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Numerical integration over arbitrary polygonal domains based on Schwarz-Christoffel conformal mapping. (English) Zbl 1176.74190
Summary: This paper presents a new numerical integration technique on arbitrary polygonal domains. The polygonal domain is mapped conformally to the unit disk using Schwarz-Christoffel mapping and a midpoint quadrature rule defined on this unit disk is used. This method eliminates the need for a two-level isoparametric mapping usually required. Moreover, the positivity of the Jacobian is guaranteed. Numerical results presented for a few benchmark problems in the context of polygonal finite elements show that the proposed method yields accurate results.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74S70 Complex-variable methods applied to problems in solid mechanics
74B05 Classical linear elasticity
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References:
[1] Long, A meshless local Petrov-Galerkin method for solving the bending problem of a thin plate, Computer Modeling in Engineering and Sciences 3 (1) pp 53– (2002) · Zbl 1147.74414
[2] Belytschko, Element-free Galerkin methods, International Journal for Numerical Methods in Engineering 37 pp 229– (1994) · Zbl 0796.73077
[3] Liu, Reproducing kernel particle method, International Journal for Numerical Methods in Engineering 20 pp 1081– (1995) · Zbl 0881.76072
[4] Nguyen, Meshfree methods: a review and computer implementation aspects, Mathematics and Computers in Simulation 79 (3) pp 763– (2008) · Zbl 1152.74055
[5] Fries TP, Matthies HG. Classification and overview of meshfree methods. InformatirkberichtReport 2003-03, Institute of Scientific Computing, Technical University Braunschweig, Hans-Sommer-Strasse 65, D38106 Braunschweig, March 2003.
[6] Trobec, Computational complexity and parallelization of the meshless local Petrov-Galerkin method, Computers and Structures 98 (1-2) pp 1– (2009)
[7] Belytschko, Crack propagation by element-free Galerkin methods, Engineering Fracture Mechanics 51 pp 295– (1995)
[8] Belytschko, Meshless methods: an overview and recent developments, Computer Methods in Applied Mechanics and Engineering 139 pp 3– (1996) · Zbl 0891.73075
[9] Rabczuk, A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics, Computational Mechanics 40 (3) pp 473– (2007) · Zbl 1161.74054
[10] Bordas, Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment, Engineering Fracture Mechanics 75 pp 943– (2008)
[11] Belytschko, Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering 45 (5) pp 601– (1999) · Zbl 0943.74061
[12] Melenk, The partition of unity finite element method: basic theory and applications, Computer Methods in Applied Mechanics and Engineering 139 pp 289– (1996) · Zbl 0881.65099
[13] Chessa J. The extended finite element method for free surface and two phase flow problems. Ph.D. Thesis, Northwestern University, December 2002.
[14] Chessa, The extended finite element method for two-phase fluids, ASME Journal of Applied Mechanics 70 (1) pp 10– (2003) · Zbl 1110.74391
[15] Chopp, Fatigue crack propagation of multiple coplanar cracks with the coupled extended finite element/fast marching method, International Journal of Engineering Science 41 (8) pp 845– (2003) · Zbl 1211.74199
[16] Duddu, A combined extended finite element and level set method for biofilm growth, International Journal for Numerical Methods in Engineering 74 pp 848– (2008) · Zbl 1195.74169
[17] Bordas, A simulation based design paradigm for complex cast components, Engineering with Computers 23 (1) pp 25– (2007)
[18] Bordas, An extended finite element library, International Journal for Numerical Methods in Engineering 71 (6) pp 703– (2007) · Zbl 1194.74367
[19] Dunant, Architecture trade-offs of including a mesher in an object-oriented extended finite element code, European Journal of Computational Mechanics 16 (16) pp 237– (2007)
[20] Menk A, Bordas S. Influence of the microstructure on the stress state of solder joints during thermal cycling. EuroSimE 2009, Delft University of Technology, Netherlands, 26-28 April 2009.
