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Adaptive refinement in defeaturing problems via an equilibrated flux a posteriori error estimator

Annalisa Buffa, Denise Grappein, Rafael Vázquez

TL;DR

This paper tackles defeaturing in PDE simulations by developing an adaptive strategy that combines standard mesh refinement with geometry refinement through feature inclusion. It builds a reliable a posteriori error estimator based on equilibrated flux reconstruction on trimmed domains, enabling error control without re-meshing when features are added via CutFEM. The estimator separates defeaturing error from numerical error, and the adaptive loop (Solve,Estimate,Mark,Refine) progressively includes features that most improve accuracy, validated by 2D tests with single and multiple features and coefficient discontinuities. The approach reduces computational cost while maintaining accuracy, and lays groundwork for extending to positive features and 3D problems.

Abstract

An adaptive refinement strategy, based on an equilibrated flux a posteriori error estimator, is proposed in the context of defeaturing problems. Defeaturing consists in removing features from complex domains in order to ease the meshing process, and to reduce the computational burden of simulations. It is a common procedure, for example, in computer aided design for simulation based manufacturing. However, depending on the problem at hand, the effect of geometrical simplification on the accuracy of the solution may be detrimental. The proposed adaptive strategy is hence twofold: starting from a defeatured geometry it allows both for standard mesh refinement and geometrical refinement, which consists in choosing, at each step, which features need to be included into the geometry in order to significantly increase the accuracy of the solution. With respect to other estimators that were previously proposed in the context of defeaturing, the use of an equilibrated flux reconstruction allows us to avoid the evaluation of the numerical flux on the boundary of features. This makes the estimator and the adaptive strategy particularly well-suited for finite element discretizations, in which the numerical flux is typically discontinuous across element edges. The inclusion of the features during the adaptive process is tackled by a CutFEM strategy, in order to preserve the non conformity of the mesh to the feature boundary and never remesh the computational domain as the features are added. Hence, the estimator also accounts for the error introduced by weakly imposing the boundary conditions on the boundary of the added features.

Adaptive refinement in defeaturing problems via an equilibrated flux a posteriori error estimator

TL;DR

This paper tackles defeaturing in PDE simulations by developing an adaptive strategy that combines standard mesh refinement with geometry refinement through feature inclusion. It builds a reliable a posteriori error estimator based on equilibrated flux reconstruction on trimmed domains, enabling error control without re-meshing when features are added via CutFEM. The estimator separates defeaturing error from numerical error, and the adaptive loop (Solve,Estimate,Mark,Refine) progressively includes features that most improve accuracy, validated by 2D tests with single and multiple features and coefficient discontinuities. The approach reduces computational cost while maintaining accuracy, and lays groundwork for extending to positive features and 3D problems.

Abstract

An adaptive refinement strategy, based on an equilibrated flux a posteriori error estimator, is proposed in the context of defeaturing problems. Defeaturing consists in removing features from complex domains in order to ease the meshing process, and to reduce the computational burden of simulations. It is a common procedure, for example, in computer aided design for simulation based manufacturing. However, depending on the problem at hand, the effect of geometrical simplification on the accuracy of the solution may be detrimental. The proposed adaptive strategy is hence twofold: starting from a defeatured geometry it allows both for standard mesh refinement and geometrical refinement, which consists in choosing, at each step, which features need to be included into the geometry in order to significantly increase the accuracy of the solution. With respect to other estimators that were previously proposed in the context of defeaturing, the use of an equilibrated flux reconstruction allows us to avoid the evaluation of the numerical flux on the boundary of features. This makes the estimator and the adaptive strategy particularly well-suited for finite element discretizations, in which the numerical flux is typically discontinuous across element edges. The inclusion of the features during the adaptive process is tackled by a CutFEM strategy, in order to preserve the non conformity of the mesh to the feature boundary and never remesh the computational domain as the features are added. Hence, the estimator also accounts for the error introduced by weakly imposing the boundary conditions on the boundary of the added features.

Paper Structure

This paper contains 15 sections, 1 theorem, 106 equations, 19 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation and model problem
  3. Error estimator via equilibrated flux reconstruction
  4. A posteriori error analysis
  5. The adaptive strategy
  6. Solve
  7. Estimate
  8. Mark
  9. Refine
  10. Numerical experiments
  11. Test 1: single internal feature
  12. Test 2: multiple internal and boundary features
  13. Test 3: multiple internal features, discontinuous coefficient
  14. Conclusions
  15. Appendix

Key Result

Proposition 1

Let $u$ be the solution to eq:weak_omega and $u_h^\star$ the solution to eq:prob_num. Then, choosing $p=1$ in eq:prob_eq_flux_nit, and under Assumption assum:hf, there exist three constants $\alpha_1,\alpha_2,\alpha_3>0$ independent on the mesh size $h$ and on the sizes of the features, such that

Figures (19)

  • Figure 1: Left, domain with positive and negative features; right, corresponding defeatured geometry.
  • Figure 2: Domain with two negative features and corresponding boundary nomenclature.
  • Figure 3: Example of a geometry with four negative features, corresponding totally defeatured geometry and partially defeatured geometry retaining only two features.
  • Figure 4: Nomenclature for patch boundaries
  • Figure 5: Example of an iteration of combined adaptivity
  • ...and 14 more figures

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 1
  • proof
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7