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Mixed-Order Meshes through rp-adaptivity for Surface Fitting to Implicit Geometries

Ketan Mittal, Veselin A. Dobrev, Patrick Knupp, Tzanio Kolev, Franck Ledoux, Claire Roche, Vladimir Z. Tomov

TL;DR

The proposed approach is purely algebraic, and extends to different types of elements (quadrilaterals/triangles/tetrahedron/hexahedra) in two- and three-dimensions, and can be used to generate mixed-order meshes that are much more efficient than high uniform-order meshes.

Abstract

Computational analysis with the finite element method requires geometrically accurate meshes. It is well known that high-order meshes can accurately capture curved surfaces with fewer degrees of freedom in comparison to low-order meshes. Existing techniques for high-order mesh generation typically output meshes with same polynomial order for all elements. However, high order elements away from curvilinear boundaries or interfaces increase the computational cost of the simulation without increasing geometric accuracy. In prior work, we have presented one such approach for generating body-fitted uniform-order meshes that takes a given mesh and morphs it to align with the surface of interest prescribed as the zero isocontour of a level-set function. We extend this method to generate mixed-order meshes such that curved surfaces of the domain are discretized with high-order elements, while low-order elements are used elsewhere. Numerical experiments demonstrate the robustness of the approach and show that it can be used to generate mixed-order meshes that are much more efficient than high uniform-order meshes. The proposed approach is purely algebraic, and extends to different types of elements (quadrilaterals/triangles/tetrahedron/hexahedra) in two- and three-dimensions.

Mixed-Order Meshes through rp-adaptivity for Surface Fitting to Implicit Geometries

TL;DR

The proposed approach is purely algebraic, and extends to different types of elements (quadrilaterals/triangles/tetrahedron/hexahedra) in two- and three-dimensions, and can be used to generate mixed-order meshes that are much more efficient than high uniform-order meshes.

Abstract

Computational analysis with the finite element method requires geometrically accurate meshes. It is well known that high-order meshes can accurately capture curved surfaces with fewer degrees of freedom in comparison to low-order meshes. Existing techniques for high-order mesh generation typically output meshes with same polynomial order for all elements. However, high order elements away from curvilinear boundaries or interfaces increase the computational cost of the simulation without increasing geometric accuracy. In prior work, we have presented one such approach for generating body-fitted uniform-order meshes that takes a given mesh and morphs it to align with the surface of interest prescribed as the zero isocontour of a level-set function. We extend this method to generate mixed-order meshes such that curved surfaces of the domain are discretized with high-order elements, while low-order elements are used elsewhere. Numerical experiments demonstrate the robustness of the approach and show that it can be used to generate mixed-order meshes that are much more efficient than high uniform-order meshes. The proposed approach is purely algebraic, and extends to different types of elements (quadrilaterals/triangles/tetrahedron/hexahedra) in two- and three-dimensions.
Paper Structure (18 sections, 15 equations, 11 figures, 1 table)

This paper contains 18 sections, 15 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Discrete Mesh Representation
  4. TMOP for Mesh Quality Improvement via $r-$adaptivity
  5. $r$-adaptivity for Mesh Quality Improvement and Surface Alignment
  6. $p$-Adaptivity Constraints
  7. Methodology
  8. Level-Set Function Representation and Error Computation for Mesh Fitting
  9. Level-Set Interpolation from $\mathcal{M}_B$ to $\mathcal{M}$
  10. $p$-Refinement Criterion for Interface Elements
  11. $p$-Derefinement Criterion
  12. Propagating $p$-refinement to Interior Elements
  13. Summary of the $p$-adaptivity Algorithm
  14. Results
  15. $p-$Refinement in 3D
  16. ...and 3 more sections

Figures (11)

  • Figure 1: Schematic representation of the major TMOP matrices.
  • Figure 2: (a) Original and (b) optimized 4th order mesh for a turbine blade.
  • Figure 3: (a) Level set function $\sigma(\mathbf{x})$ with zero isocontour in red, (b) a Cartesian mesh with material interface nodes to be aligned to the zero level set of $\sigma(\mathbf{x})$. (c) Linear mesh and (d) cubic mesh optimized to align with the level-set. Uniformly refined and optimized cubic meshes are also shown with (e) $N_E=64$ and (f) $N_E=256$ elements.
  • Figure 4: Schematic showing the DOFs for a $p$-refined mesh.
  • Figure 5: Interpolating the level-set function from (a) adaptively refined background mesh with $p_{\sigma_B}=4$ to (b) an optimized linear mesh and (c) an optimized cubic mesh.
  • ...and 6 more figures