边界积分方程的奇异性处理及其在断裂力学方面的应用
|
摘要
CAE分析技术在机械行业发挥了重要的作用。而CAE分析技术的主流数值方法—有限元法却存在一些固有缺陷,而这些缺陷刚好可以使用边界积分方程方法来弥补。在边界积分方程方法的数值实施中,近奇异积分和奇异积分是影响其计算精度的重要因素。因此,本文将重点关注边界积分方程方法中近奇异积分和奇异积分的解决方案及其在薄型结构和断裂力学方面的应用。另外,为了拓宽边界积分方程方法的工程应用,本文也提出了一系列近奇异体积分和奇异体积分的处理方案。以此为核心,本论文完成如下研究：
(1)提出了二维和三维问题的近奇异积分变换技术。和传统方法的近奇异积分求解方案不同,本文方法重点分析了近奇异积分的核心问题,即投影点位置和距离函数性质。本文利用泰勒展开得到距离函数,并根据投影点的位置和距离函数的性质将近奇异积分分为三类。从距离函数出发,利用降低被积函数梯度的思想,构造了三种对应的近奇异积分变换。与二维问题不同,在三维问题中,引入新型坐标系,构造出新坐标系下形式比较简单的变换。另外,由于投影点位置的不确定性,引入最近点,开发了一套基于投影点和最近点的近奇异积分子单元划分技术,用以保证积分子单元的良好形状,提高积分精度。数值算例充分证实了本文提出的方法可以成功地应用于薄型结构的求解。
(2)提出了全面而系统的奇异积分解决方案。本文直接从柯西主值和哈达玛有限部分积分定义出发,对弱奇异积分、强奇异积分、超奇异积分采取局部坐标近似展开,分析各类型奇异积分的性质以及相应的处理方法。另外根据主值积分和有限部分积分的区间对称性要求,开发一套用于解决三维奇异积分的自适应分块技术,有效地提高了奇异积分的精度。这些方案成功地应用于二维和三维断裂力学问题的求解。
(3)实现了二维和三维断裂力学问题的边界积分方程方法求解。针对二维断裂问题,引入裂纹张开位移,利用基本解的性质和裂纹受力平衡的边界条件,改进传统的双边界积分方程,使边界积分方程只用配置在非裂纹边界和裂纹的上表面上,从而减小矩阵规模和计算量。而对于三维问题,则进一步,只用在非裂纹边界和裂纹的上表面上配置面力边界积分方程,这样,既保留了和二维问题一样的优势,又便于奇异积分的模块化编程处理。开发了一种能够捕捉裂纹尖端位移性质的中节点奇异单元,结合裂纹尖端位移的渐近性质,建立了裂纹尖端应力强度因子和裂纹张开位移的线性插值公式。这些方案成功地应用于二维和三维的断裂力学问题求解,并得到了相当好的数值结果。
(4)开发了合理的近奇异体积分和奇异体积分技术。为了保证边界积分方程算法的通用性,针对三种常用单元开发了近奇异体积分和奇异体积分技术。对于近奇异体积分,提出以源点到单元的距离和单元尺寸的比例作为控制准则的自适应近奇异体积分方案。对于奇异体积分,首先将积分单元分为四面锥和金字塔子单元,然后对其分别做奇异积分变换技术。这些方案成功地解决了边界积分方程方法中遇到的近奇异体积分和奇异体积分,数值算例证实了本文方法的有效性。
CAE analysis technology plays an important role in the machinery industry. The mainstream numerical method of CAE analysis technology is finite element method (FEM). However, there are some inherent flaws with FEM and these flaws can be avoided by the boundary integral equation method (BIEM). In the numerical implementation of the BIEM, nearly singular integrals and singular integrals are important factors which affect the accuracy of BIEM. Thus this work focuses on the nearly singular integrals and singular integrals which arise in the BIEM and their applications in thin structures and fracture mechanics. Moreover, in order to broaden the application of BIEM, this work also includes a series of treatment for nearly singular and singular volume integrals. Treating these core contents, the work of this paper is as follows:
(1) This paper presents new transformations for nearly singular integrals in two and three dimensional BIEM. Compared with the traditional methods for nearly singular integrals, the proposed transformations focus on the core issues of nearly singular integrals in nature, namely, the location of the projection point and distance function. In this paper, employing the Taylor expansion, the distance function is obtained. Considering the location of the projection point and the distance function, the nearly singular integrals are divided into three categories. Using the distance function and the idea to lower the gradient of integrands, three different transformations for nearly singular integrals are constructed. And different from that in two dimensional BIEM, a new coordinate system is introduced and new transformation is simpler in the new system. Moreover, due to the uncertainty of the location of the projection point, another nearest point is introduced. An element subdivision technique based on the projection point and the nearest point is developed. With the help of element subdivision technique, subelements of fine shape are obtained, which can improve the integration accuracy. Numerical examples demonstrate that the proposed method can be successfully applied for problems on the thin structures.
(2) A comprehensive and systematic analysis of singular integrals in BIEM. Directly from the definition of the Cauchy principal value (C.P.V) and Hadamard finite part integrals (H.F.P), the local coordinate approximate expansion is employed to analyze the nature of weakly singular integrals, strong singular integrals, and hypersingular integrals. And the corresponding treatment is obtained. To meet the symmetry requirement of integration interval for C.P.V and H.F.P, an adaptive subdivision technique is developed to deal with singular integrals in three dimensional BIEM, which can efficiently improve the integration accuracy. These methods are successfully applied for two and three dimensional fracture mechanics.
