A fast boundary integral method for high-order multiscale mesh generation

TL;DR

This paper introduces a linear-complexity boundary integral algorithm for generating smooth, high-order multiscale meshes from arbitrary triangulated surfaces, leveraging a fast multipole method for efficiency.

Contribution

The work presents a novel boundary integral approach with controllable smoothness for multiscale surface mesh generation from arbitrary triangulations.

Findings

Linear computational complexity in input size.

Effective smoothing of complex surfaces.

Accelerated boundary integral evaluation using fast multipole method.

Abstract

In this work we present an algorithm to construct an infinitely differentiable smooth surface from an input consisting of a (rectilinear) triangulation of a surface of arbitrary shape. The original surface can have non-trivial genus and multiscale features, and our algorithm has computational complexity which is linear in the number of input triangles. We use a smoothing kernel to define a function $\Phi$ whose level set defines the surface of interest. Charts are subsequently generated as maps from the original user-specified triangles to $\mathbb R^3$. The degree of smoothness is controlled locally by the kernel to be commensurate with the fineness of the input triangulation. The expression for~$\Phi$ can be transformed into a boundary integral, whose evaluation can be accelerated using a fast multipole method. We demonstrate the effectiveness and cost of the algorithm with polyhedral and quadratic skeleton surfaces obtained from CAD and meshing software.