |
where, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\sigma }}}
stands for the macroscopic stress tensor, which turns out to be as the volume average of the microscopic stress Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\sigma }}_{\mu }}
. Equation 2.10 is also fulfilled in rate form. In addition to Eq. 2.10, the following condition emerges from the variational equation 2.9 solving for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\delta \boldsymbol{\varepsilon }}=\boldsymbol{0}}
|
(2.11) |
Eq. 2.11 defines the variational microscale equilibrium problem (or microscale virtual power principle).
2.1.3 RVE Equilibrium Problem
To derivate the equilibrium equations for the RVE, it will be supposed that inertial forces are negligible, and that the RVE is subjected in general to a body force field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{b}}={\boldsymbol{b}}(\boldsymbol{y},t)} , and an external traction field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{t}^{e}}={\boldsymbol{t}^{e}}(\boldsymbol{y},t)}
exerted upon the RVE across its external boundary, the Principle of Virtual Work establishes that the RVE is in equilibrium if and only if the variational equation hold for any time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}
|
(2.12) |
In this work, in some cases voids were consider, in order to be more specific, the previous equation can be splitted into two contributions, the solid part and its counterpart represented by a sort of randomly distributed voids, since the voids themselves are also in equilibrium. As a consequence, the internal traction field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{t}^{v}}}
(defined as the traction exerted upon the solid part of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu }}}
across de solid-void interface, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\partial \mathcal{B}^{v}_{\mu }}}
) may be taken into account, however, in the tests developed along this work, none of this internal traction fields have been tackled.
2.1.4 Variational Statement
The trial space, , the set of kinematically admissible displacement fluctuation fields, is defined formally as:
|
(2.13) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{A}_{0}}
is the discrete version of the expression 2.4, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H^1({\mathcal{B}_{\mu }})^d} stands for the Sovoleb space of functions possessing square integrable derivatives over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu }}}
. Note that this set forms a vector space. The test functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\eta }}}
appearing in the variational statement shown in the following are kinematically admissible variations (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\eta }}:={\boldsymbol{\tilde{u}}_\mu }-{\boldsymbol{\tilde v}_\mu }, \, {\boldsymbol{\tilde{u}}_\mu },{\boldsymbol{\tilde v}_\mu }\in \mathcal{\tilde V}_{\mu }}
), in this case, coincide with the space of kinematically admissible displacement fluctuation.
2.1.4.1 Formal Statement
Consider a time discretization of the interval of interest Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [t_{0},t_{f}]=\bigcup _{i=1}^{\hbox{nstp}}[t_{n},t_{n+1}]} . The current value of the microscopic stress tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\sigma }}_{\mu }}
at each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{y}\in {\mathcal{B}_{\mu }}}
is presumed to be entirely determined by, on the one hand, the current value of the microscopic strain tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\varepsilon }_\mu }_{n+1}={\boldsymbol{\varepsilon }}_{n+1}+{{\boldsymbol{\nabla }^s_{\boldsymbol{y}}}{\boldsymbol{\tilde{u}}_\mu }}}
, and, on the other hand, a set of microscopic internal variables Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\mu }}_{n+1}}
that encapsulate the history of microscopic deformations.The (incremental) RVE equilibrium problem at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_{n+1}}
can be stated as follows: given the initial data Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ {\boldsymbol{\tilde{u}}_\mu }_{n}(\boldsymbol{y}),{\boldsymbol{\varepsilon }}_{n},{\boldsymbol{\mu }}_{n}(\boldsymbol{y})\} }
and the prescribed macroscopic strain tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\varepsilon }}_{n+1}}
, find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde{u}}_\mu }_{n+1} \in \mathcal{\tilde V}_{\mu }^{u}}
such that HdezEtAlMon:2012:
|
(2.14) |
In this case, body forces and internal traction fields are considered negligible, however, those fields may be included depending on the problem without any further complexity. The actual output of interest in the microscale BVP is not the displacement fluctuation field per se, but rather the macroscopic stress tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\sigma }}_{n+1}} , which is defined as the volume average over the RVE of the microscopic stresses:
|
(2.15) |
also the constitutive homogenized tangent tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{C}_{n+1}^h}
which is composed by two parts, on the one hand, the contribution of the constitutive tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{C}^h}
, and on the other hand, the contribution given by the fluctuation displacement field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\mathbb{C}}^h}
Feijoo_multi_scale_2006, for the multiscale modeling of propagating fracture is necessary to upscale additional tensorial variables in order to guarantee the proper dissipation at both scales, and also fulfill objectivity conditions widely known in fracture mechanics, these variables will be detailed in the Section 2.2.
2.1.4.2 Finite Element Formulation
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu }}=\bigcup _{i=1}^{\hbox{n}_e}{\mathcal{B}}_{\mu }^e}
be a finite element discretization of the RVE, and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ N_1(\boldsymbol{y}) \cdots N_n(\boldsymbol{y})\} }
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
denotes the number of nodes of the discretization) be a set of shape functions associated to this discretization. Now we approximate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde{u}}_\mu }\in \mathcal{\tilde V}_{\mu }^{u}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\eta }}\in \mathcal{\tilde V}_{\mu }^{u}}
as Hughes_1987:
|
(2.16) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{U}_i \in \mathbb{R}^{d}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\eta }}_i \in \mathbb{R}^{d}}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=1,2,3 \cdots n}
) denote the nodal values of the displacement fluctuation field and test functions, respectively. Once the finite element discretization has been defined, the expressions 2.16 may be replaced into 2.14 in order to obtain the time-space discrete RVE equilibrium equation, exploiting the arbitrariness of coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\eta }}_i (i=1,2,3 \cdots n)} , one arrives at the following set of discrete equilibrium equations in index notation.
|
(2.17) |
Now, introducing Voigt's notation, and numerical integration via Gaussian quadrature zienk_taylor_2000,Belytschko_WinKL_2001, expression 2.17 can be expressed in matrix format as:
|
(2.18) |
Here, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_g=\mathcal{O}(n)}
stands for the total number of Gauss point of the mesh; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w_g}
denotes the weight associated to the g-th Gauss point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{y}_g}
- and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{B}(\boldsymbol{y}_g,:)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\sigma }}_{\mu }(\boldsymbol{y}_g,:)}
stand for the B-matrix and the stress vector at Gauss point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{y}_g}
, respectively.
2.2 Multiscale Fracture Mechanics issues
In Computational Fracture Mechanics, hierarchical multiscale methods involve additional issues. In particular:
- The existence of the RVE has been questioned in the literature, arguing that for fracture cases, the material loses its statistical homogeneity Gitman_et_al_2007,Nguyen_et_al_2010.
- The fact that the homogenized constitutive model lacks an internal length Bazant_2010, raising similar issues than in classical phenomenological monoscale problems.
Additionally, mesh-bias dependence, and the proper fracture energy dissipation issues Rots_1988 via regularized constitutive models Oliver_1989,Oliver_et_al_2002,Oliver_Huespe_2004b,oliver2015continuum are also crucial issues to be considered at each scale.
At this point, two main aspects have to be dealed in multiscale hierarchical approaches, the possible non-existence of the RVE combined with the lack of objectivity of its representative size. Nowadays, several studies have been developed in order to analyze the influence of the use of an unit cell rather than the RVE, by assessing the amount of deviation of the apparent properties obtained by the unit cell modeling Ghosh_1996,Moulinec_Suquet_1998,Michel_et_al_1999. In this work, all those previous issues have been studied and tested. The use of a complex failure cell (representing all features of the micro/meso scale) plays an important role for allowing crack nucleation and posterior coalescence. Along this section, the most important issues of this multiscale approach are summarized, being the analytical details referenced to the corresponding article.
Along this section, the most important aspects of the proposed multiscale approach are summarized. This multiscale approach is fully detailed in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{2}}
in Sec. [[#5.2 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{2}
|5.2]].
2.2.1 Multiscale modeling setting
|
Figure 2: Macroscopic (Structural scale) body Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathcal{B}}
(a) subdivision in a non-smooth domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathcal{B}_{\hbox{loc}}}(t)
, and a smooth domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathcal{B}}\setminus {\mathcal{B}_{\hbox{loc}}}(t) (b) h-regularized displacement and strain discontinuity kinematics. |
2.2.1.1 Macroscale Model
Considering the body Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}}} , at the macroscale (see Fig. 2) it is assumed that material points, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{x}} , of the macroscopic body belong, at the current time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t} , to either one of the two subdomains:
- Domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}}\setminus {\mathcal{B}_{\hbox{loc}}}(t)}
- the set of points at the macroscale, exhibiting smooth behavior. The infinitesimal strain field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\varepsilon }}(\boldsymbol{x},t)}
is described in rate form, as:
|
(2.19) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u(\boldsymbol{x},t)}
is the macroscopic displacement field, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}
stands for the time or pseudo-time parameter, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\cdot )^s}
, stands for the symmetric counterpart of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\cdot )} .
- Domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\hbox{loc}}}(t)}
- the set of points exhibiting material failure and, therefore, a non-smooth behavior. See:
Sec.2.1
Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#
2 The strain field at these points is assumed to be captured by a h-regularized strong-weak discontinuity kinematics, h being the width of the corresponding strain localization band (see Fig. 2-(b)).
|
(2.20) |
In Eq. 2.20, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\dot{\bar{\boldsymbol{\varepsilon }}}}(\boldsymbol{x},t)}
stands for the regular (smooth) counterpart of the strain, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\kappa _{{\mathcal{B}_{\hbox{loc}}}}}}
is a colocation (characteristic) function on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\hbox{loc}}}}
(See. Fig. 3), so that the term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\kappa _{{\mathcal{B}_{\hbox{loc}}}}}(\boldsymbol{x})}
becomes a h-regularized Dirac's delta function shifted to the center-line, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S(t)}
(the macroscopic discontinuity-path at the current time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}
, as shown in Fig. 2-(a)). Thus, in Eq. 2.20, the term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\delta ^{h}_{s}}(\boldsymbol{x}){\dot{\boldsymbol{\gamma }}}(\boldsymbol{x},t)}
corresponds to the non-smooth (discontinuous and h-regularized) localized counterpart of the strains; a space-discontinuous second order tensor for the weak-discontinuity case.
|
| Figure 3: Colocation function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\kappa _{{\mathcal{B}_{\hbox{loc}}}}}(\boldsymbol{x}) |
2.2.1.2 Microscale Model
Assuming that the fracture at the macroscale has arisen, in turn, by the appearance of failure mechanisms at the microscale level, originated by some type of material failure. The next step is to endow the microscale model with mechanisms to capture the onset and propagation of this material failure. Therefore, without introducing further details, it is considered that the microstructure shall be able to capture some dominant failure mechanisms of the material.
|
| Figure 4: Outline of the multiscale model for propagating fracture: a) macro and micro scales; b) microcell model accounting for material failure. |
For this purpose, a micro failure cell Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu }}} , of characteristic size Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h_{\mu }} , is considered to exist at every material point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{x}\in {\mathcal{B}}} . It accounts for the material morphology at the microscale (voids, inclusions etc.). In addition, it is endowed with a set of cohesive bands (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{coh}}}\subset {\mathcal{B}_{\mu }}} ) of very small width Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k \ll h_{\mu }} , whose position and other geometric properties (typically the normal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{\mu }} , see Fig. 4) are predefined. At the current time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t} , the activation (de-cohesion) of a number of those bands, defines the current subset of active bands Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{act}}}(t) \subset {\mathcal{B}_{\mu ,\hbox{coh}}}\subset {\mathcal{B}_{\mu }}}
which constitutes the "active" microscopic failure mechanism, for the considered point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{x}\in {\mathcal{B}}}
.
In principle, there is no intrinsic limitation on the number of the "candidate" cohesive bands to be considered at the failure cell. On one hand, their number and spatial position have to be sufficient to capture the dominant material failure mechanisms at the macroscale. On the other hand, the associated computational cost sets a limitation on the number of such bands. In this context, the following domains at the microscale are considered (see Fig. 4):
- Domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu }}\setminus {\mathcal{B}_{\mu ,\hbox{coh}}}}
- the set of points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{y}}
, which do not belong to the cohesive bands. They are compelled to exhibit a smooth behavior described by a Continuum hardening model, typically:
|
(2.21) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\sigma }}_{\mu }}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\varepsilon }_\mu }}
stand, respectively, for the stress and strain fields at the microscale point, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{y}}
, of the failure cell (corresponding to the macroscale point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{x}} ), being Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{C}^{\hbox{hard}}_{\mu }}
the microscopic inelastic constitutive tensor derived from the hardening constitutive model.
- Domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{coh}}}}
- the set of microscale cohesive bands. As for the material behavior, the disctintion of two situations has to be made in this case:
- The failure cell, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu }}}
is associated to a non-smooth material point at the macroscale (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{x}\in {\mathcal{B}_{\hbox{loc}}}}
). Without limiting the use of any other inelastic constitutive model, this domain is endowed with an isotropic continuum damage model, exhibiting inelasticity with regularized strain softening only for tensile stress - tensile-damage continuum damage model Oliver_1995b,Faria_1998,Oliver_et_al_implex_2006. Its constitutive response isSee: Sec.2.2
Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#
2 represented in a general form as:
|
(2.22) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\mu }}
stands for a set of internal variables accounting for the inelastic behavior evolution.
- The failure cell, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu }}}
is associated to a smooth material point at the macroscale, (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{x}\in {\mathcal{B}}\setminus {\mathcal{B}_{\hbox{loc}}}(t)}
). In this case, the inelastic model, in Eq. 2.22 is enforced to behave instantaneously elastic at the cohesive bands domain, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{coh}}}} , :
|
(2.23) |
where, in continuum damage models, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{C}^{\hbox{elastic}}_{\hbox{inst}} = (1-d_{\mu })\cdot \mathbb{C}_{\mu }}
with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{d}_{\mu }=0}
, and being Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{d}_{\mu }}
the rate of the damage internal variable (a scalar for isotropic damage cases).
An advantage of this methodology, in the previous setting, is that the same failure cell morphology is considered to represent the microstructure at every macroscopic point of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}}} , both for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{x}\in {\mathcal{B}_{\hbox{loc}}}(t)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{x}\in {\mathcal{B}}\setminus {\mathcal{B}_{\hbox{loc}}}(t)}
. The only difference is the considered constitutive behavior at the cohesive bands Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{coh}}}} , defined in Eqs. 2.21,2.22 and 2.23.
Now, in virtue of 2.1, and for the sake of generality, the microscopic displacement can be modified for multiscale complex problems, which involves propagation of fracture or other complex phenomena at the large, in the following way:
|
(2.24) |
where as the previous case, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u(\boldsymbol{x},t)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\varepsilon }}(\boldsymbol{x},t)}
are, respectively, the macroscale displacements and strains at point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{x}}
in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}}}
, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde{u}}_\mu }}
are the microscale displacement fluctuations.
Displacement fluctuations in the CSDA: Considering Eq. 2.1, with a local coordinate system (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi ,\eta } ) aligned with the domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{coh}}}}
(see Fig. 4), and, exhibiting the de-cohesive behavior allocated to the cohesive bands, the smooth part of the microscopic displacement fluctuation field, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\bar u}_\mu }}
, can be expressed as:
|
|
| Figure 5: Cohesive Band behavior. | |
|
(2.25) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{H}_{{\mathcal{B}_{\mu ,\hbox{coh}}}}(\xi )}}
is the k-regularized Heaviside function shifted to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{coh}}}}
, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{\boldsymbol{\beta }}_{\mu }(\eta ,t)}
is a (smooth) function arbitrarily defined except for the restriction in Eq. 2.25-(c), In Eq. 2.25 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lbrack\lbrack(\cdot )(\xi ,\eta ,t) \rbrack\rbrack^{+}_{-} \equiv (\cdot )(\xi ,\eta ,t)|_{\xi =k} - (\cdot )(\xi ,\eta ,t)|_{\xi=0}}
, is the apparent jump of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\cdot )(\xi ,\eta ,t)}
across the cohesive band.
Following these statements, the microscale displacement fluctuation is given by (see Fig. 5):
|
(2.26) |
Eq. 2.26 constitutes the displacement counterpart of a k-regularized strong discontinuity kinematics Oliver_1996a, and proves that the herein proposed cohesive-bands approach, is consistent with a k-regularized strong discontinuity at the cohesive domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{coh}}}} . In consequence, the corresponding microscopic strain fluctuation field is given by:
|
(2.27) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\delta ^{k}_{s_{\mu }}}}
stands for the k-regularized Dirac delta function, placed at the center line, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S_{\mu }}
, of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{coh}}}}
(see Fig. 5-(b)). Thus, the rate of microscopic strain field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\dot{\boldsymbol{\varepsilon }}_\mu }} can be written in terms of the rate of macroscopic strain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\dot{\boldsymbol{\varepsilon }}}}
, and the rate of microscopic displacement fluctuations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\dot{{\tilde u}}}_\mu }} , as follows:
|
(2.28) |
From Eq. 2.28, it can be concluded, that the second term at the right-hand side becomes unbounded in the limit Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k \to 0} . In multiscale modeling, this expression is equivalent to the one given, in phenomenological monoscale models, in the context of the Continuum Strong Discontinuity Approach (CSDA) of material failure Oliver_et_al_2002.
