zbMATH — the first resource for mathematics

Generalized boundary element method for Galerkin boundary integrals. (English) Zbl 1182.65182
Summary: A meshless approach to the Boundary Element Method in which only a scattered set of points is used to approximate the solution is presented. Moving Least Square approximations are used to build a Partition of Unity on the boundary and then used to construct, at low cost, trial and test functions for Galerkin approximations. A particular case in which the Partition of Unity is described by linear boundary element meshes, as in the Generalized Finite Element Method, is then presented. This approximation technique is then applied to Galerkin boundary element formulations. Finally, some numerical accuracy and convergence solutions for potential problems are presented for the singular, hypersingular and symmetric approaches.

65N38 Boundary element methods for boundary value problems involving PDEs
80M15 Boundary element methods applied to problems in thermodynamics and heat transfer
80A20 Heat and mass transfer, heat flow (MSC2010)
Full Text: DOI
[1] Nayroles, B.; Touzot, G.; Villon, P., Generalizing the finite element method: diffuse approximation and diffuse elements, Comput mech, 10, 307-318, (1992) · Zbl 0764.65068
[2] Amarantuga, K.; Williams, J.R.; Qian, S.; Weis, J., Wavelet Galerkin solutions for one-dimensional partial differential equations, Int J num meth eng, 37, 2703-2716, (1994) · Zbl 0813.65106
[3] Belytschko T, Lu YY, Gu J. Crack propagation by element free galerkin methods. In advanced computational methods for material modeling, AMD-v.180/PVP-v.268, ASME (1993), pp. 191-205.
[4] Liu, W.K.; Jun, S.; Zhang, Y., Reproducing kernel particle methods, Int J num meth eng, 20, 1081-1106, (1995) · Zbl 0881.76072
[5] Atluri, S.N.; Zhu, T., New meshless local petrov – galerkin (MPLG) approach in computational mechanics, Comput mech, 22, 117-127, (1998) · Zbl 0932.76067
[6] Sukumar, N.; Moran, B.; Belytschko, T., The natural element method, Int J num meth eng, 43, 839-887, (1998) · Zbl 0940.74078
[7] Babuška I, Melenk JM. The partition of unity method. Technical note BN-1185, Institute for physical science and technology, University of Maryland, (1995).
[8] Duarte CA, Oden JT. Hp clouds—a meshless method to solve boundary value problems. TICAM Report 95-05, University of Texas, (1995).
[9] Duarte, C.A.; Oden, J.T., H-p clouds-an h-p meshless method, Num meth partial diff eq, 1-34, (1996) · Zbl 0869.65069
[10] Oden, J.T.; Duarte, C.A.; Zienkiewicz, O.C., A new cloud-based hp finite element method, Comput methods appl mech eng, 153, 117-126, (1998) · Zbl 0956.74062
[11] Duarte, C.A.; Babuška, I.; Oden, J.T., Generalized finite element methods for the three dimensional, Struct mech probl comput struct, 77, 215-232, (2000)
[12] Sukumar, N.; Moes, N.; Moran, N.; Belytschko, T., Extended finite element method for three-dimensional crack modelling, Int J num meth eng, 48, 1549-1570, (2000) · Zbl 0963.74067
[13] Mukherjee, Y.X.; Mukherjee, S., The boundary node method for potential problems, Int J num meth eng, 40, 797-815, (1997) · Zbl 0885.65124
[14] Chati, M.K.; Mukherjee, S.; Paulino, G.H., The meshless hypersingular boundary node method for three-dimensional potential theory and linear elasticity problems, Eng anal boundary elem, 25, 639-653, (2001) · Zbl 1065.74626
[15] Chati, M.K.; Mukherjee, S.; Paulino, G.H., The meshless standard and hypersingular boundary node methods—applications to error estimation and adaptivity in three-dimensional problems, Int J numer methods eng, 50, 9, 2233-2269, (2001) · Zbl 0988.74073
[16] Zhu, T.; Zhang, J.D.; Atluri, S.N., A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach, Comput mech, 21, 223-235, (1998) · Zbl 0920.76054
[17] Sladek, J.; Sladek, V., Local boundary integral equation methods in solid mechanics, WCCM V. fifth world congress on computational mechanics, Austria, (2002) · Zbl 1062.74060
[18] Chen, W., Meshfree boundary particle method applied to Helmholtz problems, Eng anal boundary elem, 26, 577-581, (2002) · Zbl 1013.65128
[19] Wang, J.G.; Liu, G.R., On the optimal shape parameters of radial basis functions used for 2-D meshless methods, Comput methods appl mech eng, 191, 2611-2630, (2002) · Zbl 1065.74074
[20] Li, J.; Hon, Y.C.; Chen, C.S., Numerical comparisons of two meshless methods using radial basis functions, Eng anal boundary elem, 26, 205-225, (2002) · Zbl 1003.65132
[21] Mai-Duy, N.; Tran-Cong, T., Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson’s equations, Eng anal boundary elem, 26, 133-156, (2002) · Zbl 0996.65131
[22] Li, J.; Hon, Y.C.; Chen, C.S., Numerical comparisons of two meshless methods using radial basis functions, Eng anal bound elem, 26, 205-225, (2002) · Zbl 1003.65132
[23] Li, G.; Aluru, N.R., A boundary cloud method with a cloud-by cloud polynomial basis, Eng anal bound elem, 27, 57-71, (2003) · Zbl 1037.78017
[24] Lancaster, P.; Šalkauskas, R., Curve and surface Fitting, an introduction, (1986), Academic Press San Diego · Zbl 0649.65012
[25] Nicolazzi, L.C.; Duarte, C.A.; Fancello, E.A.; de Barcellos, C.S., hp clouds—a meshless method in boundary elements. part II: implementation. in: first Brazilian seminar on the boundary element method in engineering. anais… Rio de Janeiro, RJ, Brazil, August 1996, Int J boundary elem methods commun, 8, 83-85, (1997)
[26] Duarte CA, Oden JT. An h-p adaptive method using clouds. TICAM Report 96-07, University of Texas, (1996).
