We present a geometric approach for calculating integrals over irregular domains described by a level-set function. This procedure can be used to evaluate integrals over a lower dimensional interface and may be used to evaluate the contribution of singular source terms. This approach produces results that are second-order accurate and robust to the perturbation of the interface location on the grid. Moreover, since we use a cell-wise approach, this procedure can be easily extended to quadtree and octree grids. We demonstrate the second-order accuracy and the robustness of the method in two and three spatial dimensions.
Bibliographical noteFunding Information:
The authors acknowledge stimulating discussions with Peter Smereka on the approximation of the delta function, and Guofang Wei for discussion on the genus in the orthocircles’ example. The research of C. Min was supported in part by the Kyung Hee University Research Fund (KHU-20070608) in 2007. The research of F. Gibou was supported in part by a Sloan Research Fellowship in Mathematics.
- Level-set methods
- Quadtree/octree data structures