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Revision as of 09:51, 19 October 2018
Abstract
The paper describes some recent developments in finite element and particle methods for analysis of a wide range of bulk forming processes. The developments include new stabilized linear triangles and tetrahedra using finite calculus and a new procedure combining particle methods and finite element methods. Applications of the new numerical methods to casting, forging and other bulk metal forming problems and mixing processes are shown.
keywords Bulk forming processes, stabilized finite element method, particle method, particle finite element method, mixing processes.
1 INTRODUCTION
The development of efficient and robust numerical methods for analysis of bulk forming problems has been a subject of intensive research in recent years [1–7]. Many of these problems require the solution of incompressible fluid flow situations (such as in mould filling problems) whereas in other cases (such as forging, rolling, extrusion, etc.) the numerical method must be able to account for the quasi/fully incompressible behaviour induced by the large plastic deformation. The solution of these problems has motivated the development of the so called stabilized numerical methods overcoming the two main sources of instability in the analysis of incompressible continua, namely those originated by the high values of the convective terms in fluid flow situtions and those induced by the difficulty in satisfying the incompressibility condition.
Different approaches to solve both type of problems in the context of the finite element method (FEM) have been recently developed [8]. Traditionally, the underdiffusive character of the Galerkin FEM for high convection flows has been corrected by adding some kind of artificial viscosity terms to the standard Galerkin equations [8,9].
A popular way to overcome the problems with the incompressibility constraint in the FEM is by introducing a pseudo-compressibility in the continuum and using implicit and explicit algorithms ad hoc such as artificial compressibility schemes [10] and preconditioning techniques [11]. Other FEM schemes with good stabilization properties for the convective and incompressibility terms in fluid flows are based in Petrov-Galerkin (PG) techniques. The background of PG methods are the non-centred (upwind) schemes for computing the first derivatives of the convective operator in FD and FV methods [8,9,12]. A general class of Galerkin FEM has been developed where the standard Galerkin variational form is extended with adequate residual-based terms in order to achieve a stabilized numerical scheme. Among the many FEM of this kind we can name the Streamline Upwind Petrov Galerkin (SUPG) method [8,13–16], the Galerkin Least Square (GLS) method [8,17], the Taylor-Galerkin method [18], the Characteristic Galerkin method [19] and its variant the Characteristic Based Split (CBS) method [20,21], the pressure gradient operator method [22] and the Subgrid Scale (SS) method [23]. A good review of these methods can be found in [24]. Extensions of the CBS and SS methods to treat incompressible problems in solid mechanics are reported in [25,26,58] and [27–29], respectively.
In this paper a different class of stabilized FEM for quasi and fully incompressible fluid and solid materials applicable to a wide range of bulk forming problems is presented. The starting point is the modified governing differential equations of the continuous problem formulated via a finite calculus (FIC) approach [30,31]. The FIC method is based in invoking the classical balance (or equilibrium) laws in a domain of finite size. This introduces naturally additional terms in the differential equations of infinitesimal continuum mechanics which are a function of the balance domain dimensions. The new terms in the modified governing equations provide the necessary stabilization to the discrete equations obtained via the standard Galerkin FEM. One of the main advantages of the FIC formulation versus other alternative approaches (such as mixed FEM, etc.) is that it allows to solve incompressible fluid problems using low order finite elements (such as linear triangles and tetrahedra) with equal order approximations for the velocity and pressure variables [32–35]. The FIC formulation has been successfully used for analysis of fully or quasi incompressible solids [36,37].
The layout of the paper is the following. In the next section the basic FIC equations for incompressible flow problems formulated in an Eulerian frame are presented. The finite element discretization is introduced and the resulting discretized equations are detailed. A fractional step scheme for the transient solution is presented.
The stabilized Eulerian formulation is extended to account for thermal effects and the transport of the free surface which are needed for mould filling processes.
The following sections outline the FIC formulation for analysis of quasi/fully incompressible solids using a Lagrangian description and linear triangles and tetrahedra. An explicit algorithm for integrating in time the equations of motion of elasto-plastic solids in large-strain problems involving frictional contact is described. Examples of application to a casting problem and some bulk forming processes are presented.
