2D Orthogonal Grid Generation of an Irregular Region Using Modified Finite Point Method

The purpose of this study is to generate two dimensional orthogonal grids in irregular regions for further computations of grid-based numerical methods. This is because grid-based numerical methods have been fully developed and most numerical models in common uses are still coded with grid-based methods. Grid generation techniques to provide input grid information is essential.
Making use of the orthogonality of Cauchy-Riemann condition, grid generation of the forward and inverse transformations were formulated by solving Laplace equation . The numerical method in this study is a meshless numerical method, namely, modified finite point method (MFPM). Based on collocation, this method uses polynomials as the local solution form needed in the collocation approach. The advantage of this method is not only the values of the solution but also the values of its derivatives can be easily obtained. The meshless numerical method is easier to generate computational points especially in irregular regions for its flexibility in distribution of the grid. In modified finite point method, there are parameters to be determined, and Multiple Objective Programming (MOP) is used. Though the method used in this study to solve the coordinate transformation equation is a meshless one, it shows at least merits of present work with other numerical methods.
There are six benchmark problems tested in this study, including a semi-annulus, an area bounded by two triangles, a full annulus, a four-pointed star, a flower like irregular region, and the surrounding area of Taiwan. Correctness of present model is verified by checking the orthogonality of the generated results or comparing with exact solutions. In present model, except at the corner of an obtuse angle, the generated results are very accurate.