Peter Schwartz

Embedded boundary grid generation using the divergence theorem, implicit functions, and constructive solid geometry

To construct finite-volume methods for PDEs in arbitrary dimension to arbitrary accuracy in the presence of irregular boundaries, we show that estimates of moments, integrals of monomials, over various regions are all that are needed. If implicit functions are used to represent the irregular boundary, the needed moments can be computed straightforwardly and robustly by using the divergence theorem, Taylor expansions, least squares, recursion, and 1D root finding. Neither a geometric representation of the irregular boundary nor its interior is ever needed or computed. The implicit function representation is general and robust. Implicit functions can be combined via constructive solid geometry to form complex boundaries from a rich set of primitives including interpolants of sampled data, for example, 2D/3D image data and digital elevation maps.