A Triangular Grid Generation by Cauchy Riemann Equations

A Triangular Grid Generation by Cauchy Riemann Equations

In this article, we present a method that generates a smooth triangular grid around an object. A grid in a two-dimensional physical space will be generated by solving Cauchy-Riemann equations in the physical space by treating the grid as unknown. The method is in principle the same as the elliptic grid generation method.

The grid generation by elliptic partial differential equations has been a popular technique for quite sometime. The main advantage of the method is the fact that the resulting grid is smooth and orthogonal which are very important to the accuracy of the numerical solution. The idea can be easily understood by considering two Laplace’s equations with Dirichlet boundary conditions. The contours of the solutions are smooth and non intersecting, i.e. form a good grid. However, it is in reality impossible to solve the equations in the domain of interest simply because a fixed computational grid is not available in the domain. Therefore we interchange the roles of the independent and dependent variables and solve the transformed equations in the solution space. The resulting equations are highly nonlinear, and therefore they are always solved in an iterative manner starting with a reasonable initial grid. It is very important here to note that the equations we want to solve are in fact Laplace equations in the physical domain.
The method we propose in this paper is based on the fact that two Laplace’s equations are equivalent to Cauchy-Riemann equations.

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A Triangular Grid Generation by Cauchy Riemann Equations

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