Development and implementation of some BEM variants—A critical review

Due to rapid development of boundary element method (BEM), this article explores the evolution of BEM over the past half century. We here summarize the overall development and implementation of several well-known BEM variants that includes collocation BEM, galerkin BEM, dual reciprocity BEM, complex variable BEM and analog equation method. Their theoretical and mathematical backgrounds are carefully described and a generalized Laplace’s equation (and Poisson’s equation) is utilized in demonstrating the different approaches involved. An up-to-date review on characteristics and implementation for each of the five variants is presented and also highlighted their significant contributions in boundary element research. In addition, this article tries to cover whole aspect of interests including efficiency, applicability and accuracy in order to give better understanding of BEM evolution. Comparisons and techniques of improvement for these variants are also discussed.

[1]  J. Katsikadelis,et al.  Solving the plane elastostatic problem by the analog equation method , 1997 .

[2]  Attilio Frangi,et al.  3D fracture analysis by the symmetric Galerkin BEM , 2002 .

[3]  Martin Schanz,et al.  Time Domain BEM: Numerical Aspects of Collocation and Galerkin Formulations , 2009 .

[4]  Dual reciprocity boundary element formulation for potential problems with arbitrarily distributed sources , 1988 .

[5]  Stefano Miccoli,et al.  A galerkin symmetric boundary‐element method in elasticity: Formulation and implementation , 1992 .

[6]  John T. Katsikadelis,et al.  Non-linear analysis of plates by the Analog Equation Method , 1994 .

[7]  Yoshihiro Ochiai,et al.  Three-dimensional unsteady heat conduction analysis by triple-reciprocity boundary element method , 2009 .

[8]  Ernst P. Stephan,et al.  An augmented galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems , 1984 .

[9]  W. T. Ang,et al.  A complex variable boundary element method for a class of boundary value problems in anisotropic thermoelasticity , 1999, Int. J. Comput. Math..

[10]  T. A. Cruse,et al.  Recent advances in boundary element analysis methods , 1987 .

[11]  M. A. Jaswon,et al.  Integral equation methods in potential theory and elastostatics , 1977 .

[12]  A complex variable boundary element method for solving plane and plate problems of elasticity , 1991 .

[13]  Carlos Alberto Brebbia,et al.  Formulation of the boundary element method for transient problems governed by the scalar wave equation , 1982 .

[14]  P. K. Banerjee,et al.  Developments in boundary element methods , 1979 .

[15]  Yoshihiro Ochiai,et al.  Initial Stress Formulation for Elastoplastic Analysis by Improved Multiple-Reciprocity Boundary Element Method. , 1998 .

[16]  Toshiro Matsumoto,et al.  Dual reciprocity BEM for time-stepping approach to the transient heat conduction problem in nonlinear materials , 2006 .

[17]  T. Rasmussen,et al.  Determination of groundwater flownets, fluxes, velocities, and travel times using the complex variable boundary element method , 2006 .

[18]  C. Brebbia,et al.  Boundary Elements IX , 1987 .

[19]  R. Hiptmair,et al.  Galerkin Boundary Element Methods for Electromagnetic Scattering , 2003 .

[20]  Toshiro Matsumoto,et al.  A time-stepping DRBEM for transient heat conduction in anisotropic solids , 2008 .

[21]  Two-dimensional unsteady heat conduction analysis by improved multiple-reciprocity boundary element method , 1996 .

[22]  Giulio Maier,et al.  Advances in boundary element techniques , 1993 .

[23]  A. Ghassemi,et al.  Complex variable BEM for thermo- and poroelasticity , 2004 .

[24]  H. Djojodihardjo.  UNIFIED BE-FE AERODYNAMIC-ACOUSTIC-STRUCTURE COUPLING SCHEME FOR ACOUSTIC EFFECTS ON AEROELASTIC STRUCTURES , 2008 .

