Numerical grid generation in arbitrary surfaces through a second-order differential-geometric model
The Gaussian formulae for surfaces are used to derive three coupled second-order quasilinear elliptic PDEs for the Cartesian coordinates as functions of curvilinear coordinates in the surface. Let \(x^ i\), \(i=1,2,3\) be a system of curvilinear coordinates in \({\mathbb{R}}^ 3.\) Then \(X^{\alpha}\), \(\alpha\in \{1,2,3|\alpha\neq \nu\}\), are curvilinear coordinates on the surface \(x^{\nu}=\) constant and \((1)\quad \partial_{\alpha \beta}r=\gamma^{\delta}_{\alpha \beta}\partial_{\delta}r+n^{(\nu)}b_{\alpha \beta '}\) where \(r\in {\mathbb{R}}^ 3,\partial_{\alpha}\equiv \partial /\partial x^{\alpha}\), \(\gamma^{\delta}_{\alpha \beta}\) are the surface Christoffel symbols of the second kind, \(n^{(\nu)}\) is the unit surface normal and \(b_{\alpha \beta}\) are the coefficients of the second fundamental form of the surface. Letting \(g^{ij}\) be the metric tensor, denoting \(G_{\nu}=g_{\alpha \alpha}g_{\beta \beta}-(g_{\alpha \beta})^ 2\), \(\nu\), \(\alpha\), \(\beta\) cyclic, and contracting (1) with \(G_{\nu}g^{\alpha \beta}\) yields \[ (2)\quad Dr+G_{\nu}(\Delta_ 2^{(\nu)}x^{\delta})\partial_{\delta}r=n^{(\nu)}R \] where \(D\equiv G_{\nu}g^{\alpha \beta}\partial_{\alpha \beta}\) \(R\equiv G_{\nu}g^{\alpha \beta}b_{\alpha \beta}=(k_ I^{(\nu)}+k_{II}^{(\nu)}\quad G_{\nu},\Delta_ 2^{(\nu)}x^{\delta}=-g^{\alpha \beta}\gamma^{\delta}_{\alpha \beta}.\) Note that also \(k_ I^{(\nu)}+k_{II}^{(\nu)}=g^{\alpha \beta}\Gamma^{\nu}_{\alpha \beta}n^{(\nu)}\cdot \partial_{\nu}r,\) where \(\Gamma^{\nu}_{\alpha \beta}\) are the Christoffel symbols of the second kind in the embedding space. This involves only first partial derivatives in \(x^{\nu}\), which are assumed to have been evaluated at the surface. Restricting the Beltramians \(\Delta_ 2^{(\nu)}x^{\delta}\) in (2) leads to the required PDEs, the most general form being \(\Delta_ 2^{(\nu)}x^{\delta}=g^{\alpha \beta}P^{\delta}_{\alpha \beta}\) which represents 6 arbitrary control parameters. Taking specifically \(\nu =3\), \(x^ 1=\xi\), \(x^ 2=\eta\) leads to the generating PDEs (3) \({\mathcal L}r=n^{(3)}R\) where \({\mathcal L}=g_{22}\partial_{\xi \xi}-2g_{12}\partial_{\xi\eta}+ g_{11}\partial_{\eta\eta}+ \bar P\partial_{\xi}+ \bar Q\partial_{\eta}\), \(\bar P=g_{22}P^ 1_{11}- 2g_{12}P^ 1_{12}+ g_{11}P^ 1_{22},\) \(\bar Q=g_{22}P^ 2_{11}- 2g_{12}P^ 2_{12}+ g_{11}P^ 2_{22}\). It is shown in an appendix that if \(\xi\), \(\eta\), \(\zeta\) satisfy Poisson's equation and \(\zeta\) is orthogonal to the surface \(\zeta=\) constant then (3) also results. The PDEs (3) are solved using a central difference scheme, and the resulting grids are displayed for several simple quadric surfaces. The control parameters can be used to vary the grid spacings.
scientific article; zbMATH DE number 4048972
scientific article