2.2.2 Homogenized (induced) constitutive equation
One of the most specific features of the proposed multiscale approach, is that the same homogenization setting is used in points of both domains, smooth (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{x}\in {\mathcal{B}_{\mu }}\setminus {\mathcal{B}_{\mu ,\hbox{coh}}}} ), and non-smooth (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{x}\in {\mathcal{B}_{\mu ,\hbox{coh}}}} ), coinciding with the approach presented in Sec. 2.1. Other approaches Toro_al_FOMF_2014, redefine the failure cell along time, fulfilling conditions of material bifurcation induced by instabilities at the microscale. More complex approaches Kouznetsova_thesis,Geers_et_al_2010,Otero_Martinez_2015,Lesicar_et_al_2015 propose the use of second-order computational homogenization schemes in order to get better accuracy in the prediction of high strain gradients. In this work it is claimed the ability of the proposed approach to induce discrete failure in a first-order homogenization setting, giving rise to objective responses, and proper energy transfer through scales.
An issue appearing in this scenario, widely known in hierarchical multiscale approaches, is its high computational cost. In this context, the proposed model was also conceived to be combined with the use of model order reduction techniques (Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{3}} ) Oliver_Caicedo_HROM_2017. These techniques have been deeply studied in this work, and their main features are presented in Chapter 3.
In what follows, the consequences of the homogenization procedure based on the Hill-Mandel Principle of Macro-homogeneity are analyzed. The fact that the regularized strong discontinuities appear also at the microscale, being captured by the cohesive bands Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{coh}}}} , is one of the most relevant features of the proposed approach.
|
| Figure 6: Multiscale model: (a) failure cell with activated failure mode; (b) geometrical characterization of the failure mode. |
For the sake of generality, the RVE is considered composed by several components: a matrix, and randomly distributed inclusions and voids. In addition, a number of cohesive bands are considered defining the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{coh}}}}
(a sketch is presented in Fig. 6); those cohesive bands allow failure within the matrix, See:
Sec.2.4
Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \# 2 across the aggregates and at the matrix/aggregate interface.
Among them, there are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{active}}
cohesive bands, defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}^{(i)}_{\mu ,\hbox{act}}}}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=1, \cdots n_{active}}
which are in an inelastic softening state, defining a specific failure mode Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{act}}}\subset {\mathcal{B}_{\mu ,\hbox{coh}}}}
, the remaining set of bands are in unloading state, and are discarded for the purposes of this analysis. As a consequence of the previous statement, the discrete version for the middle-line of the cohesive band Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (i)}
(denoted as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S_\mu ^{(i)}}
), and the domain defining the failure mode Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{act}}}}
can be expressed as follows:
|
(2.29) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S^{(i)}_\mu }
can be regarded as the middle-line of the cohesive band segment Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (i)}
.
Following the previous domain decomposition (smooth and non-smooth subdomains) in Sec. 2.2.1.2, the Eq. 2.10 can be integrated in the two separated subdomains:
|
(2.30) |
In consonance with the definition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\dot{{\boldsymbol{\sigma }}}_{\mu }}}
(in particular, the bounded behavior of the microscopic stress field), the second term on the right hand side can be neglected assuming a small enough width of the cohesive bands (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k \to 0}
).
|
(2.31) |
Now, inserting Eq. 2.28 for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{y}\in {\mathcal{B}_{\mu }}\setminus {\mathcal{B}_{\mu ,\hbox{coh}}}}
into Eq. 2.31, yields:
|
(2.32) |
The previous equation can be rephrased in terms of the average value of the microscopic elastic constitutive tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\bar{\mathbb{C}}}} , as:
|
(2.33) |
|
Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\bar{\mathbb{C}}}}
does not depend on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu }}}
, this term can be factorized from the second term at the right hand side of Eq. 2.33, as a consequence, and taking into account the Eq. eq:MicroStrainFluc2, and the constraint eq:BVP_constraint over the fluctuation microscopic displacement field, it yields:
|
(2.34) |
being, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L_{\mu }=\left|S_\mu \right|}
the measure (length in 2D and area in 3D) of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S_\mu }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overline{(\cdot )}_{S_\mu }}
stand for the average value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overline{(\cdot )}}
in the activated microscopic failure mechanism. oliver2015continuum. The term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\dot{\boldsymbol{\chi }}}(\boldsymbol{x},t)}
vanishes, in case of components with equal material properties at the microscale, this is the trivial one also known as homogeneous case, in addition, depending on the boundary conditions, the displacement fluctuation field is also close to zero.
Finally, after some manipulations of Eq. 2.30, and following the definitions of microscale kinematics in Eq. 2.28, and the lemma in Eq. 23 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{2}}
oliver2015continuum, the resulting homogenized constitutive equation fulfills the following:
|
(2.35) |
|
(2.36) |
|
(2.37) |
where, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {l_\mu }}
stands for a characteristic length, depending on the activated microscopic failure pattern. The tensorial entities Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\dot{\boldsymbol{\chi }}}(\boldsymbol{x},t)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\dot{\boldsymbol{\varepsilon }}}^{(i)}(\boldsymbol{x},t)}
, are inelastic strains, and play the same role than internal variables in phenomenological models. However, unlike them, here, See:
Sec.2.4
Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \# 2 their evolution is determined, at every macroscopic sampling point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{x}} , by homogenized values of entities at the corresponding microscopic failure cell Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu }}} . This extends to non-smooth problems, some theoretical results already derived for smooth problems, see Michel_2003,Michel_2004. In addition, a characteristic length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {l_\mu }}
emerges naturally in Eq. 2.36, as the ratio between the measure of the failure cell (area in 2D and volume in 3D), and the measure (length/surface) of the activated microscopic failure mechanism. In consequence this length is of the order of the failure cell size. For a deeper review of the analytical results of this induced homogenized constitutive model, the reader is addressed to Sec. 2.4 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{2}}
.
The role of the characteristic length, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {l_\mu }} , naturally derived from the present formulation, is not only computational, but it has also other very relevant physical and mechanical implications. Consideration of such a characteristic length, for multiscale based approaches, has been claimed from the material mechanics community Bazant_2010, and sometimes introduced in a heuristic way in other approaches Unger_2013. This characteristic length depends on both the specific data of the problem and the local microscopic failure state. Through its consideration, the correct energy transfer between scales and mesh size objectivity can be achieved.
Remark (B): following Eq. 2.20 and Eq. 2.36, the former defines a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {l_\mu }} -regularized discontinuity kinematics of bandwidth Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h={l_\mu }} . This fact has crucial importance for a proper and meaningful modeling of the material failure propagation at the macroscale. In fact, the entities derived from this constitutive model has to be consistent with the aforementioned kinematics of regularized strong discontinuity and, more specifically, regularized with the characteristic length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {l_\mu }} .
From inspection of 2.36, it can be concluded that this expression is fulfilled under the following two circumstances:
- The fluctuations jump, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{\boldsymbol{\beta }}_{\mu }}
, is spatially constant throughout the cell:
|
(2.38) |
|
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \zeta (\boldsymbol{x})}
is a measure of the tortuosity of the activated microscale failure path Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S_{\mu }}
(for instance, for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S_{\mu }}
being a straight line, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \zeta }
=1). A similar expression for this tortuosity factor can be found in Toro_al_FOMF_2014. In this case, the inelastic strain rate (Eq. 2.36) is expressed as:
|
(2.39) |
- The activated failure mechanism at the microscale, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S_\mu }
, is a straight line (or a plane surface), with spatially constant normal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{n}_{\mu }} . However, this is an academic case, since it corresponds to a non-realistic micro-structural morphology of the failure cell, it will be discarded because of its lack of physical significance and practical interest.
In addition to the previous comments, and making an inspection of Eqs. 2.35 to 2.37, reveals that the model might exhibit some instability. Indeed, the structure of the inelastic strain in Eq. 2.36, allows the following situations:
|
(2.40) |
which can give rise to some instabilities in the microscopic failure mechanism, more details about this issue are fully detailed in Appendix A in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{2}} .
In summary, Eq. 2.35 and Eq. 2.36 retrieve the format of a constitutive model equipped with an internal length and with internal variables whose evolution is described by the microstructure behavior. Although this model will never be used for computational purposes1, it supplies relevant insights on the properties of the resulting homogenized constitutive model.
(1) Instead, the homogenized value of the stress in Eq. 2.10 is point-wise used to evaluate the current macroscopic stress in terms of the corresponding macroscopic strain.
2.2.3 Energy dissipation
Let us consider, on one hand the fracture energy, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {G_{\mu }^{f}}(\boldsymbol{y})}
corresponding to points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{y}\in {\mathcal{B}_{\mu }}}
, defined as a material property specific for every compound of the heterogeneous RVE, and, on the other hand, the macroscale fracture energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {G^{f}}(\boldsymbol{x})} , obtained as an output from the homogenization procedure. According to their definitions, those fracture energies can be computed in terms of fracture energy densities, in terms of the energy dissipation that takes place in bands with bandwidth Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}
(at the microscale) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {l_\mu }}
(at the macroscale), respectively.
|
In virtue of the Hill-Mandel Principle of Macro-Homogeneity, See:
Sec.2.6
Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \# 2 it can be concluded that the macroscopic fracture energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {G^{f}}}
is equivalent to the average of microscopic fracture energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {G_{\mu }^{f}}(\boldsymbol{y})}
, along the activated failure mechanism at the microscale Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S_{\mu }} . Replacing Eq. 2.41 into Eq. 2.9, and after some manipulations, the macroscopic fracture energy is given by the expression oliver2015continuum:
|
(2.43) |
|
(2.44) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overline{{G_{\mu }^{f}}(\boldsymbol{y})}_{S_{\mu }}}
is the mean value of the microscopic fracture energy varying along the active failure path. Eq. 2.44 provides the relationship of fracture energies at both scales. In case of an homogeneous fracture energy at the active cohesive bands of the microscale, Eq. 2.40 translates into an exact equivalence of fracture energies along the scales, :
|
(2.45) |
In the light of this result, it can be easily concluded that the fracture energies at the microscale determine, in average, the effective fracture energy at the macroscale. It is stressed the importance of the characteristic length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {l_\mu }}
in order to guarantee the proper dissipation between scales. For more details, the reader is addressed to Appendix B in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{2}}
.
2.2.4 Numerical aspects: finite element model
The proposed multiscale formulation has been implemented in a Finite Element model following the setting of a FEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^2
strategy. Accordingly, two nested finite element models are used:
- At the macroscale level, an EFEM based on the CSDA is used, aiming at capturing propagating cracks. As described in oliver2014crack, this technology consists of the insertion, during specific stages of the simulation, of goal oriented specific strain fields via mixed finite element formulations. This allows modeling propagating cracks through the macroscale finite element mesh.
- A standard Finite Element model is used at the microscale level, combining standard elements endowed with continuum hardening constitutive models and cohesive-band elements endowed with regularized constitutive softening models. These are placed in the edges of every finite element, capturing the crack onset and strain localization, similar to the cohesive interface elements in pandolfi1999finite, and more recently in rodrigues20162d. This approach benefits the simplicity of the algorithm and the non-intrusive character of its implementation.
In what follows, these two finite element models are described.
2.2.4.1 Failure cell finite element model
Standard quadrilateral finite elements are adopted for the numerical simulation of the cell response. The cohesive bands Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{coh}}}}
are also modeled by quadrilateral isoparametric finite elements of very small thickness Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k \ll {h_\mu }}
(high aspect ratio), as shown in Fig. 7-(a), endowed with constitutive models whose behavior is sketched in Fig. 7-(b) and defined through equations 2.21 to 2.23. The remaining finite elements of the cell are endowed with either elastic or inelastic hardening responses. Therefore, only elements on the cohesive bands can exhibit strain localization.
The corresponding nonlinear problem in the failure cell is then solved for the discretized version of the microscale displacement fluctuations, using Eq. 2.11. Dirichlet boundary conditions precluding rigid body motions, and minimal boundary conditions in Eq. 2.4, are also imposed.
|
|
| Figure 7: Multiscale model: finite element discretization at the microscale. | |
Material failure propagates naturally through the RVE, strain localization takes place at the finite elements defining the cohesive bands. At every time step of the analysis, those finite elements who are in loading state, define the active set of cohesive bands Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{act}}}}
conforming the active failure mechanism.
Numerically, this set can be defined as:
|
(2.46) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f}
stands for the measure of the inelastic status, corresponding to the positive evolution of the rate of strain-like internal variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{r}}
(in continuum damage models), being Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{y}_{C}}
the center of the finite element. From Eq. 2.46, the measure (length/surface) of the activated mechanism Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L_{\mu }}
, and the corresponding characteristic length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {l_\mu }}
can be determined as:
|
(2.47) |
in virtue of the strain enhancement, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {l_\mu }}
must be less than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h^{e}}
, being Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h^{e}}
the macroscopic characteristic length based on the finite element geometry at the macroscale.
2.2.4.2 Finite element model at the macroscale: material failure propagation
One of the most critical issues in computational modeling of material failure is the appropriate capture of the crack onset and propagation. When does failure trigger at a given material point? and how does it propagate?, these two questions are the cornerstone of material failure propagation algorithms.
At the microscale, where the morphology and the position of candidate propagation mechanisms are predefined, the two issues are of minor relevance due to the adopted simplified failure-bands model. However, at the macroscale, there is not a predefined failure path, and in principle, any material point may fail and propagate in any direction. To adequately solve the previous questions, the procedure for modeling onset and propagation of discontinuities recently developed for monoscale problems oliver2014crack has been extended to the multiscale setting. The proposed methodology is based on the use of the following specific techniques:
|
| Figure 8: Evolution of the injection domains for three typical stages (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t_1 < t_2 < t_3
) of the discontinuity propagation. |
- Strain injection techniques: based on the use of goal oriented assumed-strain fields injected in selected domains, via mixed formulations Simo_Hughes_1986,Simo_Riffai_1990,Reddy_Simo_1995,zienk_taylor_2000. The standard (four points) Gauss quadrature rule, corresponding to full integration of two-dimensional quadrilaterals, is complemented with two additional sampling points placed at the barycenter of the element (see Fig. 9), termed singular and regular sampling points. These two additional quadrature points sample the stresses similarly to the standard Gauss points. Therefore, for the injected elements, numerical integration (typically evaluation of the incremental internal forces in terms of the stresses), is based on those two additional sampling points by defining the weight indicated in Table 3 - in Appendix B in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{2}}
.
As for propagation purposes, two different enhanced strain injection stages, are considered1:
|
| Figure 9: Sampling points involved in the numerical integration. |
Particularly, the criterion to inject enhanced modes is based in the homogenized dissipation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{D}(\boldsymbol{x},t)} , defined as the average of the dissipation at the microscale Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{D}_\mu (\boldsymbol{y},t)}
|
(2.48) |
The time to switch the injection, from weak to strong discontinuity regime, is defined by the user based on the strain localization and the crack path stability and evolution.
- In a first stage, the weak discontinuity stage, embedded localization bands with bandwidth Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle l_{\mu }}
, at the macroscale, are incrementally injected (prior to development of displacement discontinuities) in an evolving subdomain. These embedded localization bands have no preferred orientation (they have an isotropic character), and exhibit a great ability to propagate material failure in the proper directions. This so-injected elements are used for a very short time in order to avoid stress locking effects. Once the crack propagation remains stable, and the crack path is well defined, the injection stage is switched to the second stage.
The onset of homogenized dissipation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{D}(\boldsymbol{x},t)} , is used as the criterion to inject the weak discontinuity mode. This subset (weak discontinuity domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\hbox{wd}}}} )2 is defined as:
|
(2.49) |
Eq. 2.49 characterizes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\hbox{wd}}}}
as the set of elements of the localization domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\hbox{loc}}}}
whose barycenter has not yet bifurcated3, and its corresponding homogenized dissipation take values different from zero, (and lower than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{D}(\boldsymbol{x}_{C}^{(e)},t_{SD})}
) in inelastic loading regime.
The injected strain rate at element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle e} , with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{node}}
nodes, is the following:
|
(2.50) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{i}}
are the standard shape functions, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\dot{\boldsymbol{u}}}(t)}
, the macroscale nodal displacements, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\zeta _{\mathcal{S}}^{h^{(e)},l_{\mu }^{(e)}}}}
is the regularized dipole-function in the element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (e)}
. A description of the weak enhanced mode is presented in Fig. 10, In addition, the variational problem for the weak-discontinuity regime, in rate form, is presented in Box A1 in the Appendix B in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{2}} .
|
| Figure 10: Weak discontinuity mode. Elemental regularized dipole function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\zeta _{\mathcal{S}}^{h^{(e)},l_{\mu }^{(e)}}}
. |
- In a second stage, the strong discontinuity stage, the obtained crack path field, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S}
, is used to determine the appropriate placement of an elemental embedded strong discontinuity strain field, which is incrementally injected in the appropriated set of elements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\hbox{sd}}}} .
|
(2.51) |
where, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_{SD}} , known as the second injection time, in Appendix B.2 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{2}}
is defined as, the time in which the homogenized strain field at the macroscale is compatible with the strong discontinuity kinematics, in other words, the time in which the discontinuous bifurcation in the weak-discontnuity injected element takes place. However, this might be inadequate in more complex problems, depending on the complexity of the crack-path field, this time can be shifted to ensure the stability in the propagation scheme.