In the last part of the paper a Lagrangian formulation for fluid flow analysis is presented as a straightforward extension of the formulation for solid mechanics. The procedure, called the Particle Finite Element Method (PFEM) [38–40], treats the mesh nodes in the fluid and solid domains as dimensionless particles which can freely move, an even separate from the main fluid domain, representing, for instance, the effect of liquid drops. A finite element mesh connects the nodes defining the discretized domain where the governing equations are solved in the standard FEM fashion. The main advantage of the Lagrangian flow formulation is that the convective terms do not enter in the fluid equations. The difficulty is however transferred to the problem of adequately (and efficiently) moving the mesh nodes. The final examples show that the PFEM is a promising method to solve mould filling and casting problems, material mixing processes and many other bulk metal forming problems involving the interaction between solids and fluids which can be treated with the same formulation.
2 FIC EQUATIONS FOR VISCOUS INCOMPRESSIBLE FLOWS
The FIC governing equations for a viscous incompressible fluid can be written in an Eulerian frame of reference as [30–34]
Momentum
|
(1) |
Mass balance
|
(2) |
where
|
Above Failed to parse (MathML with SVG or PNG fallback (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 }
is the analysis domain which can evolve with 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 n_d}
is the number of space dimensions (Failed to parse (MathML with SVG or PNG fallback (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_d=3}
for 3D 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 v_i}
is the velocity along the ith global axis, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho }
is the (constant) density of the fluid, Failed to parse (MathML with SVG or PNG fallback (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}
is the absolute pressure (defined positive in compression), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle b_i}
are the body forces 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 s_{ij}}
are the deviatoric stresses related to the viscosity Failed to parse (MathML with SVG or PNG fallback (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 }
by the standard expression
|
(5) |
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 _{ij}}
is the Kronecker delta and the strain rates Failed to parse (MathML with SVG or PNG fallback (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 \varepsilon _{ij}}
are
|
(6) |
The boundary conditions are written in the FIC approach as
|
(7) |
|
(8) |
and the initial condition 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 v_j =v_j^0}
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 t=t_0}
.
Summation convention for repeated indexes in products and derivatives is used unless otherwise specified.
In Eqs.(7) and (8) Failed to parse (MathML with SVG or PNG fallback (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_i^p}
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 v_j^p}
are surface tractions and prescribed velocities on the boundaries Failed to parse (MathML with SVG or PNG fallback (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 _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 \Gamma _u}
, respectively, Failed to parse (MathML with SVG or PNG fallback (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_j}
are the components of the unit normal vector to the boundary 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 \sigma _{ij}}
are the total stresses given 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 \sigma _{ij}=s_{ij}-\delta _{ij}p}
.
Eqs. (1) and (2) are obtained by invoking the classical balance equations in fluid mechanics in a domain of finite size and retaining higher order terms [30]. 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 h_i's}
in above equations are characteristic lengths of the domain where the balance of momentum and mass is enforced. In Eq.(7) these lengths define the domain where equilibrium of boundary tractions is established. In the discretized problem 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 h_{i}}
coincide with a typical element dimension, as described in Section 5. Note that by making Failed to parse (MathML with SVG or PNG fallback (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_i =0}
in these equations the standard infinitesimal form of the fluid mechanics equations is recovered [8,9,24].
Eqs.(1)–(8) are the starting point for deriving stabilized FEM for solving the incompressible Navier-Stokes equations using equal order interpolation for the velocity and pressure variables [32–35]. Application of the FIC formulation to meshless analysis of fluid flow problems using the finite point method can be found in [42].
Remark
In most metal forming processes the viscosity Failed to parse (MathML with SVG or PNG fallback (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 }
is a non linear function of the strain rate and the yield stress of the material (which also depends on the temperature) [8,43,44]. This dependence adds another non linearity to the problem.
Remark
In Eqs.(1)–(7) Failed to parse (MathML with SVG or PNG fallback (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 }
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 \Gamma _t}
respectively denote the actual volume and the boundary of the domain where the governing equations are solved at each instant of the forming processes. This domain is considered here to be fixed in space (Eulerian approach). Moving free surfaces, such as in the case of mold filling processes, can be modelled by using standard level set and volume of fluid (VOF) techniques [47,48]. An alternative is to use an arbitrary Lagragian-Eulerian (ALE) description or even a fully Lagrangian description to follow the motion of the particles during the forming process. The Lagrangian description is typical in the case of solid mechanics problems (Section 5). A particular Lagrangian formulation for fluid flow problems involving large motion of the free surface, named the Particle Finite Element Method, is presented in Section 8.
