Numerical Integrations Over Arbitrary Domains and Arbitrary Surfaces on Block-Structured Adaptive Meshes

Abstract

Abstract: We study numerical integration over arbitrary interface and arbitrary domain on block-structured adaptive mesh.Arbitrary interface and arbitrary domain are described by a level set function.We first describe numerical methods on uniform Cartesian grid.Then we extend the methods to block-structured adaptive mesh.Numerical calculations demonstrate that adaptive mesh methods are second-order accurate.Compared with uniform mesh methods, the adaptive mesh methods reduce needs on computer storage significantly.

Key words: numerical integration, adaptive mesh, Cartesian grid, level-set function

CLC Number:

O241

XU Jianjun, SHI Weidong, LI Xingwei, SHU Shi. Numerical Integrations Over Arbitrary Domains and Arbitrary Surfaces on Block-Structured Adaptive Meshes[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2017, 34(1): 10-18.

References

[1] NG Y T, MIN C, GIBOU F. An efficient fluid-solid coupling algorithm for single-phase flows[J]. J Comput Phys, 2009,228:8807-8829.
[2] PAPAC J, GIBOUF, RATSCH C.Efficient symmetric discretization for the Poisson, heat and Stefan-type problems with Robin boundary conditions[J]. J Comput Phys, 2010,229:875-889.
[3] PAPAC J, HELGADOTTIR A, RATSCH C, et al.A level set approach for diffusion and Stefan-type problem with Robin boundary conditions on quadtree/octree adaptive Cartesian grids[J]. J Comput Phys, 2013,233:241-261.
[4] XU J J, YANG Y, LOWENGRUB J. A level-set continuum method for two-phase flows with insoluble surfactant[J].J Comput Phys, 2012,231:5897-5909.
[5] XU J J, REN W. A level-set method for two-phase flows with moving contact line and insoluble surfactant[J].J Comput Phys, 2014,263:71-90.
[6] ENGQUIST B, TORNBERG AK, TSAI R. Discretization of Dirac delta function in level set methods[J]. J Comput Phys, 2005,207:28-51.
[7] OSHER S, FEDKIW R.Level set methods and dynamic implicit surfaces[M]. New York, Springer, 2006.
[8] SMEREKA P. The numerical approximation of a delta function with application to level set methods[J]. J Comput Phys,2006,211:77-90.
[9] CMIN, GIBOU F. Geometric integration over irregular domains with application to level-set methods[J]. J Comput Phys,2007, 226:1432-1443.
[10] BERGER M J, OLIGER J. Adaptive mesh refinement for hyperbolic partial differential equations[J]. J Comput Phys,1984,53:484.
[11] BERGER M J, COLELLA P.Local adaptive mesh refinement for shock hydrodynamics[J].J Comput Phys,1989,82:64-84.
[12] BERGER M J, RIGOUTSOS R. An algorithm for point clustering and grid generation[J]. IEEE Trans Syst Man Cybern, 1991,21:1278-1286.
[13] COLELLA P, GRAVES D T, LIGOCKI T J, et al. CHOMBO software package for amr applications:design document. Berkeley, USA. 2003.
[14] MITRAN S. BEARCLAW a code for multiphysics applications with embedded boundaries:user's manual.http://www.amath.unc.edu/Faculty/mitran/bearclaw.html.
[15] MACNEICE P, OLSON K M,MOBARRY C, et al. Paramesh:A parallel adaptive mesh refinement community toolkit[J]. Computer Physics Communications,2000,126:330-354.
[16] MO Z, ZHANG A, CAO X, et al. JASMIN:A parallel software infrastructure for scientific computing[J]. Front Comput Sci China,2010, 4(4):480-488.
[17] GRUNDMANN A, MOELLER M. Invariant integration formulas for the n-simplex by combinatorial methods[J]. SIAM J Numer Anal, 1978,2:282-290.