摘要
无网格方法是最近三十年才发展起来的数值方法,具有部分或者完全消除网格和前处理简单的优势,因此,在裂纹扩展,弹塑性及三维大规模分析上具有广阔的前景。本文综述了无网格数值方法的发展历史和现状,重点介绍了边界型数值方法以及杂交边界点法。而杂交边界点法是一种具有很多优良特性的边界型纯无网格方法,但是杂交边界点法在求解含体力问题或者动力问题等非齐次问题时,不可避免的需要在计算域内积分。为此,本文提出了双重互易杂交边界点法,成功的避免了域内体积分,将杂交边界点法的应用范围大大的扩大到含体力的二维、三维弹性力学问题,弹性动力学问题和非线性等非齐次问题。本文主要完成了以下几个方面的工作:
第一,将双重互易法引入到杂交边界点法当中,利用径向基插值原理,提出了双重互易杂交边界点法。对于控制方程中的非齐次项,使用双重互易法将其转化为边界积分。该方法将问题的解分为特解和通解两部分。特解采用双重互易法通过径向基插值获得。通解通过杂交边界点法求得,同时在使用杂交边界点法的过程中应用修正的边界条件。因此,此方法是一种真正的纯无网格方法,既不需要插值网格,也不需要积分网格。计算时只需离散点的信息,前处理简单。域内节点的布置只参与径向基函数插值,不参与变量的插值和背景积分。
第二,将双重互易杂交边界点法应用于二维泊松问题,二维和三维带体力的弹性力学问题,建立了这三类问题的双重互易杂交边界点法理论和数值实施方案,并编制了相应的计算程序。数值算例表明,该方法具有较高的精度和较快的收敛性,能广泛的应用于各种实际工程问题。
第三,将双重互易杂交边界点法应用于二维弹性动力学问题,建立了弹性动力学问题的双重互易杂交边界点法理论和数值实施方案。对于动力问题,仅仅依靠局部边界积分方程是不足以求解出问题的解,因此本文使用域内变量和边界变量之间的关系来构造附加方程,以达到求解该问题的目的。数值算例表明,该方法应用于弹性动力学问题中是有效的,具有较高精度的。
第四,将双重互易杂交边界点法推广到非线性势问题中去,建立了非线性问题的双重互易杂交边界点法理论,与动力问题一样,仅仅只靠局部边界积分方程是不能求解出问题的解,需要利用域内变量和边界变量之间的关系来添加方程。算例表明,该方法应用于非线性问题时是有效的而且具有较高的收敛速度。结合拟线性化理论,提出了拟线性化杂交边界点法,并使用该方法求解非线性问题,理论和算例表明,该方法具有很好的稳定性,较高的精度,具有2阶收敛速度。
第五,对整个双重互易杂交点法数值实施方案进行了专门研究,提出了一套解决弱奇异积分和近奇异积分的积分方案。为了达到该方法应用的通用性,本文提出了特解基本形式的概念,并且对于同一大类问题,只须使用相同的特解基本形式来插值特解。为此,本文探讨了特解基本形式的求解方法和求解结果。
第六,对双重互易杂交点法中的各个自由参数进行了详细研究,并对径向基函数的次数和对精度的影响进行了研究。对径向基函数插值节点的个数及其布置位置进行了研究。针对这些问题提出了一些有用的优化意见。
双重互易杂交边界点法是一种具有较多优良特性的数值方法。与杂交边界点法相比,它可以应用于非齐次问题而避免了域内积分。它保持了杂交边界点法降维的优势,要求输入的数据只是求解域上的节点信息,前处理简单,适合于大规模复杂结构的计算。同时双重互易杂交边界点法是一种纯无网格方法,无论是插值,还是积分,都不需要任何网格。
研究表明,该方法不仅计算精度高,而且收敛速度快,应用范围广。适合于自适应问题、断裂问题和接触问题的求解。
Meshless method is a new kind of numerical methods developed in the past decades. They have the advantages that no element is needed totally or partly, and their preprocess is easy. Therefore, meshless method can be widely applied to crack propagation problem, elasto-plastic analysis and large scale three-dimensional problem analysis. The original literature and recent developments of meshless methods are briefly reviewed in this dissertation. Boundary-type numerical method and Hybrid Boundary Node Method(HBNM) are described in detail in this dissertation. HBNM is a kind of boundary-type truly meshless method with many excellent characteristics. But when it is applied to the inhomogeneous problems, such as elasticity problems with body force and dynamic loading and so on, it is evitable to need domain integral. So, Dual Reciprocity Hybrid Boundary Node Method (DRHBNM) is proposed in the study, and it successfully avoids domain integral, expands the application scope of this method to inhomogeneous problems, such as two-dimensional and three-dimensional elasticity with body force, elasto-dynamic problems and nonlinear problems. The dissertation includes the following contents:
Firstly, introducing the Dual Reciprocity Method(DRM) into HBNM, and applying radial basis function interpolation, a new boundary-type meshless method—DRHBNM is proposed. Appling DRM, the domain integral of inhomogeneous term of governing equation is transformed into boundary integral. The solution in this method is divided into two parts, i. e., the complementary solution and the particular solution. The complementary solution is solved by HBNM. The particular one is obtained by radial basis function interpolation. At the same time, the modified boundary conditions are applied in hybrid boundary node method. Therefore, this method is a truly meshless method, and it does not require a 'boundary elements mesh', either for the purpose of interpolation of variable, or for the integral of the 'energy'. It only need the data of distributed nodes in the calculation, the preprocess of this method is easy. The nodes in the domain are only for the radial basis function interpolation of DRM, they are no need for the variable interpolation and background integral.