(3) BIEM for two and three dimensional fracture mechanics. For two dimensional fracture mechanics, crack opening displacements (CODs) are introduced. Using the nature of the fundamental solutions and the traction equilibrium boundary conditions of crack faces, the improved dual boundary integral equations can be obtained. The dual boundary integral equations are collocated only on one of the crack faces and the external boundary, thus reducing the matrix size and more efficient. Furthermore, for three dimensional fracture mechanics, only the traction boundary integral equation is needed to collocate on one of the crack faces and the external boundary. Thus the advantages for two dimensional fracture mechanics are retained. Moreover, it is easy for code programming to deal with singular integrals. To capture the nature of the displacement of the crack tips, special crack-front elements are proposed. Combined with the asymptotic properties of the crack tip displacement, crack tip stress intensity factors are obtained by the linear interpolation formula of the crack opening displacements. The proposed method can be successfully applied for two and three dimensional fracture mechanics. Numerical results are in good agreement with existing analytical solutions or numerical results.
(4) Reasonable schemes for nearly singular and singular domain integrals. To ensure the universality of BIEM, nearly singular and singular domain integrals techniques are developed for the three common domain cells. An adaptive cell subdivision which is based on the proportion between the distance from the source point to the domain cell and the size of domain cell for nearly singular domain integrals is proposed. For singular integrals, the domain cell is divided into several cone and pyramid subcells. Singular integrals on these subcells are regularized by transformations. The proposed method is successfully applied for the domain integrals. Numerical examples demonstrate the effectiveness of our method.
(1)提出了二维和三维问题的近奇异积分变换技术。和传统方法的近奇异积分求解方案不同,本文方法重点分析了近奇异积分的核心问题,即投影点位置和距离函数性质。本文利用泰勒展开得到距离函数,并根据投影点的位置和距离函数的性质将近奇异积分分为三类。从距离函数出发,利用降低被积函数梯度的思想,构造了三种对应的近奇异积分变换。与二维问题不同,在三维问题中,引入新型坐标系,构造出新坐标系下形式比较简单的变换。另外,由于投影点位置的不确定性,引入最近点,开发了一套基于投影点和最近点的近奇异积分子单元划分技术,用以保证积分子单元的良好形状,提高积分精度。数值算例充分证实了本文提出的方法可以成功地应用于薄型结构的求解。
(2)提出了全面而系统的奇异积分解决方案。本文直接从柯西主值和哈达玛有限部分积分定义出发,对弱奇异积分、强奇异积分、超奇异积分采取局部坐标近似展开,分析各类型奇异积分的性质以及相应的处理方法。另外根据主值积分和有限部分积分的区间对称性要求,开发一套用于解决三维奇异积分的自适应分块技术,有效地提高了奇异积分的精度。这些方案成功地应用于二维和三维断裂力学问题的求解。
(3)实现了二维和三维断裂力学问题的边界积分方程方法求解。针对二维断裂问题,引入裂纹张开位移,利用基本解的性质和裂纹受力平衡的边界条件,改进传统的双边界积分方程,使边界积分方程只用配置在非裂纹边界和裂纹的上表面上,从而减小矩阵规模和计算量。而对于三维问题,则进一步,只用在非裂纹边界和裂纹的上表面上配置面力边界积分方程,这样,既保留了和二维问题一样的优势,又便于奇异积分的模块化编程处理。开发了一种能够捕捉裂纹尖端位移性质的中节点奇异单元,结合裂纹尖端位移的渐近性质,建立了裂纹尖端应力强度因子和裂纹张开位移的线性插值公式。这些方案成功地应用于二维和三维的断裂力学问题求解,并得到了相当好的数值结果。
(4)开发了合理的近奇异体积分和奇异体积分技术。为了保证边界积分方程算法的通用性,针对三种常用单元开发了近奇异体积分和奇异体积分技术。对于近奇异体积分,提出以源点到单元的距离和单元尺寸的比例作为控制准则的自适应近奇异体积分方案。对于奇异体积分,首先将积分单元分为四面锥和金字塔子单元,然后对其分别做奇异积分变换技术。这些方案成功地解决了边界积分方程方法中遇到的近奇异体积分和奇异体积分,数值算例证实了本文方法的有效性。
CAE analysis technology plays an important role in the machinery industry. The mainstream numerical method of CAE analysis technology is finite element method (FEM). However, there are some inherent flaws with FEM and these flaws can be avoided by the boundary integral equation method (BIEM). In the numerical implementation of the BIEM, nearly singular integrals and singular integrals are important factors which affect the accuracy of BIEM. Thus this work focuses on the nearly singular integrals and singular integrals which arise in the BIEM and their applications in thin structures and fracture mechanics. Moreover, in order to broaden the application of BIEM, this work also includes a series of treatment for nearly singular and singular volume integrals. Treating these core contents, the work of this paper is as follows:
(1) This paper presents new transformations for nearly singular integrals in two and three dimensional BIEM. Compared with the traditional methods for nearly singular integrals, the proposed transformations focus on the core issues of nearly singular integrals in nature, namely, the location of the projection point and distance function. In this paper, employing the Taylor expansion, the distance function is obtained. Considering the location of the projection point and the distance function, the nearly singular integrals are divided into three categories. Using the distance function and the idea to lower the gradient of integrands, three different transformations for nearly singular integrals are constructed. And different from that in two dimensional BIEM, a new coordinate system is introduced and new transformation is simpler in the new system. Moreover, due to the uncertainty of the location of the projection point, another nearest point is introduced. An element subdivision technique based on the projection point and the nearest point is developed. With the help of element subdivision technique, subelements of fine shape are obtained, which can improve the integration accuracy. Numerical examples demonstrate that the proposed method can be successfully applied for problems on the thin structures.
(2) A comprehensive and systematic analysis of singular integrals in BIEM. Directly from the definition of the Cauchy principal value (C.P.V) and Hadamard finite part integrals (H.F.P), the local coordinate approximate expansion is employed to analyze the nature of weakly singular integrals, strong singular integrals, and hypersingular integrals. And the corresponding treatment is obtained. To meet the symmetry requirement of integration interval for C.P.V and H.F.P, an adaptive subdivision technique is developed to deal with singular integrals in three dimensional BIEM, which can efficiently improve the integration accuracy. These methods are successfully applied for two and three dimensional fracture mechanics.