Once the crack-path field is stable, the injection of the strong discontinuity induces a jump of displacements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lbrack\lbrack{\dot{\boldsymbol{u}}}\rbrack\rbrack}
across Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{S}}
, where the kinematic description is expressed as:
|
(2.52) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\dot{\bar{\boldsymbol{u}}}}}
stands for the smooth part of the displacement field, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{H}_{\mathcal{S}}}}
is the Heaviside function, shifted to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{S}}
. In the present multiscale context, the proposed second stage consists of the incremental injection of the following elemental strong discontinuity mode:
|
(2.53) |
in terms of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {l_\mu }^{(e)}} -regularized Dirac delta function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\delta ^{l_{\mu }^{(e)}}_{s}}}
(displayed in Fig. 11), being Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{n}}^{(e)}} the direction of the element normal provided by the solution of the discontinuous bifurcation problem presented in Sec. 2.5 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{2}}
. The resulting variational problem for the injection procedure is summarized in Box A2 - Appendix B in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{2}} .
|
| Figure 11: Strong discontinuity mode. Elemental regularized Dirac delta function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\delta ^{l_{\mu }^{(e)}}_{s}}
. |
- Crack-path-field techniques: their goal is the identification of the trace of the propagating crack by means of the so-called crack-path field. It is denoted as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu (\boldsymbol{x},t)}
, and obtained from a selected localized scalar variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha (\boldsymbol{x},t)} , which identifies the crack path as the locus where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha (\boldsymbol{x},t)}
takes its transversal maximum value. In order to define this locus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{S}_t}
, some alternatives have been developed in this multiscale framework, see Eqs. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 55-56}
in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{2}}
oliver2015continuum. The variational statement for the crack-path field model is fully detailed in Box 3.1 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{2}}
.
The resulting procedure is a robust and efficient technique to model propagating material failure in a finite element setting. It is especially appropriate for capturing material failure propagation in coarse meshes, in contraposition of the alternative extra elemental character techniques (phase-field, gradient or non-local damage models), where several elements span the localization band. In addition, its implementation in an existing finite element code has a little intrusive character.
In regards to the space and time integrations, as commented above, injection of weak-discontinuity and strong-discontinuity modes requires, in principle, specific integration rules in space, : a standard four-point Gauss quadrature rule, and two additional sampling points, for injected elements, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}}^{(e)} \subset {\mathcal{B}_{\hbox{wd}}}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}}^{(e)} \subset {\mathcal{B}_{\hbox{sd}}}}
so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\hbox{loc}}}= {\mathcal{B}_{\hbox{wd}}}\cup {\mathcal{B}_{\hbox{sd}}}}
. Since those domains evolve along time (see Fig. 8), some additional problems on the time-integration of the resulting equilibrium equations are found. To tackle this issue, in oliver2014crack and oliver2015continuum is proven that defining some "equivalent" stress entities at the standard Gauss points, the spatial integration can be rephrased as a standard four Gauss points integration rule in the integration domain. This space-time integration rule is fully explained in Appendix B3 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{2}} , and the corresponding stress evaluation is also summarized in Box A3.
(1) To switch between stages, a set of control variables are defined, all those detailed in Sec. 3 and Appendix B in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{2}
(2) Under the CSDA, the homogenized dissipation is evaluated at the barycenter of the finite element, denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol{x}_{C}^{(e)}
(3) For a deeper review of the bifurcation analysis, and, the definition of the corresponding bifurcation time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t_{B} , the reader is referred to Sec. 2.5 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{2} , and, for numerical aspects Oliver_bifurc_2010.
3 Model Order Reduction in Multiscale Analysis
It can be immediately noticed the high computational cost demanded by the multiscale fracture model described in the previous chapter. This computational cost becomes unaffordable, even using resources of high-performance computing (also exhibited in the classical FEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^2
approach for smooth problems).
By virtue of the uncoupling of microscale state variables between integration points in a finite element setting, and, the high data transfer between scales, commonly known in hierarchical multiscale models, the necessity of reinterpreting the problem at the small scale becomes into a potential alternative.
It is worth noting that, from a mathematical standpoint, the term model reduction is conceptually akin to the more common term model discretization, since both connote transitions from higher-dimensional to lower-dimensional solution spaces. Indeed, whereas model discretization is used to refer to the classical passage from the infinite dimensional space, to the finite element subspace, model reduction denotes a transition from this finite dimensional space to a significantly smaller manifold, known as the reduced-order space. This latter transition is not carried out directly, but in two sequential parts, namely, sampling of the parameter space also called off-line part and dimensionality reduction called on-line part.
3.1 General Framework
Along this work, some techniques for reducing the unaffordable computational cost inherent to the numerical simulation of multiscale fracture problems have been developed. Those techniques are combined to obtain a hyper-reduced order model HPROM, based on a two-stage strategy:
- First Stage: also termed ROM, consists of a Galerkin projection, via Proper Orthogonal Decomposition (POD), onto a small space (reduced-order manifold), in which the set of modes conforming the projection basis are computed off-line.
- Second Stage: also called HyPer-Reduced Order Model (HPROM). Two different techniques have been developed here. The first one is based on interpolation methods, widely applied in problems exhibiting hardening behavior (see Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{1}}
) hernandez2014high. The second one, based on a Reduced Optimal Quadrature (ROQ) rule, has been applied to fracture (non-smooth) problems. Similarly to the previous stage, the reduced modes functions used to reconstruct the state variables, are computed off-line.
- Second Stage: is the development of the HyPer-Reduced Order Model (HPROM), and consist of redefining an adequate quadrature rule , demanding low computational cost. Two alternative quadrature rules have been developed. The first one is based on interpolation methods, widely applied in problems exhibiting hardening behavior, and the second one, based on a Reduced (sub-optimal) Quadrature rule (ROQ), being applied, in most cases, to fracture (non-smooth) problems. In the same way that the previous stage, the modes used to reconstruct the state variables, are computed in the off-line part.
In what follows, these techniques have been applied to the microscale BVP, while the finite element model at the macroscale remains as the standard one.
the latter takes advantage of the unbounded character of the strain field after the bifurcation time, originally intended for fracture processes, it can be extended to smooth problems, being more general than the former one.
3.2 Reduced-order modeling (ROM) of the RVE problem
The model order reduction concept relies on the premise that, for any input parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\mu } \in \mathcal{D}}
governing the microscale displacement fluctuations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde{u}}_\mu }}
, the solution can be approximated by a set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
linearly independent basis functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\Phi }}
approximately spanning the primal variable1 space.
Following this idea, the off-line stage is devoted to determine via a POD technique, the reduced basis in which the HF solution is projected. Once this basis has been obtained, a subsequent online stage in the reduced-space is considered.
(1) Primal variable is known as the selected variable to perform the reduction process.
3.2.1 Computation of the reduced basis functions
Taking as a primal variable the displacement fluctuations, and departing from the problem depicted in Sec. 2.1, a first step consists of determining an approximation1 of the finite element space of kinematically admissible microscale displacement fluctuations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}^{h}_{\mu }} . This approximation is obtained as the span of the displacement fluctuation solutions obtained, for a judiciously chosen set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{hst}}
input strain trajectories, every trajectory being discretized into a number of steps Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{stp}}
. These set of finite element solutions are stored into the snapshot matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{X}_{u}}
as column vectors:
|
(3.1) |
In consequence, the approximating space for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}^{h}_{\mu }} , henceforth called the snapshot space, is then defined as:
|
(3.2) |
where, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{snp}=n_{stp} \cdot n_{hst}}
is the total number of snapshots.
Once the snapshot matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{X}_{u}}
has been computed, the Elastic-Inelastic decomposition technique is used to determine the reduced basis functions. The reason for it relies on the fact that the SVD applied to the whole matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{X}_{u}}
, may produce basis with a large number of elements, which makes difficult to retrieve the response of the RVE in some specific cases. Particularly, the elastic response2, might request a much larger number of basis functions, this translating into a significant waste of computational cost.
To eliminate this shortcoming, in this work, it is proposed a time partition of the space of snapshots Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}_{u}^{snp}}
into elastic (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}_{u,el}^{snp}}
), and inelastic (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}_{u,inel}^{snp}} ) subspaces.
|
(3.3) |
See:
App. B
Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \# 1obtaining the reduced basis as the combination (spatial sum) of both sub-bases. An orthonormal basis for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}_{u,el}^{snp}}
is determined by taking a low number of elastic snapshots (at a minimum, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{snp}^e=3}
for 2D problems, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{snp}^e=6}
for 3D problems), and computing the corresponding orthonormal basis.
Once this set of elastic basis is known, the orthogonal projection of each snapshot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde{u}}}^{k}}
onto the orthogonal complement of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}_{u,el}^{snp}}
is computed; with this new set of snapshots, the inelastic basis functions are obtained via SVD. Finally, the assembled basis results the following:
|
(3.4) |
and the reduced-order space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}_{u}^{*}} , spanned by this base, is:
|
(3.5) |
Placing the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m_e}
elastic modes in the first Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m_e}
positions, followed by the essential3 inelastic modes, ensures the reduced-order model to deliver linear elastic solutions with the same accuracy than the HF solutions. For more details, the reader is encouraged to sent to the Appendix B in hernandez2014high.
3.2.1.1 Formulation of the reduced order model
Once the reduced basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\boldsymbol{\Phi }]}
is computed, See:
Sec. 4
Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \# 1 the online stage consists of solving the discrete version of the microscale equilibrium equation (via FE), projected onto the reduced-order space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{\tilde V}_{u}^{*}\subseteq \mathcal{\tilde V}^{h}_{\mu }}
spanned by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\boldsymbol{\Phi }]}
. To this end, the test and trial functions, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\eta }}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde{u}}_\mu }}
, are approximated by the following linear expansions:
|
(3.6) |
|
(3.7) |
where, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde{u}}^{*}_{\mu }}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\eta }}^{*}}
stand for the low-dimensional approximations of trial and test functions, respectively.
Introducing expressions 3.6 and 3.7 into the discrete version of the microscale BVP (see Sec. 4 in hernandez2014high), and multiplying the resulting expression by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\Phi }^{T}}
(Galerkin projection), it yields:
minipage0.95 PROBLEM A (ROM) (Microscale reduced problem via POD):
Given the macroscale strain, }{\boldsymbol{\varepsilon }}\hbox{, and the reduced basis for displacement fluctuations }\boldsymbol{\Phi }\hbox{, find }\boldsymbol{c}\in \mathbb{R}^{n_{\varepsilon }}\hbox{ satisfying: equation _B_ B^*T(y) _(y,+ B^*c,) dB__i=1^n_g B^*T(y_g,:) _(y_g,:) w_g = 0
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{c}=[c_1,c_2,\cdots ,c_{n_u}] \in \mathbb{R}^{n_u}}
denotes the vector containing the coefficients associated to each basis function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\Phi }_i}
, being Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{c}}
the basic unknowns for the standard reduced-order problem. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{B}^*}
stands for the reduced strain-displacement matrix “B-matrix” defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{B}^{*}(\boldsymbol{y})=\mathbf{B}(\boldsymbol{y}) \cdot \boldsymbol{\Phi }(\boldsymbol{y})}
. When using a Gauss quadrature integration scheme, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_g=\mathcal{O}(n)}
is the total number of Gauss points of the mesh; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w_g}
denotes the weight associated to the g-th Gauss point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{y}_g}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{B}(\boldsymbol{y}_g,:)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\sigma }}_{\mu }(\boldsymbol{y}_g,:)}
stand for the reduced B-matrix and the stress vector at Gauss point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{y}_g}
, respectively hdezMONOGR_141_HROM.
light-gray Remark: The general procedure is performed into two parts, first, the offline part is devoted to compute the reduced basis functions in 3.2.1, including the sampling program to store the snapshots selected from pre-selected training trajectories, second, the online part is devoted to solve the PROBLEM A, see Eq. 2.18. The homogenization of the stress and constitutive tensors are also included in the online part.
light-gray Remark: However, the HPROM based on the interpolation of the stress field, has been proved to be effective in hardening processes, the Reduced Optimal Quadrature method (ROQ), is a more general and robust strategy, it can be applied not only for hardening but also for softening processes. For details about the interpolation method, the reader is sent to read Secs. 5 to 7 in hernandez2014high.
(1) In general, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{\tilde V}^{h}_{\mu }
cannot be precisely determined, such a task will require finite element analyses of the cell under all conceivable strain paths. Rather, one has to be content to construct an approximation of it.
(2) Under an infinitesimal strain framework, this response is exactly recovered with only three basis hernandez2014high.
(3) Essential based on a threshold given by an a-priori error estimation, see Sec. 9.4 in hernandez2014high, thus, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{\tilde V}_{u,inel}^{snp,*}
corresponds to the truncated version of the full base with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): n_u-3
dominant modes.
3.2.2 Specific issues in non-smooth (fracture) problems
3.2.2.1 Domain separation strategy
Taking advantage of the unbounded character of the microscale strain field typically observed in this kind of problems, the failure cell is splitted into a regular domain (made of elastic matrix and possible inclusions) and a cohesive domain (cohesive bands exhibiting a softening cohesive behavior). Details on this issue can be found in Sec. 3.2.2 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{3}}
Oliver_Caicedo_HROM_2017.
3.2.3 Formulation of the microscale saddle-point problem
In addition to this proposal, the ROM of the failure cell is formulated in an unconventional manner, : in terms of strain fluctuations rather than in terms of conventional displacement fluctuations.
As it will be shown later, it is convenient to rephrase the original problem, posed in terms of displacement fluctuations (PROBLEM-I in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{3}} ). The primary unknowns of the rephrased problem are now the microscale strain fluctuations instead of its displacement fluctuations, while the constrained original minimum problem (of the standard micro-cell BVP) is rewritten in terms of a Lagrange functional. The resulting formulation is a variationally consistent saddle-point formulation.
Considering the material free energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\varphi _{\mu }}}
for the isotropic damage model in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{coh}}}}
, the microscale stress field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\sigma }}_{\mu }}
can be expressed as:
|
(3.8) |
complemented by the evolution laws of the internal variables Simo_Hughes_1998. Thus, in consonance with the hierarchical multiscale approach, the following parametrized functional can be defined:
|
(3.9) |
where, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\lambda }}(t) \in \mathbb{S}^{n \times n}} , is a symmetric second order tensor Lagrange multiplier enforcing condition 2.4 on the microscale strain fluctuations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde{\varepsilon }_{\mu }}}} . With this parametrized functional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Pi _{[{\boldsymbol{\varepsilon }},\boldsymbol{\mu }]} ({\boldsymbol{\tilde{\varepsilon }_{\mu }}},{\boldsymbol{\lambda }})}} , a saddle-point problem can be stated as:
minipage0.95 PROBLEM II (HF) (Microscale saddle-point problem): Given the macroscale strain, }{\boldsymbol{\varepsilon }}\hbox{, find }{\boldsymbol{\tilde{\varepsilon }_{\mu }}}\hbox{ and }{\boldsymbol{\lambda }}\hbox{ satisfying: equation _(,),(,) =arg\Big\{min__E_ _S^n n _[,] (_,)\Big\} Such that: equation* =f(_,)
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{E}_{\mu }}
stands for the space of microscale kinematically compatible strain fluctuations and, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f}
stands for the evolution equation of the internal variables. After considering that the microscale stress field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\sigma }}_{\mu }}
is given by Eq. 3.8, the following optimality conditions emerge:
|
Eqs. 3.10 and 3.11 provide the solution of the saddle-point problem stated in Eq. 3.2.3. It can be proven that Eqs. 3.10 and 3.11 make PROBLEM II equivalent to the original problem in Eq. 2.11, but now rephrased in terms of the microscale strain fluctuations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde{\varepsilon }_{\mu }}}}
(see PROBLEM I-R in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{3}}
).
3.2.3.1 Computation of the reduced basis functions
The transition from the high-dimensional finite element space to the reduced-order space, is accomplished by applying the POD technique, now for non-smooth problems. The standard reduced order model is based on the reduction of the strain fluctuation field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde{\varepsilon }_{\mu }}}} . The first step consists of generating a collection of solutions (samples) from different trial loading cases, representatives of all possible loading cases.
In each trial case, the microscale strain fluctuation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde{\varepsilon }_{\mu }}}}
at every Gauss point, is collected and stored in the snapshot matrix as a column vector:
|
|
(3.12) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{snp}}
is the number of snapshots vectors. See:
Sec.4
Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \# 3 Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{X}}
represents a number of sampled solutions obtained with the HF model under different loading conditions. For more details, the reader is addressed to Sec. 4 in Oliver_HROM_2017.
In order to get a more accurate estimation of the dominant modes of the microscale strain fluctuations, it is convenient to separate the microscale into specific sub-blocks in accordance with the type of material response observed during the load history. The procedure is sketched in Fig. 12:
|
|
Figure 12: (a) All entries of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{X}
are partitions into two sub-blocks: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{X}_{reg}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{X}_{coh}
, the strain fluctuations from points at regular and cohesive domains, respectively. (b) Snapshots taken from the elastic regime of the failure cell correspond to the sub-block Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{X}^{E} . Snapshots taken during the inelastic regime (at least one Gauss point at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathcal{B}_{\mu ,\hbox{coh}}} is in inelastic state) correspond to the sub-block Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{X}^{I}
. |
In addition, the Elastic-Inelastic snapshot decomposition above explained hernandez2014high, See:
Sec.3.2.2
Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \# 3 is also applied to each sub-block. Thus, without loss of generality, the snapshot entries are organized so that the first, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{g,reg}}
entries correspond to Gauss points in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu }}\setminus {\mathcal{B}_{\mu ,\hbox{coh}}}}
, while the remaining Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{g,coh}}
entries correspond to Gauss points in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{coh}}}}
.
In accordance with this criterion, the snapshot matrix 3.12 can now be partitioned into sub-matrices as follows:
|
(3.13) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{X}_{\hbox{reg}}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{X}_{\hbox{coh}}}
collect the strain fluctuations located outside and inside the cohesive bands, respectively. The right hand side matrix in 3.13 emphasizes the double partition performed in accordance with elastic-inelastic regimes.
