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hr-Adaptivity for nonconforming high-order meshes with the target matrix optimization paradigm

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Abstract

We present an \(hr\)-adaptivity framework for optimization of high-order meshes. This work extends the r-adaptivity method by Dobrev et al. (Comput Fluids, 2020), where we utilized the Target-Matrix Optimization Paradigm (TMOP) to minimize a functional that depends on each element’s current and target geometric parameters: element aspect-ratio, size, skew, and rotation. Since fixed mesh topology limits the ability to achieve the target size and aspect-ratio at each position, in this paper, we augment the r-adaptivity framework with nonconforming adaptive mesh refinement to further reduce the error with respect to the target geometric parameters. The proposed formulation, referred to as \(hr\)-adaptivity, introduces TMOP-based quality estimators to satisfy the aspect-ratio target via anisotropic refinements and size target via isotropic refinements in each element of the mesh. The methodology presented is purely algebraic, extends to both simplices and hexahedra/quadrilaterals of any order, and supports nonconforming isotropic and anisotropic refinements in 2D and 3D. Using a problem with a known exact solution, we demonstrate the effectiveness of \(hr\)-adaptivity over both r- and \(h\)-adaptivity in obtaining similar accuracy in the solution with significantly fewer mesh nodes. We also present several examples that show that \(hr\)-adaptivity can help satisfy geometric targets even when \(r\)-adaptivity fails to do so, due to the topology of the initial mesh.

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Acknowledgements

The authors would like to thank Jakub Cerveny for helpful discussions regarding the nonconforming mesh refinement framework that the work in this paper is based on.

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Authors and Affiliations

  1. Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, CA, 94550, USA

    Veselin Dobrev, Tzanio Kolev, Ketan Mittal & Vladimir Tomov

  2. Dihedral LLC, Bozeman, MT, 59715, USA

    Patrick Knupp

Authors
  1. Veselin Dobrev
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  2. Patrick Knupp
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  3. Tzanio Kolev
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  4. Ketan Mittal
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  5. Vladimir Tomov
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Corresponding author

Correspondence to Ketan Mittal.

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Funding

This work was funded by the Department of Energy Office of Science.

Conflict of interest

All the authors declare that they have no conflict of interest.

Availability of data and material

All the methods developed in this work are available through the open-source code MFEM [34].

Author contributions

Veselin Dobrev: conceptualization, methodology, software, and investigation. Patrick Knupp: conceptualization, methodology, writing, and investigation. Tzanio Kolev: conceptualization, methodology, writing, and investigation. Ketan Mittal: conceptualization, methodology, software, validation, investigation, writing, and visualization. Vladimir Tomov: conceptualization, methodology, software, investigation, and writing.

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This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.

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Dobrev, V., Knupp, P., Kolev, T. et al. hr-Adaptivity for nonconforming high-order meshes with the target matrix optimization paradigm. Engineering with Computers 38, 3721–3737 (2022). https://doi.org/10.1007/s00366-021-01407-6

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  • DOI: https://doi.org/10.1007/s00366-021-01407-6

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