Geometric integration over irregular domains with application to level-set methods. (English) Zbl 1125.65021

Summary: 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.


65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
41A63 Multidimensional problems
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[1] B.P. Carnerio, C. Silva, A.E. Kaufman, Tetra-cubes: an algorithm to generate 3d isosurfaces bases upon tetrahedra, Anais do IX SIBGRAPI, 1996, pp. 205-210.
[2] Engquist, B.; Tornberg, A.K.; Tsai, R., Discretization of Dirac delta functions in level set methods, J. comput. phys., 207, 28-51, (2005) · Zbl 1074.65025
[3] Goodman, J.E.; O’Rourke, J., The handbook of discrete and computational geometry, (1997), CRC Press LL · Zbl 0890.52001
[4] Grundmann, A.; Moeller, M., Invariant integration formulas for the n-simplex by combinatorial methods, SIAM J. numer. anal., 2, 282-290, (1978) · Zbl 0376.65013
[5] Hammer, C.P.; Stroud, A.H., Numerical integration over simplexes, Math. tables other aids comput., 10, 137-139, (1956) · Zbl 0070.35405
[6] Kuhn, H.W., Some combinational lemmas in topology, IBM J. res. dev., 4, 508-524, (1960) · Zbl 0109.15603
[7] Mayo, A., The fast solution of poisson’s and the biharmonic equations on irregular regions, SIAM J. numer. anal., 21, 285-299, (1984) · Zbl 1131.65303
[8] Min, C., Simplicial isosurfacing in arbitrary dimension and codimension, J. comput. phys., 190, 295-310, (2003) · Zbl 1029.65016
[9] C. Min, F. Gibou, A second order accurate level set method on non-graded adaptive Cartesian grids, Available at UCLA CAM report (06-22) http://www.math.ucla.edu/applied/cam/index.html, 2006 (in review).
[10] Min, C.-H., Local level set method in high dimension and codimension, J. comput. phys., 200, 368-382, (2004) · Zbl 1086.65088
[11] Osher, S.; Sethian, J., Fronts propagating with curvature-dependent speed: algorithms based on hamilton – jacobi formulations, J. comput. phys., 79, 12-49, (1988) · Zbl 0659.65132
[12] Peng, D.; Merriman, B.; Osher, S.; Zhao, H.; Kang, M., A PDE-based fast local level set method, J. comput. phys., 155, 410-438, (1999) · Zbl 0964.76069
[13] Russo, G.; Smereka, P., A remark on computing distance functions, J. comput. phys., 163, 51-67, (2000) · Zbl 0964.65089
[14] Sallee, J.F., The middle-cut triangulations of the n-cube, SIAM J. alg. disc. methods, 5, 407-419, (1984) · Zbl 0543.52004
[15] Samet, H., The design and analysis of spatial data structures, (1989), Addison-Wesley New York
[16] Samet, H., Applications of spatial data structures: computer graphics, image processing and GIS, (1990), Addison-Wesley New York
[17] Smereka, P., The numerical approximation of a delta function with application to level set methods, J. comput. phys., 211, 77-90, (2006) · Zbl 1086.65503
[18] Sommerville, D.M.Y., An introduction to the geometry of N dimensions, (1958), Dover Publications · Zbl 0086.35804
[19] Strain, J., Tree methods for moving interfaces, J. comput. phys., 151, 616-648, (1999) · Zbl 0942.76061
[20] Sussman, M.; Fatemi, E., An efficient interface-preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow, SIAM J. sci. comp., 20, 1165-1191, (1999) · Zbl 0958.76070
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