Construction of Diffeomorphisms with Prescribed Jacobian Determinant and Curl

TL;DR

This paper introduces a revised variational principle for constructing diffeomorphisms with prescribed Jacobian determinant and curl, improving accuracy and invariance, especially in 3D grid generation and mismatch handling.

Contribution

A new version of the variational principle based on composition of transformations is proposed, addressing invariance issues and mismatch problems in diffeomorphism construction.

Findings

The revised VP is invariant in the Lie algebra.

It effectively computes inverse transformations of known diffeomorphisms.

Preliminary results show improved handling of mismatch issues.

Abstract

The variational principle (VP) is designed to generate non-folding grids (diffeomorphisms) with prescribed Jacobian determinant (JD) and curl. Its solution pool of the original VP is based on an additive formulation and, consequently, is not invariant in the diffeomorphic Lie algebra. The original VP works well when the prescribed pair of JD and curl is calculated from a diffeomorphism, but not necessarily when the prescribed JD and curl are not known to come from a diffeomorphism. This issue is referred as the mismatched pair problem. In spite of that, the original VP works effectively in 2D grid generations. To resolve these issues, in this paper, we describe a new version of VP (revised VP), which is based on composition of transformations and, therefore, is invariant in the Lie algebra. The revised VP seems have overcome the inaccuracy of original VP in 3D grid generations. In the following sections, the mathematical derivations are presented. It is shown that the revised VP can calculate the inverse transformation of a known diffeomorphism. Its inverse consistency and transitivity of transformations are also demonstrated numerically. Moreover, a computational strategy is formulated based on the new version of VP to handle the mismatch issue and is demonstrated with preliminary result.