2.1 Stabilized integral forms
From the momentum equations it can be obtained [32–34]
|
(9) |
where
|
(10) |
Substituting Eq.(9) into Eq.(2) and retaining the terms involving the derivatives 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 r_{m_i}}
with respect 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 x_i}
only, leads to the following expression for the stabilized mass balance equation
|
(11) |
with
|
(12) |
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 \tau _i} 's in Eq.(12), when scaled by the density, are termed in the stabilization literature intrinsic time parameters.
The weighted residual form of the momentum and mass balance equations (Eqs.(1) and (11)) is written as
|
(13) |
|
(14) |
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 v_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 q}
are arbitrary weighting functions representing virtual velocities and virtual pressure fields. Integrating by parts 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 r_{m_i}}
terms leads to
|
(15a) |
|
(15b) |
We will neglect hereonwards the third integral in Eq.(15b) by assuming 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 r_{m_i}}
is negligible on the boundaries. The deviatoric stresses and the pressure terms in the first integral of Eq.(15a) are integrated by parts in the usual manner. The resulting momentum and mass balance equations are
|
(16a) |
|
(16b) |
In the derivation of the viscous term in Eq.(16a) we have used the following identity (prior to the integration by parts)
|
(17) |
Eq.(17) is identically true for the exact incompressible 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 \left({\partial v_i \over \partial x_i}=0\right)} .
2.2 Convective and pressure gradient projections
The computation of the residual terms can be simplified if we introduce now the convective and pressure gradient projections Failed to parse (MathML with SVG or PNG fallback (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}
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 \pi _i}
, respectively defined as
|
(18) |
We can express Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_{m_i}}
in Eqs.(16a) and (16b) in terms 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 c_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 \pi _i}
, respectively which then become additional variables. The system of integral equations is now augmented in the necessary number of equations by imposing that the residual Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_{m_i}}
vanishes (in average sense) for both forms given by Eqs.(18). This gives the final system of governing equation as:
|
(19) |
|
(20) |
|
(21) |
|
(22) |
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 i,j,k=1,n_d} . In Eqs.(21) and (22) Failed to parse (MathML with SVG or PNG fallback (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 c_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 \delta \pi _i}
are appropriate weighting functions and 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 \rho }
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 \tau _i}
weights are introduced for convenience.
We note that accounting for the convective and pressure gradient projections enforces the consistency of the formulation as it ensures that the stabilization terms in Eqs.(19–22) have a residual form which vanishes for the “exact” solution. Neglecting these terms lowers the accuracy of the numerical solution and it makes the formulation more sensitive to the value of the stabilization parameters as shown in [34,36,37].
3 FINITE ELEMENT DISCRETIZATION
We choose Failed to parse (MathML with SVG or PNG fallback (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^\circ }
continuous linear interpolations of the velocities, the pressure, the convection projections Failed to parse (MathML with SVG or PNG fallback (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}
and the pressure gradient projections Failed to parse (MathML with SVG or PNG fallback (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 _i}
over three node triangles (2D) and four node tetrahedra (3D) [8]. The linear interpolations are written as
|
(23) |
where the sum goes over the number of nodes of 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 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=3/4}
for triangles/tetrahedra), Failed to parse (MathML with SVG or PNG fallback (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 {(\cdot )}^k}
denotes nodal variables 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^k}
are the linear shape functions [8].
Substituting the approximations (23) into Eqs.(19–22) and choosing the Galerking form 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 \delta v_i =q=\delta c_i=\delta \pi _i =N^i}
leads to following system of discretized equations
|
(24a) |
|
(24b) |
|
(24c) |
|
(24d) |
The form of the different matrices is given in the Appendix.
The solution in time of the system of Eqs.(24) can be written in general form as
|
(25a) |
|
(25b) |
|
(25c) |
|
(25d) |
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 {H}^{n+\theta }={H} (\bar{v}^{n+\theta })} , Failed to parse (MathML with SVG or PNG fallback (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{v}^{n+\theta }}
are the velocities evaluated 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 n+\theta }
and 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 \theta \in [0,1]}
. The direct monolitic solution of Eqs.(25) is possible using an adequate iterative scheme. However, we have found more convenient to use a fractional step method as described in the next section.
3.1 Fractional step method
A fractional step scheme is derived by noting that the discretized momentum equation (25a) can be split into the two following equations
|
(26) |
|
(27) |
In above equation Failed to parse (MathML with SVG or PNG fallback (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{v}^{n+1}}
is a predicted value of the velocity 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 n+1}
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 \alpha }
is a variable whose values of interest are zero and one. 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 \alpha =0}
(first order scheme) the splitting error is of 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 0 (\Delta t)}
, whereas 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 \alpha =1}
(second order scheme) the error is of 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 0 (\Delta t^2)}
[45].