With 3.13, the next step consists on computing the reduced basis corresponding to the elastic behavior of each subdomain. Those basis are computed via SVD, and this technique is applied, in a separate way, for each partition corresponding to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu }}\setminus {\mathcal{B}_{\mu ,\hbox{coh}}}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{coh}}}}
, resulting in the following reduced basis:
|
(3.14) |
where subindex G (standing for “Global”), is added to the matrix of modes, meaning that this matrix is built by pilling-up, in a single column, the modes evaluated at each Gauss point of the HF finite element model. The parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hbox{n}_{elas}}
stands for the number of base functions necessary to retrieve the elastic behavior for each partition (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hbox{n}_{elas}=3}
for infinitesimal strain problems).
In order to preserve the orthogonality of the full base Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Psi }}_{G}} , its inelastic part is computed with the component not contained in the space spanned by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {{\boldsymbol{\Psi }}_G^E}} . Thus, the inelastic snapshots Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{X}^{I}}
are projected onto the orthogonal complement of the subspace spanned by the basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {{\boldsymbol{\Psi }}_G^E}}
. The corresponding inelastic reduced basis functions are computed via SVD following similar procedure to the one used for the elastic reduced basis functions. The full reduced basis for the strain fluctuation field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Psi }}} , is composed by the union of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {{\boldsymbol{\Psi }}_G^E}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {{\boldsymbol{\Psi }}_G^I}}
After some additional manipulations, the corresponding set of orthonormal basis is obtained as:
|
(3.15) |
The number of reduced basis in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Psi }}_{G}}
is: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{\varepsilon }=6+n_{\varepsilon ,reg}^I+n_{\varepsilon ,coh}^I}
, where the values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{\varepsilon ,reg}^I}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{\varepsilon ,coh}^I}
are obtained from the SVD applied to the projected inelastic snapshots. Additional details can be found in Sec. 3.2.2 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{3}}
.
3.2.3.2 Formulation of the reduced order model
Once the reduced basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Psi }}_{G}}
is known, the strain fluctuations are interpolated as a linear combination of the elements of this basis as:
|
(3.16) |
where each element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Psi }}_i} , of the basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Psi }}_G} , is a microscale strain fluctuation mode and the vector of time dependent coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{c}(t)=[c_1,c_2,\cdots ,c_{n_{\boldsymbol{\varepsilon }}}]}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{c} \in \mathbb{R}^{n_{\boldsymbol{\varepsilon }}}}
) represents their corresponding amplitudes (the actual unknowns of the problem). See:
Sec.3.1
Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \# 3 In the same way, the variations of the microscale strain fluctuations are expressed similar to 3.16 as:
|
(3.17) |
The problem solved in the online stage is then the following:
minipage0.95 PROBLEM III (ROM) (RVE saddle point problem): Given the macroscale strain, }{\boldsymbol{\varepsilon }}\hbox{, find }\mathbf{c}\in \mathbb{R}^{n_{\varepsilon }}\hbox{ and }{\boldsymbol{\lambda }}\in \mathbb{R}^{n_{\sigma }}\hbox{ satisfying: align c(,),(,) = arg\Big\{min_cR_n_ _R^n_ _[,] ( c,)\Big\} equation* with _[,] ( c,)= _B_ _(+ c) dB_+ ^T \left(_B_ dB_\right)c such that: equation* =f(_,)
|
(3.18) |
|
|
The optimality conditions for the problem above yield:
|
which, expressed in matrix notation, yield:
|
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [{\boldsymbol{\sigma }}_{\mu }(\boldsymbol{c})]_{G} \in \mathbb{R}^{(n_{{\boldsymbol{\sigma }}} \cdot N_{g})}}
is the column vector constituted by piling-up the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{g}}
stress vectors, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\sigma }}_{\mu }(\boldsymbol{c}) \in \mathbb{R}^{(n_{{\boldsymbol{\sigma }}})}}
, evaluated at the integration Gauss points. The column vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [{\boldsymbol{\lambda }}]_{G}}
is also the pilled-up of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_g}
repeated values of the same constants vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\lambda }}\in \mathbb{R}^{(n_{{\boldsymbol{\sigma }}})}}
. See:
Sec. 3.2
Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \# 3 The square diagonal matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\mathbb{W}] \in \mathbb{R}^{(N_{g} n_{\boldsymbol{\sigma }}\times N_{g}n_{\boldsymbol{\sigma }})}} , and the rectangular matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [W] \in \mathbb{R}^{(n_{\boldsymbol{\sigma }}\times N_{g}n_{\boldsymbol{\sigma }})}} , collect the Gauss weights: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): w_1,w_2,...,w_{N_g} , which for plane strain cases (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{{\boldsymbol{\sigma }}} = 4} ) are distributed in sub-block matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{W}_{i} \in \mathbb{R}^{(4\times{4)}} (i=1,2,3,4,...,N_g)} , as follows:
|
(3.23) |
being Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{W}_i}
a matrix with the corresponding Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}
-th Gauss weight placed at the diagonal.
The unknowns for the reduced order model are, the modal amplitudes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{c}(t)} , and the Lagrange multiplier Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\lambda }}} .
light-gray Considering the system of equations 3.21 and 3.22 for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol{c}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\boldsymbol{\lambda }}
, it could be expected that this problem, of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): n_{{\boldsymbol{\varepsilon }}} + n_{\sigma }
equations, should be less computationally costly, than the HF model. However, this is not the case: the numerical simulations with the ROM model do not substantially reduce the computational cost, and little (or even smaller than one) speedups are obtained. This fact highlights that the actual bottleneck for fast online computation is not the solution of the balance equations but, rather, the determination of the stresses, internal forces and stiffness matrices at every integration point of the underlying finite element mesh. Therefore, an additional technique is proposed to reduce the amount of integration points in which the constitutive equation is evaluated.
light-gray Remark (B): The general procedure is performed into two parts, first, the offline part is devoted to compute the reduced basis functions in 3.2.3.1, including the sampling program to store the snapshots selected from pre-selected training trajectories, second, the online part is devoted to solve the PROBLEM III, see Eq. 3.2.3.2. The homogenization of the stress and constitutive tensors are also included in the online part.
3.3 Numerical Integration: Reduced Order Quadrature Technique (ROQ)
Next, an additional reduction step to diminish the computational burden for computing equation 2.18 or 3.21 and 3.22 is introduced. In addition, in order to guarantee the good performance of the numerical integration procedure, all possible a-priori operators1 have to be computed during the offline part. Particularly, the term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [W]\cdot [{\boldsymbol{\Psi }}_{G}]}
in Eq. 3.22 can be computed entirely in the offline part.
To reach the second stage objective, some approaches have been studied. They can be broadly classified either as interpolatory methods, or Gauss-type reduced quadrature strategies. In both type of approaches, the integrand or part of the integrand in those integral terms arising in the weak form of the microscale BVP, is approximated by a linear combination of a reduced set of empirical modes. In interpolatory approaches, the coefficients in this approximations are obtained by interpolation at a set of pre-selected sampling points; the criterion for choosing the location of such points is the minimization of the interpolation error over the finite element snapshots, or directly, in the reduced basis. In Gauss-type reduced quadrature, on the other hand, the selection of sampling points and its corresponding weight are simultaneously computed, guided by a criterion of minimum integration error compared with the exact integration scheme.
(1) A-priori operators can be interpreted as all possible algebraic operations that remain constant during the computation and that can be gathered in a global operator, computed during the offline stage. These operators are stored and called during the online stage for the global assembling of the discrete version of the optimality conditions (equilibrium equations).
3.3.1 Smooth Problems
Once the global displacement shape functions (reduced order basis functions) have been determined, the next step is to introduce an efficient method for the numerical evaluation of the integrals appearing in the weak form of the microscale BVP. Since displacement fluctuations solutions for the microscale BVP are constrained to lie in a reduced-order space of dimension Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_u \ll n} , it is reasonably to expect that the corresponding stresses, internal forces and Jacobians will also reside in reduced-order spaces of dimensions of the order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{O}(n_u)} , and consequently, only Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p=\mathcal{O}(n_u) < N_g}
sampling points would suffice, in principle, to accurately evaluate the corresponding integrals. To do so, the central idea of efficient interpolatory approaches for numerical integration of reduced-order BVPs is to replace the nonlinear term in the integrand by low-dimensional interpolants. In this case, the nonlinear term is the stress field, the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{B}^*}
in Eq. 2.18 is independent go the input parameter (macroscale strain tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\varepsilon }}}
) and hence need not be subject to approximation.
3.3.1.1 Computation of the reduced basis functions
The problem of constructing a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{O}(n_u)} -dimensional representation of the stress field is similar to the procedure previously presented for reducing the dimensionality of the displacement fluctuation field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde{u}}_\mu }} . The goal is to find a set of orthogonal basis functions, such that its span accurately approximates the set of all possible statically admissible stress solutions. In accordance with the procedure described in Sec. sec:SmoothPr_ROM, first, the finite element stress distribution for representative macroscale strain histories are computed. Then, the Elastic-Inelastic decomposition (see. Sec. 3.2.1 in hernandez2014high) is applied to the resulting set of stress solutions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [{\boldsymbol{\sigma }}_{\mu }^1(\boldsymbol{y}),{\boldsymbol{\sigma }}_{\mu }^2(\boldsymbol{y}),\dots ,{\boldsymbol{\sigma }}_{\mu }^{n_{stp}}(\boldsymbol{y})]}
in order to identify both the elastic and the essential inelastic stress modes.
The space spanned by these modes will be denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{V}_{\sigma }^{*}} , and termed reduced-order subspace of statically admissible stresses:
|
(3.24) |
With the reduced basis spanning Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{V}_{\sigma }^{*}}
at hand, the reconstruction of a low-dimensional approximation of the stress field, as a linear combination of reduced-order basis functions, is given by:
|
(3.25) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_{i} \in \mathbb{R} (i=1,2,3,\dots ,n_{\boldsymbol{\sigma }})} , the previous equation is replaced into the reduced equilibrium equation 2.18, giving rise:
|
(3.26) |
Remark: A close examination of Eq. 3.26 reveals that, the "standard" strategy proves completely fruitless, for it leads to ill-posed reduced order model equations. The reason relies on the fact that, every element of the stress field snapshot matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{X}_{\boldsymbol{\sigma }}}
is, by construction, obtained by a lineal combination of elements contained in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Psi }}}
, thus, in consonance with Eq. 2.18, in which the representative cell does not experience external forces, the space spanned by the reduced basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Psi }}}
is contained into the null space of the operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{B}^{*T}}
, namely, See:
Sec. 5.2
Paper 1 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{V}_{\sigma }^{*}\in \mathcal{N}(\mathbf{B}^{*T}} )), therefore, the integral of the second equality in 3.26, vanishes identically regardless of the value of the modal coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_i \in \mathbb{R} (i=1,2,3,\dots ,n_{\boldsymbol{\sigma }})} , and consequently, regardless the value of the reduced displacement fluctuations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde{u}}^{*}_{\mu }}} .
light-gray Remark: A close examination of Eq. 3.26 reveals that, the "standard" strategy proves completely fruitless, for it leads to ill-posed reduced order model equations. The reason relies on the fact that, every element of the stress field snapshot matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{X}_{\boldsymbol{\sigma }}
is, by construction, obtained by a lineal combination of elements contained in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\boldsymbol{\Psi }}
, thus, in consonance with Eq. 2.18, in which the representative cell does not experience external forces, the space spanned by the reduced basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\boldsymbol{\Psi }}
is contained into the null space of the operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{B}^{*T}
, namely, (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{V}_{\sigma }^{*}\in \mathcal{N}(\mathbf{B}^{*T} )), therefore, the integral of the second equality in 3.26, vanishes identically regardless of the value of the modal coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): c_i \in \mathbb{R} (i=1,2,3,\dots ,n_{\boldsymbol{\sigma }}) , and consequently, regardless the value of the reduced displacement fluctuations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\boldsymbol{\tilde{u}}^{*}_{\mu }} .
This deficiently posed mathematical problem, exhibited by the discrete formulation when adopting the standard approach using only POD modes, can be properly formulated by expanding the stress approximation spaceSee:
Sec. 5.2
Paper 1 so that it embraces also the span of the reduced strain-displacement functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{B}_{i}^{*} \in L_2({\mathcal{B}_{\mu }}) (i=1,2,3,\dots ,n_u)} , this technique was coined as the expanded basis approach1:
|
|
(3.27) |
3.3.1.2 Formulation of the Hyper-Reduced Order Model
In typical finite element implementations, both stresses and gradients of shape functions are only calculated and stored at the Gauss points of the underlying spatial discretization. For practical reasons, thus, it proves imperative to rephrase the expanded space previously mentioned, and treat both magnitudes as spatially discrete variables, defined only at such Gauss points.
Following this idea, the equilibrium equation 3.26 can be rephrased in a generalized matrix form as:
|
(3.28) |
|
|
where, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w}
stands for the integration Gauss quadrature weights. In consequence, any stress state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Sigma }}}
in the reduced-order model, See:
Sec. 5.3.2
Paper 1 can be decomposed into its admissible and inadmissible parts, namely, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Sigma }}= {\boldsymbol{\Sigma }}^{ad}+{\boldsymbol{\Sigma }}^{in}} , being Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Sigma }}^{ad} \in \mathcal{N}(\mathbb{B}^*)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Sigma }}^{in} \in \hbox{Range}(\mathbb{B^*})}
, then, the statically admissible component Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Sigma }}^{ad}}
is now approximated by a linear combination of reduced basis functions obtained via SVD from converged microscale stress snapshots. Thus, in accordance with Eq. 3.27, the low-dimensional (weighted) stress vector required in the proposed integration method is given by:
|
(3.29) |
where, the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Psi }}^{ex} = [{\boldsymbol{\Psi }}\; \mathbb{B}^*] \in \mathbb{R}^{n_{g} \cdot s \times (n_u + n_\sigma )}} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{c}=[\boldsymbol{c}^{ad} \; \boldsymbol{c}^{in}]^T \in \mathbb{R}^{(n_{\boldsymbol{\sigma }}+n_u)}}
are, the expanded basis matrix for the (weighted) stresses and the expanded vector of modal coefficients, respectively. Therefore, in consonance with Eqs. 3.28 and 3.29, the microscale equilibrium equation can be summarized as:
|
(3.30) |
The next step of the development of the proposed integration scheme is to deduce closed-form expressions for the vectors of modal coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{c}^{ad} \in \mathbb{R}^{N_\sigma }}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{c}^{in} \in \mathbb{R}^{N_u}}
in term of the stress values computed at a set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p=\mathcal{O}(n_u)}
pre-specified sampling points, to be chosen among the set of Gauss points of the underlying finite element mesh.
In the spirit of the classical polynomial quadrature, such as Newton-Cotes formulae HoffmanNMES_2001, the modal coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{c}^{ad} \in \mathbb{R}^{n_\sigma }}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{c}^{in} \in \mathbb{R}^{n_u}}
are determined by fitting the low dimensional approximation 3.29 to the weighted stresses calculated at the pre-specified sampling points. To solve this, a standard least-squares approach is used, , minimization of the squares of the deviations between “observed” (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{{\boldsymbol{\Sigma }}}}
) and fitted (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{{\boldsymbol{\Sigma }}}^{ex*} = \hat{{\boldsymbol{\Psi }}}\mathbf{a}+\hat{\mathbb{B}}^*\mathbf{b}} ) values2. This minimization problem can be stated as:
|
(3.31) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \| \cdot \| }
stands for the standard euclidean norm. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{{\boldsymbol{\Psi }}}^{ex}=\mathcal{P}_{(\mathcal{I})}{\boldsymbol{\Psi }}^{ex} = [\hat{{\boldsymbol{\Psi }}} \; \hat{\mathbb{B}}^*]}
be a sub-matrix taken from the original expanded basis matrix3, and suppose that the sampling indices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{I}}
have been chosen so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{{\boldsymbol{\Psi }}}^{ex}}
has full rank4, :
|
(3.32) |
Then, it can be shown that the solution of this standard, least-squares problem is provided by the following vector of coefficients DeVore_2001:
|
(3.33) |
where, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{{\boldsymbol{\Psi }}}^{ex \dagger }}
is the so-called pseudo-inverse of matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{{\boldsymbol{\Psi }}}^{ex}}
. By virtue of Eq. 3.30, it is convenient to find a closed expression for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{c}^{ad}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{c}^{in}}
. For doing so, taking advantage of the definition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Psi }}^{ex}}
and the solution of the least-square problem, and after some algebraic manipulations, finally:
|
(3.34) |
where, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{{\boldsymbol{\Psi }}}^{\dagger }=(\hat{{\boldsymbol{\Psi }}}^{T}\hat{{\boldsymbol{\Psi }}})^{-1}\hat{{\boldsymbol{\Psi }}}^{T}}
stands for the pseudoinverse of the gappy stress basis matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{{\boldsymbol{\Psi }}}}
, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{S}:=\mathbb{B}^{*^T}(\mathbf{I}-\hat{{\boldsymbol{\Psi }}} \hat{{\boldsymbol{\Psi }}}^{\dagger })} , See:
Sec. 6.3
Paper 1 this operator is invertible in virtue of the hypothesis presented in 3.32. In consequence, the modified microscale equilibrium equation 3.30 can be expressed in terms of the stress state associated with the sampling points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{{\boldsymbol{\Sigma }}}} , as:
minipage0.95 PROBLEM B (HPROM) (Microscale HyPer-Reduced Order Model via stress interpolation):
Given the macroscale strain, }{\boldsymbol{\varepsilon }}\hbox{, the reduced basis for the displacement fluctuations field }\boldsymbol{\Phi }\hbox{, the expanded reduced basis for the stress field }{\boldsymbol{\Psi }}^{ex}\hbox{, and the set of sampling points }\mathcal{S}=\{ \boldsymbol{y}_1,\boldsymbol{y}_2,\dots ,\boldsymbol{y}_{n_g}\} \hbox{, find }\boldsymbol{c}\in \mathbb{R}^{n_{\varepsilon }}\hbox{ satisfying: equation B^*^T(I-^)(,c,) = B^**^T(,c,) = 0
light-gray Remark (A): The “hyperreduced” qualifier - originally coined by Ryckelynck in Ryckelynck2005,Ryckelynck2009hyper is used here to indicate that Eq. 3.3.1.2 is the product of two sequentially steps of complexity reduction: first, a dimensional reduction in the number of degrees of freedom (when passing from the HF model to the ROM remaining the standard Gauss quadrature method), and, second, in the number of integration points (when passing from this ROM to the HPROM), the above mentioned process of complexity reduction can be symbolically represented as:
|
It is interesting to see how the reduction in complexity of the RVE equilibrium equation is reflected in the gradual reduction of the dimensions of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbb{B}
operators that act on the weighted vector of stresses.
light-gray Remark (B): As the ROM, the general procedure is also performed into two parts, first, the offline part is devoted to compute the reduced basis functions in 3.3.1.1, and the minimization problem to obtain the sampling points and its corresponding positions, Eq. 3.33 second, the online part is devoted to solve the PROBLEM B, see Eq. 3.3.1.2. The homogenization of the stress and constitutive tensors are also included in the online part.