[21] Rabczuk, A geometrically nonlinear three-dimensional cohesive crack method for reinforced concrete structures, Engineering Fracture Mechanics 75 pp 4740– (2008) · Zbl 1161.74054
[22] Karihaloo, Model of stationary and growing cracks in fe framework without remeshing: a state-of-the-art review, Computers and Structures 81 (3) pp 119– (2003)
[23] Yazid, A survey of the extended finite element method, Computers and Structures 86 (11-12) pp 1141– (2008)
[24] Yazid, A state-of-the-art review of the x-fem for computational fracture mechanics, Applied Mathematical Modelling (2009) · Zbl 1172.74050
[25] Belytschko, Structured extended finite element methods of solids defined by implicit surfaces, International Journal for Numerical Methods in Engineering 56 pp 609– (2003) · Zbl 1038.74041
[26] Liu, A smoothed finite element for mechanics problems, Computational Mechanics 39 (6) pp 859– (2006) · Zbl 1169.74047
[27] Nguyen-Xuan, A smoothed finite element method for plate analysis, Computer Methods in Applied Mechanics and Engineering 197 (13-16) pp 1184– (2008) · Zbl 1159.74434
[28] Nguyen, A smoothed finite element method for shell analysis, Computer Methods in Applied Mechanics and Engineering 198 pp 165– (2008) · Zbl 1194.74453
[29] Bordas, Strain smoothing in fem and xfem, Computers and Structures (2008)
[30] Nguyen-Xuan, Smooth finite elements: connecting displacement and equilibrium finite elements, applications to elasto-plasticity, Computer Methods in Applied Mechanics and Engineering (2008)
[31] Dai, Free and forced vibration analysis using the smoothed finite element method (SFEM), Journal of Sound and Vibration 301 (3-5) pp 803– (2007)
[32] Dai, An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics, Finite Element in Analysis and Design 43 (11) pp 847– (2007)
[33] Dasgupta, Interpolants within convex polygons: Wachspress’ shape functions, Journal of Aerospace Engineering 16 (1) pp 1– (2003)
[34] Ghosh, Voronoi cell finite element model-based on micropolar theory of thermoelasticity for heterogenous materials, International Journal for Numerical Methods in Engineering 38 pp 1361– (1995) · Zbl 0823.73066
[35] Sukumar, Conforming polygonal finite elements, International Journal for Numerical Methods in Engineering 61 pp 2045– (2004) · Zbl 1073.65563
[36] Bolander, Irregular lattice model for quasistatic crack propagation, Physics Review B 71 (094106) pp 1– (2005)
[37] Rashid, On a finite element method with variable element topology, Computer Methods in Applied Mechanics and Engineering 190 (11-12) pp 1509– (2000)
[38] Tabarraei, Application of polygonal finite elements in linear elasticity, International Journal of Computational Methods 3 (4) pp 503– (2006) · Zbl 1198.74104
[39] 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
[40] Sukumar, Recent advances in the construction of polygonal finite element interpolants, Archives of Computational Methods in Engineering 13 (1) pp 129– (2006) · Zbl 1101.65108
[41] Arroyo, Local Maximum-entropy Approximation Schemes pp 1– (2006)
[42] Arroyo, Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods, International Journal for Numerical Methods in Engineering 65 (13) pp 2167– (2006) · Zbl 1146.74048
[43] Wachspress, A Rational Basis for Function Approximation (1971)
[44] Ghosh, Elastic-plastic analysis of arbitrary heterogeneous materials with the Voronoi-cell finite element method, Computer Methods in Applied Mechanics and Engineering 121 pp 373– (1995) · Zbl 0853.73065
[45] Moorthy, Adaptivity and convergence in the Voronoï cell finite element method for analyzing heterogenous materials, Computer Methods in Applied Mechanics and Engineering 185 pp 37– (2000)
[46] Tiwary, Numerical conformal mapping method based Voronoï cell finite element model for analyzing microstructures with irregular heterogeneities, Finite Element in Analysis and Design 43 pp 504– (2007)
[47] Chris, Weights of links and plaquettes in a random lattice, Nuclear Physics 210 (3) pp 337– (1982)
