A Low-rank Spline Approximation of Planar Domains

TL;DR

This paper introduces a low-rank spline parameterization method for planar domains using quasi-conformal maps, reducing computational costs in numerical PDEs by producing efficient, bijective, and low-distortion representations.

Contribution

It presents a novel low-rank tensor approximation approach combined with quasi-conformal maps for efficient domain parameterization in geometric design.

Findings

Produces bijective low-rank spline representations

Reduces computational cost in solving PDEs

Outperforms previous methods in experiments

Abstract

Construction of spline surfaces from given boundary curves is one of the classical problems in computer aided geometric design, which regains much attention in isogeometric analysis in recent years and is called domain parameterization. However, for most of the state-of-the-art parameterization methods, the rank of the spline parameterization is usually large, which results in higher computational cost in solving numerical PDEs. In this paper, we propose a low-rank representation for the spline parameterization of planar domains using low-rank tensor approximation technique, and apply quasi-conformal map as the framework of the spline parameterization. Under given correspondence of boundary curves, a quasi-conformal map with low rank and low distortion between a unit square and the computational domain can be modeled as a non-linear optimization problem. We propose an efficient algorithm to compute the quasi-conformal map by solving two convex optimization problems alternatively. Experimental results show that our approach can produce a bijective and low-rank parametric spline representation of planar domains, which results in better performance than previous approaches in solving numerical PDEs.