[25]  M. Itagaki,et al.  Matrix-Type Multiple Reciprocity Boundary Element Method for Solving Three-Dimensional Two-Group Neutron Diffusion Equations , 1997 .

[26]  R. J. Whitley,et al.  Theoretical developments in the complex variable boundary element method , 2006 .

[27]  P. Rolicz.  Use of the Bubnov-Galerkin method to skin effect problems for the case of unknown boundary conditions , 1982 .

[28]  D. Poljak,et al.  Currents induced in human body exposed to the power line electromagnetic field , 1998, Proceedings of the 20th Annual International Conference of the IEEE Engineering in Medicine and Biology Society. Vol.20 Biomedical Engineering Towards the Year 2000 and Beyond (Cat. No.98CH36286).

[29]  C. Brebbia,et al.  Boundary Element Techniques in Engineering , 1979 .

[30]  C. Brebbia,et al.  The boundary element method in plasticity , 1981 .

[31]  J. Telles,et al.  A comparison between point collocation and Galerkin for stiffness matrices obtained by boundary elements , 1989 .

[32]  Song‐Ping Zhu,et al.  On the choice of interpolation functions used in the dual-reciprocity boundary-element method , 1994 .

[34]  P. Bettess,et al.  Use of wave boundary elements for acoustic computations. , 2003 .

[35]  A Boundary-only Solution to Dynamic Analysis of Non-homogeneous Elastic Membranes , 2000 .

[36]  Frank J. Rizzo,et al.  An advanced boundary integral equation method for three‐dimensional thermoelasticity , 1977 .

[37]  Alok Sutradhar,et al.  Symmetric Galerkin boundary element computation of T-stress and stress intensity factors for mixed-mode cracks by the interaction integral method , 2004 .

[38]  H. Mang,et al.  A Galerkin–type BE–formulation for Acoustic Radiation and Scattering of Structures with Arbitrary Shape , 2008 .

[39]  Dragan Poljak,et al.  Computer Aided Design of Wire Structures. Frequency and Time Domain Analysis , 2007 .

[40]  D. Poljak,et al.  GB-IBEM model of vertical antenna above, below and penetrating ground , 2008, 2008 16th International Conference on Software, Telecommunications and Computer Networks.

[41]  Alok Sutradhar,et al.  Symmetric Galerkin Boundary Element Method , 2008 .

[42]  Frank J. Rizzo,et al.  On boundary integral equations for crack problems , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[43]  Carlos Alberto Brebbia,et al.  Transformation of Domain Effects to the Boundary , 2003 .

[44]  A. Cheng,et al.  Heritage and early history of the boundary element method , 2005 .

[45]  A. Behbahani-nia,et al.  A dual reciprocity BE-based sequential function specification solution method for inverse heat conduction problems , 2004 .

[46]  Carlos Alberto Brebbia,et al.  Progress in Boundary Element Methods , 1981 .

[47]  G. Maier,et al.  Symmetric Galerkin Boundary Element Methods , 1998 .

[48]  Demosthenes Polyzos,et al.  Three-dimensional structural vibration analysis by the Dual Reciprocity BEM , 1998 .

[49]  Helmut Harbrecht,et al.  Wavelet-based boundary element methods for exterior electromagnetic shaping , 2008 .

[50]  M. Schanz,et al.  An alternative collocation boundary element method for static and dynamic problems , 2009 .

[51]  P. K. Banerjee,et al.  Boundary element methods in nonlinear fluid dynamics , 1990 .

[52]  Huy Duong Bui,et al.  An integral equations method for solving the problem of a plane crack of arbitrary shape , 1977 .

[53]  S. Sirtori.  General stress analysis method by means of integral equations and boundary elements , 1979 .

[54]  Jeng-Tzong Chen,et al.  A study on the multiple reciprocity method and complex-valued formulation for the Helmholtz equation , 1998 .

[55]  D. Arnold,et al.  On the Asymptotic Convergence of Spline Collocation Methods for Partial Differential Equations , 1984 .

