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Defining metric-aware size-shape measures to validate and optimize curved high-order meshes

Guillermo Aparicio-Estrems, Abel Gargallo-Peiró, Xevi Roca

TL;DR

To verify if the minimization of the metric-aware size-shape distortion leads to meshes approximating the target metric, the Riemannian measures of the curved high-order mesh entities are closer to unit.

Abstract

We define a regularized size-shape distortion (quality) measure for curved high-order elements on a Riemannian space. To this end, we measure the deviation of a given element, straight-sided or curved, from the stretching, alignment, and sizing determined by a target metric. The defined distortion (quality) is suitable to check the validity and the quality of straight-sided and curved elements on Riemannian spaces determined by constant and point-wise varying metrics. The examples illustrate that the distortion can be minimized to curve (deform) the elements of a given high-order (linear) mesh and try to match with curved (linear) elements the point-wise stretching, alignment, and sizing of a discrete target metric tensor. In addition, the resulting meshes simultaneously match the curved features of the target metric and boundary. Finally, to verify if the minimization of the metric-aware size-shape distortion leads to meshes approximating the target metric, we compute the Riemannian measures for the element edges, faces, and cells. The results show that, when compared to anisotropic straight-sided meshes, the Riemannian measures of the curved high-order mesh entities are closer to unit. Furthermore, the optimized meshes illustrate the potential of curved $r$-adaptation to improve the accuracy of a function representation.

Defining metric-aware size-shape measures to validate and optimize curved high-order meshes

TL;DR

To verify if the minimization of the metric-aware size-shape distortion leads to meshes approximating the target metric, the Riemannian measures of the curved high-order mesh entities are closer to unit.

Abstract

We define a regularized size-shape distortion (quality) measure for curved high-order elements on a Riemannian space. To this end, we measure the deviation of a given element, straight-sided or curved, from the stretching, alignment, and sizing determined by a target metric. The defined distortion (quality) is suitable to check the validity and the quality of straight-sided and curved elements on Riemannian spaces determined by constant and point-wise varying metrics. The examples illustrate that the distortion can be minimized to curve (deform) the elements of a given high-order (linear) mesh and try to match with curved (linear) elements the point-wise stretching, alignment, and sizing of a discrete target metric tensor. In addition, the resulting meshes simultaneously match the curved features of the target metric and boundary. Finally, to verify if the minimization of the metric-aware size-shape distortion leads to meshes approximating the target metric, we compute the Riemannian measures for the element edges, faces, and cells. The results show that, when compared to anisotropic straight-sided meshes, the Riemannian measures of the curved high-order mesh entities are closer to unit. Furthermore, the optimized meshes illustrate the potential of curved -adaptation to improve the accuracy of a function representation.
Paper Structure (34 sections, 51 equations, 23 figures, 12 tables)

This paper contains 34 sections, 51 equations, 23 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Preliminaries: shape measures for high-order Euclidean elements
  3. Size and size-shape measures for linear elements and constant metric
  4. Differentiable size and size-shape distortion for linear Euclidean elements
  5. Size-shape distortion for linear elements and constant metric
  6. Behavior of the quality measures: shape, size, and size-shape
  7. Level sets for one moving vertex
  8. Influence of element alignment
  9. Measures for curved high-order meshes with varying metric
  10. Size-shape distortion for curved high-order elements on varying metric
  11. Riemmanian measure of mesh entities
  12. Results
  13. Implementation
  14. Shape versus size-shape distortion minimization: curved high-order mesh and analytic metric
  15. Size-shape distortion minimization for high-order interpolation: 2D different degrees and 3D quadratic
  16. ...and 19 more sections

Figures (23)

  • Figure 1: Mappings between the master, the ideal, and the physical elements in the linear case.
  • Figure 2: Plots of the original and modified: (a) size distortion measure and (b) size quality measure.
  • Figure 3: Mappings between the equilateral, the ideal, the physical, and the unitary physical triangles.
  • Figure 4: Mappings between the master, the equilateral, the ideal, the physical, and the unitary physical triangles.
  • Figure 5: Level sets for the quality measures with different metrics: (a,d) shape, (b,e) size, and (c,f) size-shape; (a,b,c) isotropic and (d,e,f) anisotropic metrics.
  • ...and 18 more figures