Improved recovery of admissible stress in domain decomposition methods — application to heterogeneous structures and new error bounds for FETI‐DP

Parret-Fréaud, Augustin; Rey, Véronique; Gosselet, Pierre; Rey, Christian

doi:10.1002/nme.5462

Cited by 11 publications

(14 citation statements)

References 32 publications

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“…They used the Zienkiewicz and Zhu error estimation in the FE region and the Chung and Belytschko error estimation procedure was used in the EFG region. Parret-Fréaud et al [28] presented a moving least squares (MLS) recovery-based procedure to obtain a smoothed stress field in which the continuity of the smoothed field is provided by shape functions of the underlying mesh.…”

Section: Introductionmentioning

confidence: 99%

Techniques for Mesh Independent Displacement Recovery in Elastic Finite Element Solutions

Ahmed

2021

TFAMENA

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In this study, techniques for mesh dependent and independent displacement recovery for an a posteriori error estimation are presented. The error recovery of the field variable is made by fitting a higher order polynomial to the displacement over a mesh independent patch (support domain) using the moving least square (MLS) interpolation procedure. The mesh dependent recovery procedure is based on the recovery of the displacement over an element patch that consists of all elements surrounding the element under consideration using the least square (LS) interpolation procedure. The two-dimensional benchmark examples are analysed using linear and quadratic triangular elements to demonstrate the effectiveness and reliability of error estimations. Global and elemental errors of a finite element solution in the energy and L 2 norms are calculated directly from the post-processed displacement. The quality of error estimation obtained using the mesh independent displacement recovery technique in terms of convergence properties, effectivity and adaptive meshes under different error norms has been compared with that of the mesh dependent displacement recovery using the MLS interpolation and least square (LS) interpolation procedures. The performance of an adaptive scheme based on a mesh independent error estimator is compared with the adaptive scheme based on a mesh dependent error estimator. The numerical results show that the finite element analysis based on mesh independent recovery is very effective in converging to a predefined accuracy in a solution with a significantly smaller number of degrees of freedom.

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Section: Introductionmentioning

confidence: 99%

Techniques for Mesh Independent Displacement Recovery in Elastic Finite Element Solutions

Ahmed

2021

TFAMENA

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“…The fundamental property of guaranteed error estimation is that the final bounds do not depend on generic nonexplicit constants and that they hold regardless the size of the finite element mesh (they do not neglect higher order terms nor hold only asymptotically). Moreover, much effort has been devoted to obtain efficient guaranteed bounds …”

Section: Introductionmentioning

confidence: 99%

“…Moreover, much effort has been devoted to obtain efficient guaranteed bounds. [4][5][6][7][8][9] One of the most accurate error estimators currently available is the flux-free error estimator introduced by Parés et al 10 based on the partition of unity property to localize the error equations in nodal-patches of elements called stars. This technique has been extensively used in many applications, and even though its original form only provides asymptotic bounds for the error, it has subsequently been modified to provide guaranteed error bounds.…”

Section: Introductionmentioning

confidence: 99%

A new 3D equilibrated residual method improving accuracy and efficiency of flux‐free error estimates

Mariné

Dı́ez

2019

Numerical Meth Engineering

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Summary The paper presents a novel strategy providing fully computable upper bounds for the energy norm of the error in the context of three‐dimensional linear finite element approximations of the reaction‐diffusion equation. The upper bounds are guaranteed regardless the size of the finite element mesh and the given data, and all the constants involved are fully computable. The upper bound property holds if the shape of the domain is polyhedral and the Dirichlet boundary conditions are piecewise‐linear. The new approach is an extension of the flux‐free methodology introduced by Parés and Díez in the paper “A new equilibrated residual method improving accuracy and efficiency of flux‐free error estimates”, which introduces a guaranteed, low‐cost, and efficient flux‐free method substantially reducing the computational cost of obtaining guaranteed bounds using flux‐free methods while retaining the good quality of the bounds. Besides extending the 2D methodology, specific new modifications are introduced to further reduce the computational cost in the three‐dimensional setting. The presented methodology also provides a new strategy to obtain equilibrated boundary tractions, which improves the quality of standard techniques while having a similar computational cost.

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“…A recovery technique based upon the least square fitting of velocity field over an element patch is proposed in [14]. In [15], a moving least squares (MLS) recovery-based procedure to obtain postprocessed smoothed stresses field is presented in which the continuity of the recovered field is provided by the shape functions of the underlying mesh. Investigations are reported in [16] for getting improved recovery of stress field using domain decomposition method in heterogeneous structures.…”

Section: Introductionmentioning

confidence: 99%

Interpolation Type Stress Recovery Technique Based Error Estimator for Elasticity Problems

Ahmed¹,

Singh²,

Desmukh³

2018

mech

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Improved recovery of admissible stress in domain decomposition methods — application to heterogeneous structures and new error bounds for FETI‐DP

Cited by 11 publications

References 32 publications

Techniques for Mesh Independent Displacement Recovery in Elastic Finite Element Solutions

Techniques for Mesh Independent Displacement Recovery in Elastic Finite Element Solutions

A new 3D equilibrated residual method improving accuracy and efficiency of flux‐free error estimates

Interpolation Type Stress Recovery Technique Based Error Estimator for Elasticity Problems

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