Original geometrical stopping criteria associated to multilevel adaptive mesh refinement for problems with local singularities

Abstract

This paper introduces a local multilevel mesh refinement strategy that automatically stops relating to a user-defined tolerance even in case of local singular solutions. Refinement levels are automatically generated thanks to a criterion based on the direct comparison of the a posteriori error estimate with the local prescribed error. Singular solutions locally increase with the mesh step (e.g. load discontinuities, point load or geometric induced singularities) and are hence characterized by locally large element-wise error whatever the mesh refinement. Then, the refinement criterion may not be self-sufficient to stop the refinement process. Additional stopping criteria are required if no physical-designed estimator wants to be used. Two original geometry-based stopping criteria are proposed that consist in automatically determining the critical region for which the mesh refinement becomes inefficient. Numerical examples show the efficiency of the methodology for stress tensor approximation in \(L^2\)-relative or \(L^\infty \)-absolute norms.

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Acknowledgements

This work has been achieved in the framework of the collaboration protocol between the CEA (Commissariat à l’Énergie Atomique et aux Énergies Alternatives) and the LMA (Laboratoire de Mécanique et d’Acoustique, CNRS, Marseille). The authors are grateful to the PLEIADES project, financially supported by CEA, EDF (Électricité de France) and AREVA, that funded this research work.

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Correspondence to Isabelle Ramière.

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Appendices

A Extended results for the axisymmetric test case

See Tables 8, 9 and 10.

B Extended results for the plane strain test case

See Tables 11, 12 and 13.

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Ramière, I., Liu, H. & Lebon, F. Original geometrical stopping criteria associated to multilevel adaptive mesh refinement for problems with local singularities. Comput Mech 64, 645–661 (2019). https://doi.org/10.1007/s00466-019-01674-7

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Keywords

  • Adaptive mesh refinement
  • Local Defect Correction method
  • A posteriori error estimator
  • Stopping criteria
  • Local singular solution
  • Elastostatics