Eqs.(26) and (27) are completed with the following three equations emanating from Eqs.(25b-d)
|
(28a) |
|
(28b) |
|
(28c) |
The 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 \bar{v}^{n+1}}
obtained from Eq.(27) is substituted into Eq.(28a) to give
|
(29) |
The product Failed to parse (MathML with SVG or PNG fallback (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}^T {M}^{-1} {G}}
can be approximated by a laplacian matrix, i.e.
|
(30) |
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^{ab}}
are the element contributions 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 {L}}
(see Appendix).
The steps of the fractional step scheme chosen here are:
Step 1
The fractional nodal velocities Failed to parse (MathML with SVG or PNG fallback (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 {v}^{n+1}}
can be explicitely computed from Eq.(26) by
|
(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 {M}_d}
is the diagonal form of M obtaining by lumping the row terms into the corresponding diagonal terms.
Step 2 Compute Failed to parse (MathML with SVG or PNG fallback (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{p}^{n+1}}
from Eq.(29) as
|
(32) |
Step 3 Compute Failed to parse (MathML with SVG or PNG fallback (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{v}^{n+1}}
explicitely from Eq.(28a) as
|
(33) |
Step 4 Compute Failed to parse (MathML with SVG or PNG fallback (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{c}^{n+1}}
explicitely from Eq.(28b) as
|
(34) |
Step 5 Compute Failed to parse (MathML with SVG or PNG fallback (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{\boldsymbol \pi }^{n+1}}
explicitely from Eq.(28c) as
|
(35) |
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 {M}_d}
is the lumped form 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 \hat {M}}
. A standard diagonal lumping procedure based in summing up the terms of each row has been used.
Note that all steps can be solved explicitely except for the computation of the pressure in Eq.(32) which requires to invert the sum of two laplacian matrices. This can be effectively performed using an iterative solution scheme such as the conjugate gradient method.
Above algorithm has improved stabilization properties versus the standard segregation methods due to the introduction of the laplacian 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 {L}}
in Eq.(32). This matrix emanates from the FIC formulation.
The boundary conditions are applied as follows. No condition is applied in the computation of the fractional velocities Failed to parse (MathML with SVG or PNG fallback (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{v}^{n+1}}
in Eq.(31). The prescribed velocities at the boundary are applied when 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 \bar{v}^{n+1}}
in step 3. The prescribed pressures at the boundary are imposed by making Failed to parse (MathML with SVG or PNG fallback (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{p}^n}
equal to the prescribed pressure values when solving Eq.(32).
3.2 Stokes flow
Many metal forming processes can be simulated with the assumption that the convective terms are negligible (Stokes flow) [8,43,44]. The Stokes FIC formulation can be readily obtained simply by neglecting the convective terms in the Navier-Stokes formulation presented earlier. This also implies neglecting the convective stabilization terms in the momentum equations and, consequently, the convective projection variables are not larger necessary. Also the intrinsic time parameters Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau _i}
take now the simpler form (see Eq.(12)):
|
(36) |
We note again that in metal forming problems the viscosity Failed to parse (MathML with SVG or PNG fallback (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 }
will typically be a function of the strain rate and the yield stress of the material [8,43,44].
The resulting discretized system of equations can be written as (see Eqs.(28))
|
(37) |
The algorithm of previous section can now be implemented.
The steady-state form of Eqs.(37) can be expressed in matrix form as
|
(38) |
The system is symmetric and always positive definite and therefore leads to a non singular solution. This property holds for any interpolation function chosen 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 \bar {v},\bar {p}}
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 \bar {\boldsymbol \pi }}
, therefore overcoming the Babuŝka-Brezzi (BB) restrictions [8].
A reduced velocity-pressure formulation can be obtained by elliminating 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 \bar \pi }
variables from the last row of Eq.(38) [37].
The FIC formulation for Stokes flows is applicable for analysis of quasi/fully incompressible solids. An analogous formulation based on solid mechanics concepts is derived in Section 6.
4 THERMAL-COUPLED FLOWS. TREATMENT OF THE FREE BOUNDARY
4.1 Thermal coupled flow
The effect of temperature can be easily introduced by solving the equation for heat transport coupled to the fluid flow equations.