Secondly, DRHBNM is implemented successfully for solving problems in two dimensional Poisson's equations, two and three dimensional linear elasticity with body force. The formulation of this method in these problems is developed, and the numerical implementation scheme is obtained. The relative programs are complied. The numerical examples are shown that the present method possesses not only high accuracy, but also good performance of convergence, and it can be widely applied to the practical problems.
Thirdly, DRHBNM is applied to elasto-dynamic problems, the formulation of this method in elasto-dynamic problems is developed, and numerical implementation details are given in detail. For the elasto-dynamic problems, only boundary integral equations can not solved the solution of this problem, so the relations between the domain variable and boundary variable are applied to form some new equations. Based on the above analysis, the computer codes are written for elasto-dynamic problem, and the numerical examples are shown that the present method is effective for two dimensional elasto-dynamic problems, and it can achieve high accuracy.
Fourthly, DRHBNM is applied to nonlinear potential problems, formulations for these kinds of problems are achieved. As same as the dynamic problems, the relations between the boundary variables and domain variables are applied to obtain the additional equations. Numerical examples are given to show that the present method is effective for the nonlinear potential problems, and it can achieve much high convergence. Applied Generalized Quasilinearization Method(GQM), Quasilinear Hybrid Boundary Node Method(QHBNM) is proposed, and employed to solve nonlinear problems. Theroy and numerical examples are shown that this method is stable and has high accuracy, and the convergence is quadratic.
Fifthly, some specialized studies are done for numerical implementation details. A series of integral schemes for weakly singular integral and nearly singular integral are proposed. In order to achieve the universal of the present method, the basic form of particular solution is proposed, so the particular solution can be interpolated by the same form of basic form of particular solution for the same kind of problems. The solving method and the results of the basic form of particular solution are presented in detail in this dissertation.
Sixthly, the free parameters of the present method have been studied in this dissertation, and the relations between the influence to the precision and the order of the radial basis function are discussed. Also, the nodes number and the distributed position for the radial basis function interpolation are studied. Finally, some optimization proposals are presented for above problems.
DRHBNM is is a new boundary-type truly meshless method with many excellent characteristics. Compared to HBNM, it can be applied to the inhomogeneous problems and without domain integral. It keeps the dimension-reduction advantages of HBNM, and only requires data information of nodes in the calculation. Therefore, the preprocess is easy, and suitable for the large scale complex structure analysis. Besides, DRHBNM is a truly meshless method, it does not require 'elements mesh' not only for variable interpolation, but also for background integral.
The study shows that the present method possesses the not only high accuracy, but also excellent performance of convergence. It can be widely applied to the practical problems, such as adaptive problem, crack propagation and contact analysis and so on.
