Goal-oriented error estimation and adaptivity in MsFEM computations

Chamoin, Ludovic; Legoll, Frédéric

doi:10.1007/s00466-021-01990-x

Cited by 15 publications

(10 citation statements)

References 64 publications

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“…• Goal-oriented error estimation and adaptivity Minimize the error on a given quantity of engineering interest through goal-oriented error estimation for smart point clouds [64,65];…”

Section: Discussionmentioning

confidence: 99%

Smart Cloud Collocation: Geometry-Aware Adaptivity Directly From CAD

Jacquemin

Suchde

Bordas

2023

Computer-Aided Design

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“…• Goal-oriented error estimation and adaptivity Minimize the error on a given quantity of engineering interest through goal-oriented error estimation for smart point clouds [64,65];…”

Section: Discussionmentioning

confidence: 99%

Smart Cloud Collocation: Geometry-Aware Adaptivity Directly From CAD

Jacquemin

Suchde

Bordas

2023

Computer-Aided Design

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“…The periodic assumption of the perforated domain is the classical treatment for the asymptotic analysis. Maybe it is not so practical because there are often randomly distributed configurations in the composite materials, which can be well described and effectively dealt with numerically by the MsFEM 22,23 . However, if the coefficients of the governing equations are randomly defined in that case, the homogenized eigenfunctions and eigenvalues are not uniquely defined according to the homogenization theory.…”

Section: Discussionmentioning

confidence: 99%

“…Maybe it is not so practical because there are often randomly distributed configurations in the composite materials, which can be well described and effectively dealt with numerically by the MsFEM. 22,23 However, if the coefficients of the governing equations are randomly defined in that case, the homogenized eigenfunctions and eigenvalues are not uniquely defined according to the homogenization theory. Since the eigensolutions invlove the global behaviors of a structure, it may be hard to analyze the Steklov eigenvalue problems when the cavities are randomly located.…”

Section: Discussionmentioning

confidence: 99%

“…17,18 Theoretically, the concept of two-scale convergence are introduced and developed, 19,20 which also validates the multi-scale expansion techniques. Sequently, the multi-scale finite element method (MsFEM) [21][22][23] and the heterogeneous multi-scale Method (HMM) 24 are proposed for the practical computations and simulations. This asymptotic analysis can be also applied to study the Spectral properties of the composite domain, and asymptotic expansions of both the eigenfunctions and eigenvalues are obtained by Kesaven.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Multiscale asymptotic analysis and computations for Steklov eigenvalue problem in periodically perforated domain

et al. 2021

Math Methods in App Sciences

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A second‐order asymptotic analysis method is developed for the Steklov eigenvalue problem in periodically perforated domain. By the two‐scale expansions of the eigenfunctions and eigenvalues, the first‐ and second‐order cell functions defined on the representative cell are obtained successively, the homogenized elliptic eigenvalue problem is formulated, and effective coefficients are derived. The first‐ and second‐order correctors of the eigenvalues are expressed in terms of the integrations of the cell functions and homogenized eigenfunctions. The error estimations of the expansions of eigenvalues are established, and the corresponding finite element algorithm is proposed. Numerical examples are carried out, and both qualitative and quantitative comparisons with the solutions by classical finite element computations are performed. It is demonstrated that this asymptotic model is effective to capture the local details of the eigenfunctions by considering the second‐order expansion terms, and the algorithm presented in this work is efficient to obtain the accurate spectral properties of the porous material at lower cost.

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“…As for any numerical method, a crucial issue is to control the accuracy of the numerical solution provided by the MsFEM approach. In that spirit, we have recently developed an a posteriori error estimate for the global error (in energy norm) [1] as well as for quantities of interest [2], along with the associated adaptive procedure. The estimates are based on the concept of Constitutive Relation Error (CRE), which we extend to the multiscale framework.…”

mentioning

confidence: 99%

Certified Computations with PGD Model Reduction in the MsFEM Framework

Legoll¹,

Chamoin²

2021

10th International Conference on Adaptative Modeling and Simulation

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The Multiscale Finite Element Method (MsFEM) is a Finite Element type approach for multiscale PDEs, where the basis functions used to generate the approximation space are precomputed and are specifically adapted to the problem at hand. The computation is performed in a two-stage procedure: (i) a offline stage, in which local basis functions are computed, and (ii) a online stage, in which the global problem is solved using an inexpensive Galerkin approximation.

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Goal-oriented error estimation and adaptivity in MsFEM computations

Cited by 15 publications

References 64 publications

Smart Cloud Collocation: Geometry-Aware Adaptivity Directly From CAD

Smart Cloud Collocation: Geometry-Aware Adaptivity Directly From CAD

Multiscale asymptotic analysis and computations for Steklov eigenvalue problem in periodically perforated domain

Certified Computations with PGD Model Reduction in the MsFEM Framework

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