[56]  George C. Hsiao,et al.  Boundary element methods: an overview , 2006 .

[57]  Wolfgang Dahmen,et al.  Compression Techniques for Boundary Integral Equations - Asymptotically Optimal Complexity Estimates , 2006, SIAM J. Numer. Anal..

[58]  R. Harijono Djojodihardjo,et al.  A Numerical Method for the Calculation of Nonlinear, Unsteady Lifting Potential Flow Problems , 1969 .

[59]  S. L. Crouch,et al.  Complex variable boundary integral method for linear viscoelasticity: Part I—basic formulations , 2006 .

[60]  Paulo Sollero,et al.  Dual reciprocity boundary element method in Laplace domain applied to anisotropic dynamic crack problems , 2003 .

[61]  E. Balaras,et al.  A dual reciprocity boundary element formulation using the fractional step method for the incompressible Navier–Stokes equations , 2009 .

[62]  D. Frayce,et al.  A DUAL RECIPROCITY BOUNDARY ELEMENT APPROACH TO THREE-DIMENSIONAL TRANSIENT HEAT CONDUCTION AS APPLIED TO MATERIALS PROCESSING , 1996 .

[63]  G. Maier,et al.  Symmetric Galerkin boundary element method in plasticity and gradient plasticity , 1995 .

[64]  R. Attarnejad,et al.  A novel approach to solve linear systems arising from BEM using fast wavelet transforms , 2010 .

[65]  Dragan Poljak,et al.  Wire antenna model for transient analysis of simple grounding systems. Part II : the horizontal grounding electrode , 2006 .

[66]  P. Juhl.  A NOTE ON THE CONVERGENCE OF THE DIRECT COLLOCATION BOUNDARY ELEMENT METHOD , 1998 .

[67]  John T. Katsikadelis,et al.  The 2D elastostatic problem in inhomogeneous anisotropic bodies by the meshless analog equation method (MAEM) , 2008 .

[68]  Martin Schanz,et al.  Wave Propagation in Viscoelastic and Poroelastic Continua: A Boundary Element Approach , 2001 .

[69]  P. C. Dumir,et al.  Complex variable boundary element method for torsion of anisotropic bars , 1993 .

[70]  Christian Soize,et al.  Structural Dynamics EURODYN 2005 , 2005 .

[71]  P. Rolicz.  Bubnov-Galerkin method for solving exterior alternating field problems , 1982 .

[72]  O. von Estorff,et al.  Boundary Elements in Acoustics : Advances and Applications , 2000 .

[73]  John T. Katsikadelis,et al.  Large deflection analysis of beams with variable stiffness , 2003 .

[74]  P. L. Levin,et al.  A Galerkin formulation of the boundary element method for two-dimensional and axi-symmetric problems in electrostatics , 1992 .

[75]  G. Schmid,et al.  Three-dimensional transient half-space dynamics using the dual reciprocity boundary element method , 2008 .

[76]  Jin-you Xiao,et al.  A-posteriori compression of wavelet-BEM matrices , 2009 .

[77]  Mark Ainsworth,et al.  Topics in Computational Wave Propagation: Direct and Inverse Problems , 2003 .

[78]  T. Hromadka.  A multi-dimensional complex variable boundary element method , 2002 .

[79]  On the implementation of the galerkin approach in the boundary element method , 1989 .

[80]  Shiliang Ping,et al.  A complex variable boundary element method for solving interface crack problems , 1999 .

[81]  Jukka Saranen,et al.  On the collocation method for a nonlinear boundary integral equation , 1989 .

[82]  M. H. Aliabadi,et al.  A symmetric Galerkin boundary element method for dynamic frequency domain viscoelastic problems , 2001 .

[83]  An iterative CVBEM solution of nonlinear heat transfer problems , 1993 .

[84]  L. Shampine,et al.  A collocation method for boundary value problems , 1972 .

