Construction of differentiable transformations

Abstract

a b s t r a c tConstruction of invertible transformations using differential equations is an interesting and challenging mathematical problem with important applications. We briefly review the existing method by means of harmonic maps in 2D and propose a method of constructing differentiable, invertible transformations between domains in two and three dimensions. Preliminary numerical results demonstrate the effectiveness of the method.

A New Method for Triangular Mesh Generation

“…A uniform Cartesian mesh on the background domain [1,9] by [1,12] is shown in Figure 1a below. The domain D is the interior of the contour described by points #1 to #18 on the curved boundary, and by the nodes (1,5), (1,6), (1,7), (1,8), (1,9), and (1, 10) on the vertical boundary, see Figure 1a. We select a set of 18 nodes on the Cartesian mesh that are close to the boundary of domain D, and move them to the points #1 through #18, respectively (see Appendix).…”

“…In the example, we take n = 10. The following equations are solved by the least squares finite element method (as described in [Cai et al 2004] [7] and [Liao et al 2009…”

Planar Immersions with Prescribed Curl and Jacobian Determinant are Unique

“…A noteworthy subset of techniques for this purpose are known as the "variational methods" (see e.g. [8,1,6,3,9] and references therein), which aim to control characteristic quantities of grid transformations such as their divergence, curl, and Jacobian determinant. In essence, such variational methods are numerical techniques for minimizing nonlinear and often nonconvex functionals whose critical points represent mappings with desirable properties.…”

New Variational Method of Grid Generation with Prescribed Jacobian Determinant and Prescribed Curl

“…The deformation method [4], [5], [7] has its origin in differential geometry [1]. Its main theoretical advantages is that the grid transformation generated has prescribed Jacobian determinant.…”

“…a 1 , a 2 by(6), then solve Poisson's equation(7) to get g 1 , g 2 3. Update f by f i,new = f i,old − g i × tstep,where tstep is an optimization parameter 4.…”