A dual-mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions

Gatica, Gabriel N.; Gatica, Luis F.; Stephan, Ernst P.

doi:10.1016/j.cma.2007.03.007

Cited by 15 publications

(19 citation statements)

References 28 publications

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“…Sufficient conditions for solvability and abstract error analysis are also given in [21] and [25]. This theory has been applied to many problems in elasticity and fluids as well, see [27,29,24,15,22,23,26] for some examples. Problems of type (1.2) were also studied in [16,17].…”

Section: Related Resultsmentioning

confidence: 99%

“…Twofold saddle point problems arise ubiquitously when mixed finite element formulations are used to approximate the stress in an incompressible fluid or solid [2,9,27,29,24,15,22,23,6,26]. When the usual linear relation S 0 = νD(u) for the viscous stress is replaced with a more general relation S 0 = νA(D(u)) the equations for the creeping (Stokes) flow of a fluid in a domain Ω ⊂ R d become, −div(S) = f , div(u) = 0, S = −pI + A(D(u)).…”

Section: Examples Of Twofold Saddle Point Problemsmentioning

confidence: 99%

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Inf–sup conditions for twofold saddle point problems

Howell

Walkington

2011

Numer. Math.

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Summary Necessary and sufficient conditions for existence and uniqueness of solutions are developed for twofold saddle point problems which arise in mixed formulations of problems in continuum mechanics. This work extends the classical saddle point theory to accommodate nonlinear constitutive relations and the twofold saddle structure. Application to problems in incompressible fluid mechanics employing symmetric tensor finite elements for the stress approximation is presented.

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Section: Related Resultsmentioning

confidence: 99%

Section: Examples Of Twofold Saddle Point Problemsmentioning

confidence: 99%

Inf–sup conditions for twofold saddle point problems

Howell

Walkington

2011

Numer. Math.

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“…This error analysis uses the computed solution and a higher order approximation to a linear dual problem to estimate the error in a given norm or functional of the solution. Furthermore, it also identifies areas of the computational domain where the contribution to the error is greatest, highlighting regions of this domain where mesh refinement may significantly improve accuracy, thus driving mesh adaptation strategies [12,13,20,21,[26][27][28][29]. A posteriori error analysis has been attempted before for nonlinear elasticity in the special case where the strain tensor has been linearised [26,27,29,30].…”

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confidence: 99%

Error estimation and adaptivity for incompressible hyperelasticity

Whiteley

Tavener

2014

Numerical Meth Engineering

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SUMMARYA Galerkin FEM is developed for nonlinear, incompressible (hyper) elasticity that takes account of nonlinearities in both the strain tensor and the relationship between the strain tensor and the stress tensor. By using suitably defined linearised dual problems with appropriate boundary conditions, a posteriori error estimates are then derived for both linear functionals of the solution and linear functionals of the stress on a boundary, where Dirichlet boundary conditions are applied. A second, higher order method for calculating a linear functional of the stress on a Dirichlet boundary is also presented together with an a posteriori error estimator for this approach. An implementation for a 2D model problem with known solution, where the entries of the strain tensor exhibit large, rapid variations, demonstrates the accuracy and sharpness of the error estimators. Finally, using a selection of model problems, the a posteriori error estimate is shown to provide a basis for effective mesh adaptivity.

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“…In addition, it is well known that for low-order finite element approximations of elasticity problems subject to the volume preserving constraint, the undesirable so-called volume locking phenomenon, in which the displacement decreases severely in a non-physical fashion may show up [12]. Several remedies have been proposed in the literature, including for instance resorting to different kinds of mixed and double-mixed formulations [4,6,10,15]. Herein, following the framework in [5,27], we formulate a suitable discontinuous Galerkin (DG) method.…”

Section: Introductionmentioning

confidence: 99%

Convergence of a stabilized discontinuous Galerkin method for incompressible nonlinear elasticity

2012

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In this paper we present a discontinuous Galerkin method applied to incompressible nonlinear elastostatics in a total Lagrangian deformationpressure formulation, for which a suitable interior penalty stabilization is applied. We prove that the proposed discrete formulation for the linearized problem is well-posed, asymptotically consistent and that it converges to the corresponding weak solution. The derived convergence rates are optimal and further confirmed by a set of numerical examples in two and three spatial dimensions.

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A dual-mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions

Cited by 15 publications

References 28 publications

Inf–sup conditions for twofold saddle point problems

Inf–sup conditions for twofold saddle point problems

Error estimation and adaptivity for incompressible hyperelasticity

Convergence of a stabilized discontinuous Galerkin method for incompressible nonlinear elasticity

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