The equation of balance of heat is written in the FIC formulation as [30]
|
(39a) |
with
|
(39b) |
In above Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi }
is the temperature, Failed to parse (MathML with SVG or PNG fallback (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}
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 k}
are the specific heat and the thermal conductivity of the material, respectively, Failed to parse (MathML with SVG or PNG fallback (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}
is the heat source 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_j}
are the characteristic length distances which are typical of the FIC formulation [30].
Eq.(39a) is completed with the Dirichlet and Neuman boundary conditions for the heat problem. For details see [30,46].
The convective velocities Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v_i}
in Eq.(39b) are provided by the solution of the fluid flow problem. As usual in metal forming processes, the heat source Failed to parse (MathML with SVG or PNG fallback (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}
is a function of the mechanical work generated in the flow of the material during the forming process. The temperature field affects in turn the flow viscosity via its dependence with the yield stress which is very sensitive to the temperature changes. The solution of the heat transfer equation is therefore fully coupled with that of the fluid flow problem [44].
4.2 Treatment of the free boundary motion
For the treatment of the free boundary we use a standard volume of fluid (VOF) technique, also known as pseudo-concentration method or level set technique [22,47,48]. In the VOF method the motion of the free boudary is followed by solving the following transport equation (written using the FIC formulation)
|
(40a) |
where
|
(40b) |
In above Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta }
is an auxiliary variable which takes a value equal to one on the free surface and zero elsewhere. The solution of Eq.(40a) in time in the discretized fluid-air domain allows to track the free surface which is characterized by the contours 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 \beta }
which have a unit value.
The underlined term in Eq.(40a) emerges from the FIC formulation [33]. Again, 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 \bar h_j} 's in Eq.(40a) are characteristic length distances of the order of the element size. This term introduces the necessary stabilization in the transient solution of Eq.(40a).
4.3 Fully coupled algorithm for mould filling problems
The analysis of mould filling problems typically requires the solution of the coupled-thermal flow problem accounting for the transport of the free surface.
The simplest explicit algorithm is as follows:
- For each time step:
- Compute the velocities and the pressure in the fluid Failed to parse (MathML with SVG or PNG fallback (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{v}^{n+1}_i, \bar{p}^{n+1})}
using the fractional step scheme of Section 3.1.
- Compute the nodal temperatures Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\phi ^{n+1})}
solving Eq.(39a).
- Compute the new position of the free surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\beta ^{n+1})}
solving Eq.(40a).
A number of implicit versions of the algorithm are possible and they all involve an iteration loop within each time step until convergence for the flow variables, the temperature and the free surface position is found. For details see [47,48].
5 COMPUTATION OF THE CHARACTERISTIC LENGTHS
The evaluation of the stabilization parameters is one of the crucial issues in stabilized methods. Excellent results have been obtained in all problems solved using linear tetrahedra with the characteristic length vector defined by
|
(41a) |
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 v=\vert {v}\vert }
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_s}
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_{c}}
are the “streamline” and “cross wind” contributions given by
|
(41b) |
|
(41c) |
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}_j}
are the vectors defining the element sides (Failed to parse (MathML with SVG or PNG fallback (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_s=6}
for tetrahedra).
As for the free surface equation the following value of the characteristic length 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 \bar{h}}
in Eq.(40a) has been taken
|
(42a) |
The streamline 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 \bar h_s}
has been obtained by Eq.(41b) whereas the cross wind 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 \bar h_c}
has been computed by
|
(42b) |
The cross-wind terms in eqs.(41a) and (42a) account for the effect of the gradient of the solution in the stabilization parameters. This is a standard assumption in most “shock-capturing” stabilization procedures [8,24].
6 FIC FORMULATION FOR QUASI/FULLY INCOMPRESSIBLE SOLIDS
6.1 Equilibrium equations
As usual the governing equations of solid mechanics are written in a Lagrangian reference frame. Following the arguments of Section 2 the equilibrium equations for a solid are written using the FIC technique as [37]
|
(43) |
where for the dynamic case
|
(44) |
In Eqs.(43) and (44) Failed to parse (MathML with SVG or PNG fallback (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_i}
are the 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/":): {\sigma }_{ij}
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/":): {b}_i are the stresses and the body forces, respectively 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/":): {h}_k are characteristic length distances of an arbitrary prismatic domain where equilibrium of forces is considered.
