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Quasi-optimal mesh generation for the virtual element method: A fully adaptive remeshing procedure

Daniel van Huyssteen, Felipe Lopez Rivarola, Guillermo Etse, Paul Steinmann

TL;DR

A novel fully adaptive remeshing procedure for the virtual element method is presented and novel procedures are proposed for the identification of elements qualifying for refinement or coarsening based on user-defined targets.

Abstract

The mesh flexibility offered by the virtual element method through the permission of arbitrary element geometries, and the seamless incorporation of `hanging' nodes, has made the method increasingly attractive in the context of adaptive remeshing. There exists a healthy literature concerning error estimation and adaptive refinement techniques for virtual elements while the topic of adaptive coarsening (i.e. de-refinement) is in its infancy. The creation of a quasi-optimal mesh is based on the principle of quasi-even error distribution over the elements which inherently relies on localized refinement and coarsening techniques. Thus, necessitating a fully adaptive remeshing procedure. In this work a novel fully adaptive remeshing procedure for the virtual element method is presented. Additionally, novel procedures are proposed for the identification of elements qualifying for refinement or coarsening based on user-defined targets. Specifically, a target percentage error, target number of elements, or target number of nodes can be selected. Numerical results demonstrate that the adaptive remeshing procedure can meet any prescribed target while creating a quasi-optimal mesh. The proposed fully adaptive procedure is of particular interest in engineering applications requiring an efficient simulation of a given accuracy, or desiring a simulation with the maximum possible accuracy for a given computational constraint.

Quasi-optimal mesh generation for the virtual element method: A fully adaptive remeshing procedure

TL;DR

A novel fully adaptive remeshing procedure for the virtual element method is presented and novel procedures are proposed for the identification of elements qualifying for refinement or coarsening based on user-defined targets.

Abstract

The mesh flexibility offered by the virtual element method through the permission of arbitrary element geometries, and the seamless incorporation of `hanging' nodes, has made the method increasingly attractive in the context of adaptive remeshing. There exists a healthy literature concerning error estimation and adaptive refinement techniques for virtual elements while the topic of adaptive coarsening (i.e. de-refinement) is in its infancy. The creation of a quasi-optimal mesh is based on the principle of quasi-even error distribution over the elements which inherently relies on localized refinement and coarsening techniques. Thus, necessitating a fully adaptive remeshing procedure. In this work a novel fully adaptive remeshing procedure for the virtual element method is presented. Additionally, novel procedures are proposed for the identification of elements qualifying for refinement or coarsening based on user-defined targets. Specifically, a target percentage error, target number of elements, or target number of nodes can be selected. Numerical results demonstrate that the adaptive remeshing procedure can meet any prescribed target while creating a quasi-optimal mesh. The proposed fully adaptive procedure is of particular interest in engineering applications requiring an efficient simulation of a given accuracy, or desiring a simulation with the maximum possible accuracy for a given computational constraint.
Paper Structure (27 sections, 35 equations, 27 figures, 1 table)

This paper contains 27 sections, 35 equations, 27 figures, 1 table.

Table of Contents

  1. Introduction
  2. Governing equations of linear elasticity
  3. Weak form
  4. The virtual element method
  5. The consistency term
  6. The stabilization term
  7. Mesh generation, refinement and coarsening
  8. Mesh generation
  9. Mesh refinement
  10. Mesh coarsening
  11. Error estimation and prediction
  12. Global error estimation
  13. Local (element-level) error estimation
  14. Local (common node) error prediction
  15. Super-convergent patch recovery
  16. ...and 12 more sections

Figures (27)

  • Figure 1: Arbitrary elastic body $\Omega$ with boundary ${\partial \Omega}$ subject to body ${\hbox{\boldmath{$b$}}}$ force and traction ${\bar{\hbox{\boldmath{$t$}}}}$.
  • Figure 2: Sample virtual element $E$ with $n_{\rm v}$ vertices where edge $e_{i}$ connects vertices $V_{i}$ and $V_{i+1}$.
  • Figure 3: Mesh generation procedure for structured and unstructured/Voronoi meshes.
  • Figure 4: Refinement procedure for structured and unstructured/Voronoi elements.
  • Figure 5: Coarsening procedure for structured and unstructured/Voronoi elements.
  • ...and 22 more figures