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Simple multidimensional integration of discontinuous functions with application to level set methods. (English) Zbl 1352.65084
Summary: We present a simple, tree-based approach for the numerical integration over volumes and surfaces defined by the zero iso-contour of a level set function. The work is motivated by a variant of the discontinuous Galerkin method that is characterized by discontinuous enrichments of the polynomial basis. Although numerical results suggest that the presently achieved accuracy is comparable with methods based on discretized delta functions and on the geometric reconstruction of the interface, the presented approach is conceptually simpler and applicable to almost arbitrary grid types, which we demonstrate by means of numerical experiments on triangular, quadrilateral, tetrahedral and hexahedral meshes.

MSC:
65D30 Numerical integration
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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[1] Osher, Level Set Methods and Dynamic Implicit Surfaces (2002)
[2] Mittal, Immersed boundary methods, Annual Review of Fluid Mechanics 37 pp 239– (2005) · Zbl 1117.76049
[3] Düster, The finite cell method for three-dimensional problems of solid mechanics, Computer Methods in Applied Mechanics and Engineering 197 (45-48) pp 3768– (2008)
[4] Melenk, The partition of unity finite element method: basic theory and applications, Computer Methods in Applied Mechanics and Engineering 139 (1-4) pp 289– (1996) · Zbl 0881.65099
[5] Moës, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 (1) pp 131– (1999) · Zbl 0955.74066
[6] Kummer F The extended DG discretization of discontinuous PDE’s and the BoSSS software framework PhD Thesis 2011
[7] Ventura, On the elimination of quadrature subcells for discontinuous functions in the eXtended Finite Element method, International Journal for Numerical Methods in Engineering 66 (5) pp 761– (2006) · Zbl 1110.74858
[8] Mousavi, Generalized Gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method, Computer Methods in Applied Mechanics and Engineering 199 (49-52) pp 3237– (2010) · Zbl 1225.74099
[9] Tornberg, Multi-dimensional quadrature of singular and discontinuous functions, BIT Numerical Mathematics 42 (3) pp 644– (2002) · Zbl 1021.65010
[10] Tornberg, Regularization techniques for numerical approximation of PDEs with singularities, Journal of Scientific Computing 19 pp 527– (2003) · Zbl 1035.65085
[11] Tornberg, Numerical approximations of singular source terms in differential equations, Journal of Computational Physics 200 (2) pp 462– (2004) · Zbl 1115.76392
[12] Engquist, Discretization of Dirac delta functions in level set methods, Journal of Computational Physics 207 pp 28– (2004)
[13] Smereka, The numerical approximation of a delta function with application to level set methods, Journal of Computational Physics 211 pp 77– (2006) · Zbl 1086.65503
[14] Towers, Two methods for discretizing a delta function supported on a level set, Journal of Computational Physics 220 (2) pp 915– (2007) · Zbl 1115.65028
[15] Zahedi, Delta function approximations in level set methods by distance function extension, Journal of Computational Physics 229 (6) pp 2199– (2010) · Zbl 1186.65018
[16] Wen, High order numerical methods to a type of delta function integrals, Journal of Computational Physics 226 (2) pp 1952– (2007) · Zbl 1125.65024
[17] Wen, High order numerical quadratures to one dimensional delta function integrals, SIAM Journal on Scientific Computing 30 pp 1825– (2008) · Zbl 1170.65008
[18] Wen, High order numerical methods to two dimensional delta function integrals in level set methods, Journal of Computational Physics 228 (11) pp 4273– (2009) · Zbl 1167.65008
[19] Wen, High order numerical methods to three dimensional delta function integrals in level set methods, SIAM Journal on Scientific Computing 32 pp 1288– (2010) · Zbl 1410.65042
[20] Strain, Tree methods for moving interfaces, Journal of Computational Physics 151 (2) pp 616– (1999) · Zbl 0942.76061
[21] Min, Geometric integration over irregular domains with application to level-set methods, Journal of Computational Physics 226 (2) pp 1432– (2007) · Zbl 1125.65021
[22] Min, Robust second-order accurate discretizations of the multi-dimensional Heaviside and Dirac delta functions, Journal of Computational Physics 227 (22) pp 9686– (2008) · Zbl 1153.65014
[23] Grooss, A level set discontinuous Galerkin method for free surface flows, Computer Methods in Applied Mechanics and Engineering 195 (25-28) pp 3406– (2006) · Zbl 1121.76035
[24] Šolín, Higher-order finite element methods (2004)
[25] Min, Local level set method in high dimension and codimension, Journal of Computational Physics 200 (1) pp 368– (2004) · Zbl 1086.65088
[26] Grundmann, Invariant integration formulas for the n-simplex by combinatorial methods, SIAM Journal on Numerical Analysis 2 pp 282– (1978) · Zbl 0376.65013
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