Equations (43) and (44) are completed with the boundary conditions on the 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 u_i}
|
(45) |
and the equilibrium of surface tractions
|
(46) |
In the above Failed to parse (MathML with SVG or PNG fallback (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}_i^p}
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 t_i}
are prescribed displacements and tractions over the boundaries Failed to parse (MathML with SVG or PNG fallback (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 _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 \Gamma _t} , respectively 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_i}
are the components of the unit normal vector.
Note that for consistency with the fluid flow equations of previous section and differently from the tradition in solid mechanics, the compression pressure has been taken as positive.
6.2 Stabilized pressure constitutive equations
For simplicity the treatment of the constitutive equations will be explained for the linear elastic model. The approach extends naturally to the non linear elasto-plastic/viscoplastic constitutive equations typical of metal forming problems.
As usual in quasi-incompressible problems the stresses are split into deviatoric and volumetric (pressure) parts
|
(47) |
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/":): {\delta }_{ij} is the Kronnecker delta function. The linear elastic constitutive equations for the deviatoric stresses Failed to parse (MathML with SVG or PNG fallback (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_{ij}}
are written as
|
(48) |
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 G} is the shear modulus,
|
(49) |
The constitutive equation for the pressure Failed to parse (MathML with SVG or PNG fallback (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}
can be written for an arbitrary domain of finite size of 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 V}
as
|
(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 K}
is the bulk modulus of the material 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 p^{av}}
is the average value of the pressure over 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 V}
.
The 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 p^{av}}
can be approximated as [37]
|
(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 p}
is the pressure at an arbitrary point within 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 V}
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_k}
are characteristic lengths of such a domain.
The ratio Failed to parse (MathML with SVG or PNG fallback (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 V\over V}}
can be expressed as
|
(52) |
Substituting Eqs.(51) and (52) into Eq.(50) and neglecting second order terms 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 h_k}
gives the FIC constitutive equation for the pressure as
|
(53) |
Note that 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 h_k \to 0}
the standard relationship between the pressure and the volumetric strain of the infinitesimal theory Failed to parse (MathML with SVG or PNG fallback (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=-K\varepsilon _v )}
is found.
For an incompressible material Failed to parse (MathML with SVG or PNG fallback (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 \infty }
and Eq.(53) yields
|
(54) |
Eq.(54) expresses the limit incompressible behaviour of the solid. This equation is identical to Eq.(2) for incompressible flow problems and there arises from the mass continuity condition [30,32].
By combining Eqs. (43), (44), (47), (48) and (53) a mixed displacement–pressure formulation can be written as
|
(55) |
|
(56) |
From the observation of Eq.(55) we can obtain after some algebra [37]
|
(57) |
Substituting Eq.(57) into (53) the mass balance equation can be written as
|
(58) |
In the derivation of Eq.(58) we have neglected the terms involving products Failed to parse (MathML with SVG or PNG fallback (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_ih_j}
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 h_i \not = h_j}
.
The pressure gradient projections Failed to parse (MathML with SVG or PNG fallback (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 _i}
are introduced now as
|
(59) |
The final system of governing equations is
|
(60) |
|
(61) |
|
(62) |
In Eqs.(60)–(62) we note the following:
- The stabilization terms have been neglected in Eq.(60). These terms were useful to derive the stabilized form of Eq.(61). However they are not longer necessary as the convective terms do not appear in the equilibrium equations.
- The constitutive equation for the pressure (Eq.(61)) has been written in an incremental form. This is more convenient for non linear material behaviour typical of metal forming situations. Here, appropriate elasto-plastic or elasto-viscoplastic constitutive laws relating stresses and strains in incremental or rate forms must be used [1–5,43,44].
6.3 Finite element equations
The weighted residual form of the governing equations can be written as (after integration by parts of the relevant terms)
|
(63a) |
|
(63b) |
|
(63c) |
where again 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 \tau _i}
are introduced in Eq.(63c) for symmetry reasons.
The finite element discretization of the displacements, the pressure and the pressure gradient projections is written by expressions identical to Eqs.(23) with the nodal variables now being a function of the 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} . Substituting the approximations into eqs.(63) and using the Galerkin form gives the following system of discretized equations
|
(64a) |
|
(64b) |
|
(64c) |
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 \ddot{\bar {u}}}
is the nodal acceleration vector,
|
(65) |
is the mass matrix,
|
(66) |
is the internal nodal force vector and the rest of matrices and vectors are defined in the Appendix. Note that the expression 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 {g}}
of eq.(66) is adequate for non linear analysis.