[85]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[86]  Wolfgang L. Wendland,et al.  Asymptotic Accuracy and Convergence for Point Collocation Methods , 1985 .

[87]  Christian Lage,et al.  Wavelet Galerkin algorithms for boundary integral equations , 1997 .

[88]  D. Arnold,et al.  The convergence of spline collocation for strongly elliptic equations on curves , 1985 .

[89]  Carlos Alberto Brebbia,et al.  The dual reciprocity boundary element formulation for diffusion problems , 1987 .

[90]  Chintu Lai,et al.  The Complex Variable Boundary Element Method in Engineering Analysis , 1987, Springer New York.

[91]  Dual reciprocity boundary element analysis for the laminar forced heat convection problem in concentric annulus , 1999 .

[92]  S. Amini.  Boundary integral solution of the exterior Helmholtz problem , 1993, Computational Mechanics.

[93]  L. Wrobel,et al.  On the convergence of the dual reciprocity boundary element method , 1994 .

[94]  Masataka Tanaka,et al.  Dual reciprocity boundary element analysis of inverse heat conduction problems , 2001 .

[95]  Y. Ochiai,et al.  Meshless convection–diffusion analysis by triple-reciprocity boundary element method , 2009 .

[96]  Yoshihiro Ochiai,et al.  Two‐dimensional unsteady heat conduction analysis with heat generation by triple‐reciprocity BEM , 2001 .

[97]  Stephan T. Grilli,et al.  Dual-reciprocity BEM based on global interpolation functions , 1994 .

[98]  Helmut Harbrecht,et al.  Fast Methods for Three-dimensional Inverse Obstacle Scattering Problems , 2007 .

[99]  W. S. Venturini.  Boundary Element Method in Geomechanics , 1983 .

[100]  Alok Sutradhar,et al.  On hypersingular surface integrals in the symmetric Galerkin boundary element method: application to heat conduction in exponentially graded materials , 2005 .

[101]  Paul William Partridge,et al.  Towards criteria for selecting approximation functions in the Dual Reciprocity Method , 2000 .

[102]  J. M. Baldasano,et al.  BEM Solution for Design of Trenches in Multilayered Landfills , 1998 .

[103]  W. S. Hall,et al.  Boundary element methods for soil-structure interaction , 2004 .

[104]  J. Hess,et al.  Calculation of potential flow about arbitrary bodies , 1967 .

[105]  Zhiye Zhao,et al.  Numerical implementation of the symmetric Galerkin boundary element method in 2D elastodynamics , 2003 .

[106]  A. Kassab,et al.  Application of the complex variable boundary element method to solving potential problems in doubly connected domains , 1990 .

[107]  Computational techniques for differential equations , 1985 .

[108]  Boundary element method applied to the analysis of thin plates , 1987 .

[109]  F. Hartmann,et al.  The direct boundary element method in plate bending , 1986 .

[110]  Dragan Poljak,et al.  Comparison of wire antenna and modified transmission line approach to the assessment of frequency response of horizontal grounding electrodes , 2008 .

[111]  N. Kamiya,et al.  A note on multiple reciprocity integral formulation for the Helmholtz equation , 1993 .

[112]  Leonard J. Gray,et al.  Direct Evaluation of Hypersingular Galerkin Surface Integrals , 2004, SIAM J. Sci. Comput..

[113]  C. Brebbia,et al.  On the application of the boundary element method to plasticity , 1979 .

[114]  John T. Katsikadelis,et al.  The ponding problem on elastic membranes: an analog equation solution , 2002 .

[115]  P. Bettess,et al.  Plane wave interpolation in direct collocation boundary element method for radiation and wave scattering: numerical aspects and applications , 2003 .

[116]  Carlos Alberto Brebbia,et al.  Boundary Element Methods in Heat Transfer , 1992 .

[117]  Douglas N. Arnold,et al.  On the asymptotic convergence of collocation methods , 1983 .

