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Reproducing polynomial particle methods for boundary integral equations. (English) Zbl 1273.65189

Summary: Since meshless methods have been introduced to alleviate the difficulties arising in conventional finite element method, many papers on applications of meshless methods to boundary element method have been published. However, most of these papers use moving least squares approximation functions that have difficulties in prescribing essential boundary conditions. Recently, in order to strengthen the effectiveness of meshless methods, Oh et al. developed meshfree reproducing polynomial particle (RPP) shape functions, patchwise RPP and reproducing singularity particle (RSP) shape functions with use of flat-top partition of unity. All of these approximation functions satisfy the Kronecker delta property. In this paper, we report that meshfree RPP shape functions, patchwise RPP shape functions, and patchwise RSP shape functions effectively handle boundary integral equations with (or without) domain singularities.

MSC:

65N75 Probabilistic methods, particle methods, etc. for boundary value problems involving PDEs
65N38 Boundary element methods for boundary value problems involving PDEs
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[1] Arad M, Yosibash Z, Ben-Dor G, Yakhot A (1998) Computing flux intensity factors by a boundary metod for elliptic equations with singularities. Commum Numer Methods Eng 14: 657–670 · Zbl 0911.65103 · doi:10.1002/(SICI)1099-0887(199807)14:7<657::AID-CNM180>3.0.CO;2-K
[2] Babuška I, Banerjee U, Osborn JE (2003) Survey of meshless and generalized finite element methods: a unified approach. Acta Numer 12: 1–125 · Zbl 1048.65105 · doi:10.1017/S0962492902000090
[3] Babuška I, Banerjee U, Osborn JE (2003) Meshless and generalized finite element methods: survey of some major results. In: Meshfree methods for partial differential equations. Lecture notes in computational science and engineering, vol 26. Springer, New York
[4] Ciarlet PG (1978) The finite element method for elliptic problems. North-Holland · Zbl 0383.65058
[5] Duarte CA, Oden JT (1996) An hp adaptive method using clouds. Comput Methods Appl Mech Eng 139: 237–262 · Zbl 0918.73328 · doi:10.1016/S0045-7825(96)01085-7
[6] Duffy MG (1982) Quadrature over a pyramid or cube of integrands with a singularity at a vertex. SIAM J Numer Anal 19: 1260–1262 · Zbl 0493.65011 · doi:10.1137/0719090
[7] Griebel M, Schweitzer MA (2007) A particle-partition of unity methods. Part VII: adaptivity. In: Meshfree methods for partial differential equations III. Lecture notes in computational science and engineering, vol 57. Springer, New York
[8] Han W, Meng X (2001) Error analysis of reproducing kernel particle method. Comput Methods Appl Mech Eng 190: 6157–6181 · Zbl 0992.65119 · doi:10.1016/S0045-7825(01)00214-6
[9] Hunter P, Pullan A (2001) FEM and BEM notes. Department of Engineering Science, University of Auckland
[10] Li S, Liu WK (2004) Meshfree particle methods. Springer-Verlag, Berlin · Zbl 1073.65002
[11] Li X, Zhu J (2009) A Galerkin boundary node method and its convergence analysis. J Comput Appl Math 230: 314–328 · Zbl 1189.65291 · doi:10.1016/j.cam.2008.12.003
[12] Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20: 1081–1106 · Zbl 0881.76072 · doi:10.1002/fld.1650200824
[13] Liu WK, Liu SJ, Li S, Adee J, Belytschko T (1995) Reproducing kernel particle methods for structural dynamics. Int J Numer Methods Eng 38: 1655–1679 · Zbl 0840.73078 · doi:10.1002/nme.1620381005
