Colloquiam: Research Reports of the International Centre for Numerical Methods in Engineering (CIMNE)

A finite point method for elasticity problems

María Jesús Samper — Tue, 02 Jul 2019 09:59:03 +0200

The basis of the finite point method (FPM) for the fully meshless solution of elasticity problems in structural mechanics is described. A stabilization technique based on a finite calculus procedure is used to improve the quality of the numerical solution. The efficiency and accuracy of the stabilized FPM in the meshless analysis of simple linear elastic structural problems is shown in some examples of applications.

Finite element analysis of membrane structures

María Jesús Samper — Tue, 02 Jul 2019 09:52:51 +0200

This report summarizes the formulation for a large displacement formulation of a membrance composed of three-node triangular elements. A formulation in terms of the deformation gradient is first constructed in terms of nodal variables. In particular, the use of the right Caunchy-Green deformation tensor is shown to lead to a particulary simple representation in terms of nodal quantities. This may then be used to construct general models for use in static and transient analyses.

On discontinuous Galerkin methods

María Jesús Samper — Tue, 02 Jul 2019 09:44:37 +0200

Discontinuous Galerkin methods have received considerable attention in recent years for applications to many problems in which convection and diffusion terms are present. Several alternatives for treating the diffusion flux effects have been introduced, as well as, for treatment of the convective flux terms. This report summarizes some of the treatments that have been proposed. It also considers how elementary finite volume methods may be considered as the most primative form of a discontinuous Galerkin method as well as how it may be formed as a finite element method. Several numerical examples are included in the report which summarize results for discontinuous Galerkin solutions of one-dimensional problems with a scalar variable. Results are presented for diffusion-reaction problems, convection-diffusion problems, and a special problem with a turning point. We identify aspects which relate to accuracy as well as stability of the method.

Possibilities of finite calculus in computational mechanics

María Jesús Samper — Tue, 02 Jul 2019 09:36:07 +0200

The expression ‘finite calculus’ refers to the derivation of the governing differential equations in mechanics by invoking balance of fluxes, forces, etc. in a space–time domain of finite size. The governing equations resulting from this approach are different from those of infinitesimal calculus theory and they incorporate new terms which depend on the dimensions of the balance domain. The new governing equations allow the derivation of naturally stabilized numerical schemes using any discretization procedure. The paper discusses the possibilities of the finite calculus method for the finite element solution of convection–diffusion problems with sharp gradients, incompressible fluid flow and incompressible solid mechanics problems and strain localization situations.

A finite element methodology for local global damage evaluation in civil engineering structures

María Jesús Samper — Tue, 02 Jul 2019 09:28:53 +0200

The paper introduces a new global damage evaluation method, thus obtaining a meaningful global damage index (GDI). A numerical procedure for predicting a local and global damage in civil engineering structures using the finite element method and a continuum damage model is presented. The method is adequate for computing the limit load in reinforced concrete structures and for predicting the failure mechanisms. Details of the damage model used a given together with a description of the finite element implementation and the procedure for computing the global damage parameters. Examples of applications to the non-linear analysis of a range of reinforced concrete structures presented.

Posibilidades de los métodos numéricos en el mundo industrial

María Jesús Samper — Tue, 02 Jul 2019 09:17:10 +0200

Se presentan en el artículo las ideas básicas de qué son los métodos numéricos, cuáles son los métodos numéricos más populares, cómo se aplican para resolver ecuaciones diferenciales de interés práctico en ingeniería y qué posibilidades y limitaciones tienen para ayudarnos a entender mejor el mundo que nos rodea. El contenido del artículo se completa con diversas aplicaciones prácticas del método de elementos finitos a diversos problemas de ingeniería.

A finite element method for fluid-structure interaction with surface waves using finite calculus formulation

María Jesús Samper — Tue, 02 Jul 2019 09:10:06 +0200

A stabilized semi-implicit fractional step finite element method for solving coupled fluid-structure interaction problems involving free surface waves is presented. The stabilized governing equations for the viscous incompressible fluid and the free surface are derived at a differential level via a finite calculus procedure. A mesh updating technique based on solving a fictitious elastic problem on the moving mesh is described. Examples of the efficiency of the stabilized semi-implicit algorithm for the analysis of fluid-structure interaction problems in totally or partially submerged bodies is presented.

Recovering lower bounds of the error postprocessing implicit residual a posteriori error estimates

María Jesús Samper — Tue, 02 Jul 2019 08:54:05 +0200

Classical Residual type error estimators approximate the error flux around the elements and yield upper bounds of the exact (or references) error. Lower bounds of the error are also needed in goal oriented adaptivity and for bounds on functional outputs. This work introduces a simple and cheap strategy to recover a lower bound estimate from standard upper bound estimates. This lower bound mat be also to assess the effectivity of the former estimate and to improve it.