引文
[1]马晓青,韩峰.高速碰撞动力学.北京:国防工业出版社.1998
[2]袁龙蔚.流变力学.北京:科学出版社.1986
[3]匡震邦,马法尚.裂纹端部场.西安:西安交通大学出版社.2002
[4]胡文武.现代接触动力学.南京:东南大学出版社.2003
[5]Chen Y P.Lee J D.and Eskandarian A.Meshless Methods in Solid Mechanics.Washinggton,DC,USA:Springer Press,2006
[6]Liu G R.Mesh Free Methods:Moving Beyond the Finite Element Method.Boca Raton,USA:CRC Press,2002
[7]张雄,刘岩.无网格法.北京:清华大学出版社,2004
[8]刘更,刘天祥.无网格法及其应用.西安:西北工业大学出版社,2005
[9]顾元通,丁桦.无网格法及其最新进展.力学进展,2005,35(3):323-337
[10]Mukherjee Y X,Mukherjee S.The boundary node method for potential problems.International Journal for Numerical Methods in Engineering,1994,140:797-815
[11]Chati M K,Mukherjee S,Mukherjee Y X.The boundary node method for three-dimensional linear elasticity.International Journal for Numerical Methods in Engineering,1999,46:1163-1184
[12]Zhang J M,Yao Z H,Li H.A Hybrid boundary node method.International Journal for Numerical Methods in Engineering,2002,53:751-763
[13]张见明,姚振汉,李宏.二维势问题的杂交边界点法.重庆建筑大学学报,2000,22(6):105-107
[14]苗雨,王元汉等.岩土工程中的杂交边界点方法.岩土力学,2005,26(9):1452-1455
[15]苗雨.奇异杂交边界点法理论研究及应用[博士学位论文].武汉:华中科技大学,2005
[16]Brebbia C A,Nardini D.Dynamic analysis in solid mechanics by an alternative boundary element procedure.International Journal of Soil Dynamics and Earthquake Engineering,1983,2(4):228-233
[17]Brebbia C A.The Solution of Time Dependent Problems Using Boundary Elements,in The Mathematics of Finite Elements V.London,UK:Academic Press,1985
[18]Belytschko T,Krongauz Y,Organ D,Fleming M,Krysl P.Meshless methods:An overview and recent developments.Computer Methods in Applied Mechanics Engineering,1996,139(1-4):3-47
[19]Lucy L B.A numerical approach to the testing of the fission hypothesis.The Astronomical Journal,1977,8(12):1013-1024
[20]Gingold R A,Monaghan J J.Smoothed particle hydrodynamics:theory and applications to nonspherical stars.Monthly Notices of the Royal Astronomical Society,1977,18:375-389
[21]Monaghan J J.An introduction to SPH.Comput.Phys.Comm.,1988,48:89-96
[22]Monaghan J J.Smoothed particle method work.Annual.Review Astronomics and Astrophysics,1992,30:543-574
[23]Swegle J W,Hicks D L,Attaway S W.Smoothed particle hydrodynamics stability analysis.Journal of Computational Physics,1995,116:123-134
[24]Dyka C T.Addressing tension instability in SPH methods.Technical Report NRL/MR/6384,NRL,1994
[25]Johnson G R,Stryk R A,Beissel S R.SPH for high velocity impact computations.Computer Methods in Applied Mechanics and Engineering,1996,139:347-373
[26]Liu W K,Jun S,Zhang Y F.Reproducing kernel particle methods.Int.J.Numer.Methods in Fluids,1995,20(8-9):1081-1106
[27]Liu W K,Jun S,Li S,et al.Reproducing kernel particle methods for structural dynamics.International Journal for Numerical Methods in Engineering,1995,38(10):1655-1679
[28]Jun S,Liu W K,Belytschko T.Explicit reproducing kernel particle methods for large deformation problems.International Journal for Numerical Methods in Engineering,1998,41(1):137-166
[29]Li S F,Liu W K.Meshfree and particle methods and their applications.Applied Mechanics Review,2002,55(1):1-34
[30]Li S F,Liu W K.Moving least square reproducing kernel methods Part Ⅱ:Fourier analysis. Computer Methods in Applied Mechanics and Engineering, 1996, 139(14): 159-193
[31] Nayroles, Touzot, Villon. Generalizing the finite element method: diffuse approximation and diffuse elements. Computational Mechanics, 1992,10: 307-318
[32] Lancaster P, Salkauskas K. Surfaces generated by moving least suqres method. Mathematics Computation, 1981,37: 141-158
[33] Liu W K. An introduction to wavelet reproducing kernel particle methods. USACM Bulletin, 1995, 8(1): 3-16
[34] Belytschko T, Liu Y Y, Gu L. Element-free Galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37: 229-256
[35] Lu Y Y, Belytschko T, et al. A new implementation of the element free Galerkin method. Computer Methods in Applied Mechanics and Engineering, 1994, 113: 397-414
[36] Chung H. J, Belytschko T. An error estimate in the EFG method. Computational Mechanics, 1998,21:91-100