[14] Liu WK, Li S, Belytschko T (1997) Moving least square reproducing kernel method. Part I: methodology and convergence. Comput Methods Appl Mech Eng 143: 422–453 · Zbl 0883.65088 · doi:10.1016/S0045-7825(96)01132-2
[15] Liu WK, Li S, Belytschko T (1996) Moving least square reproducing kernel method. Part II: Fourier analysis. Comput Methods Appl Mech Eng 139: 159–193 · Zbl 0883.65089 · doi:10.1016/S0045-7825(96)01082-1
[16] Li S, Liu WK (1999) Reproducing kernel hierarchical partition of unity. Part I: formulations. Int J Numer Methods Eng 45: 251–288 · Zbl 0945.74079 · doi:10.1002/(SICI)1097-0207(19990530)45:3<251::AID-NME583>3.0.CO;2-I
[17] Lucas TR, Oh H-S (1993) The method of auxiliary mapping for the finite element solutions of elliptic problems containing singularities. J Comput Phys 108: 327–342 · Zbl 0797.65083 · doi:10.1006/jcph.1993.1186
[18] Melenk JM, Babuška I (1996) The partition of unity finite element method: theory and application. Comput Methods Appl Mech Eng 139: 239–314 · Zbl 0881.65099 · doi:10.1016/S0045-7825(96)01087-0
[19] Mukherjee YX, Mukherjee S (1997) Boundary node method for potential problems. Int J Numer Methods Eng 40: 797–815 · Zbl 0885.65124 · doi:10.1002/(SICI)1097-0207(19970315)40:5<797::AID-NME89>3.0.CO;2-#
[20] Oh H-S, Babuška I (1995) The method of auxiliary mapping for the finite element solutions of plane elasticity problems containing singularities. J Comput Phys 121: 193–212 · Zbl 0833.73061 · doi:10.1016/S0021-9991(95)90017-9
[21] Oh H-S, Jeong JW, Hong WT (2010) The generalized product partition of unity for the meshless methods. J Comput Phys 229: 1600–1620 · Zbl 1180.65152 · doi:10.1016/j.jcp.2009.10.047
[22] Oh H-S, Jeong JW (2009) Almost everywhere partition of unity to deal with essential boundary conditions in meshless methods. Comput Methods Appl Mech Eng 198: 3299–3312 · Zbl 1230.74231 · doi:10.1016/j.cma.2009.06.013
[23] Oh H-S, Jeong JW (2009) Reproducing polynomial (singularity) particle methods and adaptive meshless methods for two-dimensional elliptic boundary value problems. Comput Methods Appl Mech Eng 198: 933–946 · Zbl 1229.65227 · doi:10.1016/j.cma.2008.11.005
[24] Oh H-S, Kim JG, Hong WT (2008) The piecewise polynomial partition of unity shape functions for the generalized finite element methods. Comput Methods Appl Mech Eng 197: 3702–3711 · Zbl 1197.65185 · doi:10.1016/j.cma.2008.02.035
[25] Oh H-S, Kim JG, Jeong JW (2007) The closed form reproducing polynomial particle shape functions for meshfree particle methods. Comput Methods Appl Mech Eng 196: 3435–3461 · Zbl 1173.65303 · doi:10.1016/j.cma.2007.03.012
[26] Oh H-S, Kim JG, Jeong JW (2007) The smooth piecewise polynomial particle shape functions corresponding to patch-wise non-uniformly spaced particles for meshfree particles methods. Comput Mech 40: 569–594 · Zbl 1165.74353 · doi:10.1007/s00466-006-0126-x
[27] Oh H-S, Jeong JW, Kim JG (2007) The reproducing singularity particle shape function for problems containing singularities. Comput Mech 41: 135–157 · Zbl 1162.74505 · doi:10.1007/s00466-007-0174-x
[28] Ong ET, Lim KM (2005) Three-dimensional singular boundary elements for corner and edge singularities in potential problems. Eng Anal Bound Elem 29: 175–189 · Zbl 1182.78023 · doi:10.1016/j.enganabound.2004.10.004
[29] Szabo B, Babuska I (1991) Finite element analysis. Wiley, New York
[30] Xenophontos C, Elliotis M, Georgiou G (2006) A singular function boundary integral methods for Laplacian problems with boundary singularities. SIAM J Sci Comput 28: 517–532 · Zbl 1120.65121 · doi:10.1137/050622742
[31] Yun BI (1999) The boundary element method for potential problems with singularities. J KSIAM 3: 17–28
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