(3) BIEM for two and three dimensional fracture mechanics. For two dimensional fracture mechanics, crack opening displacements (CODs) are introduced. Using the nature of the fundamental solutions and the traction equilibrium boundary conditions of crack faces, the improved dual boundary integral equations can be obtained. The dual boundary integral equations are collocated only on one of the crack faces and the external boundary, thus reducing the matrix size and more efficient. Furthermore, for three dimensional fracture mechanics, only the traction boundary integral equation is needed to collocate on one of the crack faces and the external boundary. Thus the advantages for two dimensional fracture mechanics are retained. Moreover, it is easy for code programming to deal with singular integrals. To capture the nature of the displacement of the crack tips, special crack-front elements are proposed. Combined with the asymptotic properties of the crack tip displacement, crack tip stress intensity factors are obtained by the linear interpolation formula of the crack opening displacements. The proposed method can be successfully applied for two and three dimensional fracture mechanics. Numerical results are in good agreement with existing analytical solutions or numerical results.
(4) Reasonable schemes for nearly singular and singular domain integrals. To ensure the universality of BIEM, nearly singular and singular domain integrals techniques are developed for the three common domain cells. An adaptive cell subdivision which is based on the proportion between the distance from the source point to the domain cell and the size of domain cell for nearly singular domain integrals is proposed. For singular integrals, the domain cell is divided into several cone and pyramid subcells. Singular integrals on these subcells are regularized by transformations. The proposed method is successfully applied for the domain integrals. Numerical examples demonstrate the effectiveness of our method.
引文
[1]崔俊芝.计算机辅助工程(CAE)的现在和未来.计算机辅助设计与制造,2000,6(3)：3-7
[2]张洪武,顾元宪,关振群,等.用于有限元分析与优化设计的JIFEX软件.计算机集成制造系统,2003 9：160-166.
[3]钟万勰,陆仲绩.CAE：事关国家竞争力和国家安全的战略技术—关于发展我国CAE软件产业的思考.中国科学院季刊,2007,22(2)：115-119
[4]纪爱敏.机械CAE分析原理及工程实践.北京：机械工业出版社,2009
[5]崔俊芝.CAE——推动工程和产品创新的生产力.科技创新和品牌,2010,3：18-20
[6]王勖成.有限单元法.北京：清华大学出版社,2003
[7]曾攀.有限元分析及应用.北京：清华大学出版社,2004
[8]杜庆华.边界积分方程方法.北京：高等教育出版社,1989
[9]Brebbia C A等著,龙述尧等译.边界单元理论和工程应用.长沙：国防业出版社,1988
[10]龙述尧编著.边界单元法概论.中国科学文化出版社,2002
[11]姚振汉.边界元法.北京：高等教育出版社,2010
[12]茹忠亮,朱传锐,张友良,等.断裂问题的扩展有限元法研究.岩土力学,2011,32(7)：2171-2176
[13]吴圣川,吴玉程.断裂分析及CAE软件的现状与发展.计算机辅助工程,2011,20(1)：1-2
[14]姚振汉,杜庆华.边界元法应用的若干近期研究及国际新进展.清华大学学报：自然科学版,2009,5：89-93
[15]姚振汉.三十年边界元法研究的心得.力学与工程应用(第十三卷),2010
[16]Zhang J M, Qin X Y, Han X, Li G Y. A boundary face method for potential problems in three dimensions. International Journal for Numerical Methods in Engineering,2009,80:320-337
[17]张见明.基于边界面法的完整实体应力分析理论与应用.计算机辅助工程,2010,19(3)：5-10
[18]Qin X Y, Zhang J M, Li G Y, et al. An element implementation of the boundary face method for 3D potential problems. Engineering Analysis with Boundary Elements,2010,34:934-943
[19]Greengard L, Rokhlin V. A new version of the fast multipole method for the Laplace equation in three dimensions. Acta Numerica,1997,6:229-269
[20]姚振汉,王海涛,王朋波,等.固体力学中快速多极边界元法研究进展.中国科学技术大学学报,2008,38(1)：1-17
[21]Wang H T, Lei T, Li J, et al. A parallel fast multipole accelerated integral equation scheme for 3D Stokes equations. International Journal for Numerical Methods in Engineering,2007,70:812-839
[22]Liu Y J. Fast mutipole boundary element method theory and application in engineering. Cambridge:Cambridge University Press,2009
[23]Zhang J M, Tanaka Masa, Endo M. The hybrid boundary node method accelerated by fast multipole method for 3D potential problems. International Journal for Numerical Methods in Engineering,2005,63:660-680
[24]Zhang J M, Tanaka M. Fast HdBNM for large-scale thermal analysis of CNT-reinforced composites. Computational Mechanics,2008,41:777-787
[25]Zhang J M, Tanaka M. Adaptive spatial decomposition in fast multipole method. Journal of Computational Physics,2007,226:17-28
[26]Zhang J M, Tanaka M. Systematic study of thermal properties of CNT composites by a fast multipole hybrid boundary node method. Engineering Analysis with Boundary Elements,2007,31:388-401
[27]Zhang J M, Tanaka M. An effective tree data structure in Fast Multipole Method. Transactions of the Japan Society for Computational Methods in Engineering,2006,6:17-22
[28]Xiao J Y, Tausch J. A fast wavelet-multipole method for direct BEM. Engineering Analysis with Boundary Elements,2010,34(7):673-679.
