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A surface moving mesh method based on equidistribution and alignment

Avary Kolasinski, Weizhang Huang

TL;DR

The paper introduces a surface moving mesh method that operates directly on simplicial meshes on general surfaces, avoiding reliance on surface parameterizations. It extends the moving mesh PDE framework by linking surface element area to the (generally non-square) Jacobian of the reference-to-surface mapping and formulating a combined equidistribution/alignment energy $I_h$ whose gradient drives a surface-projected MMPDE. The authors prove key properties: $I_h$ decreases along mesh trajectories, element areas and altitudes are bounded away from zero, and nonsingular limiting meshes exist; they also provide explicit matrix-based expressions for mesh velocities, enabling straightforward implementation. Numerical experiments on curves and surfaces in 2D and 3D demonstrate improved mesh quality under both Euclidean and curvature-based metrics, with the curvature metric concentrating nodes where surface curvature is large. The approach is robust to surfaces without explicit parameterizations and offers a practical tool for adaptive mesh generation and PDE solutions on surfaces, with potential extensions to spline-based surface representations and anisotropic monitors.

Abstract

A surface moving mesh method is presented for general surfaces with or without explicit parameterization. The method can be viewed as a nontrivial extension of the moving mesh partial differential equation method that has been developed for bulk meshes and demonstrated to work well for various applications. The main challenges in the development of surface mesh movement come from the fact that the Jacobian matrix of the affine mapping between the reference element and any simplicial surface element is not square. The development starts with revealing the relation between the area of a surface element in the Euclidean or Riemannian metric and the Jacobian matrix of the corresponding affine mapping, formulating the equidistribution and alignment conditions for surface meshes, and establishing a meshing energy function based on the conditions. The moving mesh equation is then defined as the gradient system of the energy function, with the nodal mesh velocities being projected onto the underlying surface. The analytical expression for the mesh velocities is obtained in a compact, matrix form, which makes the implementation of the new method on a computer relatively easy and robust. Moreover, it is analytically shown that any mesh trajectory generated by the method remains nonsingular if it is so initially. It is emphasized that the method is developed directly on surface meshes, making no use of any information on surface parameterization. It utilizes surface normal vectors to ensure that the mesh vertices remain on the surface while moving, and also assumes that the initial surface mesh is given. The new method can apply to general surfaces with or without explicit parameterization since the surface normal vectors can be computed even when the surface only has a numerical representation. A selection of two- and three-dimensional examples are presented.

A surface moving mesh method based on equidistribution and alignment

TL;DR

The paper introduces a surface moving mesh method that operates directly on simplicial meshes on general surfaces, avoiding reliance on surface parameterizations. It extends the moving mesh PDE framework by linking surface element area to the (generally non-square) Jacobian of the reference-to-surface mapping and formulating a combined equidistribution/alignment energy whose gradient drives a surface-projected MMPDE. The authors prove key properties: decreases along mesh trajectories, element areas and altitudes are bounded away from zero, and nonsingular limiting meshes exist; they also provide explicit matrix-based expressions for mesh velocities, enabling straightforward implementation. Numerical experiments on curves and surfaces in 2D and 3D demonstrate improved mesh quality under both Euclidean and curvature-based metrics, with the curvature metric concentrating nodes where surface curvature is large. The approach is robust to surfaces without explicit parameterizations and offers a practical tool for adaptive mesh generation and PDE solutions on surfaces, with potential extensions to spline-based surface representations and anisotropic monitors.

Abstract

A surface moving mesh method is presented for general surfaces with or without explicit parameterization. The method can be viewed as a nontrivial extension of the moving mesh partial differential equation method that has been developed for bulk meshes and demonstrated to work well for various applications. The main challenges in the development of surface mesh movement come from the fact that the Jacobian matrix of the affine mapping between the reference element and any simplicial surface element is not square. The development starts with revealing the relation between the area of a surface element in the Euclidean or Riemannian metric and the Jacobian matrix of the corresponding affine mapping, formulating the equidistribution and alignment conditions for surface meshes, and establishing a meshing energy function based on the conditions. The moving mesh equation is then defined as the gradient system of the energy function, with the nodal mesh velocities being projected onto the underlying surface. The analytical expression for the mesh velocities is obtained in a compact, matrix form, which makes the implementation of the new method on a computer relatively easy and robust. Moreover, it is analytically shown that any mesh trajectory generated by the method remains nonsingular if it is so initially. It is emphasized that the method is developed directly on surface meshes, making no use of any information on surface parameterization. It utilizes surface normal vectors to ensure that the mesh vertices remain on the surface while moving, and also assumes that the initial surface mesh is given. The new method can apply to general surfaces with or without explicit parameterization since the surface normal vectors can be computed even when the surface only has a numerical representation. A selection of two- and three-dimensional examples are presented.

Paper Structure

This paper contains 16 sections, 123 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Equidistribution and alignment for surface meshes
  3. Area and affine mappings for surface elements
  4. Area of surface elements in a Riemannian metric
  5. Equidistribution and alignment conditions
  6. Energy function for combined equidistribution and alignment
  7. Moving mesh equations for surface meshes
  8. Gradient of meshing energy
  9. Derivatives of the meshing function with respect to the physical coordinates
  10. Surface moving mesh equations
  11. Nonsingularity of surface moving meshes
  12. Equivalent measure of minimum height
  13. Mesh nonsingularity
  14. Numerical examples
  15. Conclusions and further comments
  16. ...and 1 more sections

Figures (13)

  • Figure 1: Example \ref{['circle']}. Meshes of $N=80$ are obtained for $\Phi(x,y) = x^2+y^2-1$.
  • Figure 2: Example \ref{['ellipse']}. Meshes of $N = 60$ are obtained for $\Phi(x,y) = \dfrac{x^2}{64}+y^2-1$.
  • Figure 3: Example \ref{['sin2d']}. Meshes of $N = 60$ are obtained for $\Phi(x,y) = 4\sin(x) - y$.
  • Figure 4: Example \ref{['sin2d']}. $I_h$ and $K_{\min}$ plotted as functions of $t$ for $\Phi(x,y) = 4\sin(x) - y$.
  • Figure 5: Example \ref{['lemni']}. Meshes of $N = 60$ are obtained for the lemniscate $\Phi(x,y) = (x^2 + y^2)^2 - 4(x^2 - y^2)$.
  • ...and 8 more figures

Theorems & Definitions (6)

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