[37] Dolbow J, Belytschko T. Numerical integration of the Galerkin weak form in meshless method. Computational Mechanics, 1999, 23: 219-230
[38] Belytschko T, Lu Y Y, Gu L. Fracture and crack growth by element free galerkin methods. Modelling and Simulation in Material Science and Engineering, 1994,2(3a): 519-534
[39] Belytschko T, Lu Y Y, Gu L. Crack propagation by element free galerkin methods. Engineering Fracture Mechanics, 1995, 51(2): 295-315
[40] Belytschko T, Lu Y Y, Gu L. element free galerkin methods for static and dynamics fracture. International Journal of Solids and Structures, 1995,32(17-18): 2547-2570
[41] Lu Y Y, Belytschko T, Tabbara M. Element free Galerkin methods for wave propagation and dynamic fracture. Computer Methods in Applied Mechanics and Engineering, 1995, 126(1-2): 131-153
[42] Krysl P, Belytschko T. Analysis of thin shells by element-free Galerkin method. Computational Mechanics, 1995,17: 26-35
[43] Liu G R, Gu Y T. Coupling of element free Galerkin and hybrid boundary element methods using modified variational formulation.Computational Mechanics,2000,26(2):166-173
[44]Belytschko T,Organ D,Krongnauz Y.A Coupled finite element-element free Galekin method.Computational Mechanics,.1995,17:186-195
[45]Beissel S,Belytschko T.Nodal integration of the element-free Galerkin method.Computer Methods in Applied Mechanics and Engineering,1996,139:49-74
[46]Smolinski P,Palmer T.Procedures for multi-time step integration of element-free Galerkin methods for diffusion problems.Computers & Structures,2000,77:171-183
[47]周维垣,寇晓东.无单元法及其工程应用.力学学报,1998,30(2):193-202
[48]张伟星,庞辉.弹性地基板计算的无单元法.工程力学,2000,17(3):138-144
[49]庞作会,葛修润,王水林.无网格伽辽金法(EFGM)在边坡开挖问题中的应用.岩土力学,1999,20(1):61-64
[50]Duarte C A,Oden J T.Hp clouds:a h-p meshless method.Numerical Method for Partial Differential Equations,1996,12:673-705
[51]Duarte C A,Oden J T.An h-p adaptive method using clouds.Computer Methods in Applied Mechanics and Engineering,1996,139:237-262
[52]Mendoncca P T R,Barcellos C S,Duarte A.Investigations on the hp-Cloud Method by solving Timoshenko beam problems.Computational Mechanics,2000,25:286-295
[53]Garcia O,Fancello E A.hp-clouds in Mindlin's thick plate model.International Journal for Numerical Methods in Engineering,2000,47:1381-1400
[54]刘欣,陆明万等.平面裂纹问题的h,p,hp型自适应无网格方法的研究.力学学报,2000,32(3):308-318
[55]Oden J T,Duarte C A,Zienkiewicz O C.A new cloud-based hp finite element method.International Journal for Numerical Methods in Engineering,1998,50:160-170
[56]Liszka T J,Duarte C A,et al.Hp-meshless method.Computer Methods in Applied Mechanics and Engineering,1996,139:263-288
[57]Duarte C A,Babuska I,Oden J T.Generalized finite element method for three dimensional structural mechanics problems. Computers & Structures, 2000, 77: 215-232
[58] Strouboulis T, Babuska I and Copps K. The design and analysis of the Generalized Finite Element Method. Computer Methods in Applied Mechanics and Engineering, 2000,181:43-69
[59] Onate E, Idelsohn S, Zienkiewicz O C, et al. A finite point method in computational mechanics: Applications to convective transport and fluid flow. International Journal for Numerical Methods in Engineering, 1996, 39: 3839-3866
[60] Onate E, Perazzo F, Miquel J. A finite point method for elasticity problems. Computers & Structures, 2001, 79: 2151-2163
[61] Song Z K, Zhang X, Lu M W. Meshless method based on collocation for elasto-plastic analysis. In: Proceedings of Internal Conference on Computational Engineering & Science[C], August 20-25, 2000, Los Angeles, USA
[62] Zhu T, Zhang J, Atluri S N. A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach. Computational Mechanics, 1998,21(3): 223-235
[63] Atluri S N, Sladek J. The Local boundary integral equation (LBIE) and it's meshless implementation for linear elasticity. Computational Mechanics, 2000,25(2): 180-198
[64] Zhu T L. A new meshless regular local boundary integral equation (MRLBIE) approach. International Journal for Numerical Methods in Engineering, 1999, 46(8): 1237-1252