[29]Hackbusch W. A sparse matrix arithmetic based on H-matrices. I. Introduction to H-matrices. Computing,1999,62:89-108
[30]Bebendorf M. Hierarchical matrices. Berlin Heidelberg:Springer-Verlag Press, 2008
[31]Ostrowskil J, Andjelic Z, Bebendof M, et al. Fast BEM-solution of Laplace problems with H-Matrices and ACA. IEEE Transactions on Magnetics,2006, 42(2):627-630
[32]Smajic J, Andjelic Z, Bebendof M. Fast BEM for eddy-current problems using H-Matrices and Adaptive Cross Approximation. IEEE Transactions on Magnetics,2007,43(4):1269-1272
[33]Zhang J M., Zheng X S, Lu C J, et al. A Geometric Mapping Cross Approximation method. Engineering Analysis with Boundary Elements,2013, 37(12):1668-1673
[34]王静,高效伟.基于单元子分法的结构多尺度边界单元法.计算力学学报,2010,27(2)：258-263
[35]Gao X W, Davies T G. Adaptive integration in elasto-plastic boundary element analysis. Journal of the Chinese Institute of Engineers,2000,23:349-356
[36]霍同如,姚振汉.基于边界元法的弹性结构边界点和近边界点力学量的计算.数值计算与计算机应用,1993,14(1)：38-47
[37]申光宪,肖宏.物体边界层域内位移及应力的边界元法新公式.固体力学学报,1989,10(3)：279-284
[38]陈海波,金建峰,张培强,等Multi-variable non-singular BEM for 2-D potential problems清华大学学报(自然科学英文版),2005,10：43-50
[39]王有成,李洪求,陈海波,等.奇性校正特解场法计算任意点应力和位移.力学学报,1994,26(2)：222-232
[40]Chen, H B, Lu P, Huang M G, et al. An effective method for finding values on and near boundaries in the elastic BEM. Computers and Structures,1998,69(4), 421-431
[41]Niu Z R, Wendland W L, Wang X X, et al. A sim-analytic algorithm for the evaluation of the nearly singular integrals in three-dimensional boundary element methods. Computer methods on applied mechanics and engineering, 2005,31:949-64
[42]周焕林,牛忠荣,王秀喜.三维位势问题边界元法中几乎奇异积分的正则化.计算物理,2006,22(6)：501-506.
[43]牛忠荣,胡宗军,葛仁余.二维边界元法高阶元几乎奇异积分半解析算法.力学学报,2013,45(6)：897-907
[44]Zhou H L, Niu Z R, Cheng C, et al. Analytical integral algorithm in the BEM for orthotropic potential problems of thin bodies. Engineering Analysis with Boundary Elements,2007,31(9):739-748
[45]Zhou H L, Niu Z R, Cheng C Z. Analytical integral algorithm applied to boundary layer effect and thin body effect in BEM for anisotropic potential problems. Computers and Structures,2008,86:1656-1671
[46]Niu Z R, Zhou H L. The natural boundary integral equation in potential problems and regularization of the hypersingular integral. Computers and Structures,2004,82(2):315-323
[47]周焕林.边界元法中边界层效应和薄体问题的研究：[中国科技大学博士学位 论文].合肥：中国科技大学力学与机械工程系,2003
[48]Jun L, Beer G, Meek J L. Efficient evaluation of integrals of order 1/r,1/r 2,1/r 3 using Gauss quadrature. Engineering Analysis with Boundary Elements,1985, 2(3):118-123
[49]Telles J C F. A self-adaptive co-ordinates transformation for efficient numerical evaluation of general boundary element integrals. International Journal for Numerical Methods in Engineering,1987,24:959-973
[50]Johnston B M, Johnston P R, Elliott D. A sinh transformation for evaluating two-dimensional nearly singular boundary element integrals. International Journal for Numerical Methods in Engineering,2007,69:1460-1479
[51]Gu Y, Chen W, Zhang C. Stress analysis for thin multilayered coating systems using a sinh transformed boundary element method. International journal of solids and structures,2013,50(20-21):3460-3471
[52]Hayami K. Variable transformations for nearly singular integrals in the boundary element method. Publications of the Research Institute for Mathematical Sciences,2005,41:821-842
[53]Zhang Y M, Gu Y, Chen J T. Boundary layer effect in BEM with high order geometry elements using transformation. Computer Modeling in Engineering and Sciences,2009,45(3):227-247.
[54]Zhang Y M, Gu Y, Chen J T. Boundary element analysis of the thermal behaviour in thin-coated cutting tools. Engineering analysis with boundary elements,2010,34:775-784.
[55]张耀明,孙翠莲,谷岩.边界积分方程中近奇异积分计算的一种变量替换法.力学学报,2008,40(2)：207-214
[56]张耀明,谷岩,陈正宗.位势边界元法中的边界层效应与薄体结构.力学学报,2010,42(2)：220-226
[57]Xie G Z, Zhang J M, Qin X Y, et al. New variable transformations for evaluating nearly singular integrals in 2D boundary element method, Engineering Analysis with Boundary Elements,2011,35:811-817
[58]Xie G Z, Zhou F L, Zhang J M, et al. New variable transformations for evaluating nearly singular integrals in 3D boundary element method. Engineering Analysis with Boundary Elements,2013,37(9):1169-1178
[59]Ma H, Kamiya N. A general algorithm for accurate computation of field variables and its derivatives near boundary in BEM. Engineering Analysis with Boundary Elements,2001,25:833-841
[60]Ma H, Kamiya N. A general algorithm for the numerical evaluation of nearly singular boundary integrals of various orders for two-and three-dimensional elasticity. Computational mechanics,2002,29:277-288
[61]Ma H, Kamiya N. Distance transformation for the numerical evaluation of near singular boundary integrals with various kernels in boundary element method. Engineering Analysis with Boundary Elements,2002,26:329-339
[62]Ma H, Kamiya N. Nearly singular approximations of CPV integrals with end and corner-singularities for the numerical solution of hypersingular boundary integral equations. Engineering Analysis with Boundary Elements,2003,27: 625-637