[118]  J. H. Kane,et al.  Symmetric Galerkin fracture analysis , 1995 .

[119]  Božidar Šarler,et al.  Dual reciprocity boundary element method for convective-diffusive solid-liquid phase change problems, Part 1. Formulation , 1998 .

[120]  D. Poijak,et al.  Transient response of a thin wire buried in a real ground , 2001, IEEE Antennas and Propagation Society International Symposium. 2001 Digest. Held in conjunction with: USNC/URSI National Radio Science Meeting (Cat. No.01CH37229).

[121]  J. Watson.  Singular boundary elements for the analysis of cracks in plane strain , 1995 .

[122]  M. H. Aliabadi,et al.  A boundary element method for three‐dimensional elastoplastic problems , 1998 .

[123]  S. L. Crouch,et al.  A fast and accurate algorithm for a Galerkin boundary integral method , 2005 .

[124]  G. Gudehus,et al.  Finite Elements in Geomechanics , 1978 .

[125]  John T. Katsikadelis,et al.  Ponding on floating membranes , 2003 .

[126]  C. Katz,et al.  Boundary elements and symmetry , 1985 .

[127]  Reza Attarnejad,et al.  Fast solution of BEM systems for elasticity problems using wavelet transforms , 2010, Int. J. Comput. Math..

[128]  A symmetric Galerkin BEM for plate bending analysis , 2009 .

[129]  J. Katsikadelis,et al.  Buckling of plates with variable thickness—an analog equation solution , 1996 .

[130]  Comparison of the dual reciprocity and the multiple reciprocity methods , 1992 .

[131]  Boyce E. Griffith,et al.  A faster galerkin boundary integral algorithm , 1998 .

[132]  K. Ruotsalainen.  On the convergence of the collocation method for nonlinear boundary integral equations , 1994 .

[133]  Carlos Alberto Brebbia,et al.  Boundary element methods for potential problems , 1977 .

[134]  H. Power,et al.  Multi-domain mass conservative dual reciprocity method for the solution of the non-Newtonian Stokes equations , 2002 .

[135]  M. H. Aliabadi,et al.  Symmetric Galerkin BEM for multi‐connected bodies , 2001 .

[136]  S. Amini,et al.  Multiwavelet Galerkin boundary element solution of Laplace's equation , 2006 .

[137]  Theodore V. Hromadka,et al.  The Complex Variable Boundary Element Method , 1984 .

[138]  M. Denda,et al.  Complex variable approach to the BEM for multiple crack problems , 1997 .

[139]  S. Mukherjee,et al.  Boundary element techniques: Theory and applications in engineering , 1984 .

[140]  Luiz C. Wrobel,et al.  Boundary Integral Methods in Fluid Mechanics , 1995 .

[141]  Complex variable boundary element method for torsion of hollow shafts , 1992 .

[142]  Carlos Alberto Brebbia,et al.  Dynamic analysis in solid mechanics by an alternative boundary element procedure , 2000 .

[143]  Carlos Alberto Brebbia,et al.  Advances in boundary elements , 1989 .

[144]  M. Aliabadi,et al.  The Boundary Element Method , 2002 .

[145]  A. Nowak,et al.  STEADY‐STATE THERMOELASTICITY BY MULTIPLE RECIPROCITY METHOD , 1992 .

[146]  Helmut Harbrecht,et al.  Fast wavelet BEM for 3d electromagnetic shaping , 2005 .

[147]  C. Gáspár.  Multigrid technique for biharmonic interpolation with application to dual and multiple reciprocity method , 2004, Numerical Algorithms.

[148]  H. Power,et al.  Multi-domain dual reciprocity BEM approach for the Navier–Stokes system of equations , 2000 .

[149]  J.A.M. Carrer,et al.  Stability of Galerkin and collocation time domain boundary element methods as applied to the scalar wave equation , 2000 .

[150]  A direct boundary integral equation method for the acoustic scattering problem , 1993 .