3.3.1.3 Online Stage:
The online stage for smooth problems is devoted to solve the HPROM model described in Eqs. 3.3.1.2 the input parameters for this stage are the reduced bases for the displacement fluctuations and the Cauchy stress fields sumarized in the operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{\mathbb{B}}^{*^T}} , and the position of the sampling points in which the constitutive equation has to be evaluated. In addition, by virtue of the use of the computational homogenization approach, the homogenized Cauchy stress and the homogenized tangent constitutive tensors are also computed at every iteration.
Attention is then focused on reducing the computational cost arisen by the use of a classical Gauss quadrature for the numerical integration of the optimality conditions (Eqs. 3.19 and 3.20).
For this purpose, a reduced integration technique has been developed by resorting to a nonconventional method, termed Reduced Optimal Quadrature (ROQ), to integrate the term involving the microscale free energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\varphi _{\mu }}({\boldsymbol{\varepsilon }}+{\boldsymbol{\Psi }}\boldsymbol{c})}
in 3.2.3.2:
|
(3.39) |
Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \int _{*}(\cdot )\, d{\mathcal{B}_{\mu }}}
stands for he ROQ.
The ROQ technique is based on selecting, from the initial set of “Gauss” sampling points, and through an adequate algorithm, an equivalent subset of sampling points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z_j; \, j=1,2,\dots ,N_r} , and their new corresponding weights Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w_j; \, j=1,2,\dots ,N_r} . The success of the reduced integration numerical scheme, in front of the conventional Gauss quadrature, lies on the fact that it is possible to reduce notably the number of involved quadrature points to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_r \ll N_g} , being Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_g}
the number of integration points for the Gauss quadrature scheme, keeping under strict control, or even reducing to zero, the numerical error introduced by the reduced quadrature rule. Then, the microscale potential energy in Eq. 3.39, is re-expressed as:
|
(3.40) |
In consequence, the corresponding optimality conditions (equilibrium equations) to be solved during the online stage are:
minipage0.95 PROBLEM IV (HPROM) (Microscale reduced saddle-point problem): Given the macroscale strain, }{\boldsymbol{\varepsilon }}\hbox{, find }\boldsymbol{c}\in \mathbb{R}^{n_{\varepsilon }}\hbox{ and }{\boldsymbol{\lambda }}\in \mathbb{R}^{n_{\sigma }}\hbox{ satisfying: align _* ^T_(+ c) dB__Reduced o. quadrature + \left(_B_ ^T dB_\right)_Gauss quadrature = 0
\left(_B_ ^T dB_\right)_Gauss quadrature c =0
A similar procedure could also be used for the integral terms (underlined as “Gauss quadrature”) in Eqs. 3.3.1.3 and 3.3.1.3. However, this would not produce a substantial computational cost gain due to the fact that those terms are constant (not depending neither on the unknowns of the problem nor on the constitutive internal variables). They are required to be integrated only once, via the standard Gauss quadrature, and the result can be stored, and retrieved when necessary, during the online stage execution.
(1) For a deeper review of this approach, the reader is encouraged to read the Sec. 5.3 in hernandez2014high
(2) In this context, observed means calulated through the pertinent constitutive equation.
(3) The operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{P}_{(\mathcal{I})}
is the so-called selection operator associated to sampling indices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{I}
, for instance, the restricted matrix of weighted strain modes is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \hat{\mathbb{B}}^*=\mathcal{P}_{(\mathcal{I})}\mathbb{B}^*
(4) The selection criterion used for the set of sampling points in this approach, is fully explained in Sec. 7 in hernandez2014high, details about the optimality criteria, and its corresponding accuracy are deeply detailed in this section.
3.3.2 Reduced Optimal Quadrature
In spite that the goal of the ROQ is to develop a reduced cost interpolation scheme as a general framework for both static and dynamic problems, attention is focussed here on the multiscale quasi-static fracture problems. The minimum number of quadrature points providing an admissible integration error in the free energy integral, 3.39 is based on the optimal linear expansion of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\varphi _{\mu }}}
in terms of the free energy modes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Phi }}_i \, (i=1,2,3,\dots ,n_\varphi )}
and its corresponding amplitudes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_i}
. Thus, a similar expression to Eq. 3.16) can be adopted for constructing the reduced microscale strain fluctuations, as follows:
|
(3.41) |
With the previous approximation in hand, the total microscale free energy can be expressed as:
|
(3.42) |
Summarizing, two approximations have been introduced with respect to the HF integral of microscopic free energy. First, in 3.41, the approximation of the microscale energy based in the reduced basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Phi }}} . Second, the approximated numerical integration for every energy mode Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Phi }}_i} . In what follows, this second approximation is described.
3.3.3 A Greedy algorithm for obtaining a reduced quadrature rule
In order to obtain the reduced optimal numerical quadrature rule, the following optimization problem is considered:
minipage0.95 OPTIMIZATION PROBLEM : Given the expanded reduced basis }{\boldsymbol{\Phi }}\hbox{, and the set of sampling points }\mathcal{S}=\{ \boldsymbol{y}_1,\boldsymbol{y}_2,\dots ,\boldsymbol{y}_{N_g}\} \hbox{, find }{\boldsymbol{\omega }}\in \mathbb{R}^{N_r}_{+}\hbox{ and }\mathbf{\mathcal{Z}}\in \mathbb{N}^{N_r}\hbox{ satisfying: equation (,Z) = arg min_(w R_+^m,Z_g B_) _i=1^n_(e_i)^2 + (e_vol)^2
being:
|
(3.43) |
Where, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle e_i}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle e_{vol}}
stand for the error committed through the reduced integration of every free energy reduced basis function, and the error in the integration of the volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Omega _\mu }}
, respectively. The resulting algorithm (described in the flowchart of Box IV See:
Sec. 5
Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \# 4 in Sec. 5 of Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{4}}
Hernandez_HPROM_2017) returns a sub-set of optimal Gauss points, and the corresponding weights, that integrate exactly the basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Phi }}(\boldsymbol{y})}
and, therefore, the free energy in Eq. 3.41.
Apart from integrating the microscale energy modes with the ROC, See:
Sec. 4.1
Paper 4 an additional, necessary, condition has to be fulfilled in this approach. It is the integration of the volume1. The Eq. 3.3.3 is equivalent to Eq. 55 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{3}} .
light-gray Remark: The shortcoming presented for standard approaches in Sec. 4.3.1 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{4} , does not play any role in this case, the reason relies on the fact that, the integral of each microscale energy mode is not zero, indeed, the modes given by the SVD (regardless the precedence of the corresponding snapshots) form an orthonormal basis.
However, procedure depicted in Sec. 4.3.3 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{4}} 2 is now utilized, the reason is that, with the EBA, the additional condition to guarantee that the reduced cubature integrates properly the volume, emerges naturally, as a consequence of the expansion of the modified microscale energy modes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\hat{\Phi }}}}
with the modes that span the column space3 of the integral operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{R}(\mathcal{I})}
. In what follows, the procedure is detailed.
In the EBA, every energy mode Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Phi }}_i} , is projected onto the kernel of the integral operator, denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{N}(\mathcal{I})} , this projection can be computed by subtracting to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Phi }}_i}
its average value over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu }}}
, :
|
(3.44) |
Thus, the desired set of basis functions for the integrand is constructed as the union of the zero-average basis set, and a constant function (for instance, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g(\boldsymbol{y}) =1, \; \forall \boldsymbol{y}\in {\mathcal{B}_{\mu }}} ):
|
(3.45) |
With the expanded basis at hand, the minimization problem 3.3.3 can be cast in matrix format as follows:
|
(3.46) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{\boldsymbol{J}}_{\bar{\mathbf{\mathcal{Z}}}} \in \mathbb{R}^{n_{\varphi } \times N_r}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{\boldsymbol{b}} \in \mathbb{R}^{n_{\varphi }}}
are defined by:
|
(3.47) |
whereas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{1}^T=[1,1,\dots ,1]}
(an all-ones row vector of dimension Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_r}
)See:
Sec. 4.3.3
Paper 4, taking into account the definitions of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{J}_{\bar{\mathbf{\mathcal{Z}}}}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{b}}
, the objective function is rephrased as:
|
(3.48) |
It can be noticed that the problem stated in Eq. 3.48 is equivalent to the one presented in 3.3.3 and in Eq. 55 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{3}} .
By virtue of the use of a Finite Element technique, used to obtain a discrete formulation, integrals are computed by numerical quadrature, particularly, the classical Gauss quadrature. In order to define the new integration scheme, the minimization problem 3.46 is defined by choosing a small subset of all Gauss points, computing its corresponding optimal weights.
For doing so, the scheme described in Sec. 4.4.1 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{4}}
is utilized with slightly changes. It is worth nothing that, in consonance with the Remark 4.2 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{4}}
, for the use of the EBA, some changes in the microscale energy reduced basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Phi }}} , have to be introduced as follows:
|
(3.49) |
where:
|
(3.50) |
The basis matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde \Phi }}\in \mathbb{R}^{N_g \times n_{\varphi }}} See:
Sec. 4.4.1
Paper 4 is the projection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Phi }}\in \mathbb{R}^{N_g \times n_{\varphi }}}
onto the kernel of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sqrt{\boldsymbol{W}}^T \in \mathbb{R}^{N_g}}
. With Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde \Phi }}}
at hand, and applying an additional dimensionality reduction technique such as the SVD to obtain an approximated basis of rank Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_\varphi }
for the range of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde \Phi }}}
, :
|
(3.51) |
Thus, the expanded basis for the discrete minimization problem Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Gamma }}} , is defined as:
|
(3.52) |
The minimization problem to obtain the optimum placement of quadrature points and their corresponding weights, is quite similar to the continuous problem 3.46:
minipage0.95 DISCRETE OPTIMIZATION PROBLEM : Given the expanded reduced basis }{\boldsymbol{\Gamma }}\hbox{, and the set of sampling points }\mathcal{S}=\{ \boldsymbol{y}_1,\boldsymbol{y}_2,\dots ,\boldsymbol{y}_{N_g}\} \hbox{, find }\boldsymbol{\alpha } \in \mathbb{R}^{N_r}_{+}\hbox{ and }\boldsymbol{z}\in \mathbb{N}^{N_r}\hbox{ satisfying: equation (,z) = arg min_,z J_z-b ^2 Such that equation* 0
The heuristic method employed in the present work is described in the flowchart of Box IV See:
Sec. 5
Paper 4 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{4}} . This method is based in the Greedy algorithm of An and co-workers in their seminal paper An_2009. Two features distinguishes this algorithm: a) the definition of the matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{J}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{b}}
in the objective function, b) the use of the unrestricted least-squares scheme.
3.3.3.1 Computation of the reduced basis functions
Regarding the computation of the microscale energy reduced basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Phi }}} , a SVD-based strategy is used in the off-line stage, similar to that described in Sec. 3.2.1.
The method is again based on the construction of a snapshots matrix, in this case, for the free energy, and the computation of its corresponding reduced basis via SVD. For this purpose, two options appear:
- To construct the microscale energy snapshots by collecting solutions of the ROM in 3.2.3.2. This strategy implies the following actions: (a) the ROM is constructed considering a number of microscale strain fluctuation modes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_\varepsilon }
. In consequence, the obtained energy snapshots matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{X}^{\varphi }}
provides a reduced basis, which spans a space determined by the selected set of strain fluctuation modes.
- To construct the microscale energy snapshots by collecting solutions of the HF model in 3.2.3. This strategy computes simultaneously the microscale energy and the strain fluctuation snapshots. The resulting reduced bases are consequently independent from each other, but converging to the HF solution as the number of strain basis functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{\boldsymbol{\varepsilon }}}
, and energy basis functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_\psi } , increase.
light-gray The first method is considered as the algorithmically consistent strategy. However, it is also more expensive than the second one. The reason relies on the fact that, in order to get the reduced basis for the microscale energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\boldsymbol{\Phi }} , training trajectories have to be computed twice: a) First using the HF model to obtain the strain modes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\boldsymbol{\Psi }} , b) Second, using the ROM model to obtain the corresponding free-energy modes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\boldsymbol{\Phi }} .
Both strategies have been tested and both provide accurate results. However, the later, being the cheaper and simpler one, was adopted as the most convenient.
In summary, both the strains and the free energies of the microscale are sampled simultaneously at the off-line stage, for different sampling trajectories with the HF model, and a series of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q_{snp}}
snapshots of energy, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\varphi _{\mu }}}
, are evaluated and collected for each Gauss point. Then, the microscale energy snapshot matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{X}^{\varphi }}
is built as:
|
|
(3.53) |
In accordance with the position of the Gauss point,See:
Sec. 3.4
Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \# 3 in the finite element mesh, and following a similar procedure to that adopted in Eq. 3.13, this snapshot matrix is also partitioned into components associated to the domains Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu }}\setminus {\mathcal{B}_{\mu ,\hbox{coh}}}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{coh}}}}
as:
|
(3.54) |
and the SVD technique is then separately applied to both partitions of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathbf{X}}_\varphi ^{E}}
to obtain two distinct (orthogonal) bases, for the elastic regime of both subdomains:
|
(3.55) |
The corresponding inelastic reduced basis functions are also computed via SVD, following a procedure similar to the one described in Sec. 3.2.1. The complete reduced basis for the energy field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Phi }}} , is made of the union of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {{\boldsymbol{\Phi }}^{E}}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {{\boldsymbol{\Phi }}^{I}}}
|
(3.56) |
The number of basis vectors in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Phi }}}
is: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{\varphi }=12+n_{\varphi ,reg}^I+n_{\varphi ,coh}^I}
, where the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{\varphi ,reg}^I}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{\varphi ,coh}^I}
are obtained from the solution of the SVD applied to the inelastic projected snapshots.
In addition to the computation of the reduced basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\Phi }}} , in the offline part the ROQ based on the discrete minimization problem in Eq. 54 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{4}} , for both domains (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu }}\setminus {\mathcal{B}_{\mu ,\hbox{coh}}}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{B}_{\mu ,\hbox{coh}}}}
) is also computed.
light-gray Remark: As the ROM, the general procedure is also performed into two parts, first, the offline part is devoted to compute the reduced basis functions in 3.3.3.1, and the Discrete Optimization Problem to obtain the sampling points, its corresponding positions and optimal weights, Eq. 3.3.3 second, the online part (Sec. 3.5 in Paper 3) is devoted to solve the PROBLEM IV, see Eqs. 3.3.1.3 and 3.3.1.3. The homogenization of the stress and constitutive tensors are also included in the online part.
3.3.3.2 Online Stage:
The online stage for non-smooth cases is devoted to solve the HPROM model described in Eqs. 3.3.1.3 and 3.3.1.3,See:
Sec. 3.5
Paper 3 the input parameters for this stage are the reduced bases for the microscale energy field and the corresponding ROC, for both domains. In addition, by virtue of the use of the computational homogenization approach, the homogenized Cauchy stress and the homogenized tangent constitutive tensors are also computed at every iteration.
This approach is developed under a small strain framework, the equality of internal power at both scales is guaranteed via Hill-Mandell Macro-Homogeneity principle. In virtue of the finite element method, the dissipative processes that occur at the mesoscale are modeled using cohesive bands, represented by quadrilateral elements endowed with a regularized continuum damage model. These bands are characterized by a high aspect ratio (width smaller than its length and, in turn, that width being much thinner than the representative cell dimensions). In addition, scattered within the matrix, the aggregates and the interfaces between them are also included. In this way, they can model a set of predefined crack patterns including several mechanisms such as percolation of the crack through the matrix (necessary for softening behavior), mortar/aggregate decohesion and rupture between aggregates. Depending on the loading process at the large scale, these crack patterns are loading and unloading until the full consolidation, finally, a dominant mechanism naturally prevails, thus representing the final pattern of the micro-crack. This mechanism is now referred to as mesoscopic failure mechanism. That mechanism has several features, its form and orientation will be as precise as the richness of the lower scale, and is closely related to the crack orientation obtained at the large scale.