A four steps semi-explicit time integration algorithm can be derived as follows
step
Step step.step
- Compute the nodal velocities Failed to parse (MathML with SVG or PNG fallback (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 {u}}^{n+1/2}}
from Eq.(64a)
|
(67a) |
- Compute the nodal 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 \bar {u}^{n+1}}
|
(67b) |
- Compute the nodal pressures Failed to parse (MathML with SVG or PNG fallback (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 {p}^{n+1}}
from Eq.(64b)
|
(67c) |
- Compute the nodal projected pressure gradients Failed to parse (MathML with SVG or PNG fallback (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 {\boldsymbol \pi }^{n+1}}
from Eq.(64c)
|
(67d) |
In above, all matrices are evaluated 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/":): {\textstyle t^{n+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 (\cdot )_d }
denotes a lumped diagonal matrix and
|
(68) |
where the stresses Failed to parse (MathML with SVG or PNG fallback (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}
are obtained by consistent integration of the adequate (non linear) constitutive law.
Note that steps 1, 2 and 4 are fully explicit as a diagonal form of 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 {M}}
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 \bar{C}}
has been chosen. The solution of step 3 requires invariably the inverse of a Laplacian matrix. This can be an inexpensive process using an iterative equation solution method (e.g. a preconditioned conjugate gradient method).
For the full incompressible 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 K=\infty }
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 {C}=0 }
in all above equations.
The critical time step Failed to parse (MathML with SVG or PNG fallback (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 t}
is taken as that of the standard explicit dynamic scheme (see Section 6.5 and [9,37]).
Fully explicit algorithm
A fully explicit four steps algorithm can be obtained by computing Failed to parse (MathML with SVG or PNG fallback (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 {p}^{n+1}}
from step 3 in eq.(67c) as follows
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(69) |
Note that the explicit algorithm is not applicable in the full incompressible limit as the solution of Eq.(69) breaks down 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 K=\infty }
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 {C}=0}
. The explicit form can however be used with success in problems where quasi-incompressible regions exist adjacent to standard “compressible” zones. Examples of this kind are shown in the next section. In both cases the semi-explicit and fully explicit schemes gave identical results with important savings in both computer time and memory storage requirements obtained when using the explicit form.
6.4 Thermal coupled effects
The effect of temperature can be easily accounted for by solving the heat transfer equation formulated in a Lagrangian frame as
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(70) |
As usual, the source Failed to parse (MathML with SVG or PNG fallback (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}
is dependent of the mechanical work generated during the forming process.
Note that the convective terms do not enter into Eq.(70) This also eliminates the need to stabilize the numerical solution.
The coupled thermal-mechanical problem requires the computation of the temperature Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi }
at each time step using a transient solution scheme [47,48,50].
Remark
Above formulation is similar to that developed by Chiumenti et al. [27,29] and Cervera et al. [28] for analysis of incompressible problems in solid mechanics using a sub-grid scale approach.
6.5 Computation of the intrinsic time parameter for quasi/fully incompressible solids
In solid mechanics applications it is usual to accept that all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau _i}
are identical and constant within each element and given by
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(71) |
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 V^{(e)}}
is the element volume (or the element area for 2D problems). This 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 \tau ^{(e)}}
does not take into account the element distorsions along a particular direction during the deformation process.
The correct value of the shear modulus in the expression 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 \tau ^{(e)}}
is another sensitive issue as, obviously, for non linear problems the 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 G}
will differ from the elastic modulus. This fact has been identified by Cervera et al. [28] for non linear analysis of incompressible problems using linear triangles.
A useful alternative to compute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau ^{(e)}}
for explicit non linear transient situations is to make use of the value of the speed of sound in an elastic solid, defined by
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(72) |
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}
is the Young modulus. The stability condition for explicit dynamic computations is given by the Courant condition defined as [8]
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(73) |
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 t^{(e)}_c}
is the critical time step for the element.
Accepting 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 G\simeq {E\over 3}}
for the incompressible case and using Eqs.(72) and (73) (assuming the identity in Eq.(73)) an alternative expression for the element intrinsic time parameter in terms of the critical time step can be found as
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(74) |
Eq.(74) shows clearly that the intrinsic time parameter varies across the mesh as a function of the critical time step for each element.
7 APPLICATIONS TO CASTING AND BULK METAL FORMING PROCESSES
7.1 Aluminium casting simulation
A numerical simulation of an aluminium casting process is presented as a demonstration of the accuracy of the stabilized formulation. The computations are performed with the finite element code VULCAN where the stabilized FEM presented has been implemented [56].