[151]  G. Paulino,et al.  Symmetric Galerkin boundary integral fracture analysis for plane orthotropic elasticity , 1997 .

[152]  Plane wave coupling to finite length cables buried in a lossy ground , 2002 .

[153]  J. Sládek,et al.  Axial symmetric stationary thermoelastic analysis by triple-reciprocity BEM , 2005 .

[154]  M. A. Jaswon.  Integral equation methods in potential theory. I , 1963, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[155]  John T. Katsikadelis,et al.  The boundary element method for nonlinear problems , 1999 .

[156]  C. Rajakumar,et al.  The Boundary Element Method: Applications in Sound and Vibration , 2004 .

[157]  Hubert Maigre,et al.  Inverse Problems in Engineering Mechanics , 1994 .

[158]  Giulio Maier,et al.  A Galerkin approach to boundary element elastoplastic analysis , 1987 .

[159]  K. W. Man,et al.  Contact Mechanics Using Boundary Elements , 1994 .

[160]  Complex variable boundary element methods for the solution of potential problems in simply and multiply connected domains , 1991 .

[161]  Carlos Alberto Brebbia,et al.  BOUNDARY ELEMENT METHODS IN ENGINEERING , 1982 .

[162]  Extending The Complex Variable Boundary Element Method Of ThreeDimensions , 2003 .

[163]  Giovanni Monegato,et al.  New Numerical Integration Schemes for Applications of Galerkin Bem to 2-D Problems , 1997 .

[164]  Some developments of the CVBEM-application to the mixed boundary value problem for the Laplace equation , 1987 .

[165]  L. J. Gray,et al.  Evaluation of singular and hypersingular Galerkin integrals: direct limits and symbolic computation , 1998 .

[166]  L. Wen,et al.  Combined equivalent charge formulations and fast wavelet Galerkin BEM for 3‐D electrostatic analysis , 2009 .

[167]  Ted Belytschko,et al.  Advances in computational mechanics , 1992 .

[168]  L. Wen,et al.  Fully discrete Alpert multiwavelet Galerkin BEM in 2D , 2008 .

[169]  Martin Costabel,et al.  An improved boundary element Galerkin method for three-dimensional crack problems , 1987 .

[170]  Carlos Alberto Brebbia,et al.  Numerical implementation of the boundary element method for two dimensional transient scalar wave propagation problems , 1982 .

[171]  P. W. Partridge,et al.  The dual reciprocity boundary element method , 1991 .

[172]  John T. Katsikadelis,et al.  Nonlinear dynamic analysis of heterogeneous orthotropic membranes by the analog equation method , 2003 .

[173]  M. Brun,et al.  Integral representations at the boundary for Stokes flow and related symmetric Galerkin formulation , 2005 .

[174]  C. Brebbia,et al.  A new approach to free vibration analysis using boundary elements , 1983 .

[175]  J. O. Watson,et al.  Boundary Elements from 1960 to the Present Day , 2007 .

[177]  J. Napier,et al.  Symmetric‐Galerkin BEM simulation of fracture with frictional contact , 2003 .

[178]  A Symmetric Boundary Element Method/Finite Element Method Coupling Procedure for Two-Dimensional Elastodynamic Problems , 2003 .

[179]  G. T. Symm,et al.  Integral equation methods in potential theory. II , 1963, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[180]  Yuzhe Liu,et al.  Lower bound shakedown analysis by the symmetric Galerkin boundary element method , 2005 .

[181]  F. Moslehy,et al.  Modal analysis of plates using the dual reciprocity boundary element method , 1994 .

[182]  Boundary Elements VIII , 1986 .

[183]  J. Bernard Minster,et al.  A numerical boundary integral equation method for elastodynamics. I , 1978 .

[184]  S. Syngellakis,et al.  Dual reciprocity in boundary element formulations of the plate buckling problem , 1992 .