[65] Zhu T, Zhang J, Atluri S N. A meshless local boundary integral equation (LBIE) method for solving nonlinear problems. Computational Mechanics, 1998, 22(2): 174-186
[66] Sladek V, Sladek J, Atluri S N and Vankeer R. Numerical integration of singularities in meshless implementation of local boundary integral equations. Computational Mechanics, 2000,25(4): 394-430
[67] Sladek J, Sladek V and Vankeer R. Meshless Local Boundary Integral Equation (LBIE) method for 2D elastodynamic problems. International Journal for Numerical Methods in Engineering, 2003, 57(2): 235-249
[68]Atluri S N,Zhu T.A new Meslaless Local Petrov-Galerkin(MLPG) approach in computational mechanics.Computational Mechanics,1998,22:117-127
[69]Atluri S N,Zhu T.The Meshless Local Petrov-Galerkin(MLPG) approach for solving problems in elasto-statics.Computational Mechanics,2000,25:169-179
[70]Atluri S N,Kim H G,Cho J Y.A critical assessment of the truly Meshless Local Petrov-Galerkin(MLPG),and Local Boundary Integral Equation(LBIE) methods.Computational Mechanics,1999,24:348-372
[71]Mukherjee Y X,Mukherjee S.The boundary node method for potential problems.Journal for Numerical Methods in Engineering,1997,40:797-815
[72]Chati M K,Mukherjee S.The boundary node method for three-dimensional problems in potential theory.International Journal for Numerical Methods in Engineering,2000,47:1523-1547
[73]Zhang J M,and Yao Z H.Analysis of 2D thin structure by the meshless regular hybrid boundary node method.Acta Mechanica Solida Sinica,2002,15(1):36-44,763
[74]Zhang J M,Masataka T,Toshiro M.Meshless analysis of potential problems in three dimensions with the hybrid boundary node method.Journal for Numerical Methods in Engineering,2004,59:1147-1168
[75]Zhang J M,Yao Z H,Masataka T.The meshless regular hybrid boundary node method for 2-D linear elasticity.Engineering Analysis with Boundary Elements,2003,127:259-268
[76]Zhang J M,Yao Z H.The regular hybrid boundary node method for three-dimensional linearelasticity.Analysis with Boundary Elements,2004,28:525-534
[77]Miao Y,Wang Y H.An improved hybrid boundary node method in two-dimensional solids.Acta Mechanica of Solida.2005,18(4):307-315
[78]Miao Y,Wang Y H.Development of hybrid boundary node method in two dimensional elasticity.Analysis with Boundary Elements,2005,29:703-712
[79]Miao Y,Wang Y H.Meshless analysis for three-dimensional elasticity with singular hybrid boundary node method.Applied Mathematics and Mechanics.2006,27(5): 673-681
[80] Nardini D, Brebbia C A. Boundary Integral Formulation of Mass Matrics for Dynamic Analysis, in Topics in Boundary Element Research, vol. 2, Springer-Verlag, Berlin and New York, 1985
[81] Wrobel L C, Brebbia C A. The dual reciprocity boundary element formulation for non-Linear diffusion problems, Computer Methods in Applied Mechanics and Engineering, 1987, 5(1): 147-164
[82] Partridge P W, Brebbia C A. Computer implementation of the BEM dual reciprocity method for the solution of poisson type equations. Software for Engineering Workststions, 1989, 5(4): 199-206
[83] Partridge P W, Brebbia C A. Computer implementation of the BEM dual reciprocity method for the solution of general field problems. Communications in Applied Numerical Methods, 1990, 6(2): 83-92
[84] Golberg M A, Chen C S. The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations. Boundary Elements Communications, 1994, 5: 57-61
[85] Bridges T R, Wrobel L C. A DRM formulation for elasticity problems with body forces using augmented thin plate splines. Communications in Numerical Methods Engineering, 1996,12: 209-220
[86] Partridge P W, Sensale B. Hybrid approximation functions in the dual reciprocity BEM. Communications in Numerical Methods Engineering, 1997,13: 83-94
[87] Neves A C, Brebbia C A. The multiple reciprocity boundary element method in elasticity: a new approach for transforming domain integrals to the boundary. International Journal for Numerical Methods in Engineering, 1991, 31: 709-727
