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Fast variable density node generation on parametric surfaces with application to mesh-free methods

Urban Duh, Gregor Kosec, Jure Slak

TL;DR

The paper introduces a dimension-independent algorithm for generating variable-density nodes on parametric surfaces to support mesh-free PDE discretizations. By mapping parameter-domain offsets to the surface through the Jacobian, it places nodes so that projected surface spacing approximates a user-defined $h(p)$ while maintaining $O(N\log N)$ complexity via a kd-tree-based proximity search. It compares the approach with naive sampling and supersampling-decimation, demonstrates favorable local regularity and stability across challenging surfaces, and validates the method through a mesh-free Poisson problem using RBF-FD with monomial augmentation. The proposed method supports irregular domains, variable density, and patchwise surfaces, offering a robust, scalable alternative to traditional mesh generation in mesh-free simulations and paving the way for automatic CAD-model discretization. The work also provides theoretical spacing-error bounds and discusses practical considerations such as patch joints and parallelization for future enhancements.

Abstract

Domain discretization is considered a dominant part of solution procedures for solving partial differential equations. It is widely accepted that mesh generation is among the most cumbersome parts of the FEM analysis and often requires human assistance, especially in complex 3D geometries. When using alternative mesh-free approaches, the problem of mesh generation is simplified to the problem of positioning nodes, a much simpler task, though still not trivial. In this paper we present an algorithm for generation of nodes on arbitrary $d$-dimensional surfaces. This algorithm complements a recently published algorithm for generation of nodes in domain interiors, and represents another step towards a fully automated dimension-independent solution procedure for solving partial differential equations. The proposed algorithm generates nodes with variable density on surfaces parameterized over arbitrary parametric domains in a dimension-independent way in $O(N\log N)$ time. It is also compared with existing algorithms for generation of surface nodes for mesh-free methods in terms of quality and execution time.

Fast variable density node generation on parametric surfaces with application to mesh-free methods

TL;DR

The paper introduces a dimension-independent algorithm for generating variable-density nodes on parametric surfaces to support mesh-free PDE discretizations. By mapping parameter-domain offsets to the surface through the Jacobian, it places nodes so that projected surface spacing approximates a user-defined while maintaining complexity via a kd-tree-based proximity search. It compares the approach with naive sampling and supersampling-decimation, demonstrates favorable local regularity and stability across challenging surfaces, and validates the method through a mesh-free Poisson problem using RBF-FD with monomial augmentation. The proposed method supports irregular domains, variable density, and patchwise surfaces, offering a robust, scalable alternative to traditional mesh generation in mesh-free simulations and paving the way for automatic CAD-model discretization. The work also provides theoretical spacing-error bounds and discusses practical considerations such as patch joints and parallelization for future enhancements.

Abstract

Domain discretization is considered a dominant part of solution procedures for solving partial differential equations. It is widely accepted that mesh generation is among the most cumbersome parts of the FEM analysis and often requires human assistance, especially in complex 3D geometries. When using alternative mesh-free approaches, the problem of mesh generation is simplified to the problem of positioning nodes, a much simpler task, though still not trivial. In this paper we present an algorithm for generation of nodes on arbitrary -dimensional surfaces. This algorithm complements a recently published algorithm for generation of nodes in domain interiors, and represents another step towards a fully automated dimension-independent solution procedure for solving partial differential equations. The proposed algorithm generates nodes with variable density on surfaces parameterized over arbitrary parametric domains in a dimension-independent way in time. It is also compared with existing algorithms for generation of surface nodes for mesh-free methods in terms of quality and execution time.

Paper Structure

This paper contains 16 sections, 1 theorem, 37 equations, 16 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Node generation algorithm
  3. Time complexity analysis
  4. Remarks on generalizations
  5. Discussion and comparison
  6. Existing algorithms
  7. Naive parametric sampling
  8. Supersampling-decimation
  9. Setup for comparison
  10. Local regularity
  11. Minimal and maximal spacing requirements
  12. Spatial variability
  13. Computational complexity and execution time
  14. Mesh-free numerical analysis example
  15. Conclusions
  16. ...and 1 more sections

Key Result

Proposition 3.1

\newlabelprop:bounds0 The following estimates hold for the error of local node spacing radius due to linear approximation in eq:tay-lin: where and $\sigma_i(A)$ denotes the $i$-th largest singular value of $A$. In particular, this means that the relative error in spacing $|\Delta h|/h$ decreases linearly with $h$ for well behaved $\boldsymbol{r}$ and the algorithm for placing points on surfaces

Figures (16)

  • Figure 1: Illustration of candidate generation by the proposed algorithm in parametric space $\Xi$ (left) and main domain $\partial \Omega$ (top), around parameter $\boldsymbol{\xi} = (8.42, 5.99)$ with spacing $h = 0.23$.
  • Figure 1: Comparison of different algorithms on a 2D polar curve from \ref{['eq:polar-curve']} sampled with $h = 0.06$
  • Figure 1: Convergence of Poisson's equation $u(x, y) = \sin(\pi x) \cos(2 \pi y)$ with different boundary conditions and monomials up to order $m$ on nodes generated by the proposed algorithm and the algorithm presented in slak2019generation on 2D polar curve from \ref{['eq:polar-curve']}. The values $k$ represent the estimated line slopes until the error starts diverging.
  • Figure 2: Execution illustration of the proposed algorithm algorithm (left to right) in parametric domain $\Xi$ (bottom) and main domain $\partial \Omega$ (top). Part of a unit sphere was sampled with nodal spacing $h = 0.08$.
  • Figure 2: Comparison of different algorithms on a 3D heart-like surface from \ref{['eq:heart-surf']} sampled with $h = 0.05$
  • ...and 11 more figures

Theorems & Definitions (2)

  • Proposition 3.1
  • Proof 1