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

TL;DR

This paper introduces a fast, dimension-independent algorithm for generating variable density nodes on parametric surfaces, enhancing mesh-free methods for solving PDEs with improved automation and efficiency.

Contribution

The paper presents a novel, efficient algorithm for node generation on parametric surfaces that is dimension-independent and complements existing interior node algorithms.

Findings

Generates nodes with variable density on surfaces in O(N log N) time.

Compared favorably with existing algorithms in quality and speed.

Advances towards fully automated PDE solution procedures.

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.