3-D Quasi-Conformal Mappings and Grid Generation

Abstract

The difficulties of creating three-dimensional (3-D) analogs of conformal mappings are related to the topological and analytical features of 3-D space. The most interesting quasi-conformal mappings of 3-D regions are obtained with help of hydrodynamic analogy. For the steady irrotational flow of incompressible inviscid fluid together with the potential of velocities, two streamline functions are introduced. Any solenoid vector can be represented as a vector product of the gradients of two streamline functions. We obtain the connection of the velocity components with the streamline functions. These transformations are the basis for Lavrentiev's type of harmonic mappings. On the other hand, these conditions can be considered as a generalization of the Cauchy-Riemann conditions in the 3-D case. As shown in our work, generalized 3-D Cauchy-Riemann conditions for harmonic mappings are reduced to Cauchy-Riemann conditions for two functions of a usual complex variable. An analog of 3-D quasi-conformal mappings is obtained as combination of two ordinary functions of a plane complex variable. Examples of grid generation obtained by the theory of 3-D quasi-conformal mappings are given. The best proof of these results is their visualization.