Theoretical Overview

The goal of most grid generation methods is to find the transformation from a regular computational mesh defined by $\{ \xi^i \}$ on a domain $\Xi^n$ to an irregular grid defined by $\{ x^k \}$ on domain X^n+l. Usually, the only a priori information is the location of the mesh boundary points (i.e. $\phi^i(x^j)$ on $\partial \Xi^N$). Hence, grid generation is a two-fold task: propagate the boundary values into the interior in a physically consistent manner, and generate meshes with appropriate attributes (e.g. well-formed, smooth and non-singular). Thus, two important questions are by what physical principles are the boundary values propagated into the interior? And by what manner is the mesh regularized to ensure that it has acceptable attributes?

An answer to the first query is found by recasting grid generation as a Dirchelet BVP: given the boundary values of the mesh, solve an elliptic Laplace's equation on the mesh's interior. Mathematically, this requires solving the following system of equations,

$$D^{\xi}[s^j] = 0, \quad \; \quad \Gamma [\xi^i] = \left. \xi... ...right\vert _{\partial S} = \phi^i[s^j], \quad \; \quad i,j=1,n$$ (1)

where $D^{\xi}[v]$ is a generalized Laplacian operator acting on field v in coordinates $\{ \xi^i \}$, and $\Gamma$ is a projection operator that maps boundary values $\phi^i[s^j]$ of intermediate domain Sⁿ onto the boundaries of initial domain $\Xi^n$. More specifically, a BVP is formed for each coordinate component $\{ s^j \}$, which requires finding N solutions. Generally, this is an iterative process that continues until the coordinate fields in $\{ s^j \}$ converge to those of $\{ x^j \}$ to within some level of tolerance.

Division of the coordinate fields into independent BVPs, though, does not permit mesh regularization because coordinate fields are subject to geometric coupling. Because we wish to generate meshes over generalized spaces, we must represent this coupling through differential geometry. Importantly, this provides us with a powerful set of tools for mesh generation because it: i) specifies a metric (literally!) for evaluating mesh characteristics (e.g. extent and orientation of grid clustering); and ii) permits the introduction of monitor metrics that enable local mesh regularization. These two topics are discussed in the following two sections.

 

• Differential N-D Mesh Generation
• Regularization through Monitor Functions