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.
翻译:变式原则( VP) 旨在生成非叠叠格格( 硬度) 。 它最初的 VP 解决方案库基于添加剂配方, 因此, 在二变形变代中, 不存在变化。 当指定的 JD 和 curl 的配方由二变形论计算出来时, 最初的 VP 效果良好, 但当指定的 JD 和 curl 并非来自二变形论, 这个问题被称为不匹配配对问题 。 尽管如此, 最初的 VP 解决方案库基于添加剂配方, 因而在二代的电网中, 其最初的解决方案库基于添加剂配方, 因此, 在二变代中, 我们描述一个新的 VP 版本( 修订版 VP ), 其基于变异变的构成, 因此, 在 Lie algebra 中, 的配方。 经修订的 VP 似乎克服了 了 3D 电网代中原 VP 的不准确性。 在以下各节中, 提供了数学衍生结果。 显示, 经修订的 VP 和 版本的变式的代代代代的变换代 。