[63]Qin X Y, Zhang J M, Xie G Z, et al. A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element. Journal of Computational and Applied Mathematics,2011,235(14):4174-4186
[64]Liu Y J, Fan H. Analysis of the thin piezoelectric solids by the boundary element method. Computer methods on applied mechanics and engineering, 2002,191:2297-2315
[65]Liu Y J. Analysis of shell-like structures by the boundary element method based on 3-D elasticity:formulation and verification. International Journal for Numerical methods in Engineering,1998,41:541-558
[66]Mi Y, Aliabadi M H. A Taylor expansion algorithm for integration of 3D near-hypersingular integrals. Communications in numerical methods in engineering,1996,12(1):51-62
[67]Sladek V, Sladek J, Tanaka M. Regularization of hypersingular and nearly singular integrals in the potential theory and elasticity. International Journal for Numerical methods in Engineering,1993,36(10):1609-1628
[68]Kutt H R. The numerical evaluation of principal value integrals by finite-part integration. Numerische Mathematik,1975,24(3):205-210
[69]Kutt H R. On the numerical evaluation of finite-part integrals involving an algebraic singularity. Stellenbosch:Stellenbosch University,1975
[70]Pan E N, Yuan F G. Boundary element analysis of three-dimensional cracks in anisotropic solids. International Journal for Numerical Methods in Engineering, 2000,48(2):211-237
[71]Lachat J C, Watson J O. Effective numerical treatment of boundary integral equations:A formulation for three-dimensional elastostatics. International Journal for Numerical Methods in Engineering,1976,10(5):991-1005
[72]Liu Y J. On the simple-solution method and non-singular nature of the BIE/BEM-a review and some new results. Engineering Analysis with Boundary Elements,2000,24(10):789-795
[73]Liu Y J, Rudolphi T J. Some identities for fundamental solutions and their applications to weakly-singular boundary element formulations. Engineering Analysis with Boundary Elements,1991,8(6):301-311
[74]Dominguez J, Ariza M P. A direct traction BIE approach for three-dimensional crack problems. Engineering analysis with boundary elements,2000,24(10): 727-738
[75]Gao X W, Feng W Z, Zhao J. A direct method for evaluating arbitrary high-order singular curved boundary integrals. Boundary Elements and Other Mesh Reduction Methods XXXVI,2013,56:335
[76]Gao X W. An effective method for numerical evaluation of general 2D and 3D high order singular boundary integrals. Computer methods in applied mechanics and engineering,2010,199(45):2856-2864
[77]董春迎,谢志成,姚振汉.边界积分方程中超奇异积分的解法.力学进展,1990,25(3)：424-429
[78]董春迎.弹塑性边界元法的若干基础性研究及在接触问题上的应用：[清华大学博士学位论文].北京：清华大学工程力学系,1992
[79]汤任基,秦太验.三维断裂力学的超奇异积分方程方法.力学学报,1993,25(6)：665-675
[80]Qin T Y., Noda N.A. Application of hypersingular integral equation method to a three-dimensional crack in piezoelectric materials. JSME international journal. Series A, Solid mechanics and material engineering,2004,47(2):173-180
[81]陈梦成.三维断裂力学问题求解超奇异积分方程方法.四川：西南交通大学出版社,2007
[82]Guiggiani M, Gigante N. A general algorithm for multi-dimensional Cauchy principal value integrals in the boundary element method. ASME Journal of applied Mechanics,1990,57:906-915
[83]Guiggiani M. Formulation and numerical treatment of boundary integral equations with hypersingular kernels. Singular integrals in boundary element methods,1998:85-124
[84]Aliabadi M H. Boundary element formulations in fracture mechanics. Applied Mechanics Reviews,1997,50(2):83-96
[85]Mi Y M, Aliabadi M H. Dual boundary element method for three-dimensional fracture mechanics analysis. Engineering Analysis with Boundary Elements, 1992,10(2):161-171
[86]Wang Q, Noda N A, Honda M A. Variation of stress intensity factor along the front of a 3D rectangular crack by using a singular integral equation method. International journal of fracture,2001,108(2):119-131
[87]Murakami Y. Analysis of stress intensity factors of modes Ⅰ, Ⅱ and Ⅲ for inclined surface cracks of arbitrary shape. Engineering Fracture Mechanics, 1985,22(1):101-114
[88]Guidera J T, Lardner R W. Penny-shaped cracks. Journal of Elasticity,1975, 5(1):59-73
[89]Cruse T A. Boundary element analysis in computational fracture mechanics. Berlin:Springer,1988
[90]Weaver J. Three-dimensional crack analysis. International Journal of Solids and Structures,1977,13(4):321-330
[91]Balas J, Sladek J. Method of boundary integral equations for analysis of three dimensional crack problems. In:Brebbia CA ed. Boundary Element Methods. Berlin:Springer Berlin Heidelberg,1981,183-205
[92]Ioakimidis N I. Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity. Acta mechanica,1982,45(1-2):31-47
[93]赵明皞,刘元杰.三维裂纹问题的第二类边界积分方程.机械强度,1992,14(1)：17-21
[94]秦太验.三维断裂力学的有限部积分边界元法研究：[上海交通大学博士学位论文].上海：上海交通大学工程力学系,1993
[95]乐金朝,冯新,韩连元.双材料平面裂纹问题的超奇异积分方程方法.固体力学学报,1999,20：34-37
[96]Cruse T A, Vanburen W. Three-dimensional elastic stress analysis of a fracture specimen with an edge crack. International Journal of Fracture Mechanics,1971, 7(1):1-15