[27] Kane, J.H., Boundary element analysis in engineering continuum mechanics, (1994), Prentice Hall Inc. USA
[28] de Barcellos CS, Mendonça PTR, Duarte CA. Investigations on timoshenko beam problems using the HP-Cloud Meshless FEM. In: IV world congress on computation mechanics. Anais… Buenos Aires, Argentine, (1998).
[29] Babuška, I.; Melenk, J.M., The particion of unity finite element method: basic theory and applications, Comput methods appl mech eng, 139, 1-4, 289-314, (1996) · Zbl 0881.65099
[30] Demkowicz, L.; Oden, J.T.; Rachowicz, W.; Hardy, O., Toward a universal h-p adaptative finite element strategy, part 1. constrained approximation and data structure, Comput methods appl mech eng, 77, 79-112, (1989) · Zbl 0723.73074
[31] Oden, J.T.; Demkowicz, L.; Rachowicz, W.; Westermann, T.A., Toward a universal h-p adaptative finite element strategy, part 2. A posteriori error estimation, Comput methods appl mech eng, 77, 113-180, (1989) · Zbl 0723.73075
[32] Rachowicz, W.; Oden, J.T.; Demkowicz, L., Toward a universal h-p adaptative finite element strategy, part 3. design of h-p meshes, Comput methods appl mech eng, 77, 181-212, (1989) · Zbl 0723.73076
[33] Duarte CAM. A Study of the p-version of finite elements for elasticity and potential problems. Brazil. Dissertação de Mestrado—UFSC. (1991). (In Portuguese).
[34] Jorge, A.B.; Ribeiro, G.O.; Fisher, T.S., Error estimators for BEM based on partial gradient recovery for higher order elements, WCCM V, fifth world congress on computational mechanics, Austria, (2002)
[35] Postell, F.V.; Stephan, E.P., On the h, p and hp versions of boundary element method—numerical results, Comput methods appl mech eng, 83, 69-89, (1990) · Zbl 0732.65101
[36] Jorge, A.B.; Ribeiro, G.O.; Cruse, T.A.; Fisher, T.S., Self-regular boundary integral equation formulations for Laplace’s equation in 2-D, Int J numer methods eng, 51, 1-29, (2001) · Zbl 0987.65124
[37] Sladek, V.; Sladek, J., Regularization of hypersingular and nearly singular integrals in potential theory and elasticity, Int J numer methods eng, 36, 1609-1628, (1993) · Zbl 0772.73091
[38] Tanaka, M.; Sladek, V.; Sladek, J., Regularization techniques applied to boundary element methods, Appl mech rev, 47, 10, (1994) · Zbl 0795.73077
[39] Ghosh, N.; Rajiyah, H.; Ghosh, S.; Mukherjee, S., A new boundary element method formulation for linear elasticity, J appl mech trans ASME, 53, 69-76, (1986) · Zbl 0592.73115
[40] Telles, J.C.F., A self-adaptative co-ordinate transformation for efficient numerical evaluation of general boundary element integrals, Int J numer methods eng, 24, 959-973, (1987) · Zbl 0622.65014
[41] Frangi, A.; Novati, G., Symmetric BE method in two-dimensional elasticity: evaluation of double integrals for curved elements, Comput mech, 19, 58-68, (1996) · Zbl 0888.73069
[42] Holzer, S.M., A p-extension of the symmetric boundary element method, Comput methods appl mech eng, 71, 339-357, (1993)
[43] Li, Z.C.; Manthon, R.; Sermer, P., Boundary methods for solving elliptic problems with singularities and interfaces SIAM, J numer anal, 24, 3, 487-498, (1987) · Zbl 0631.65103
[44] Holzer, S.M., The h-, p- and hp- version of the BEM in elasticity: numerical results, Commun numer methods eng, 11, 255-265, (1995) · Zbl 0815.73069
[45] Schwatz, A.H.; Thomee, V.; Wendland, W.L., Mathematical theory of finite and boundary element methods, (1990), Birhauser Verlag Basel Germany
[46] Stephan, E.P.; Suri, M., On the convergence of the p-version of the boundary element Galerkin method, Math comput, 52, 185, 1-48, (1989) · Zbl 0661.65118
[47] De-hao, Yu, Mathematical foundation of adaptative boundary element method, Comput methods appl mech eng, 91, 1237-1243, (1991)
[48] Kolmogorov, A.N.; Fomin, S.V., Introductory real analysis, (1994), Dover Publications Inc. USA
[49] Babuška, I.; Strouboulis, T.; Copps, K., Hp optimization of finite element approximations: analysis of the optimal mesh sequences in one dimension, Comput methods appl mech eng, 150, 89-108, (1997) · Zbl 0908.65073
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.