Differential 2-D Mesh Generation

Generating a 2-D coordinate system through differential methods requires solving for coordinates $\{x^1,x^2\}$ within domain X². Incorporating l monitor functions for grid regularization expands the dimensionality of the mapping to,

$$\mathbf{x}(\mathbf{s}) \; : \; S^2 \rightarrow X^{2+l},\;\;\... ...hbf{s}) = \{ s^1,s^2,f^1(\mathbf{s}),..., f^l (\mathbf{s}) \}.$$ (9)

Coordinate system $\{ s^j \}$ is related to an underlying Cartesian grid, which is chosen to be a unit square defined by $\Xi^2 = \{0 \le \xi^1,\xi^2 \le 1 \}$. Transformation $s^j(\xi^i)$ is assumed to be piece-wise smooth and known on the boundary of $\Xi^2$ such that: $\phi ( \mathbf{\xi} ) \; : \; \partial \Xi^2 \rightarrow \partial S^2, \quad \mathbf{\phi}=(\phi^1,\phi^2)$. Within this framework, the 2-D gridding equations become,

$$\begin{eqnarray} D^{\xi}[s^j] = - D^{\xi} [ f^k ] \frac{\partial f^k}{\partial s... ...l \Xi^2} = \mathbf{\phi}(\mathbf{\xi}), \quad j=1,2, \quad k=1,l,\end{eqnarray}$$ (10)

where Laplacian operator $D^{\xi}[\cdot]$ and metric tensor g_ij are written explicitly Liseikin (2004),

$$\begin{eqnarray} D^{\xi}[v] = & g_{22}^{\xi} \frac{\partial^2 v}{\partial \xi^1 ... ...}(\mathbf{\xi})] }{\partial \xi^j}, \quad i,j,k=1,2, \quad m=1,l.\end{eqnarray}$$ (11) (12)

One convenient way to solve the set of elliptical equations 10 is by transforming them to a set of parabolic equations (i.e. include time-dependence) that have a common steady-state solution. Thus, equations 10 are reformulated to include time-dependence - $\{ s^i(\xi^1,\xi^2,t) \}$ - leading to the six governing equations,

$$\begin{eqnarray} \frac{ \partial s^1}{ \partial t} = D[s^1] + D[ f^k] \frac{\par... ...1,\xi^2), & t=0 \ s^2(\xi^1,\xi^2,0) = s^2_0(\xi^1,\xi^2), & t=0\end{eqnarray}$$ (13) (14) (15) (16) (17) (18)

Solutions $s^1 (\xi^1,\xi^2,t)$ and $s^2 (\xi^1,\xi^2,t)$ satisfying equations 13-18 will converge to the solutions $\{x^1,x^2\}$ of equations 10 as $t \rightarrow \infty$. Hence, the answer to within tolerance factor $\epsilon$ occurs at some T_n. Details of an iterative scheme and an algorithm for computation to solve equations 13 are provided in Appendix A.