In this approach, the macroscopic constitutive response is proven to be point-wise equivalent to an inelastic law (in an incremental fashion) as a function of the homogenized elastic tangent tensor, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{C}^{hom}} , and the incremental homogenized inelastic strain rate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{{\boldsymbol{\varepsilon }}}^{(i)}}
i.e.:
|
(3.57) |
Where, the inelastic strain component Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{{\boldsymbol{\varepsilon }}}^{(i)}}
is expressed as a function of the homogenized variables taken from the lower scale, and represent the average value of the symmetrical tensor product between the strong discontinuity normal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{n}}
, and the rate of displacement jump Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{\boldsymbol{\beta }}}
of each cohesive band, belonging to the manifold of the mesoscopic failure mechanism Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{S}_{\mu }}
, i.e. the mesoscopic crack. In addition, the so-called material characteristic length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle l_{\mu }}
is defined as the ratio between the measure (volume or area) of the representative volume and the measure (surface or length) of the mesoscopic failure mechanism. The equations that govern the lower scales are the next:
PROBLEM I: Given a macroscale strain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{{\boldsymbol{\varepsilon }}}} , Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\tilde{u}}_\mu }}
such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\varepsilon }_\mu }= {\boldsymbol{\varepsilon }}+ \nabla ^{s} {\boldsymbol{\tilde{u}}_\mu }}
and:
|
(3.58) |
Regarding the large scale (macro-scale), it is modeled via the finite element method. The strain injection technique [oliver_crack-path_2014] is used in order to provide a robust and efficient model that can capture failure propagation even in high strain localization scenarios. In addition, the crucial matter of positioning strong discontinuities is tackled by a parallel technique termed crack-path field. This technique uses a directional derivative of a scalar field, based on a location variable (in our case, the average of mesoscale dissipated energy) whose zero level set defines the crack path.
(1) A natural restriction in numerical integration, is that the sum of the integration weights must be equivalent to the volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\Omega _\mu } , thus, in case of integrate constant functions, the result must be the total volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\Omega _\mu }
multiplied by a constant value.
(2) In that Section, the continuum setting is presented, the discrete version applied to a Finite Element framework, is presented in Sec. 4.4
(3) The column space is also known as the range of the operator.
3.3.4 Standard reduced order model (ROM) of the RVE problem
The reduction of the mesoscale strain field is based on the projection of the weak form of the discrete mechanical problem into a reduced manifold (reduced-order space), this reduced space is spanned by Ritz (globally supported) basis functions obtained via Singular Value Decomposition (SVD) of a set of snapshots taken from training tests computed during the offline part. Following this reasoning, the mesoscale strain fluctuation can be expressed as:
|
(3.59) |
Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{c}(t)=\{ c_{1},c_{2},c_{3},\dots ,c_{n_{{\boldsymbol{\varepsilon }}}}\} }
is time dependent (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{c}\in \mathbb{R}^{n_{\boldsymbol{\varepsilon }}}}
) and represents the amplitude of the corresponding mesoscale strain mode updated during the online part. Now, introducing (3.59) and (eq:EnergyPotential) into the PROBLEM IB and, after some straightforward manipulations, results into a new model written in terms of the reduced basis:
PROBLEM II: Given a macro-scale strain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{{\boldsymbol{\varepsilon }}}} , find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{c}\in \mathbb{R}^{n_{\boldsymbol{\varepsilon }}}}
satisfying:
|
(3.60) |
Solving the system of equations (3.60) for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{c}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\lambda }}
(Lagrange multiplier to ensure the equality of internal power at both scales via Hill-Mandel Macro-Homogeneity principle), it can be immediately noticed that this problem with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{{\boldsymbol{\varepsilon }}} + n_{\sigma }}
equations will be cheaper, (in computational cost terms), than the standard (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle FE^2}
) framework. However, the matricial form of PROBLEM II has to be computed (in a standard way) prior its projection onto the reduced-order space. This fact highlights that the actual bottleneck for fast online computation is not the solution of the discrete balance equations but, rather, the determination of the stresses, internal forces and stiffness matrices at all the integration points of the underlying finite element mesh and its posterior projection. Alternatively, this approach proposes a second stage based on the PROBLEM II, that intends to reduce the amount of integration points in which the constitutive equation is computed.
3.3.5 Hyperreduced-order model (HP-ROM) of the RVE problem
As pointed out in the previous section, the next objective is to introduce an additional reduction step to diminish the computational burden for equation (3.60-a). In addition, in order to guarantee the good performance for the second stage, all possible operators have to be computed during the offline part. Particularly, the term (3.60-b) can be computed entirely in the offline part. To persue the main objective of the second stage, we develop a Hyperreduced Order Model (HPROM) via Reduced Optimized Cubature (ROC), this technique is based on a discrete minimization problem that allows determining the optimized location of integration points and the corresponding weights. Once these positions and weights are at one's disposal, the equation (3.60-a) can be easily determined as:
|
(3.61) |
The success of our proposed scheme, relies on the fact that it is possible to find a set of integration points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_r} , substantially smaller than the ones given by the Gauss standard quadrature, minimizing the error in the assessment of (3.60). Introducing the expression (3.61) into the PROBLEM II, we get:
PROBLEM III: Given the macro-scale strain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{{\boldsymbol{\varepsilon }}}} , find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{c}\in \mathbb{R}^{n_{\boldsymbol{\varepsilon }}}}
satisfying:
|
(3.62) |
3.4 Numerical assessment and approximation errors
The accuracy of the reduced models, ROM and HPROM, depends on several aspects. In order to assess it, three different sets of tests are done:
- Consistency tests: A set of trajectories already sampled with the HF model during the microscale sampling process in the off-line stage, are re-evaluated using the ROM and HPROM strategies. This kind of assessment provides an estimation of the quality and richness of the basis to reproduce the stored snapshots, and the accuracy of the ROQ scheme. It is expected that the error with respect to the HF solutions (consistency error) tends to zero as the number of considered modes, for each reduction strategies, are increased.
- Accuracy tests: Similarly to the aforementioned consistency tests, the representative cell is subjected to a unsampled loading trajectory. In these cases, the quality of the reduced bases and the ROQ scheme is also tested. In contrast with the previous case, unsampled trajectories during the off-line stage, are not supposed to be exactly captured, due to the underlying sampling error.
- Multiscale structural tests: These kind of tests are based on multiscale benchmarks. The aim is to evaluate the accuracy of the solutions when the proposed overall HPROM strategy is applied, and to obtain the corresponding speed-ups.
Details on this issue can be found in Sec. 5 in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{3}} .
3.5 Representative example
A squared microscale model, made of a matrix and randomly distributed aggregates, is devised and tested (see figure 14) to simulate the microstructure of a cementitious-like material (concrete). Relevant details about the finite element model are presented in Table 1. To mimic the concrete material response, the failure cell is modeled with three components: aggregates, which are assumed to be elastic, bulk matrix, also assumed elastic, and interfaces (matrix-matrix and matrix-aggregates), which are modeled with cohesive-band equipped with an isotropic damage constitutive law. The properties of the components in the microscale are defined in Table 2.
|
|
| Figure 13: Failure cell |
| Number | Number | Number of | Total number of |
| of elements | of D.o.f. | Cohesive Bands | Gauss points (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{g}}
) |
| 5409 | 14256 | 2189 | 21636 |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): E_{\mu }
[MPa] |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \nu _{\mu } | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sigma _{\mu u}
[MPa] |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): G_{\mu f}
[N/m] | |
| Elastic matrix | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1.85\times{10}^4 | 0.18 | –- | –- |
| Elastic aggregate | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 3.70\times{10}^4 | 0.18 | –- | –- |
| Cohesive bands of | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1.85\times{10}^4 | 0.18 | 2.60 | 140 |
| matrix-matrix interface | ||||
| Cohesive bands of | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1.85\times{10}^4 | 0.18 | –- | –- |
| matrix-aggregate interface |
3.5.1 Design of the HPROM Strategy
Figure 14 shows the summary of a number of results obtained by running the HPROM strategy in a number of cases for the microstructure in Fig. 13.
In general terms, Figure 14 can be used as an “abacus” for a-priori selection by the user of the HPROM strategy in a multiscale problem (for a given microstructure at the RVE). For instance, by selecting the admissible error ( 3,5%) in the top figure, the number of strain modes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{\varepsilon }=80)}
is obtained. Entering in the lower plot, with this result (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{\varepsilon }=80}
), one obtains the suitable number of integration points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle OQN\simeq{200}}
and the expected speedup Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \simeq{110}}
.
The availability of a catalog (constructed off-line) for a specific RVE microstructure, allows the user's a-priori selection of the appropriated HPROM strategy, by balancing the admissible error vs. the desired speedup.
|
|
Figure 14: HPROM design diagrams. Top: HPROM error in terms of the number of strain modes. Bottom: OQN and obtained speedup in terms of the number of strain modes. By selecting the admissible error (say 3,5%) in the upper diagram, one obtains the requested number of strain modes, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): n_{\varepsilon }=80.
Entering with this result in the lower diagram one obtains the suitable number of integration points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (OQN=200)
and the resulting speedup (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): speedup=110)
. |
3.5.2 Multiscale crack propagation problem: L-shaped panel
The test shown in Figure 15 is a benchmark commonly used for testing macroscale propagating fracture models. This concrete-like specimen is considered here to test the qualitative results and convergence properties of the proposed HPROM approach, when utilized in real FEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^2
multiscale crack propagation problems.
|
|
| Figure 15: L-shaped panel: a) Specimen geometry; b) Finite element mesh |
The geometry of the simulated specimen is depicted in Figure 15-a. As shown in Figure 15-b, the domain of the L-shaped panel is split into two domains: 1) the multiscale domain (with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 721}
elements) corresponding to the region where the crack is expected to propagate, modeled with the HPROM of the microstructure depicted in Figure 13, and 2) the remaining part of the panel, which is modeled with an elastic monoscale approach (using 1709 elements), where the elasticity tensor is obtained through an elastic homogenization of the micro-structure elastic properties. Even for this (rather coarse) multiscale problem, the high fidelity HF computational solution is extremely costly to handle, until the point that, with the available computational resources1, it was not possible to display the complete action-response curve (in Fig. 16).
However, the remaining structural responses in Figure 16, obtained through a number of HPROM strategies, involve very reasonable computational costs, and they were obtained in advance with no previous knowledge of the HF results. The accuracy is very good, and a response indistinguishable from the HF can be obtained Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 60}
times faster (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle speedup=60}
). A less accurate response, but with a fairly good agreement with the HF can be obtained with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle speedup=130} .
|
|
| Figure 16: L-shaped panel: Structural responses in terms of force P vs. vertical displacement Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta
, for different RVE HPROM strategies, and obtained speed-ups. |
In Fig. 17, the evolution of the microscale crack opening is shown. It is worth noting that, both, the microscale failure mechanism and displacement jump vary along the macroscale in agreement with the crack propagation direction observed at the macroscale.
However, the remaining structural responses in figure 15-d, obtained through a number of HPROM strategies, involve very reasonable computational costs, and they were obtained in advance with no previous knowledge of the HF results. The accuracies are very good, and a response indistinguishable from the HF can be obtained 60 times faster (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): speedup=60 ). A less accurate response, but with a fairly good agreement with the HF can be obtained with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): speedup=130
This illustrates the new paradigm and computational possibilities open by HPROM strategies in computational multiscale modeling.
|
|
Figure 17: L-shaped panel: microscale crack activation along the crack-path field, using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): n_{\boldsymbol{\varepsilon }}=80
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): n_r=258
. |
This illustrates the new paradigm that is set and the computational possibilities open by the HPROM strategies in computational multiscale modeling explored in this work.
(1) A cluster of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 500
cores, is used. The multiscale finite element code is written in Matlab©environment.
3.5.3 Consistency tests
Three sources of error are involved in the consistency tests. First, by considering that the reduced basis for the POD procedure used in the ROM, neglects modes associated with singular values smaller than a given limit. This error is identified as the a-priori error of the ROM, and can be determined during the offline stage. A second source of error is associated with the snapshot sampling technique, in the sense only a few snapshots, of each sampled trajectory, are taken to build the snapshot matrix, this error applies for both models (ROM and HPROM).
The ROM reproducing sampled trajectories has associated an error, identified as the a -posterori error, which is considered as the combination of the a-priori and sampling errors. Moreover, the HPROM involves an additional source of error, which is governed by the interpolation or the reduced quadrature schemes.
3.5.4 A-priori consistency error
Once the POD basis is at one's disposal, the a-priori (percentaje) error in approaching every snapshot can be estimated through the following formula:
|
(3.63) |
where, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Theta }
stands for the reduced basis, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M}
stands for the model from which the snapshot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{X}_i^{M}}
has been taken, , HF model or ROM, and the operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \| \cdot \| }
corresponds to the euclidean norm. Clearly, if the space spanned by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Theta }
is rich enough to reproduce Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{X}_i^{M}}
, the error 3.63 is zero. Then, the maximum error is given by:
|
(3.64) |
and gives one a global view about the richness of the reduced basis for capturing all the snapshot matrix. In case of use the domain partition strategy, being every partition dealed in a separated way, a-priori error must be computed separately.
3.5.5 A-posteriory consistency error
The a-posteriori error can be computed in a similar way, however, to obtain this error, is necessary to compute the online stage. In this case, not only the primal variables can be compared with the ones obtained by the HF model, the general expression is the following:
|
(3.65) |
where, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M}
in this case change between models, from costly to cheaper, in computational cost terms, this clever notation was adopted in order to show that the error assessment must be consistent, for instance, in case of compare the accuracy of the first reduction, the ROM will adopt the index (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M+1}
), and the HF model will adopt the index Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M} , in case of compare the accuracy of the second reduction, the HPROM adopts the index (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M+1} ), and the HF model or ROM adopt the index (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M} ).
Particularly, for fracture problems, it is convenient to have control on the error committed in the traction vector over the crack plane, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{t}} , this vector has sense only after the bifurcation time, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_b} , this error can also be integrated in time:
|
(3.66) |
3.5.6 Speedup
Not only the previous measures of error are controlled in model order reduction, other important parameter, perhaps the most important one, is the so-called speedup, which aims at comparing the computation time of the reduced order model with the HF model.
However, since the computational time depends on several factors, particularly, the computer architecture, the possible sources of additional tasks occupying the RAM memory at the same time, the speedup must be considered as the maximum value obtained from a set of representative tests in the same machine. It can be expressed as:
|
(3.67) |
It can also be computed for both, ROM and HPROM, fixing in the denominator the corresponding time.
A very important factor for successful thesis writing is the organization of the material. This template suggests a structure as the following:
You can use these margins for summaries of the text body
-
FrontBackMatter/is where all the stuff goes that surrounds the “real” content, such as the acknowledgments, dedication, etc. -
gfx/is where you put all the graphics you use in the thesis. Maybe they should be organized into subfolders depending on the chapter they are used in, if you have a lot of graphics. -
Bibliography.bib: the BibTeX database to organize all the references you might want to cite. -
classicthesis.sty: the style definition to get this awesome look and feel. Does not only work with this thesis template but also on its own (see folderExamples). Bonus: works with both and pdfand . -
Classicwork.tcpa TeXnicCenter project file. Great tool and it's free! -
Classicwork.tex: the main file of your thesis where all gets bundled together. -
classicthesis-config.tex: a central place to load all nifty packages that are used. In there, you can also activate backrefs in order to have information in the bibliography about where a source was cited in the text (, the page number).
Make your changes and adjustments here. This means that you specify here the options you want to load classicthesis.sty with. You also adjust the title of your thesis, your name, and all information.
This had to change as of version 3.0 in order to enable an easy transition from the “basic” style to .
In total, this should get you started in no time.
There are a couple of options for classicthesis.sty that allow for a bit of freedom concerning the layout: or your supervisor might use the margins for some comments of her own while reading.
- General:
-
drafting: prints the date and time at the bottom of each page, so you always know which version you are dealing with. Might come in handy not to give your Prof. that old draft.
- Parts and Chapters:
-
parts: if you use Part divisions for your document, you should choose this option. (Cannot be used together withnochapters.)
-
nochapters: allows to use the look-and-feel with classes that do not use chapters, , for articles. Automatically turns off a couple of other options:eulerchapternumbers,linedheaders,listsseparated, andparts.
-
linedheaders: changes the look of the chapter headings a bit by adding a horizontal line above the chapter title. The chapter number will also be moved to the top of the page, above the chapter title.
- Typography:
-
eulerchapternumbers: use figures from Hermann Zapf's Euler math font for the chapter numbers. By default, old style figures from the Palatino font are used.
-
beramono: loads Bera Mono as typewriter font. (Default setting is using the standard CM typewriter font.) -
eulermath: loads the awesome Euler fonts for math. (Palatino is used as default font.)
-
pdfspacing: makes use of pdftex' letter spacing capabilities via themicrotypepackage.1 This fixes some serious issues regarding math formul etc. (, ``) in headers.
-
minionprospacing: uses the internaltextssccommand of theMinionPropackage for letter spacing. This automatically enables theminionprooption and overrides thepdfspacingoption.
- Table of Contents:
-
tocaligned: aligns the whole table of contents on the left side. Some people like that, some don't.
-
dottedtoc: sets pagenumbers flushed right in the table of contents.
-
manychapters: if you need more than nine chapters for your document, you might not be happy with the spacing between the chapter number and the chapter title in the Table of Contents. This option allows for additional space in this context. However, it does not look as “perfect” if you use \parts for structuring your document.
- Floats:
-
listings: loads thelistingspackage (if not already done) and configures the List of Listings accordingly.
-
floatperchapter: activates numbering per chapter for all floats such as figures, tables, and listings (if used).
-
subfig(ure): is passed to thetocloftpackage to enable compatibility with thesubfig(ure) package. Use this option if you want use classicthesis with thesubfigpackage.