[88] Banerjee P K. The Boundary Element Methods in Engineering. McGraw Hill Europe, Maidenhead, Berkshire, UK: Academic Press, 1994
[89] Bonnet M. Boundary Integral Equation Method for Solid and Fluids. Chichester, UK: Wiley Press, 1995
[90] Brebbia C A. and Dominguez J. Boundary Elements: An Introductory Course. 2nd edition. Computional Mechanics Publications, Southampton, UK, and McGraw Hill, New York.1992
[91]Chandra A.and Mukherjee S.Boundary Element methods in Manufacturing.New York,USA:Oxford University Press,1997
[92]Krishnasamy G,Rizzo F J and Rudolphi T J.Hypersingular boundary integral equations:their occurrence,interpretation,regularization and computation.Developments in Boundary Element Methods-7.P.K.Banerjee and S.Kobayashi eds,.London,UK:Elsevier Applied Science,1992
[93]Tanaka M,Sladek V and Sladek J.Regularization techniques applied to boundary element methods.ASME Applied Mechanics Reviews,1994,47:457-499
[94]Guiggiani M.Hypersingular formulation for noundary stress evaluation.Engineering Analysis with Boundary Elements,1994,13:169-179
[95]Wilde A J and Aliabadi M H.Direct evaluation of boundary stress in 3D BEM of elastostatics.Communications in Numerical methods in engineering,1998,14:505-517
[96]Cruse T A.Boundary Element Analysis in Computational Fracture Mechanics.Dordrecht,The Netherlands:Kluwer,1988
[97]Gray L J,Martha L F and Ingraffea A R.Hypersingular integrals in boundary element fracture analysis.International Journal for Numerical Methods in Engineering,1990,29:1135-1158
[98]Kothnur V S.Mukherjee S.and Mukherjee Y X.Two-dimensional linear elasticity by the boundary node method.International Journal of Solids and Structures,1998,36(8):1129-1147
[99]Chati M K,Muldaerjee S and Paulino G H.The meshless hypersingular boundary node method for three-dimensional potential theory and linear elasticity problems.Engineering Analysis with Boundary Elements,2001,25:639-653
[100]Li G and Alttru N R.Boundary cloud method:a combined scattered point/boundary integral approach for boundary only analysis.Computer Method in Applied Mechanics and Engineering,2002,191:2337-2370
[101]Li G and Aiuru N R.A boundary cloud method with a cloud-by-cloud polynomial basis.Engineering Analysis with Boundary Elements,2003,27:57-71
[102]Chen W and Tanaka M.A meshless,integration-free,boundary-only RBF technique.Computers and Mathematics with Applications,2002,43:379-391
[103]Sladek J,Sladek V and Atluri S N.Local boundary integral equation(LBIE) method for solving problems of elasticity with nonhomogeneous material properties.Computational Mechanics,2000,24(6):456-462
[104]Lutz E D.Numerical method for hypersingular and near-singular boundary integrals in fracture mechanics.Ph.D.Dissertation,New York,USA:Cornell University,Ithaca,1991
[105]Nagarajan A,Lutz E D and Mukherjee S A novel boundary element method for linear elasticity with no numerical integration for 2D and line integrals for 3-D problems.ASME Journal of Applied Mechanics,1994,61:264-269
[106]Nagarajan A,Mukherjee S.and Lutz E D.The boundary contour method for three-dimensional linear elasticity.ASME Journal of Applied Mechanics,1996,63:278-286
[107]Phan A V,Mukherjee S.and Mayer J R.The hypersingular boundary contour method for two-dimensional linear elasticity,Acta Mechanics,1998,35:209-225
[108]Mukherjee S and Mukherjee Y X.The hypersingular boundary contour method for three-dimensional linear elasticity.ASME Journal of Applied Mechanics 1998,65:300-309
[109]Schnack E.A hybrid BEM model.International Journal for Numerical Methods in Engineering,1987,24:1015-1025
[110]Dumont N A.The hybrid boundary element method.Proceedings 9~(th) International Conference On BEM.Stuttgart,Computational Mechanics Publications.Southampton and Springer Verlag,Berlin,1987
[111]DeFigueredo T G and Brebbia C A.A new hybrid displacement variational formulation of BEM for elastostatics,in Brebbia C A,Conner J J(eds.),Advances in Boundary Elements,Computational Mechanics Publications,Southampton,1989,1:47-57
