Low-rank Parameterization of Volumetric Domains for Isogeometric Analysis

Abstract

Constructing a volumetric spline parameterization of a three dimensional domain with given boundary is a fundamental problem in isogeometric analysis. Recently it is shown that the rank of the parameterization has great impact in subsequent numerical simulation, and lower rank leads to higher computational efficiency. In this paper, we propose a method for low-rank spline parameterization of volumetric domains using low-rank tensor approximation technique. We model the problem as a non-linear optimization problem, the objective function of which consists of the MIPS (Most Isometric Parameterizations) term and the low-rank regularization term. An algorithm based on the alternating direction method of multipliers (ADMM) is presented to solve the optimization problem efficiently. To further obtain a low-rank spline representation of the parameterization, a modified CANDECOMP/PARAFAC (CP) decomposition algorithm is proposed. Experimental examples demonstrate that our approach can produce low-rank and bijective parameterizations with high quality.