[185]  Y. Ochiai.  Two-dimensional steady heat conduction in functionally gradient materials by triple-reciprocity boundary element method , 2004 .

[186]  Alessandra Aimi,et al.  Hypersingular kernel integration in 3D Galerkin boundary element method , 2002 .

[187]  G. L. Guymon,et al.  A complex variable boundary element method: Development , 1984 .

[188]  W. Yeih,et al.  Applications of the complete multiple method reciprocity method for solving the ID Helmholtz equation of a semi-infinite domain , 2001 .

[189]  V. Popov,et al.  On the optimal implementation of the boundary element dual reciprocity method—multi-domain approach for 3D problems , 2007 .

[190]  Dragan Poljak,et al.  New numerical approach in the analysis of a thin wire radiating over a lossy half-space , 1995 .

[191]  Theodore V. Hromadka,et al.  Advances in the Complex Variable Boundary Element Method , 2002 .

[192]  Giovanni Monegato,et al.  Numerical integration schemes for the BEM solution of hypersingular integral equations , 1999 .

[193]  P. W. Partridge,et al.  Hybrid approximation functions in the dual reciprocity boundary element method , 1997 .

[194]  Carlos Alberto Brebbia,et al.  The multiple-reciprocity method. A new approach for transforming BEM domain integrals to the boundary , 1989 .

[195]  J. Sládek,et al.  Multiple reciprocity method for harmonic vibration of thin elastic plates , 1993 .

[196]  J. Sládek,et al.  Transient heat conduction analysis by triple-reciprocity boundary element method , 2006 .

[197]  S. Marburg,et al.  Computational acoustics of noise propagation in fluids : finite and boudary element methods , 2008 .

[198]  G. Tsiatas,et al.  The Analog Equation Method For LargeDeflection Analysis Of Heterogeneous AnisotropicMembranes: A Boundary-only Solution , 2000 .

[199]  J. Telles,et al.  A numerical Green's function and dual reciprocity BEM method to solve elastodynamic crack problems , 2005 .

[200]  T. A. Cruse,et al.  Numerical solutions in three dimensional elastostatics , 1969 .

[201]  Frank J. Rizzo,et al.  An integral equation approach to boundary value problems of classical elastostatics , 1967 .

[202]  R. Hickman,et al.  A scheme to randomly generate surface dust points for the 3D complex variable boundary element method (CVBEM) , 2006 .

[203]  T. Cruse,et al.  A direct formulation and numerical solution of the general transient elastodynamic problem. II , 1968 .

[204]  Y. Shiah,et al.  Multiple reciprocity boundary element analysis of two-dimensional anisotropic thermoelasticity involving an internal arbitrary non-uniform volume heat source , 2003 .

[205]  Indirect Galerkin–Bubnov boundary element method for solving integral equations in electromagnetics , 2004 .

[206]  A symmetric Galerkin BEM for multi-connected bodies: a new approach , 2001 .

[207]  C. Schwab,et al.  Boundary Element Methods , 2010 .

[208]  W. Ang.  Non-steady state heat conduction across an imperfect interface: A dual-reciprocity boundary element approach , 2006 .

[209]  J. Watson,et al.  Effective numerical treatment of boundary integral equations: A formulation for three‐dimensional elastostatics , 1976 .

[210]  A Symmetric Galerkin BEM for Harmonic Problems and Multiconnected Bodies , 2001 .

[211]  Paul William Partridge,et al.  Dual reciprocity BEM: local versus global approximation functions for diffusion, convection and other problems , 1994 .

[212]  B. Šarler,et al.  Dual reciprocity boundary element method for convective-diffusive solid-liquid phase change problems, Part 2. Numerical examples , 1998 .

[213]  R. D. Ciskowski,et al.  Boundary element methods in acoustics , 1991 .

[214]  Carlos Alberto Brebbia,et al.  The Boundary Element Method for Engineers , 1978 .

