A mixed finite element method for nonlinear elasticity: two-fold saddle point approach and a-posteriori error estimate

Barrientos, Mauricio A.; Gatica, Gabriel N.; Stephan, Ernst P.

doi:10.1007/s002110100337

Cited by 50 publications

(55 citation statements)

References 44 publications

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“…Next, the boundedness ofb is straightforward and the proof of the continuous inf-sup condition forb reduces to combine the corresponding proof for b (see Lem. 4.3 in [6]) with the following result. …”

Section: Mixed Boundary Conditionsmentioning

confidence: 67%

“…: η + η t = 0 }, and the bilinear forms a : 6) for all ζ, τ ∈ H and for all (v, η) ∈ Q. It is easy to see from (2.5) and (2.6) that for any (τ ,…”

Section: Dirichlet Boundary Conditionsmentioning

confidence: 99%

“…Similarly, it is easy to see from (2.5) that a is bounded with a constant given by 1 2µ . For a detailed proof of the continuous inf-sup condition satisfied by b we refer to Lemma 4.3 in [6], while the boundedness of b is clear from (2.6). Alternatively, the well posedness of (2.10) is also proved in [4].…”

Section: Dirichlet Boundary Conditionsmentioning

confidence: 99%

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Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb{RT}_0$-$\mathbb{P}_1$-$\mathbb{P}_0$ approximations

Gatica¹

2006

ESAIM: M2AN

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Abstract. We present a new stabilized mixed finite element method for the linear elasticity problem in R 2 . The approach is based on the introduction of Galerkin least-squares terms arising from the constitutive and equilibrium equations, and from the relation defining the rotation in terms of the displacement. We show that the resulting augmented variational formulation and the associated Galerkin scheme are well posed, and that the latter becomes locking-free and asymptotically locking-free for Dirichlet and mixed boundary conditions, respectively. In particular, the discrete scheme allows the utilization of Raviart-Thomas spaces of lowest order for the stress tensor, piecewise linear elements for the displacement, and piecewise constants for the rotation. In the case of mixed boundary conditions, the essential one (Neumann) is imposed weakly, which yields the introduction of the trace of the displacement as a suitable Lagrange multiplier. This trace is then approximated by piecewise linear elements on an independent partition of the Neumann boundary whose mesh size needs to satisfy a compatibility condition with the mesh size associated to the triangulation of the domain. Several numerical results illustrating the good performance of the augmented mixed finite element scheme in the case of Dirichlet boundary conditions are also reported.Mathematics Subject Classification. 65N12, 65N15, 65N30, 74B05.

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Section: Mixed Boundary Conditionsmentioning

confidence: 67%

“…: η + η t = 0 }, and the bilinear forms a : 6) for all ζ, τ ∈ H and for all (v, η) ∈ Q. It is easy to see from (2.5) and (2.6) that for any (τ ,…”

Section: Dirichlet Boundary Conditionsmentioning

confidence: 99%

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Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb{RT}_0$-$\mathbb{P}_1$-$\mathbb{P}_0$ approximations

Gatica¹

2006

ESAIM: M2AN

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“…In such an event, the convergence performance can be improved by resorting to a robust meshrefinining algorithm for an efficient automatic mesh-design. A list of contributions proposing and analyzing robust and effective adaptive finite element methods in compressible, nearly incompressible and pure incompressible solid and fluid mechanics includes references [4,11,12,14,13,20,28,29,5,22,23].…”

Section: Introductionmentioning

confidence: 99%

A posteriori dual-mixed adaptive finite element error control for Lamé and Stokes equations

2005

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A unified and robust mathematical model for compressible and incompressible linear elasticity can be obtained by rephrasing the Herrmann formulation within the Hellinger-Reissner principle. This quasi-optimally converging extension of PEERS (Plane Elasticity Element with Reduced Symmetry) is called Dual-Mixed Hybrid formulation (DMH). Explicit residual-based a posteriori error estimates for DMH are introduced and are mathematically shown to be locking-free, reliable, and efficient. The estimator serves as a refinement indicator in an adaptive algorithm for effective automatic mesh generation. Numerical evidence supports that the adaptive scheme leads to optimal convergence for Lamé and Stokes benchmark problems with singularities.

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“…To this respect, concerning the combination of the usual FEM with BEM, we may refer to [10,13,14], where mainly reliable a-posteriori error estimates are provided. More recently, this kind of result has been extended to the coupling of dual-mixed FEM and BEM for linear and nonlinear problems (see [5,6,12,19,21,23]). Here, the estimates for the linear problems are of explicit residual type, and those for the nonlinear ones are based on the classical Bank-Weiser implicit approach.…”

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confidence: 96%

A posteriorierror estimates for linear exterior problemsviamixed-FEM and DtN mappings

2002

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Abstract. In this paper we combine the dual-mixed finite element method with a Dirichlet-toNeumann mapping (given in terms of a boundary integral operator) to solve linear exterior transmission problems in the plane. As a model we consider a second order elliptic equation in divergence form coupled with the Laplace equation in the exterior unbounded region. We show that the resulting mixed variational formulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derive the usual Cea error estimate and the corresponding rate of convergence. In addition, we develop two different a-posteriori error analyses yielding explicit residual and implicit Bank-Weiser type reliable estimates, respectively. Several numerical results illustrate the suitability of these estimators for the adaptive computation of the discrete solutions.Mathematics Subject Classification. 65N15, 65N30, 65N38, 65N50.

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A mixed finite element method for nonlinear elasticity: two-fold saddle point approach and a-posteriori error estimate

Cited by 50 publications

References 44 publications

Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb{RT}_0$-$\mathbb{P}_1$-$\mathbb{P}_0$ approximations

Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb{RT}_0$-$\mathbb{P}_1$-$\mathbb{P}_0$ approximations

A posteriori dual-mixed adaptive finite element error control for Lamé and Stokes equations

A posteriorierror estimates for linear exterior problemsviamixed-FEM and DtN mappings

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