[112]Mukherjee S.and Mukherjee Y X.Boundary methods elements,contours,and nodes.Boca Raton,USA:CRC Press,2005
[113] Newman J N. Distributions of sources and normal dipoles over a quadrilateral panel. Journal of Engineering Mathematics, 1986,20: 113-126
[114] Gaul L, Kogl M and Wagner M. Boundary element methods for engineering and scientists: an introductory course with advanced topics. Berlin, Hong Kong: Springer-Verlag, 2003
[115] Medeiros G C, Partridge P W and Brandao J O. The method of fundamental solutions with dual reciprocity for some problems in elasticity, Engineering Analysis with Boundary Elements, 2004; 28: 453-461
[116] Li SC, Chen YM. Numerical manifold method for crack tip fields. China Civil Engineering Journal, 2005, 38(7): 96-102
[117] Dong YW, Yu TT, Ren QW. Extended finite element method for direct evaluation of strength intensity factors. Chinese Journal of Computational Mechanics, 2008, 25(1): 72-77
[118] Chang CC, Mear ME. A boundary element method for two dimensional linear elastic fracture analysis. International Journal of Fracture, 1995, 74: 219-251
[119] Pan E. A general boundary element analysis of 2-D linear elastic fracture mechanics. International Journal of Fracture, 1997, 88: 41-59
[120] Perez-Gavilan J J and Aliabadi M H. A symmetric Galerkin formulation and dual reciprocity for 2D elastostatics, Engineering Analysis with Boundary Elements, 2001, 25: 229-235
[121] Partridge P W and Sensale B. The method of fundamental solutions with dual reciprocity for diffusion and diffusion-convection using subdomains[J], Engineering Analysis with Boundary Elements, 2000,24(9): 633-641
[122] Ren C B and He G Z. A meshless precise integration method to solve a 2-D structural vibration problem, Journal of Vibration and Shock, 2007,26: 126-129
[123] Partridge P W, Brebbia C A and Wrobel L C. The Dual Reciprocity Boundary Element Method. London, New York: Computational Mechanics Publications, Southampton Boston Co-published with Elsevier Applied Science, 1992
[124] EI-Zafrany A, Cookson R A and Iqbal M. Boundary Element Stress Analysis withDomain Type Loading, in Advances in the Use of Boundary Element Method for Stress Analysis, London, UK: Mechanical Engineering Publication, 1986
[125] Youcef S and Martin H S. GMRES: A Generalized minmal residual algorithm for solving nonsymmetric linear systems. Society for Industrial and Applied Mathematics, 1986,7(3): 856-869
[126] Ladde G S, Lakshmikantham V, and Vatsala A S. Monotone iterative techniques for nonlinear differential equations. Boston, London, Melbourne: Pitman Publishing Company, 1985
[127] Amman H. On the existence of positive solutions of nonlinear elliptic boundary value problems. Indian University Mathematical Journal, 1971, 21: 125-146
[128] Byszewski L. Monotone iterative method for a system of nonlocal initial-boundary parabolic problems. Journal of Mathematical Analysis and Application, 1993, 177: 45-458
[129] Byszewski L and Lakshmikantham V. Monotone iterative technique for nonlocal hyperbolic differential problem. Journal of Mathematical Physical Science, 1992, 26(4): 345-359
[130] Deo S G and Pandit S G Method of generalized quasilinearization for hyperbolic initial-boundary value problems. Nonlinear World, 1996, 3: 267-275
[131] Lakshmikantham V and Carl S. Generalized quasilinearization method for reaction-diffusion equations under nonlinear and nonlocal flux conditions. Journal of Mathematical Analysis and Application, 2002,271: 182-205
[132] Lakshmikantham V and Carl S. Generalized quasilinearization and semilinear parabolic problems. Nonlinear Analysis, 2002,48(7): 947-960
[133] Lakshmikantham V and Vatsala A S. Generalized quasilinearization and semilinear elliptic boundary value problems. Journal of Mathemathical Analysis and Application, 2000, 249: 199-220
[134] Kasab J J, Karur S R and Ramachandran P A. Quasilinear boundary element method for nonlinear poisson type problems. Engineering Analysis with Boundary Elements, 1995,15: 277-282