A finite element method for fluid-structure interaction with surface waves using a finite calculus formulation

María Jesús Samper — Thu, 11 Jul 2019 14:57:46 +0200

A stabilized semi-implicit step finite element method for solving coupled fluid-structure interaction problems involving free surface waves is presented. The stabilized governing equations for the viscous fluid and the free surface are derived at a differential level via finite increment calculus procedure. A mesh updating technique based on solving a fictitious elastic problem on the moving mesh is described. Examples of the efficiency of the stabilized semi-implicit algorithm for the analysis of fluid-structure interaction problems in totally or partially submerged bodies is presented.

Stabilization techniques for finite element analysis of convection-diffusion problems

María Jesús Samper — Thu, 11 Jul 2019 14:41:09 +0200

The accurate solution of convection type problems on practical grids has been ever a challenging issue, and invariably some sort of stabilization is needed in order to get a physical solution. This has pushed researchers to develop various stabilization algorithms used in every day practice by numerical analysts. In this chapter some methods are presented along with a new finite increment calculus approach to obtain the different algorithms using higher order conservation equations.

The Monte Carlo method. Application to the stochastic analysis of sheet stamping processes

María Jesús Samper — Thu, 11 Jul 2019 14:22:35 +0200

Stochastic Mechanics is a rapidly growing area of research, whose importance is being recognized not only in academic circles but also in industrial practice. This is no doubt due to the fact that most structural properties and loads are either random or uncertain. The first term refers to a natural chaotic variation of the parameter, while the second is associated to the human lack of knowledge about it. Both kinds of unpredictability work together in rendering doubtful the results of a (usually single) deterministic mechanical analysis. When thinking about the randomness and uncertainty linked to all physical parameters and phenomena a big question mark closes the large list of numbers produced by a finite element calculation.

In Stochastic Mechanics there are several techniques to analyse the natural scatter of strains and stresses caused by the dispersion in the given loads and/or the structural parameters. The most general one is the Monte Carlo method. However, it must be recognized that is as well the most costly in computational terms. Nevertheless, this cost has becoming feasible with the advance in Computer Science, specially with the advent of parallel computing, due to the fact that a Monte Carlo calculation is intrinsically a task that can be performed in parallel.

The present report is intended to provide the reader an introduction to the Monte Carlo method in the context of Computational Mechanics. The technique is applied to the analysis of the uncertainty spread in a stamping process. The first chapter summarises the Monte Carlo method and its theoretical backgrounds. The second chapter is devoted to the case study, namely, the stochastic analysis of a square cup deep drawing problem. Finally, the basic equations governing the mechanical modelling of the stamping process are summarized in the appendix.

Stabilized finite element approximation of transient incompressible flows using orthogonal subscales

María Jesús Samper — Thu, 11 Jul 2019 13:31:43 +0200

In this paper we present a stabilized finite element formulation to solve the Oseen equations as a model problem involving both convection effects and the incompressibility restriction. The need for stabilization techniques to solve this problem arises because of the restriction in the possible choices for the velocity and pressure spaces dictated by the inf-sup condition, as well as the instabilities encountered when convection is dominant. Both can be overcome by resorting from the standard Galerkin method to a stabilized formulation. The one presented here is based on the subgrid scale concept, in which unresolvable scales of the continuous solution are approximately accounted for. In particular, the approach developed herein is based on the assumption that unresolved subscales are orthogonal to the finite element space. The motivation of the method is fully described. It is also shown that this formulation is table and optimally convergent for an adequate choice of the algorithmic on which the method depends.

Pressure stability in fractional step finite element methods for incompressible flows

María Jesús Samper — Thu, 11 Jul 2019 13:00:25 +0200

The objective of this paper is to analyze the pressure stability of fractional step finite element methods for incompressible flows. For the classical first order projection method, it is shown that there is a pressure control which depends on the time step size, and therefore there is a lower bound for this time step for stability reasons. The situation is much worse for a second order scheme in which part of the extremely weak. To overcome these shortcomings, a stabilized fractional step finite element method is also considered, and its stability is analyzed. Some simple numerical examples are presented to support the theoretical results.

