Practical lowest distortion mapping

TL;DR

This paper introduces a numerical optimization method for constructing minimal-distortion mesh deformations in 2D and 3D, improving the reliability and stability of mesh untangling and parameterizations.

Contribution

It extends previous mesh untangling techniques by applying a polyconvex functional approach to achieve optimal quasi-isometric deformations with minimal distortion.

Findings

Successfully computes deformations with smallest known quasi-isometry constants.

Provides stable quasi conformal parameterizations for stiff problems.

Builds on finite element approximations of hyperelastic functionals.

Abstract

Construction of optimal deformations is one of the long standing problems of computational mathematics. We consider the problem of computing quasi-isometric deformations with minimal possible quasi-isometry constant (global estimate for relative length change).We build our technique upon [Garanzha et al. 2021a], a recently proposed numerical optimization scheme that provably untangles 2D and 3D meshes with inverted elements by partially solving a finite number of minimization problems. In this paper we show the similarity between continuation problems for mesh untangling and for attaining prescribed deformation quality threshold. Both problems can be solved by a finite number of partial solutions of optimization problems which are based on finite element approximations of parameter-dependent hyperelastic functionals. Our method is based on a polyconvex functional which admits a well-posed variational problem. To sum up, we reliably build 2D and 3D mesh deformations with smallest known distortion estimates (quasi-isometry constants) as well as stable quasi conformal parameterizations for very stiff problems.