Multiblock grid generation. Part I: Elliptic grid generation methods for structured grids

The paper is organized as follows. In Section 2, the Laplace equations, the parameter space and the algebraic transformation are presented for domains in two dimensional physical space. The resulting Poisson equations are derived together with the appropriate expressions of the control functions. The relationship with other methods is explained. The discretization and solution of the nonlinear elliptic equations is discussed and also the orthogonalization of the grid at boundaries. Examples of grids in 2D domains are given. Surface grid generation on minimal surfaces is discussed in Section 3. It is shown that grid generation on minimal surface is in fact the same problem as grid generation in a domain in 2D physical space. llustrations of grids on minimal surfaces are given. Surface grid generation on surfaces with a prescribed shape is treated in Section 4. It is assumed that such surfaces are parametrized and that the parametrization is a differentiable one-to-one mapping from a unit square onto the surface. The generated surface grids are independent of the parametrization. The solution method to generate the grids in the interior of parametrized surfaces is different from that used for minimal surfaces. It is much easier to solve the two linear elliptic partial differential equations defined by the LaplaceBeltrami equations directly, instead of interchanging the dependent and independent variables which leads to an nonlinear elliptic system of partial differential equations. An inversion problem must then be solved afterwards. Such a simple solution method is only possible for parametrized surfaces. This is due to the fact that an initial grid folding free surface grid on a parametrized surface can be easily generated because the given parametrization is one-to-one.