The analysis simulates the casting process of an aluminium (AlSi7Mg) specimen in a steel (X40CrMoV5) mould. Material behavior of aluminium casting has been modeled by a fully coupled thermo-viscoplastic model, while the steel mould has been modeled by a simpler thermo-elastic model [51]. Geometrical and material data were provided by the foundry RUFFINI. Figure 1 shows the finite element mesh used for the part and the cooling system. The full mesh, including the mould has 380.000 four node tetrahedra. The pouring temperature is 650Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^\circ } C. Initial temperature for the mould is obtained through a thermal die-cycling simulation. Figure 2 shows the evolution of the mould temperature after 6 cycles. The cooling system has been kept at 20Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^\circ } C. Filling evolution has been simulated as in a pressure die-casting process using the stabilized VOF technique described in [47,48]. Figure 3 shows different time steps of the simulation.
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| Figure 1: Finite element discretization of the aluminium casting. |
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| Figure 2: Temperature die-cycling. |
The final temperature field obtained after the filling simulation is taken as the initial condition for the solidification and cooling analysis. Temperature and liquid-fraction distributions during solidification are shown in Figures 4 and 5, respectively. The heat transfer coefficient takes into account the air-gap resistance due to the casting shrinkage during the solidification process. Figure 6 shows volumetric and von Mises deviatoric stress distributions in a x-y section. The figures also show the air-gap between the part and the mould, responsible of a non-uniform heat flux at the contact interface.
Other examples of application of the stabilized formulation to casting problems can be found in [47–51].
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| Figure 3: Filling evolution: pressure die-casting simulation. | |
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| Figure 4: Temperature evolution during cooling phase. | ||
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| Figure 5: Liquid-fraction evolution during phase-change. | |
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| Figure 6: Aluminium casting. Stress-trace and von Mises deviatoric stress indicator during phase-change (plane xy). |
7.2 Side pressing of a cylinder
A cylinder 100 mm long with a radius of 100 mm is subjected to sidepressing between two plane dies. It is compressed to 100 mm. The material properties are the following: Failed to parse (MathML with SVG or PNG fallback (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 =217}
GPa, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nu =0.3}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho =7830}
kg/mFailed to parse (MathML with SVG or PNG fallback (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}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _0=170}
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/":): {\textstyle H=30}
MPa, friction coefficient = 0.2. The die velocity is assumed to be 2 m/s. Initial set-up is shown in Figure 7a. A quarter of a cylinder was discretized with tri-linear hexahedra and linear tetrahedra meshes.
Figure 7 shows the results obtained using the hexahedral mesh and a standard mixed formulation. The results show the distribution of the effective plastic strain and pressure on the deformed shape. The sensitivity of the FIC results with the expression of the intrinsic time parameter of Eq.(71) was studied by defining Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau ^{(e)} =a {[h^{(e)}]^2\over G}}
and solving the problem for different 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 a}
. The results obtained with the FIC method are shown in Figs. cylt-fine-fast-fic01 and cylt-fine-fast-fic 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 a =0.1}
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 0.03}
, respectively. The alternative 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 \tau ^{(e)}}
of Eq.(74) has also been studied. The results for this case are shown in Fig. cylt-fine-fast-fic-dtcr. Quite a good agreement can be seen between the FIC solutions and the reference solution with the best results for the effective plastic strain obtained 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 \tau ^{(e)}}
calculated according to Eq. (71) 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 a=0.03}
. The results for the two alternative formulae 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 \tau ^{(e)}}
are similar, but those obtained using Eq. (71) seem to be slightly better. In any case, the results on the pressure distribution are quite insentitive to the 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 \tau ^{(e)}}
. This is also confirmed in Fig. cylt-liniab, which displays the distribution of the pressure along the 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 ABCDEA}
defined in Fig. cylt-liniaa. A small perturbation can be seen at the sharp edges of the deformed body.
All the calculations here have been carried out using fully explicit version of the algorithm which is more efficient than the semi-implicit one with giving practically the same results.
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(a) Sidepressing of a cylinder: (a) initial tetrahedral mesh; (b) initial hexahedral mesh |
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| Figure 7: Sidepressing of a cylinder, mixed formulation, hexahedral mesh: (a) effective plastic strain; (b) pressure distribution | ||
Document information
Published on 01/01/2006
DOI: 10.1016/j.cma.2004.10.018
Licence: CC BY-NC-SA license
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