The best way to figure these options out is to try the different possibilities and see, what you and your supervisor like best.
In order to make things easier in general, classicthesis-config.tex contains some useful commands that might help you.
This section will give you some hints about how to adapt classicthesis to your needs.
The file classicthesis.sty contains the core functionality of the style and in most cases will be left intact, whereas the file classicthesis-config.tex is used for some common user customizations.
The first customization you are about to make is to alter the document title, author name, and other thesis details. In order to do this, replace the data in the following lines of classicthesis-config.tex: Modifications in classicthesis-config.tex
% **************************************************
% 2. Personal data and user ad-hoc commands
% **************************************************
\newcommand{\myTitle}{A Classic work Style\xspace}
\newcommand{\mySubtitle}{An Homage to...\xspace}
Further customization can be made in classicthesis-config.tex by choosing the options to classicthesis.sty
\PassOptionsToPackage{eulerchapternumbers,drafting,listings,subfig,eulermath,parts}{classicthesis}
If you want to use backreferences from your citations to the pages they were cited on, change the following line from:
\setboolean{enable-backrefs}{false} % true false
to
\setboolean{enable-backrefs}{true} % true false
Many other customizations in classicthesis-config.tex are possible, but you should be careful making changes there, since some changes could cause errors.
Finally, changes can be made in the file classicthesis.sty, NOTE: Modifications in classicthesis.sty although this is mostly not designed for user customization. The main change that might be made here is the text-block size, for example, to get longer lines of text.
(1) Use microtype's DVIoutput option to generate DVI with pdftex.
Outline
This section will list some information about problems using classicthesis in general or using it with other packages.
Beta versions of classicthesis can be found at the following Google code repository:
http://code.google.com/p/classicthesis/
There, you can also post serious bugs and problems you encounter.
Compatibility with the glossaries Package
If you want to use the glossaries package, take care of loading it with the following options:
\usepackage[style=long,nolist]{glossaries}
Thanks to Sven Staehs for this information.
Compatibility with the (Spanish) babel Package
Spanish languages need an extra option in order to work with this template:
\usepackage[spanish,es-lcroman]{babel}
Thanks to an unknown person for this information (via Google Code issue reporting).
In order to get the bibliography style right, you can use the following:
\bibliographystyle{babplain}
For this, it is necessary to load the package:
\usepackage[spanish,fixlanguage]{babelbib}
\selectbiblanguage{spanish}
If there are issues changing \tablename, , using this:
\renewcommand{\bibname}{Referencias}
\renewcommand{\tablename}{Tabla}
This can be solved by passing es-tabla parameter to babel:
\PassOptionsToPackage{es-tabla,spanish,es-lcroman,english}{babel}
\usepackage{babel}
But it is also necessary set spanish in the \documentclass.
Thanks to Alvaro Jaramillo Duque for this information.
Compatibility with the pdfsync Package
Using the pdfsync package leads to linebreaking problems with the graffito command. Thanks to Henrik Schumacher for this information.
4 Discussion, Conclusions and Future Work
4.1 Discussion and Conclusions
4.1.1 Overview of the work
Multiscale modeling is foreseen to become a key approach to enable the next wave of design paradigms for engineering materials and structures. Indeed, it has an excellent potential to account for the physical links between different scales, involving the diverse phenomenologies intervening in the mechanical response of materials (grains, particles, defects, inclusions, etc.).
Quoting from a report by a group of experts to the US National Science Foundation ReportNSF:
". . . . In recent years, a large and growing body of literature in physics, chemistry, biology, and engineering has focused on various methods to fit together simulation models of two or more scales, and this has led to the development of various multi-level modeling approaches. . . .. To date, however, progress on multiscale modeling has been agonizingly slow. Only a series of major breakthroughs will help us establish a general mathematical and computational framework for handling multiscale events and reveal to us the commonalities and limitations of existing methods . . . .".
In this sense, the effort invested in developing and using multiscale models, has been, in many cases, fruitless, due to the involved computational cost in this kind of methodologies. This limitation becomes a bottleneck for multiscale modeling, usually discarded, or, relegated to the availability of supercomputers, and, therefore, not always accessible to the whole computational mechanics community.
In addition, while multiscale models exhibiting material hardening behavior have widely been studied, multiscale models dealing with material softening behavior are in an early stages of development.
Therefore, the development of a reliable, minimally intrusive multiscale fracture models becomes a crucial task, not only in order to have a robust and consistent multiscale fracture numerical tools, but also for developing their related reduced order models that allow their use in complex cases that can be used for industrial purposes, with an affordable cost. These are the fundamental reasons for the research and development about this issues.
In accordance with this idea, the main objectives of this work can be summarized as follows:
To develop, implement and validate a set of computational tools to reduce, considerably, the unaffordable computational burden induced by models which involve interaction between scale levels. The so-developed model should be enough general, to be applied to solid mechanics problems, exhibiting either, hardening or softening behavior.
Currently, many multiscale models have been developed using different approaches to transfer information between scales. The decision adopted in this work to use the FEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^2
method, relies on the following premises:
- There is no limitations related with the scale separation and mesh density of the sub-scales. In contrast with , concurrent models, in which its applicability is feasible for materials with small scale-separation (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {h_\mu }\approx {L}}
).
- In virtue of the coupling between scales handled by this technique, and the fact that the microscale equilibrium BVP can be separately solved from the macroscale problem, then, the use of computational parallelization procedures via MPI protocol are suitable, and straightforwardly applied.
- For reduced order modeling purposes, the training trajectories are selected based on the input parameters of the microscale BVP, in this case, the macroscopic strain tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol{\varepsilon }}}
. An important issue for defining an optimum training program lies on the fact that this strain tensor can be parametrized by three components, in 2D plane state problems.
A sketch of the overall work carried out in this work is shown in Figure 18. In there, contributions are chronologically numbered and highlighted with a blue arrow. Contributed papers are numbered from P1, corresponding to the the first contribution (Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{1}} ), to P6 (Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{6}} ) the last one; in this context, CB means Chapter in Book.
In what follows, they are specifically commented, and the corresponding conclusions and achievements, are presented.
|
| Figure 18: Global Flow Chart of the work |
4.1.2 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{1}
- J. A. Hernández, J. Oliver, A. E. Huespe, M. A. Caicedo, J. C. Cante. High-performance model reduction techniques in computational multiscale homogenization, Computer Methods in Applied Mechanics and Engineering - 2014, Volume 276, Pages 149–189
4.1.2 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{1}
J. A. Hernández, J. Oliver, A. E. Huespe, M. A. Caicedo, J. C. Cante. High-performance model reduction techniques in computational multiscale homogenization, Computer Methods in Applied Mechanics and Engineering - 2014, Volume 276, Pages 149–189
This Article presents the first research developments in this work on MOR techniques applied to multiscale modeling. The scope of this publication is limited to smooth problems and exclude fracture processes. Techniques like interpolation methods via HPROM have been studied.
The concept of a two-stage reduction (ROM-HPROM) is presented. The first reduction, denoted as ROM, is performed via POD, taking the displacement fluctuation field as a primal variable. The second reduction, denoted as HPROM, is performed via interpolation techniques (DEIM) of the microscale stress field.
It is shown that the interpolation-based HPROM obtained in this way, leads to an ill-posed mathematical formulation when the reduction process involves an interpolant constructed using POD modes provided by the primal variable (microscale stress field). This issue has been studied in the paper, and a robust and consistent solution has been proposed.
An additional aspect in this contribution, is the selection of the interpolation points for the stress field. These interpolation points are chosen guided, not only by accuracy requirements, but also by stability considerations.
The method of selection of the interpolation points (Greedy Algorithm) is, at the present, an intensive research field. However, although in the literature there are several alternative algorithms, none of them offers a robust and general treatment to handle this purposes.
Different measures of error have been presented to test the accuracy and the convergence. The work is assessed by the homogenization of a highly complex porous metal material. The results show that, the speed-up factor is about three orders of magnitude, for an error in stress smaller than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 10%}
.
As conclusions of this work, it can be stated that:
- The hyper-reduced form of the RVE equilibrium equation has a conceptual simplicity, and the corresponding solution scheme is also very simple to implement. Taking as departure point an existing FE code, one only has to replace the typical loop over elements in the FE code by a loop over the pre-selected sampling points.
- Storage of history data (internal variables) is only required at the pre-selected sampling points.
- Consistency with respect to the HF solution is achieved when the amount of reduced order basis functions, for both reductions, is increased.
- Can the model order reduction techniques capture the RVE solution in problems displaying crack propagation processes?
- Will the number of modes necessary to accurately replicate its solution, increase with the number of potential crack paths (i.e., with the geometrical complexity of the RVE)?
4.1.3 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{2}
- J. Oliver, M. Caicedo, E. Roubin, A. E. Huespe, J. A. Hernández. Continuum approach to computational multiscale modeling of propagating fracture, Computer Methods in Applied Mechanics and Engineering - 2015, Volume 294, Pages 384–427
4.1.3 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{2}
J. Oliver, M. Caicedo, E. Roubin, A. E. Huespe, J. A. Hernández. Continuum approach to computational multiscale modeling of propagating fracture, Computer Methods in Applied Mechanics and Engineering - 2015, Volume 294, Pages 384–427
This work presents a novel approach to two-scale modeling of propagating fracture, based on computational homogenization FEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^2
. The specific features of this contribution are:
- Extends the homogenization paradigms for smooth problems presented in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{1}}
- In both scales of analysis, a continuum (stress–strain) constitutive relationship is considered, instead of making use of the most common discrete traction/separation-law. This contributes to provide a unified setting for smooth and non-smooth, problems. This is achieved by resorting to the well-established Continuum Strong Discontinuity Approach (CSDA).
- As for the multiscale modeling issue, it involves a new and crucial additional entity: a characteristic length, which is point-wise obtained from the geometrical features of the failure mechanism developed at the low scale. As a specific feature of the presented approach, this characteristic length is exported, in addition to the homogenized stresses and the tangent constitutive operator, to the macroscale, and considered as the bandwidth of a propagating strain localization band, at that scale.
- Consistently with the characteristic length, a specific computational procedure is used for modeling the onset and propagation of this localization band at the macro-scale. It is based on the crack-path-field and strain injection techniques, developed oliver2014crack. This computational procedure ensures the macroscale mesh-size and microscale RVE-size objectivity of the results, and a consistent energy dissipation at both scales.
- From the computational point of view, the proposed technique is minimally invasive with regards to procedures well established in the literature on multiscale modeling of materials. In fact, in terms of the computational homogenization, the proposed approach displays no substantial difference with respect to the ones used for smooth (continuous) problems. In terms of material failure propagation, existing algorithms for monoscale crack propagation modeling can be easily extended to this multiscale case. In addition, this multiscale approach is extensible to other families of propagation schemes.
- Consistency has been assessed by comparison, with a number of representative cases, through results obtained with the proposed FEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^2
- Inclusion of non-linear hardening behavior, before the onset of material failure can be trivially included in the considered damage model. In the same way, consideration of other families of constitutive behavior, like plasticity, rate dependence etc., can be simply done by including these effects in the basic constitutive model.
- Using other propagating crack models at the micro-scale, either based on continuum strain-localization methods (CSDA, non-local models or gradient-regularized models) or discrete methods (based on cohesive interfaces equipped with traction-separation laws) where the crack onset and propagation is not predefined (just as it is done here at the macro-scale) also fits in the approach at the cost of some additional sophistication.
4.1.3.1 HPROM for hardening processes applied to quasi-brittle fracture
The reduced order model described in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{1}}
was used as a first attempt. However, the results were very unsatisfactory. The conclusions of this interpolation-based approach to multiscale reduced order modeling in fracture cases were:
- The reduced basis for the microscale displacement fluctuations obtained via SVD does not make a clear distinction between smooth and non-smooth domains. Hence, a large set of displacement modes (considerably larger than the one requested in hardening problems) has to be used to retrieve accurate solutions.
- The stress snapshots, taken from high localized strain stages with released near-to-zero stresses, are numerically neglected by the SVD1, this taking interpolation-based HPROM methods to fail in reproducing the post-critical stages.
- To obtain a good approximation with the HPROM, it is necessary to largely increase the number of displacement and stress modes, but, in this scenario, the interpolation method is not longer robust.
This suggests additional research and exploration of specific model order reduction techniques for multiscale fracture problems.
(1) The SVD strategy, gives importance to repeated snapshots, and mainly, snapshots which euclidean norm is considerably high.
4.1.4 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{3}
- J. Oliver, M. Caicedo, A. E. Huespe, J. A. Hernández, E. Roubin. Reduced Order Modeling strategies for Computational Multiscale Fracture, Computer Methods in Applied Mechanics and Engineering - 2016, Volume 313, Pages 560–595
4.1.4 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{3}
J. Oliver, M. Caicedo, A. E. Huespe, J. A. Hernández, E. Roubin. Reduced Order Modeling strategies for Computational Multiscale Fracture, Computer Methods in Applied Mechanics and Engineering - 2016, Volume 313, Pages 560–595
This article proposes a set of new computational techniques to solve multiscale problems via HPROM techniques. These techniques have been applied to the multiscale model described in [[#4.1.3 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{2}
- J. Oliver, M. Caicedo, E. Roubin, A. E. Huespe, J. A. Hernández. Continuum approach to computational multiscale modeling of propagating fracture, Computer Methods in Applied Mechanics and Engineering - 2015, Volume 294, Pages 384–427|4.1.3]], and they are summarized next:
- J. Oliver, M. Caicedo, E. Roubin, A. E. Huespe, J. A. Hernández. Continuum approach to computational multiscale modeling of propagating fracture, Computer Methods in Applied Mechanics and Engineering - 2015, Volume 294, Pages 384–427|4.1.3]], no reduction have been performed. In consequence, all parameters needed for solving the equilibrium problem, and the failure propagation scheme, are computed once the microscale HPROM has been solved.
- J. A. Hernández, J. Oliver, A. E. Huespe, M. A. Caicedo, J. C. Cante. High-performance model reduction techniques in computational multiscale homogenization, Computer Methods in Applied Mechanics and Engineering - 2014, Volume 276, Pages 149–189|4.1.2]], two simultaneously reductions are also used, the ROM is also performed via POD, while, the HPROM is improved using a ROQ technique, to compute the microscale integration equations.
- A domain separation strategy. The RVE is split into the regular domain (made of the elastic matrix and possible inclusions) and the singular domain (the cohesive bands exhibiting a softening cohesive behavior). These are designed to provide a sufficiently good representation of the microscopic fracture and of its effects on the homogenized material behavior oliver2015continuum. The distinct constitutive behavior of both domains suggests a specific ROM strategy for each of them, in order to obtain a reduction strategy with information on the mechanical variables in every specific sub-domain. Therefore, selection of the ROM low-dimensional projection space is made independently for each of these domains.
- In combination with the previous strategy, the ROM for the RVE is formulated in an unconventional manner : in terms of the strain fluctuations rather than in terms of the conventional displacement fluctuations. The reduced strain fluctuation space is spanned by basis functions satisfying, by construction, the strain compatibility conditions, this guaranteeing that, after reduction, the solution in the strain fluctuation space also satisfies the strain compatibility.
- A specific Reduced Optimal Quadrature (ROQ) is used as a key technique to obtain relevant computational cost reduction from the ROM. This technique consists of replacing the standard Gauss integration rule by an optimal quadrature, involving much less sampling points, has been proposed in other works Farhat_2015,Hernandez_HPROM_2017 as an ingredient of HyPer-Reduced Order Modeling (HPROM) strategies. In these works, the reduced numerical integration technique is applied to the variational equations of the problem (i.e. internal forces, involving n-dimensional vector entities) whereas, in the herein proposed approach, a similar reduced integration technique is applied, again unconventionally, to the primitive problem, i.e: the functional (a scalar entity) in the micro-scale saddle-point problem that supplies the RVE variational equations. In the present proposal, this functional turns out to be the stored energy (free energy) at the RVE, which, being a scalar entity, is much less demanding in terms of the integration rule.
- The RVE domain separation technique: to account for distinct constitutive models used at the RVE and take the maximum advantage of this distinction.
- A strain-based formulation of the variational RVE problem allowing a simpler application of the previous technique, without the need of introducing compatibility constraints.
- A specific sampling program, for the construction of the sets of snapshots in the off-line stage of the HPROM procedure, in accordance with the rest of elements of the proposed strategy.
- The Reduced Optimal Quadrature (ROQ) technique, which resorts to the primitive formulation of the RVE problem as a saddle-point problem.
|
| Figure 19: Speedup scalability. |
4.1.5 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{4}
- J. A. Hernández, M. A. Caicedo, A. Ferrer. Dimensional hyper-reduction of nonlinear finite element models via empirical cubature, Journal of Computer Methods in Applied Mechanics and Engineering - 2016, Volume 313, Pages 560–595
4.1.5 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{4}
J. A. Hernández, M. A. Caicedo, A. Ferrer. Dimensional hyper-reduction of nonlinear finite element models via empirical cubature, Journal of Computer Methods in Applied Mechanics and Engineering - 2016, Volume 313, Pages 560–595
This work has been developed in combination with the reduced order model for non-smooth problems (see [[#4.1.4 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{3}
- J. Oliver, M. Caicedo, A. E. Huespe, J. A. Hernández, E. Roubin. Reduced Order Modeling strategies for Computational Multiscale Fracture, Computer Methods in Applied Mechanics and Engineering - 2016, Volume 313, Pages 560–595|4.1.4]]). The main objective is to develop the algorithmic procedure in a general setting to be applied to different problems involving integral operators that can be sampled. Not only problems involving multiple scales can be analyzed, but also monoscale (static and dynamic) problems based on the Finite Element method.