Implementation of a stabilized finite element formulation for the incompressible Navier-Stokes equations based on a pressure gradient projection

María Jesús Samper — Thu, 11 Jul 2019 12:51:08 +0200

We discuss in this paper some implementation aspects of a finite element formulation for the incompressible Navier-Stokes which allows the use of equal order velocity-pressure interpolations. The method consists in introducing the project of the pressure gradient and adding the difference between the pressure Laplacian and divergence of this new field to the incompressibility equations, both multiplied by suitable algorithmic parameters. The main purpose of this paper is to discuss how to deal with the new variable in the implementation of the algorithm. Obviously, it could be treated as one extra unknown, either explicitly or as a condensed variable. However, we take for granted that the only way or another. Here we discuss some iterative schemes to perform this uncoupling of the pressure gradient projection (PGP) from the calculation of the velocity and the pressure, both for stationary and the transient of the linearization loop and the iterative segregation of the PGP, whereas in the second the main dilemma concerns the explicit or implicit treatment of the PGP.

Prediction of damage and failure in civil engineering structures using a finite element model.

María Jesús Samper — Thu, 11 Jul 2019 12:13:09 +0200

The paper describes a finite element damage model for non linear analysis of concrete or reinforced structures. It is show how can be effectively used to predict local and global damage up to structural failure. Examples of applications of the model to the analysis of different structures such as a nuclear containment shell, a housing building and the domes of St. Mark Basilica are presented.

Rigid-flexible body analysis without rotation parameters

María Jesús Samper — Fri, 05 Jul 2019 11:21:44 +0200

This paper addresses the coupled flexible and rigid body response of solids undergoing large motions and deformations. The formulation is presented in a form which is free of rotation parameters and thus avoids the need of finding compatible integration formulae for translations and rotations. The motion of each rigid body is represented in terms of the nodal parameters of a simplex element subject to constraints which ensure rigid motions. Flexible structural members for rods and shells are then expressed in terms of displacement and relative displacement parameters leading to total system of equations involving only translation degrees of freedom.

The motions are integrated using classical energy and momentum conserving schemes, thus leading to systems which are unconditionally stable for Hamiltonian (elastic-rigid) systems. The only requirement for absolute stability is that the non-linear algebraic equations representing the solution at each time step must converge.

The formulation is illustrated in two dimensions by representative numerical problems which involve both rigid and flexible parts or multi-rigid body situations. In all cases the conservation properties are observed.

Locking in the incompressible limit for the element free galerkin method

María Jesús Samper — Fri, 05 Jul 2019 10:59:33 +0200

Volumetric locking (locking in the incompressible limit) for linear elastic isotropic materials is studied in the context of the Element Free Galerkin method. The modal analysis developed here shows here shows that the number of non-physical locking modes is independent of the dilation parameter (support of the interpolation functions). Thus increasing the dilation parameter does not suppress locking. Nevertheless, an increase in the dilation parameter does reduce the energy associated to the non-physical locking behavior. Although more locking modes are present in the Element Free Galerkin method with quadratic consistency than with the standard biquadratic finite element method. Finally, numerical examples are shown.

Límites de los métodos numéricos

María Jesús Samper — Fri, 05 Jul 2019 10:40:56 +0200

Se plantean en el artículo los límites de los métodos numéricos para resolver cualquier problema que afecte a la vida del hombre. Se discute la capacidad de la razón para expresar todos los problemas del universo en forma matemática, y la posibilidad de encontrar su solución en forma numérica. Finalmente, se especula sobre el alcance y posibilidad de los métodos numéricos en el amplio campo de las ciencias sociales

El bucle de los números

María Jesús Samper — Fri, 05 Jul 2019 10:31:36 +0200

El artículo presenta una sucinta panorámica sobre la evolución de los métodos numéricos dese la antigua Babilonia hasta nuestros días. Se destaca como la máxima Pitagórica de que “todo es número”, adquiere plena actualidad en nuestros días, en que, con la ayuda de los ordenadores, podemos dar respuestas numéricas a prácticamente cualquier problema que afecte a la vida del hombre. Recorriendo ese bucle de los números se relata brevemente en el artículo como la humanidad ha evolucionado paralelamente en su aspiración de cuantificar los fenómenos de la naturaleza, y como el paso de los pueblos ha ido de la mano de los avances en expresar numéricamente la solución de sus problemas más cotidianos.

Simulación por ordenador del comportamiento resistente de estructuras: El laboratorio virtual de estructuras

María Jesús Samper — Fri, 05 Jul 2019 10:23:25 +0200

Se presenta el concepto de Laboratorio Virtual de Estructuras (LAVE) para evaluar por ordenador la capacidad resistente de estructuras de hormigón y el coeficiente de seguridad al colapso utilizando modelos de cálculo no lineal basados en la teoría del daño y el método de elementos finitos. Se describen diversos ejemplos de aplicación del LAVE al análisis de la resistencia última de estructuras de hormigón en masa y armado.