Regularization through Monitor Functions

Monitor functions are a useful tool for regularizing meshes because they can establish a metric of minimal quality that prevents problematic grid clustering. (A simple scalar analog is preventing division by zero by adding a small number to the denominator.) Monitor functions can be introduced into equations 4 and 5 by adding additional components to the metric tensor,  

$$g_{ij}^{s} = z(\mathbf{s})\,g^{xs}_{ij} + v^k(\mathbf{s})\,f^k(\mathbf{s}) \quad i,j=1,n,\quad k=1,l,$$ (6)

where g_ij^s is the regularized metric tensor, g^xs_ij is the unregularized metric tensor calculated by equations 3, $f^k(\mathbf{s})$ are functions of coordinate $\{ s^i \}$ that provide metric stabilization, and $z(\mathbf{s})$ and $v^k(\mathbf{s})$ are weighting functions. Hence, where g_ij^xs tends to zero, functions $f^k(\mathbf{s})$ are set to non-zero values. Note that the functions and their corresponding weights are specified on a point-by-point basis allowing for localized mesh regularization, and that the functions should have zero values and derivatives on the boundary so as to not regularize the boundary geometry.

Incorporating monitor functions into the Laplacian equation framework requires altering equations 4. Accordingly, the N-D differential method gridding equations incorporating monitor functions are given by,

$$\begin{eqnarray} D^{\xi}[s^j] = - D^{\xi} [ f^k ] \frac{\partial f^k}{\partial s... ...rtial S^n} = \phi^i \left[ \mathbf{s} \right], & \quad i=1,n, & \,\end{eqnarray}$$ (7) (8)

where $D^{\xi}$ is specified in equation 4 above. Liseikin (2004) provides theoretical justification of a number of different approaches to control grid clustering through the manipulation of monitor functions. In this paper, I use a fairly basic approach where the monitoring function is specified by a scaled spatially variant metric determinant.