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High-Order Meshfree Surface Integration, Including Singular Integrands

Daniel R. Venn, Steven J. Ruuth

TL;DR

This work addresses high-order numerical integration on surface point clouds without mesh generation, including handling singular integrands. It introduces two meshfree strategies: Method 1 uses Poisson solvability to derive integral ratios, and Method 2 uses the divergence theorem to reduce surface integrals to boundary (or line) integrals, both preserving super-algebraic convergence. Theoretical results connect solvability, boundedness, and convergence within a norm-minimizing, symmetric meshfree framework, while extensions to singular integrals are achieved via a generalized Hilbert-space approach. Numerical tests on genus-two surfaces and singular kernels demonstrate rapid convergence, robustness to irregular point distributions, and substantial accuracy improvements over traditional triangulation in several scenarios, highlighting the practical impact for PDE solvers and boundary-integral computations.

Abstract

We develop and test high-order methods for integration on surface point clouds. The task of integrating a function on a surface arises in a range of applications in engineering and the sciences, particularly those involving various integral methods for partial differential equations. Mesh-based methods require a curved mesh for high-order convergence, which can be difficult to reliably obtain on many surfaces, and most meshfree methods require the ability to integrate a set of functions (such as radial basis functions) exactly on the domain of interest; these integrals are generally not known in closed form on most surfaces. We describe two methods for integrating on arbitrary, piecewise-smooth surfaces with or without boundary. Our approaches do not require a particular arrangement of points or an initial triangulation of the surface, making them completely meshfree. We also show how the methods can be extended to handle singular integrals while maintaining high accuracy without changing the point density near singularities.

High-Order Meshfree Surface Integration, Including Singular Integrands

TL;DR

This work addresses high-order numerical integration on surface point clouds without mesh generation, including handling singular integrands. It introduces two meshfree strategies: Method 1 uses Poisson solvability to derive integral ratios, and Method 2 uses the divergence theorem to reduce surface integrals to boundary (or line) integrals, both preserving super-algebraic convergence. Theoretical results connect solvability, boundedness, and convergence within a norm-minimizing, symmetric meshfree framework, while extensions to singular integrals are achieved via a generalized Hilbert-space approach. Numerical tests on genus-two surfaces and singular kernels demonstrate rapid convergence, robustness to irregular point distributions, and substantial accuracy improvements over traditional triangulation in several scenarios, highlighting the practical impact for PDE solvers and boundary-integral computations.

Abstract

We develop and test high-order methods for integration on surface point clouds. The task of integrating a function on a surface arises in a range of applications in engineering and the sciences, particularly those involving various integral methods for partial differential equations. Mesh-based methods require a curved mesh for high-order convergence, which can be difficult to reliably obtain on many surfaces, and most meshfree methods require the ability to integrate a set of functions (such as radial basis functions) exactly on the domain of interest; these integrals are generally not known in closed form on most surfaces. We describe two methods for integrating on arbitrary, piecewise-smooth surfaces with or without boundary. Our approaches do not require a particular arrangement of points or an initial triangulation of the surface, making them completely meshfree. We also show how the methods can be extended to handle singular integrals while maintaining high accuracy without changing the point density near singularities.
Paper Structure (23 sections, 3 theorems, 66 equations, 5 figures, 4 tables)

This paper contains 23 sections, 3 theorems, 66 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Norm-Minimizing Hermite-Birkhoff Interpolants
  4. High-Order Bounds on Functions with Scattered Zeros
  5. Hilbert Spaces for Meshfree Methods
  6. Methods and Analysis
  7. Method 1: Poisson Solvability
  8. Analysis
  9. Method 2: Dimension Reduction
  10. Line Integrals
  11. Solving Surface Poisson Problems
  12. Creating Surface Subdomains
  13. Voronoi Cells
  14. Planar Boundary
  15. Numerical Tests
  16. ...and 8 more sections

Key Result

Proposition 2.1

Let $\left\{ V_{n}\right\} _{n=1}^{\infty}$ be a collection of closed, non-empty, convex sets in a Hilbert space $\mathcal{H}$ such that $V_{1}\supseteq V_{2}\supseteq\ldots$ and define Assume $\left\{ \left\Vert \tilde{u}_{n}\right\Vert _{\mathcal{H}}\right\} _{n=1}^{\infty}$ is bounded by a constant $B>0$, then $\tilde{u}_{n}$ converges in $\mathcal{H}$ to some $\tilde{u}_{\infty}\in\bigcap

Figures (5)

  • Figure 1: Convergence plot for estimates of the average value of $x^{2}$ on a genus-two surface. Two shape parameters are shown for the meshfree method ($q=7/3,10/3$).
  • Figure 2: Three point clouds on a genus-two surface with 800 points. Left: Regularly-spaced. Centre: Random. Right: Irregularly-spaced.
  • Figure 3: Left: Voronoi cell subdomains on a genus-two surface. Right: Planar subdomain boundary for the genus-two surface.
  • Figure 4: Relative error for computing $-\frac{1}{2\pi}\int_{S}\ln\left\Vert \boldsymbol{x}\right\Vert _{2}\,\text{d}\boldsymbol{x}$ on $\tilde{N}$ scattered points on the unit disk.
  • Figure 5: A computed solution to $\Delta_{S}u=-\frac{1}{4\pi}\left\Vert \boldsymbol{x}-\boldsymbol{x}_{0}\right\Vert _{2}^{-1}$ on the paraboloid using the augmented method with $\tilde{N}=2560$ points ($\boldsymbol{x}_{0}=\left(0,0,1\right)$).

Theorems & Definitions (9)

  • Proposition 2.1
  • Proof 1
  • Corollary 2.2
  • Proof 2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Example 2.6
  • Example 2.7