Zhou Zicong, Liao Guojun
Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China.
Math Department, University of Texas at Arlington, Arlington, Texas 76019, USA.
ICGG 2022 (2022). 2023;146:598-611. doi: 10.1007/978-3-031-13588-0_52. Epub 2022 Aug 13.
The Variational Principle (VP) is designed to generate non-folding grids (diffeomorphisms) with prescribed Jacobian determinant (JD) and curl. The 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 unknown to come from a diffeomorphism. In spite of that, the original VP works effectively in 2D grid generations. To resolve this issue, in this paper, we describe a new version of VP (revised VP), which is based on the composition of transformations and, therefore, is invariant in the Lie algebra. The revised VP seems to have overcome the inaccuracy of the 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. Finally, a new definition of averaging diffeomorphisms based on the revised VP is proposed.
变分原理（VP）旨在生成具有规定雅可比行列式（JD）和旋度的非折叠网格（微分同胚）。原始VP的解池基于加法公式，因此在微分同胚李代数中不是不变的。当规定的JD和旋度对由微分同胚计算得出时，原始VP效果良好，但当规定的JD和旋度未知是否来自微分同胚时则不一定。尽管如此，原始VP在二维网格生成中有效。为了解决这个问题，在本文中，我们描述了一种新的VP版本（修订后的VP），它基于变换的组合，因此在李代数中是不变的。修订后的VP似乎克服了原始VP在三维网格生成中的不准确性。在接下来的部分中，将给出数学推导。结果表明，修订后的VP可以计算已知微分同胚的逆变换。还通过数值验证了其变换的逆一致性和传递性。最后，基于修订后的VP提出了平均微分同胚的新定义。