[97]Luchi M L, Poggialini A. Computation of 3-dimensional stress intensity factors using special boundary element. In:Proc 5th Int Conf BEM. Berlin,1983: 461-470
[98]Luchi M L, Rizzuti S. Boundary elements for three-dimensional elastic crack analysis. International journal for numerical methods in engineering,1987, 24(12):2253-2271
[99]Chen J T, Hong H K. Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series. Applied Mechanics Reviews, 1999,52(1):17-33
[100]Chen J T, Hong H K, Chyuan S W. Boundary element analysis and design in seepage problems using dual integral formulation. Finite Elements in Analysis and Design,1994,17(1):1-20
[101]Hong H K, Chen J T. Derivations of integral equations of elasticity. Journal of Engineering Mechanics,1988,114(6):1028-1044
[102]Portela A, Aliabadi MH, Rooke DP. Dual boundary element incremental analysis of crack propagation. Computers and Structures,1993,46(2):237-247
[103]Portela A, Aliabadi MH, Rooke DP. The dual boundary element method: effective implementation for crack problems. International Journal for Numerical Methods in Engineering,1992,33(6):1269-1287
[104]Mi Y, Aliabadi M H. Discontinuous crack-tip elements:application to 3D boundary element method. International journal of fracture,1994,67(3): R67-R71
[105]王朋波,姚振汉.快速多极对偶边界元法在大规模裂纹分析中的应用.见：北京力学会第11届学术年会论文摘要集.北京：北京力学会,2005,116-117
[106]孙玉周,王银邦.反平面裂纹问题的边界元解法.兰州大学学报：自然科学版,2001,37(4)：25-30
[107]Sun Y Z, Yang S S, Wang Y B. A new formulation of boundary element method for cracked anisotropic bodies under anti-plane shear. Computer methods in applied mechanics and engineering,2003,192(22):2633-2648
[108]Wang Y B, Sun Y Z. A new boundary integral equation method for cracked 2-D anisotropic bodies. Engineering fracture mechanics,2005,72(13): 2128-2143
[109]Pan E N. A general boundary element analysis of 2-D linear elastic fracture mechanics. International Journal of Fracture,1997,88(1):41-59
[110]Yue Z Q, Xiao H T, Tham L G. Boundary element analysis of crack problems in functionally graded materials. International journal of solids and structures,2003,40(13):3273-3291
[111]肖洪天.梯度材料断裂力学的新型边界元法分析.北京：高等教育出版社,2007
[112]方志民,柴国钟.三维线弹性断裂分析的混合边界元法.浙江工业大学学报,1994,3：8-17
[113]柴国钟.线弹性断裂力学的混合边界元法：[华东理工大学博士学位论文].上海：华东理工大学化工过程机械系,1993
[114]Nowak A J, Neves A C. The multiple reciprocity boundary element method. Southampton and Boston:Computational Mechanics Publications,1994
[115]Gao X W. The radial integration method for evaluation of domain integrals with boundary-only discretization. Engineering Analysis with Boundary Elements,2002,26:905-916
[116]Yang K, Gao X W. Radial integration BEM for transient heat conduction problems. Engineering analysis with boundary elements,2010,34(6):557-563
[117]Yang K, Gao X W, Liu Y F. Using analytical expressions in radial integration BEM for variable coefficient heat conduction problems. Engineering Analysis with Boundary Elements,2011,35(10):1085-1089
[118]Partridge P W, Brebbia C A, Wrobel L C. The dual reciprocity boundary element method. Southampton and Springer-Verlag:Computational Mechanics Publications,1992
[119]Scuderi, Letizia. On the computation of nearly singular integrals in 3D BEM collocation. International journal for numerical methods in engineering,2008, 74(11):1733-1770
[120]Hayami K, Brebbia CA. A new coordinate transformation method for singular and nearly singular integrals over general curved boundary elements. In:Proc of 9th Int Con BEM. Berlin,1987,375-399
[121]Kane J H. Boundary element analysis in engineering continuum mechanics. Englewood Cliffs:Prentice-Hall Inc Press,1994
[122]Hughes T J, Cottrell J A, Bazilevs Y. Isogeometric analysis:CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer methods in applied mechanics and engineering,2005,194(39):4135-4195
[123]Hadamard J. Lectures on Cauchy's problem in linear partial differential equations. Courier Dover Publications,2003
[124]黄玲珍,聂毓琴.计算断裂力学研究的现状与进展.吉林工业大学学报,1995,25(1)：120-124
[125]范天佑.断裂力学基础.南京：江苏科学技术出版社,1978
[126]Murakami Y. Stress intensity factors handbook. Oxford:Pergarnon Press, 1987
[127]Gupta G D, Erdogan F. The problem of edge cracks in an infinite strip. Journal of Applied Mechanics,1974,41(4):1001-1006
[128]黎在良,王乘.高等边界元法.北京：科学出版社,2008
[129]Wen P H, Aliabadi M H, Rooke D P. Mixed-mode weight functions in three-dimensional fracture mechanics static. Engineering fracture mechanics, 1998,59(5):563-575
[130]Tada H, Paris P C, Irwin G R. The Analysis of Cracks Handbook. New York: ASME Press,2000
[131]Raju I S, Newman J C. Three dimensional finite-element analysis of finite-thickness fracture specimens. New York:National Aeronautics and Space Administration,1977
[132]Raju I S, Newman J C. Stress-intensity factors for a wide range of semi-elliptical surface cracks in finite-thickness plates. Engineering Fracture Mechanics,1979,11(4):817-829
[133]李亚莎.高精度快速边界元法及其在绝缘子电场计算中应用研究：[华北电力大学博士学位论文].北京：华北电力大学电气与电子工程学院,2007
[134]Zhang L P, Cui T, Liu H. A set of symmetric quadrarure rules on triangles and tetrahedra. Journal of Computational Mathematics,2009,27(1):89-96
[2]张洪武,顾元宪,关振群,等.用于有限元分析与优化设计的JIFEX软件.计算机集成制造系统,2003 9：160-166.