[215]  Douglas N. Arnold,et al.  Collocation Versus Galerkin Procedures for Boundary Integral Methods , 1982 .

[216]  Seppo Karrila,et al.  INTEGRAL EQUATIONS OF THE SECOND KIND FOR STOKES FLOW: DIRECT SOLUTION FOR PHYSICAL VARIABLES AND REMOVAL OF INHERENT ACCURACY LIMITATIONS , 1989 .

[217]  U. Meissner,et al.  Finite elements in water resources. , 1982 .

[218]  Carlos Alberto Brebbia,et al.  Boundary Elements: An Introductory Course , 1989 .

[219]  L. Ram-Mohan.  Finite Element and Boundary Element Applications in Quantum Mechanics , 2002 .

[220]  Carlos Alberto Brebbia,et al.  Boundary element method for fluid flow , 1979 .

[221]  Carlos Alberto Brebbia,et al.  Boundary Element Methods For Electrical Engineers , 1984 .

[222]  Peter Schiavone,et al.  Integral Methods in Science and Engineering , 1994 .

[223]  菊地 文雄.  J.J.Connor and C.A.Brebbia, Finite Element Techniques for Fluid Flow, Newnes-Butterworths, London and Boston, 1976, viii+310ページ, 22.5×14.5cm, 7,600円. , 1977 .

[224]  A. Nowak.  Solving Linear Heat Conduction Problems by the Multiple Reciprocity Method , 1992 .

[225]  C. A. Brebbia,et al.  The multiple Reciprocity boundary element method in elasticity: A new approach for transforming domain integrals to the boundary , 1991 .

[226]  Wenqing Wang,et al.  A dual reciprocity boundary element approach for the problems of large deflection of thin elastic plates , 2000 .

[227]  L. Gray,et al.  Direct evaluation of hypersingular Galerkin surface integrals II , 2007 .

[228]  T. Hromadka,et al.  A collocation CVBEM using program Mathematica , 2010 .

[229]  John T. Katsikadelis,et al.  A boundary element solution to the soap bubble problem , 2001 .

[230]  TRANSIENT THERMOELASTICITY BY MULTIPLE RECIPROCITY METHOD , 1993 .

[231]  J. T. Katsikadelis.  Keynote AddressThe Analog Equation Method - A PowerfulBEM-based Solution Technique For SolvingLinear And Nonlinear Engineering Problems , 1970 .

[232]  H. Harbrecht,et al.  Wavelet Galerkin Schemes for 2D-BEM , 2001 .

[233]  B. Ozgener,et al.  The application of the multiple reciprocity method to the boundary element formulation of the neutron diffusion equation , 1994 .

[234]  Carlos Alberto Brebbia,et al.  Formal equivalence of direct and indirect boundary element methods , 1978 .

[235]  Multiple reciprocity method in BEM formulations for solution of plate bending problems , 1996 .

[236]  Leonard J. Gray,et al.  Galerkin boundary integral method for evaluating surface derivatives , 1998 .

[237]  M. Bonnet,et al.  A GALERKIN SYMMETRIC AND DIRECT BIE METHOD FOR KIRCHHOFF ELASTIC PLATES : FORMULATION AND IMPLEMENTATION , 1998 .

[238]  R. Whitley,et al.  Approximating three-dimensional steady-state potential flow problems using two-dimensional complex polynomials , 2005 .

[239]  J. Domínguez.  Boundary elements in dynamics , 1993 .

[240]  T. Ha-Duong,et al.  A Galerkin BEM for transient acoustic scattering by an absorbing obstacle , 2003 .

[241]  S. L. Crouch,et al.  Complex variable boundary integral method for linear viscoelasticity: Part II—application to problems involving circular boundaries , 2006 .

[242]  Carlos Alberto Brebbia,et al.  Recent advances in boundary element methods , 1978 .

[243]  Kozo Sato,et al.  Complex variable boundary element method for potential flow with thin objects , 2003 .