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A unifying moving mesh method for curves, surfaces, and domains based on mesh equidistribution and alignment

Min Zhang, Weizhang Huang

TL;DR

This work introduces a unifying moving mesh method that simultaneously handles curves, surfaces, and domains as $m$-dimensional objects in $\mathbb{R}^d$ by exploiting mesh equidistribution and alignment within a metric tensor framework. It derives a gradient-flow based MMPDE for mesh movement, provides closed-form analytical velocity expressions via scalar-by-matrix differentiation, and proves nonsingularity of the evolving mesh. The approach does not require an analytical parameterization of the geometry, instead leveraging normals and tangents (or curvature) computed from the mesh to keep points on $S$ while controlling concentration through a curvature- or identity-based metric. Numerical experiments on a range of 2D and 3D curves and surfaces demonstrate robust, singularity-free mesh movement and curvature-driven concentration patterns, illustrating practical applicability to geometric PDEs and simulations on general manifolds.

Abstract

A unifying moving mesh method is developed for general $m$-dimensional geometric objects in $d$-dimensions ($d \ge 1$ and $1\le m \le d$) including curves, surfaces, and domains. The method is based on mesh equidistribution and alignment and does not require the availability of an analytical parametric representation of the underlying geometric object. Mathematical characterizations of shape and size of $m$-simplexes and properties of corresponding edge matrices and affine mappings are derived. The equidistribution and alignment conditions are presented in a unifying form for $m$-simplicial meshes. The equation for mesh movement is defined based on the moving mesh PDE approach, and suitable projection of the nodal mesh velocities is employed to ensure the mesh points stay on the underlying geometric object. The analytical expression for the mesh velocities is obtained in a compact matrix form. The nonsingularity of moving meshes is proved. Numerical results for curves ($m=1$) and surfaces ($m=2$) in two and three dimensions are presented to demonstrate the ability of the developed method to move mesh points without causing singularity and control their concentration.

A unifying moving mesh method for curves, surfaces, and domains based on mesh equidistribution and alignment

TL;DR

This work introduces a unifying moving mesh method that simultaneously handles curves, surfaces, and domains as -dimensional objects in by exploiting mesh equidistribution and alignment within a metric tensor framework. It derives a gradient-flow based MMPDE for mesh movement, provides closed-form analytical velocity expressions via scalar-by-matrix differentiation, and proves nonsingularity of the evolving mesh. The approach does not require an analytical parameterization of the geometry, instead leveraging normals and tangents (or curvature) computed from the mesh to keep points on while controlling concentration through a curvature- or identity-based metric. Numerical experiments on a range of 2D and 3D curves and surfaces demonstrate robust, singularity-free mesh movement and curvature-driven concentration patterns, illustrating practical applicability to geometric PDEs and simulations on general manifolds.

Abstract

A unifying moving mesh method is developed for general -dimensional geometric objects in -dimensions ( and ) including curves, surfaces, and domains. The method is based on mesh equidistribution and alignment and does not require the availability of an analytical parametric representation of the underlying geometric object. Mathematical characterizations of shape and size of -simplexes and properties of corresponding edge matrices and affine mappings are derived. The equidistribution and alignment conditions are presented in a unifying form for -simplicial meshes. The equation for mesh movement is defined based on the moving mesh PDE approach, and suitable projection of the nodal mesh velocities is employed to ensure the mesh points stay on the underlying geometric object. The analytical expression for the mesh velocities is obtained in a compact matrix form. The nonsingularity of moving meshes is proved. Numerical results for curves () and surfaces () in two and three dimensions are presented to demonstrate the ability of the developed method to move mesh points without causing singularity and control their concentration.
Paper Structure (14 sections, 19 theorems, 123 equations, 13 figures)

This paper contains 14 sections, 19 theorems, 123 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Simplicial meshes and edge matrices
  3. Properties of edge matrices
  4. Properties of affine mapping
  5. Simplexes in Riemannian metric
  6. Mathematical characterization of nonuniform meshes
  7. Equidistribution and alignment conditions
  8. Mesh energy functions
  9. Unifying moving mesh equations
  10. Scalar-by-matrix differentiation
  11. Analytical formulas for mesh velocities
  12. Mesh nonsingularity
  13. Numerical examples
  14. Conclusions and remarks

Key Result

Lemma 2.1

For an $m$-simplex $K\subset\mathbb{R}^d$, the size $|K|$ for $K$ can be expressed as where $\det (\cdot )$ denotes the determinant of a matrix.

Figures (13)

  • Figure 1: Mesh examples for a torus curve and a torus surface in $\mathbb{R}^3$.
  • Figure 2: Example\ref{['ex:ellipse']}. Meshes ($N=100$) of a unit circle are obtained by the unifying MMPDE method.
  • Figure 3: Example\ref{['ex:ellipse']}. Meshes ($N=120$) of an ellipse are obtained by the unifying MMPDE method.
  • Figure 4: Example\ref{['ex:lemniscate']}. Meshes ($N=100$) of the lemniscate curve are obtained by the unifying MMPDE method.
  • Figure 5: Example\ref{['ex:cardioid']}. Meshes ($N=70$) of the cardioid curve are obtained by the unifying MMPDE method.
  • ...and 8 more figures

Theorems & Definitions (43)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 33 more