[3]钟万勰,陆仲绩.CAE：事关国家竞争力和国家安全的战略技术—关于发展我国CAE软件产业的思考.中国科学院季刊,2007,22(2)：115-119
[4]纪爱敏.机械CAE分析原理及工程实践.北京：机械工业出版社,2009
[5]崔俊芝.CAE——推动工程和产品创新的生产力.科技创新和品牌,2010,3：18-20
[6]王勖成.有限单元法.北京：清华大学出版社,2003
[7]曾攀.有限元分析及应用.北京：清华大学出版社,2004
[8]杜庆华.边界积分方程方法.北京：高等教育出版社,1989
[9]Brebbia C A等著,龙述尧等译.边界单元理论和工程应用.长沙：国防业出版社,1988
[10]龙述尧编著.边界单元法概论.中国科学文化出版社,2002
[11]姚振汉.边界元法.北京：高等教育出版社,2010
[12]茹忠亮,朱传锐,张友良,等.断裂问题的扩展有限元法研究.岩土力学,2011,32(7)：2171-2176
[13]吴圣川,吴玉程.断裂分析及CAE软件的现状与发展.计算机辅助工程,2011,20(1)：1-2
[14]姚振汉,杜庆华.边界元法应用的若干近期研究及国际新进展.清华大学学报：自然科学版,2009,5：89-93
[15]姚振汉.三十年边界元法研究的心得.力学与工程应用(第十三卷),2010
[16]Zhang J M, Qin X Y, Han X, Li G Y. A boundary face method for potential problems in three dimensions. International Journal for Numerical Methods in Engineering,2009,80:320-337
[17]张见明.基于边界面法的完整实体应力分析理论与应用.计算机辅助工程,2010,19(3)：5-10
[18]Qin X Y, Zhang J M, Li G Y, et al. An element implementation of the boundary face method for 3D potential problems. Engineering Analysis with Boundary Elements,2010,34:934-943
[19]Greengard L, Rokhlin V. A new version of the fast multipole method for the Laplace equation in three dimensions. Acta Numerica,1997,6:229-269
[20]姚振汉,王海涛,王朋波,等.固体力学中快速多极边界元法研究进展.中国科学技术大学学报,2008,38(1)：1-17
[21]Wang H T, Lei T, Li J, et al. A parallel fast multipole accelerated integral equation scheme for 3D Stokes equations. International Journal for Numerical Methods in Engineering,2007,70:812-839
[22]Liu Y J. Fast mutipole boundary element method theory and application in engineering. Cambridge:Cambridge University Press,2009
[23]Zhang J M, Tanaka Masa, Endo M. The hybrid boundary node method accelerated by fast multipole method for 3D potential problems. International Journal for Numerical Methods in Engineering,2005,63:660-680
[24]Zhang J M, Tanaka M. Fast HdBNM for large-scale thermal analysis of CNT-reinforced composites. Computational Mechanics,2008,41:777-787
[25]Zhang J M, Tanaka M. Adaptive spatial decomposition in fast multipole method. Journal of Computational Physics,2007,226:17-28
[26]Zhang J M, Tanaka M. Systematic study of thermal properties of CNT composites by a fast multipole hybrid boundary node method. Engineering Analysis with Boundary Elements,2007,31:388-401
[27]Zhang J M, Tanaka M. An effective tree data structure in Fast Multipole Method. Transactions of the Japan Society for Computational Methods in Engineering,2006,6:17-22
[28]Xiao J Y, Tausch J. A fast wavelet-multipole method for direct BEM. Engineering Analysis with Boundary Elements,2010,34(7):673-679.
[29]Hackbusch W. A sparse matrix arithmetic based on H-matrices. I. Introduction to H-matrices. Computing,1999,62:89-108
[30]Bebendorf M. Hierarchical matrices. Berlin Heidelberg:Springer-Verlag Press, 2008
[31]Ostrowskil J, Andjelic Z, Bebendof M, et al. Fast BEM-solution of Laplace problems with H-Matrices and ACA. IEEE Transactions on Magnetics,2006, 42(2):627-630
[32]Smajic J, Andjelic Z, Bebendof M. Fast BEM for eddy-current problems using H-Matrices and Adaptive Cross Approximation. IEEE Transactions on Magnetics,2007,43(4):1269-1272
[33]Zhang J M., Zheng X S, Lu C J, et al. A Geometric Mapping Cross Approximation method. Engineering Analysis with Boundary Elements,2013, 37(12):1668-1673
[34]王静,高效伟.基于单元子分法的结构多尺度边界单元法.计算力学学报,2010,27(2)：258-263
[35]Gao X W, Davies T G. Adaptive integration in elasto-plastic boundary element analysis. Journal of the Chinese Institute of Engineers,2000,23:349-356
[36]霍同如,姚振汉.基于边界元法的弹性结构边界点和近边界点力学量的计算.数值计算与计算机应用,1993,14(1)：38-47
[37]申光宪,肖宏.物体边界层域内位移及应力的边界元法新公式.固体力学学报,1989,10(3)：279-284
[38]陈海波,金建峰,张培强,等Multi-variable non-singular BEM for 2-D potential problems清华大学学报(自然科学英文版),2005,10：43-50
[39]王有成,李洪求,陈海波,等.奇性校正特解场法计算任意点应力和位移.力学学报,1994,26(2)：222-232
[40]Chen, H B, Lu P, Huang M G, et al. An effective method for finding values on and near boundaries in the elastic BEM. Computers and Structures,1998,69(4), 421-431
[41]Niu Z R, Wendland W L, Wang X X, et al. A sim-analytic algorithm for the evaluation of the nearly singular integrals in three-dimensional boundary element methods. Computer methods on applied mechanics and engineering, 2005,31:949-64
[42]周焕林,牛忠荣,王秀喜.三维位势问题边界元法中几乎奇异积分的正则化.计算物理,2006,22(6)：501-506.
[43]牛忠荣,胡宗军,葛仁余.二维边界元法高阶元几乎奇异积分半解析算法.力学学报,2013,45(6)：897-907
[44]Zhou H L, Niu Z R, Cheng C, et al. Analytical integral algorithm in the BEM for orthotropic potential problems of thin bodies. Engineering Analysis with Boundary Elements,2007,31(9):739-748
[45]Zhou H L, Niu Z R, Cheng C Z. Analytical integral algorithm applied to boundary layer effect and thin body effect in BEM for anisotropic potential problems. Computers and Structures,2008,86:1656-1671
[46]Niu Z R, Zhou H L. The natural boundary integral equation in potential problems and regularization of the hypersingular integral. Computers and Structures,2004,82(2):315-323
[47]周焕林.边界元法中边界层效应和薄体问题的研究：[中国科技大学博士学位 论文].合肥：中国科技大学力学与机械工程系,2003
[48]Jun L, Beer G, Meek J L. Efficient evaluation of integrals of order 1/r,1/r 2,1/r 3 using Gauss quadrature. Engineering Analysis with Boundary Elements,1985, 2(3):118-123
[49]Telles J C F. A self-adaptive co-ordinates transformation for efficient numerical evaluation of general boundary element integrals. International Journal for Numerical Methods in Engineering,1987,24:959-973
[50]Johnston B M, Johnston P R, Elliott D. A sinh transformation for evaluating two-dimensional nearly singular boundary element integrals. International Journal for Numerical Methods in Engineering,2007,69:1460-1479
[51]Gu Y, Chen W, Zhang C. Stress analysis for thin multilayered coating systems using a sinh transformed boundary element method. International journal of solids and structures,2013,50(20-21):3460-3471
[52]Hayami K. Variable transformations for nearly singular integrals in the boundary element method. Publications of the Research Institute for Mathematical Sciences,2005,41:821-842
[53]Zhang Y M, Gu Y, Chen J T. Boundary layer effect in BEM with high order geometry elements using transformation. Computer Modeling in Engineering and Sciences,2009,45(3):227-247.
