Equilibrated Stress Reconstructions for Linear Elasticity Problems with Application to a Posteriori Error Analysis

Riedlbeck, Rita; Pietro, Daniele Antonio Di; Ern, Alexandre

doi:10.1007/978-3-319-57397-7_22

Cited by 7 publications

(13 citation statements)

References 14 publications

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“…This reconstruction uses the Arnold-Falk-Winther mixed finite element spaces [4], leading to weakly symmetric tensors . In [37] this reconstruction is compared to a similar reconstruction introduced in [38] using the Arnold-Winther finite element spaces [5], yielding a symmetric tensor, and very good agreement was observed while saving substantial computational effort. In Section 3 we apply this reconstruction to the nonlinear case by constructing two stress tensors: one playing the role of the discrete stress and one expressing the linearization error.…”

Section: Introductionmentioning

confidence: 93%

“…Let us for now suppose that u h solves (2.13) exactly, before considering iterative linearization methods such as (2.14) in Section 3.2. For the stress reconstruction we will use mixed finite element formulations on patches around mesh vertices in the spirit of [37,38]. The mixed finite elements based on the dual formulation of (1.1a) will provide a stress tensor lying in Hpdiv, Ωq.…”

Section: Patchwise Construction In the Arnold-falk-winther Mixed Finimentioning

confidence: 99%

“…In [38] the stress tensor is reconstructed in the Arnold-Winther finite element space [5], directly providing symmetric tensors, but requiring high computational effort. In this work, as in [37], we weaken the symmetry constraint and impose it weakly, as proposed in [4]: for each element T P T h , the local Arnold-Falk-Winther mixed finite element spaces of degree q ě 1 hinge on the Brezzi-Douglas-Marini mixed finite element spaces [9] for each line of the stress tensor and are defined by…”

Section: Patchwise Construction In the Arnold-falk-winther Mixed Finimentioning

confidence: 99%

“…It is well known that, in contrast to the analytical solution, the discrete stress tensor resulting from the conforming finite element method does not have continuous normal components across mesh interfaces, and that its divergence is not locally in equilibrium with the source term f on mesh elements. In this paper we consider the stress tensor reconstruction proposed in [37] for linear elasticity to restore these two properties. This reconstruction uses the Arnold-Falk-Winther mixed finite element spaces [4], leading to weakly symmetric tensors .…”

Section: Introductionmentioning

confidence: 99%

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Equilibrated Stress Tensor Reconstruction and A Posteriori Error Estimation for Nonlinear Elasticity

Botti

Riedlbeck²

2018

Computational Methods in Applied Mathematics

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We consider hyperelastic problems and their numerical solution using a conforming finite element discretization and iterative linearization algorithms. For these problems, we present equilibrated, weakly symmetric, Hpdivq-conforming stress tensor reconstructions, obtained from local problems on patches around vertices using the Arnold-Falk-Winther finite element spaces. We distinguish two stress reconstructions, one for the discrete stress and one representing the linearization error. The reconstructions are independent of the mechanical behavior law. Based on these stress tensor reconstructions, we derive an a posteriori error estimate distinguishing the discretization, linearization, and quadrature error estimates, and propose an adaptive algorithm balancing these different error sources. We prove the efficiency of the estimate, and confirm it on a numerical test with analytical solution for the linear elasticity problem. We then apply the adaptive algorithm to a more application-oriented test, considering the Hencky-Mises and an isotropic damage models.

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Section: Introductionmentioning

confidence: 93%

Section: Patchwise Construction In the Arnold-falk-winther Mixed Finimentioning

confidence: 99%

Section: Patchwise Construction In the Arnold-falk-winther Mixed Finimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Equilibrated Stress Tensor Reconstruction and A Posteriori Error Estimation for Nonlinear Elasticity

Botti

Riedlbeck²

2018

Computational Methods in Applied Mathematics

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“…This is particularly true in three dimensions where the lowest-order member of the symmetric H(div)-conforming finite element space constructed in [4] already involves polynomials of degree 4 and possesses 162 degrees of freedom per tetrahedron. Equilibrated stress reconstructions with weak symmetry are also considered in [18], [3], [23]. These approaches utilize special stress finite element spaces and are therefore less general than the one presented in this work.…”

mentioning

confidence: 99%

Equilibrated Stress Reconstruction and a Posteriori Error Estimation for Linear Elasticity

Bertrand

Kober

Moldenhauer

et al. 2019

Novel Finite Element Technologies for Solids and Structures

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A stress equilibration procedure for linear elasticity is proposed and analyzed in this paper with emphasis on the behavior for (nearly) incompressible materials. Based on the displacement-pressure approximation computed with a stable finite element pair, it constructs an H(div)-conforming, weakly symmetric stress reconstruction. Our focus is on the Taylor-Hood combination of continuous finite element spaces of polynomial degrees k + 1 and k for the displacement and the pressure, respectively. Our construction leads then to reconstructed stresses by Raviart-Thomas elements of degree k which are weakly symmetric in the sense that its anti-symmetric part is zero tested against continuous piecewise polynomial functions of degree k. The computation is performed locally on a set of vertex patches covering the computational domain in the spirit of equilibration [11]. Due to the weak symmetry constraint, the local problems need to satisfy consistency conditions associated with all rigid body modes, in contrast to the case of Poisson's equation where only the constant modes are involved. The resulting error estimator is shown to constitute a guaranteed upper bound for the error with a constant that depends only on the shape regularity of the triangulation. Local efficiency, uniformly in the incompressible limit, is deduced from the upper bound by the residual error estimator. ). The authors gratefully acknowledge support by the Deutsche Forschungsgemeinschaft in the Priority Program SPP 1748 'Reliable simulation techniques in solid mechanics. Development of nonstandard discretization methods, mechanical and mathematical analysis' under the project numbers BE 6511/1-1 and STA 402/12-2.

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Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity

Bertrand

Kober

Moldenhauer

et al. 2021

Numerical Methods Partial

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This paper proposes and analyzes a posteriori error estimator based on stress equilibration for linear elasticity with emphasis on the behavior for (nearly) incompressible materials. It is based on an H(div)-conforming, weakly symmetric stress reconstruction from the displacement-pressure approximation computed with a stable finite element pair. Our focus is on the Taylor-Hood combination of continuous finite element spaces of polynomial degrees k + 1 and k for the displacement and the pressure, respectively. This weak symmetry allows us to prove that the resulting error estimator constitutes a guaranteed upper bound for the error with a constant that depends only on local constants associated with the patches and thus on the shape regularity of the triangulation. It does not involve global constants like those from Korn's in equality which may become very large depending on the location and type of the boundary conditions. Local efficiency, also uniformly in the incompressible limit, is deduced from the upper bound by the residual error estimator. Numerical results for the popular Cook's membrane test problem confirm the theoretical predictions. KEYWORDSa posteriori error estimation, incompressible linear elasticity, Raviart-Thomas elements, Taylor-Hood elements, weakly symmetric stress equilibrationThis is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

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Equilibrated Stress Reconstructions for Linear Elasticity Problems with Application to a Posteriori Error Analysis

Cited by 7 publications

References 14 publications

Equilibrated Stress Tensor Reconstruction and A Posteriori Error Estimation for Nonlinear Elasticity

Equilibrated Stress Tensor Reconstruction and A Posteriori Error Estimation for Nonlinear Elasticity

Equilibrated Stress Reconstruction and a Posteriori Error Estimation for Linear Elasticity

Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity

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