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Mixed Stabilized Finite Element Methods in Nonlinear Solid Mechanics.

Part I: Formulation

M. Cervera, M. Chiumenti and R. Codina International Center for Numerical Methods in Engineering (CIMNE)

Technical University of Catalonia (UPC)

Edicio C1, Campus Norte, Jordi Girona 1-3, 08034 Barcelona, Spain.

Keywords: mixed nite element interpolations, stabilization methods, algebraic sub-grid scales, orthogonal sub-grid scales, nonlinear solid mechanics.

Abstract

This paper exploits the concept of stabilized nite element methods to formulate stable mixed stress/displacement and strain/displacement nite elements for the solution of nonlinear solid mechanics prob- lems. The di¤erent assumptions and approximations used to derive the methods are exposed. The proposed procedure is very general, applicable to 2D and 3D problems. Implementation and computa- tional aspects are also discussed, showing that a robust application of the proposed formulation is feasible. Numerical examples show that the results obtained compare favourably with those obtained with the corresponding irreducible formulation.

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1 Introduction

The term mixed methods has been used in the nite element method litera- ture since the mid 1960s to denote formulations in which both the displace- ment and stress elds are approximated as primary variables [1]. Despite the doubtless interest of mixed methods from the theoretical point of view, their practical application is greatly outnumbered by the implementation of irreducible methods, in which only the displacement eld is considered pri- mary variable of the problem and the stress eld is obtained a posteriori by di¤erentiation. However, there are several elds of application in computational solid

mechanics in which mixed methods are well established and regularly used in practice. For instance, it is well known that standard irreducible low order nite elements perform miserably in nearly incompressible situations, producing solutions which are almost completely locked by the incompress- ibility constraint. Remedies for this undesirable behavior have been actively sought for decades. In fact, the purely incompressible problem (Stokes prob- lem) does not admit an irreducible formulation and, consequently, a mixed framework in terms of displacements and pressure is necessary for these sit- uations. Over the years, and particularly in the 1990s, di¤erent strategies were proposed and tested to reduce or avoid volumetric locking and pres- sure oscillations in nite element solutions with di¤erent degrees of success ([2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]). Many of these methods, while resembling displacement methods, have been shown to be equivalent to more general mixed methods. Another common application of mixed methods is plate bending and other

fourth order problems ([15, 5, 16, 17]). Here, the motivation is the avoidance of C1-continuity in the denition of the interpolation functions, required if the primal variational functional is used. Alternatively, the mixed functional only involves second derivatives and, after integration by parts, C0-continuous elements may be used. Another alternative is the use of non-conforming elements. The reasons for the limited popularity of mixed methods in computa-

tional solid mechanics are twofold: computational cost and lack of stability [18, 19, 20]. On one hand, because mixed methods approximate both dis- placements and stresses simultaneously, the corresponding discrete systems of equations involve many more degrees of freedom than the correspond- ing irreducible formulations. Concurrently, the mixed system of equations

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is very often indenite, which makes most of the direct and iterative solu- tion methods inapplicable. These di¢ culties may be avoided with a suitable implementation. On the other hand, many choices of the individual inter- polation elds for the mixed problem yield meaningless, not stable, results. This is due to the strictness of the inf-sup condition [19] when the stan- dard Galerkin nite element method is applied straightforwardly to mixed elements, as it imposes severe restrictions on the compatibility of the inter- polations used for the displacement and the stress elds. This di¢ culty, if not circumvented, is severely restrictive (see [21] and [22, 23] for the analysis of admissible elements in linear elasticity). In parallel, mixed methods have also been the focus of attention in com-

putational uid dynamics. In [24] and [25], the variational multiscale (VMS) formulation was proposed as a new way of circumventing the di¢ culties posed by the inf-sup condition. In the case of incompressible problems, the reason- ing behind was not new, as it consisted of modifying the discrete variational form to attain control on the pressure eld. The result was the possibility of using equal order interpolations for displacements and pressures and to con- struct stable low order elements. Since then, the sub-grid concept underlying the VMS approach has been extensively and fruitfully used in uid dynamics. In [26] and [27], the concept of orthogonal subscale stabilization (OSS) was introduced, which leads to well sustained and better performing stabilization procedures. The analysis of the formulation can be found in [28] for the lin- earized incompressible Navier-Stokes equations and, in subjects closer to the topic of this paper, in [29] for the stress-displacement-pressure formulation of the Stokes problem (equivalent to the linear elastic incompressible problem) and in [30] for Darcys problem. In previous works, the authors have applied stabilized mixed displace-

ment/pressure methods (see [31, 32, 33, 34, 35] and [36]) to the solution of incompressible J2-plasticity and damage problems with strain localization us- ing linear/linear simplicial elements in 2D and 3D. These procedures lead to a discrete problem which is fully stable, free of pressure oscillations and vol- umetric locking and, thus, results obtained are practically mesh independent. This translates in the achievement of two important goals: (a) the position and orientation of the localization band is independent of the directional bias of the nite element mesh and (b) the global post-peak load-deection curves are independent of the size of the elements in the localization band. Similar ideas have been used in [37, 38] and [39]. In the present work we apply this approach in order to derive stable mixed

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stress-displacement and strain-displacement formulations using linear/linear interpolations in triangular elements and bilinear/bilinear interpolations in quadrilateral elements. It is noteworthy that, from the numerical point of view, the di¢ culties encountered in this problem are very di¤erent to those found in incompressible situations, analyzed in previous works. The treat- ment of the incompressible case in the stress/displacement formulation would require considering the pressure as an additional independent variable and appropriate stabilization techniques (see [29]). The incompressible limit will not be treated here, and the following formulation is limited to compressible nonlinear solid mechanics. The basic motivation for this work is to show that the di¢ culties en-

countered when solving solid mechanics problems involving the creation and propagation of strain localization bands using standard elements and local constitutive models are due to the approximation error inherent to the spatial discretization, as well as to the poor stability in the stresses and/or strains. When using the basic, irreducible, formulation of the problem, the stresses (or strains), which are the variables of most interest for the satisfaction of the highly nonlinear constitutive behavior, are not the fundamental unknowns of the problem and they are obtained by di¤erentiation of the displacement eld, a process which entails an important loss of accuracy, particularly where strong displacement gradients occur. The local approximation error commit- ted makes propagation of the localization bands strongly dependent on the nite element mesh used. Contrariwise, when using a mixed formulation in which the stress (or the strain) eld is selected as primary variable, together with the displacement eld, the added accuracy and stability achieved are enough to overcome the mesh dependency problem satisfactorily. The outline of the present paper is as follows. In Section 2 the mixed

stress/displacement nite element formulation for linear elasticity is sum- marized. The sub-grid scale approach is used to derive two stabilized for- mulations. Results concerning stability and convergence of these schemes are discussed. In Section 3 the stabilization is extended to nonlinear prob- lems, proposing both stress-displacement and strain-displacement formula- tions. The later can be considered more suitable for the implementation of nonlinear constitutive models. Implementation and computational aspects are discussed next. Finally, some numerical benchmarks and examples are presented to assess the present formulation and to compare its performance with the standard irreducible elements. The problem of strain localization is discussed in a companion paper [36].

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2 Mixed stabilized stressdisplacement for- mulation in linear elasticity

2.1 Continuous problem

The formulation of the solid mechanics problem can be written considering the stress as an independent unknown, additional to the displacement eld. In this case, the strong form of the continuous problem can be stated as: given a eld of prescribed body forces f and a constant constitutive tensor C, nd the displacemen