[48] Belikov, The non-Sibsonian interpolation: a new method of interpolation of the values of a function on an arbitrary set of points, Computational Mathematics and Mathematical Physics 37 (1) pp 9– (1997) · Zbl 0948.65005
[49] Hiyoshi, Two generalizations of an interpolant based on Voronoi diagrams, International Journal of Shape Modeling 5 (2) pp 219– (1999)
[50] Sibson, A vector identity for Dirichlet tessellation, Mathematical Proceedings of the Cambridge Philosophical Society 87 pp 151– (1980) · Zbl 0466.52010
[51] Cueto, Overview and recent advances in natural neighbour Galerkin methods, Archives of Computational Methods in Engineering 10 (4) pp 307– (2003) · Zbl 1050.74001
[52] Yoo, Stabilized conforming nodal integration in the natural-element method, International Journal for Numerical Methods in Engineering 60 pp 861– (2004) · Zbl 1060.74677
[53] Sukumar, The natural element method in solid mechanics, International Journal for Numerical Methods in Engineering 43 (5) pp 839– (1998) · Zbl 0940.74078
[54] Sukumar N. The natural element method in solid mechanics. Ph.D. Thesis, Theoretical and Applied Mechanics, Northwestern University, Evanston, IL, U.S.A., 1998. · Zbl 0940.74078
[55] Sukumar, Maximum entropy approximation, Bayesian Inference and Maximum Entropy Methods in Science and Engineering 803 pp 337– (2005)
[56] Sukumar, Overview and construction of meshfree basis functions: from moving least squares to entropy approximants, International Journal for Numerical Methods in Engineering 70 pp 181– (2007) · Zbl 1194.65149
[57] Sukumar, Construction of polygonal interpolants: a maximum entropy approach, International Journal for Numerical Methods in Engineering 61 (12) pp 2159– (2004) · Zbl 1073.65505
[58] Meyer, Generalized barycentric coordinates for irregular n-gons, Journal of Graphics Tools 7 (1) pp 13– (2002) · Zbl 1024.68109
[59] Warren, Barycentric coordinates for convex sets, Advances in Computational Mechanics 27 (3) pp 319– (2007) · Zbl 1124.52010
[60] Warren J. On the uniqueness of barycentric coordinates. Proceedings of AGGM02 2003; 93-99. · Zbl 1043.52009
[61] Farin, Surfaces over Dirichlet tessellations, Computer Aided Geometric Design 7 (1-4) pp 281– (1990) · Zbl 0728.65013
[62] Tobin Driscoll, Schwarz-Christoffel Mapping 8 (2002) · Zbl 1003.30005
[63] Driscoll, Algorithm 756: a matlab tool box for Schwarz Christoffel mapping, ACM Transactions on Mathematical Software 22 (2) pp 168– (1996) · Zbl 0884.30005
[64] Howell LH. Computation of conformal maps by modified Schwarz-Christoffel transformations. Ph.D. Thesis, Department of Mathematics, Massachusetts Institute of Technology, MA, U.S.A., 1990.
[65] Trefethen, Numerical conformal mapping, SIAM Journal on Scientific and Statistical 1 (1) pp 82– (1980)
[66] De, The method of finite spheres with improved numerical integration, Computers and Structures 79 (22) pp 2183– (2001)
[67] Peirce, Numerical integration over the planar annulus, Journal of the Society for Industrial and Applied Mathematics 5 (2) pp 66– (1957) · Zbl 0079.34101
[68] Dunavant, High degree efficient symmetrical gaussian quadrature rules on triangle, International Journal for Numerical Methods in Engineering 23 pp 397– (1985) · Zbl 0589.65021
[69] Sukumar, The natural element method in solid mechanics, International Journal for Numerical Methods in Engineering 43 pp 839– (1998) · Zbl 0940.74078
[70] Timoshenko, Theory of Elasticity (1970)
[71] de Veubeke, Displacement and equilibrium models in the finite element method, International Journal for Numerical Methods in Engineering 52 pp 287– (2001) · Zbl 1065.74625
[72] Debongnie, Dual analysis with general boundary conditions, Computer Methods in Applied Mechanics and Engineering 122 pp 183– (1994)
[73] Richardson, The approximate arithmetical solution by finite differences of physical elements, Transactions of the Royal Society of London A210 pp 307– (1910)
[74] Beckers, Numerical comparison of several a posteriori error estimators for 2D stress analysis, European Journal of Finite Element Method 2 (2) pp 155– (1993) · Zbl 0813.73059
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