- J. A. Hernández, J. Oliver, A. E. Huespe, M. A. Caicedo, J. C. Cante. High-performance model reduction techniques in computational multiscale homogenization, Computer Methods in Applied Mechanics and Engineering - 2014, Volume 276, Pages 149–189|4.1.2]] and [[#4.1.4 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{3}
- J. Oliver, M. Caicedo, A. E. Huespe, J. A. Hernández, E. Roubin. Reduced Order Modeling strategies for Computational Multiscale Fracture, Computer Methods in Applied Mechanics and Engineering - 2016, Volume 313, Pages 560–595|4.1.4]]), the reduction process is divided into two sequential stages, the first consists of a Galerkin projection of the strain fluctuations, via POD, and the second consists of a novel cubature rule also used in [[#4.1.4 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{3}
- J. Oliver, M. Caicedo, A. E. Huespe, J. A. Hernández, E. Roubin. Reduced Order Modeling strategies for Computational Multiscale Fracture, Computer Methods in Applied Mechanics and Engineering - 2016, Volume 313, Pages 560–595|4.1.4]]. In this case, this method is deeply studied and analyzed. The distinguish features of the proposed method to be highlighted are:
- The minimization method is set in terms of orthogonal basis vectors (obtained via Singular Value Decomposition SVD) rater than in terms of snapshots taken from the integrand.
- The volume of the domain is exactly integrated.
- The selection algorithm does not require solve, in all iterations, a non-negative least-squares problem to obtain positive weights.
- J. Oliver, M. Caicedo, A. E. Huespe, J. A. Hernández, E. Roubin. Reduced Order Modeling strategies for Computational Multiscale Fracture, Computer Methods in Applied Mechanics and Engineering - 2016, Volume 313, Pages 560–595|4.1.4]], attention was focused on use the free energy to determine the reduced integration rule.
4.1.6 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{5}
- J. Oliver, M. Caicedo, E. Roubin, A. E. Huespe. Continuum Approach to Computational Multi-Scale Modeling of Fracture, Key Engineering Materials - 2014, Volume 627, Pages 349–352
4.1.6 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{5}
J. Oliver, M. Caicedo, E. Roubin, A. E. Huespe. Continuum Approach to Computational Multi-Scale Modeling of Fracture, Key Engineering Materials - 2014, Volume 627, Pages 349–352
This work presents a brief summary of the two-scale approach for modeling failure propagation, providing details about propagation at the macro and micro levels. This publication is centered in exploring the applicability of the method to structural problems. The four-point bending and the Nooru-Mohamed problems have been chosen as benchmarks, taking the material properties form experimental tests.
In the case of the Nooru-Mohamed test, it has been shown, the influence of the horizontal load (shear force) in the microscale behavior, and the activation of different crack patterns, representing the macroscale changes in the crack propagation scheme. In the four-point bending test, it is displayed the influence on the macroscale propagation scheme, when critical failure mechanisms at the microscale are precluded.
4.1.7 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{6}
- M. Caicedo, J. Oliver, A. E. Huespe, O. Lloberas-Valls. Model Order Reduction in computational multiscale fracture mechanics, Key Engineering Materials - 2016, Volume 713, Pages 248–253
4.1.7 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{6}
M. Caicedo, J. Oliver, A. E. Huespe, O. Lloberas-Valls. Model Order Reduction in computational multiscale fracture mechanics, Key Engineering Materials - 2016, Volume 713, Pages 248–253
This work has a similar objective than the previous one. A brief summary about the reduced order model based on the two-scale approach for modeling failure propagation, has been presented. This work also presents a summary about the results obtained in the L-Shaped Panel, and the influence of the size of reduced order basis functions (for strain fluctuations and free energy) is presented and analyzed.
4.1.8 Chapter In Book (CB): J. Oliver, M. Caicedo, E. Roubin, J. A. Hernández, A. E. Huespe. Multi-scale (FEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^2
) analysis of material failure in cement-aggregate-type composite structures, Computational Modeling of Concrete Structures - 2014, Pages 39–494.1.8 Chapter In Book (CB): J. Oliver, M. Caicedo, E. Roubin, J. A. Hernández, A. E. Huespe. Multi-scale (FEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^2
) analysis of material failure in cement-aggregate-type composite structures, Computational Modeling of Concrete Structures - 2014, Pages 39–49
This work focuses on exploring different issues of the two-scale approach for modeling failure propagation. Particularly, the total energy dissipation and its relation at both scales is analyzed in some specific fracture problems.
4.2 Ongoing work and future research lines
4.2.1 Ongoing work
- Most of the real industrial problems require 3D modelling. This is the reason because an immediate task is to extend all procedures developed in this work to 3D problems. This extension is carried out in Kratos Multi-Physics, an open-source code developed at CIMNE (International Center for Numerical Methods in Engineering).
- The approach developed in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{3}}
is being extended to nonlinear geometrical multiscale problems. Considering an elasto-plastic constitutive model endowed, with hardening behavior, the goal is to study and analyze geometric bifurcation at the macroscale. Some early results have been obtained, exhibiting the potential uses of this methods in nonlinear geometric multiscale problems.
- Fracture processes of composite materials via multiscale modeling, are being studied by using the formulations developed in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{2}}
and Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{3}}
. The main goals are: to reproduce the experimental behavior of composite sublaminates of ultra-thin plies Arteiro_2014 and, to devise some design alternatives based on the material failure analysis, taking advantage of the reduced order model techniques.
- In order to obtain an optimum performance of the reduced order models developed in this work, it is convenient to optimize the tasks performed in the offline stage, particularly, the Singular Value Decomposition performed after sampling the training trajectories. This method can be highly computational demanding in very fine meshes. Therefore, the study of SVD partitioned procedures, and iterative strategies are presently explored.
4.2.2 Future research lines
- Extension of the multiscale aproach to other family of finite elements at the macroscale, in view of the unaffordable computational burden, an additional reduction can be obtained using stabilized finite elements with reduced integration rules at the macroscale.
- Extension of the developed multiscale model to propagating fracture in non-linear dynamic cases. This includes modeling more complex phenomena like branching and multiscale dynamic processes. This field was also studied via monoscale phenomenological modeling in Belytschko_et_al_2003,Prabel_Combescure_2007,Linder_2009,LloberasVallsDin2016. In addition, inclusion of non-linear hardening behavior, before the onset of material failure in the considered damage model, and consideration of other families of constitutive behavior, like plasticity, rate dependence etc., should be studied.
- The use, in the developed multiscale model for propagating fracture, of other crack propagation models at the microscale, either based on continuum methods (CSDA, non-local models or gradient-regularized models), or discrete methods (cohesive interfaces equipped with traction-separation laws).
- Extension of the reduced order model described in Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \#{3}}
to other microscale failure methodologies, , gradient damage models, Enhanced finite element methods (EFEM), etc. In the same way, the extension of the reduced order model to other multiscale strategies involving fracture processes, , concurrent models lloberas2012multiscale.
- Extension of the reduced order techniques to problems in computational fluid dynamics, coupled thermo-mechanical problems, and wave propagation processes.
- The development of the concept of Material-Catalog of the reduced order modeled RVE. Where, every RVE is modeled for specific materials and morphologies, developing a reduced order model. This data, typically, for a given finite element mesh, are conformed by the strain fluctuation reduced basis functions, the reduced integration rule, and a performance HPROM plot, aiming at allowing the user to design its own strategy for reduce the RVE problem.
This data, stored in a catalog, is understood as a box to be used for particular users and commercial codes, for research or industrial purposes.
4.3 Scientific Contributions
in process
4.3.1 Main original contributions
in process
4.3.2 Other original contributions
in process
dummy toc
BIBLIOGRAPHY
Appendix
5 Related Articles
5.1 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{1}
Title: High-performance model reduction techniques in computational multiscale homogenization.
Authors:
- J. A. Hernández: Assistant Professor of Structural Engineering and Strength of Materials at the School of Industrial and Aeronautic Engineering of Terrassa, of the Technical University of Catalonia. Senior researcher at the International Center for Numerical Methods in Engineering (CIMNE).
- J. Oliver: Professor of Continuum Mechanics and Structural analysis at the Escola Tecnica Superior d'Enginyers de Camins, Canals i Ports (Civil Engineering School) of the Universitat Politecnica de Catalunya (Technical University of Catalonia BarcelonaTech). Senior researcher at the International Center for Numerical Methods in Engineering (CIMNE).
- A. E. Huespe: Professor of Mechanics at the Faculty of Chemical Engineering, Dept. of Materials, National University of Litoral, Santa Fe, Argentina. Independent researcher of Conicet at CIMEC (Centro de Investigaciones en Mecánica Computacional), National University of Litoral (UNL).
- M. Caicedo: PhD Candidate in Structural Analysis in UPC BarcelonaTech and International Center for Numerical Methods in Engineering (CIMNE).
- J. C. Cante: Associate Professor of Computational Engineering at the Escola Tecnica Superior d'Enginyeries Industrial i Aeronautica de Terrassa – Universitat Politecnica de Catalunya (Technical University of Catalonia, BarcelonaTech). Associate researcher at the International Center for Numerical Methods in Engineering (CIMNE).
Journal of Computer Methods in Applied Mechanics and Engineering
Editors: Thomas J.R. Hughes, J. Tinsley Oden, Manolis Papadrakakis
ISSN: 0045-7825
Elsevier Editors
http://dx.doi.org/10.1016/j.cma.2014.03.011
5.2 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{2}
Title: Continuum approach to computational multiscale modeling of propagting fracture.
Authors:
- J. Oliver: Professor of Continuum Mechanics and Structural analysis at the Escola Tecnica Superior d'Enginyers de Camins, Canals i Ports (Civil Engineering School) of the Universitat Politecnica de Catalunya (Technical University of Catalonia BarcelonaTech). Senior researcher at the International Center for Numerical Methods in Engineering (CIMNE).
- M. Caicedo: PhD Candidate in Structural Analysis in UPC BarcelonaTech and International Center for Numerical Methods in Engineering (CIMNE).
- E. Roubin: Maitre de conférence at the 3SR (Sols, Solides, Structures et Risques) and the IUT DGGC in Grenoble.
- A. E. Huespe: Professor of Mechanics at the Faculty of Chemical Engineering, Dept. of Materials, National University of Litoral, Santa Fe, Argentina. Independent researcher of Conicet at CIMEC (Centro de Investigaciones en Mecánica Computacional), National University of Litoral (UNL).
- J. A. Hernández: Assistant Professor of Structural Engineering and Strength of Materials at the School of Industrial and Aeronautic Engineering of Terrassa, of the Technical University of Catalonia. Senior researcher at the International Center for Numerical Methods in Engineering (CIMNE).
Journal of Computer Methods in Applied Mechanics and Engineering
Editors: Thomas J.R. Hughes, J. Tinsley Oden, Manolis Papadrakakis
ISSN: 0045-7825
Elsevier Editors
http://dx.doi.org/10.1016/j.cma.2015.05.012
5.3 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{3}
Title: Reduced Order Modeling strategies for Computational Multiscale Fracture.
Authors:
- J. Oliver: Professor of Continuum Mechanics and Structural analysis at the Escola Tecnica Superior d'Enginyers de Camins, Canals i Ports (Civil Engineering School) of the Universitat Politecnica de Catalunya (Technical University of Catalonia BarcelonaTech). Senior researcher at the International Center for Numerical Methods in Engineering (CIMNE).
- M. Caicedo: PhD Candidate in Structural Analysis in UPC BarcelonaTech and International Center for Numerical Methods in Engineering (CIMNE).
- A. E. Huespe: Professor of Mechanics at the Faculty of Chemical Engineering, Dept. of Materials, National University of Litoral, Santa Fe, Argentina. Independent researcher of Conicet at CIMEC (Centro de Investigaciones en Mecánica Computacional), National University of Litoral (UNL).
- J. A. Hernández: Assistant Professor of Structural Engineering and Strength of Materials at the School of Industrial and Aeronautic Engineering of Terrassa, of the Technical University of Catalonia. Senior researcher at the International Center for Numerical Methods in Engineering (CIMNE).
- E. Roubin: Maitre de conférence at the 3SR (Sols, Solides, Structures et Risques) and the IUT DGGC in Grenoble.
Journal of Computer Methods in Applied Mechanics and Engineering
Editors: Thomas J.R. Hughes, J. Tinsley Oden, Manolis Papadrakakis
ISSN: 0045-7825
Elsevier Editors
5.4 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{4}
Title: Dimensional hyper-reduction of nonlinear finite element models via empirical cubature.
Authors:
- J. A. Hernández: Assistant Professor of Structural Engineering and Strength of Materials at the School of Industrial and Aeronautic Engineering of Terrassa, of the Technical University of Catalonia. Senior researcher at the International Center for Numerical Methods in Engineering (CIMNE).
- M. Caicedo: PhD Candidate in Structural Analysis in UPC BarcelonaTech and International Center for Numerical Methods in Engineering (CIMNE).
- A. Ferrer: PhD Candidate in Structural Analysis in UPC BarcelonaTech and International Center for Numerical Methods in Engineering (CIMNE).
Journal of Computer Methods in Applied Mechanics and Engineering
Editors: Thomas J.R. Hughes, J. Tinsley Oden, Manolis Papadrakakis
ISSN: 0045-7825
Elsevier Editors
5.5 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{5}
Title: Continuum Approach to Computational Multi-Scale Modeling of Fracture.
Authors:
- J. Oliver: Professor of Continuum Mechanics and Structural analysis at the Escola Tecnica Superior d'Enginyers de Camins, Canals i Ports (Civil Engineering School) of the Universitat Politecnica de Catalunya (Technical University of Catalonia BarcelonaTech). Senior researcher at the International Center for Numerical Methods in Engineering (CIMNE).
- M. Caicedo: PhD Candidate in Structural Analysis in UPC BarcelonaTech and International Center for Numerical Methods in Engineering (CIMNE).
- E. Roubin: Maitre de conférence at the 3SR (Sols, Solides, Structures et Risques) and the IUT DGGC in Grenoble.
- A. E. Huespe: Professor of Mechanics at the Faculty of Chemical Engineering, Dept. of Materials, National University of Litoral, Santa Fe, Argentina. Independent researcher of Conicet at CIMEC (Centro de Investigaciones en Mecánica Computacional), National University of Litoral (UNL).
Key Engineering Materials Vol. 627
Advances in Fracture and Damage Mechanics XIII
Editors: J. Alfaiate and M.H. Aliabadi
ISSN: 1662-9795
Trans Tech Publications
DOI: 10.4028/www.scientific.net/KEM.627.349
5.6 Paper Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \#{6}
Title: Model Order Reduction in computational multiscale fracture mechanics.
Authors:
- M. Caicedo: PhD Candidate in Structural Analysis in UPC BarcelonaTech and International Center for Numerical Methods in Engineering (CIMNE).
- J. Oliver: Professor of Continuum Mechanics and Structural analysis at the Escola Tecnica Superior d'Enginyers de Camins, Canals i Ports (Civil Engineering School) of the Universitat Politecnica de Catalunya (Technical University of Catalonia BarcelonaTech). Senior researcher at the International Center for Numerical Methods in Engineering (CIMNE).
- A. E. Huespe: Professor of Mechanics at the Faculty of Chemical Engineering, Dept. of Materials, National University of Litoral, Santa Fe, Argentina. Independent researcher of Conicet at CIMEC (Centro de Investigaciones en Mecánica Computacional), National University of Litoral (UNL).
- O. Lloberas-Valls: Postdoctoral researcher at the International Center for Numerical Methods in Engineering (CIMNE).
Key Engineering Materials Vol. 713
Advances in Fracture and Damage Mechanics XV
Editors: Jesús Toribio, Vladislav Mantic, Andrés Sáez, M.H. Ferri Aliabadi
ISSN: 1662-9795
Trans Tech Publications
DOI: 10.4028/www.scientific.net/KEM.713.248
5.7 Chapter in Book
Title: Multi-scale (FEFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^2} ) analysis of material failure in cement/aggregate-type composite structures
Authors:
- J. Oliver: Professor of Continuum Mechanics and Structural analysis at the Escola Tecnica Superior d'Enginyers de Camins, Canals i Ports (Civil Engineering School) of the Universitat Politecnica de Catalunya (Technical University of Catalonia BarcelonaTech). Senior researcher at the International Center for Numerical Methods in Engineering (CIMNE).
- M. Caicedo: PhD Candidate in Structural Analysis in UPC BarcelonaTech and International Center for Numerical Methods in Engineering (CIMNE).
- A. E. Huespe: Professor of Mechanics at the Faculty of Chemical Engineering, Dept. of Materials, National University of Litoral, Santa Fe, Argentina. Independent researcher of Conicet at CIMEC (Centro de Investigaciones en Mecánica Computacional), National University of Litoral (UNL).
- E. Roubin: Maitre de conférence at the 3SR (Sols, Solides, Structures et Risques) and the IUT DGGC in Grenoble.
- J. A. Hernández: Assistant Professor of Structural Engineering and Strength of Materials at the School of Industrial and Aeronautic Engineering of Terrassa, of the Technical University of Catalonia. Senior researcher at the International Center for Numerical Methods in Engineering (CIMNE).
Computational Modelling of Concrete Structures
Proceedings of EURO–C 2014
Editors: N. Biani; H Mang; Gunther Meschke; Reneé de Borst
ISBN: 978-1-138-00145-9
Taylor & Francis Group: CRC Press–Balkema, 2014
Document information
Published on 01/01/2017
Licence: CC BY-NC-SA license
Share this document
claim authorship
Are you one of the authors of this document?