[54]Zhang Y M, Gu Y, Chen J T. Boundary element analysis of the thermal behaviour in thin-coated cutting tools. Engineering analysis with boundary elements,2010,34:775-784.
[55]张耀明,孙翠莲,谷岩.边界积分方程中近奇异积分计算的一种变量替换法.力学学报,2008,40(2)：207-214
[56]张耀明,谷岩,陈正宗.位势边界元法中的边界层效应与薄体结构.力学学报,2010,42(2)：220-226
[57]Xie G Z, Zhang J M, Qin X Y, et al. New variable transformations for evaluating nearly singular integrals in 2D boundary element method, Engineering Analysis with Boundary Elements,2011,35:811-817
[58]Xie G Z, Zhou F L, Zhang J M, et al. New variable transformations for evaluating nearly singular integrals in 3D boundary element method. Engineering Analysis with Boundary Elements,2013,37(9):1169-1178
[59]Ma H, Kamiya N. A general algorithm for accurate computation of field variables and its derivatives near boundary in BEM. Engineering Analysis with Boundary Elements,2001,25:833-841
[60]Ma H, Kamiya N. A general algorithm for the numerical evaluation of nearly singular boundary integrals of various orders for two-and three-dimensional elasticity. Computational mechanics,2002,29:277-288
[61]Ma H, Kamiya N. Distance transformation for the numerical evaluation of near singular boundary integrals with various kernels in boundary element method. Engineering Analysis with Boundary Elements,2002,26:329-339
[62]Ma H, Kamiya N. Nearly singular approximations of CPV integrals with end and corner-singularities for the numerical solution of hypersingular boundary integral equations. Engineering Analysis with Boundary Elements,2003,27: 625-637
[63]Qin X Y, Zhang J M, Xie G Z, et al. A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element. Journal of Computational and Applied Mathematics,2011,235(14):4174-4186
[64]Liu Y J, Fan H. Analysis of the thin piezoelectric solids by the boundary element method. Computer methods on applied mechanics and engineering, 2002,191:2297-2315
[65]Liu Y J. Analysis of shell-like structures by the boundary element method based on 3-D elasticity:formulation and verification. International Journal for Numerical methods in Engineering,1998,41:541-558
[66]Mi Y, Aliabadi M H. A Taylor expansion algorithm for integration of 3D near-hypersingular integrals. Communications in numerical methods in engineering,1996,12(1):51-62
[67]Sladek V, Sladek J, Tanaka M. Regularization of hypersingular and nearly singular integrals in the potential theory and elasticity. International Journal for Numerical methods in Engineering,1993,36(10):1609-1628
[68]Kutt H R. The numerical evaluation of principal value integrals by finite-part integration. Numerische Mathematik,1975,24(3):205-210
[69]Kutt H R. On the numerical evaluation of finite-part integrals involving an algebraic singularity. Stellenbosch:Stellenbosch University,1975
[70]Pan E N, Yuan F G. Boundary element analysis of three-dimensional cracks in anisotropic solids. International Journal for Numerical Methods in Engineering, 2000,48(2):211-237
[71]Lachat J C, Watson J O. Effective numerical treatment of boundary integral equations:A formulation for three-dimensional elastostatics. International Journal for Numerical Methods in Engineering,1976,10(5):991-1005
[72]Liu Y J. On the simple-solution method and non-singular nature of the BIE/BEM-a review and some new results. Engineering Analysis with Boundary Elements,2000,24(10):789-795
[73]Liu Y J, Rudolphi T J. Some identities for fundamental solutions and their applications to weakly-singular boundary element formulations. Engineering Analysis with Boundary Elements,1991,8(6):301-311
[74]Dominguez J, Ariza M P. A direct traction BIE approach for three-dimensional crack problems. Engineering analysis with boundary elements,2000,24(10): 727-738
[75]Gao X W, Feng W Z, Zhao J. A direct method for evaluating arbitrary high-order singular curved boundary integrals. Boundary Elements and Other Mesh Reduction Methods XXXVI,2013,56:335
[76]Gao X W. An effective method for numerical evaluation of general 2D and 3D high order singular boundary integrals. Computer methods in applied mechanics and engineering,2010,199(45):2856-2864
[77]董春迎,谢志成,姚振汉.边界积分方程中超奇异积分的解法.力学进展,1990,25(3)：424-429
[78]董春迎.弹塑性边界元法的若干基础性研究及在接触问题上的应用：[清华大学博士学位论文].北京：清华大学工程力学系,1992
[79]汤任基,秦太验.三维断裂力学的超奇异积分方程方法.力学学报,1993,25(6)：665-675
[80]Qin T Y., Noda N.A. Application of hypersingular integral equation method to a three-dimensional crack in piezoelectric materials. JSME international journal. Series A, Solid mechanics and material engineering,2004,47(2):173-180
[81]陈梦成.三维断裂力学问题求解超奇异积分方程方法.四川：西南交通大学出版社,2007
[82]Guiggiani M, Gigante N. A general algorithm for multi-dimensional Cauchy principal value integrals in the boundary element method. ASME Journal of applied Mechanics,1990,57:906-915
[83]Guiggiani M. Formulation and numerical treatment of boundary integral equations with hypersingular kernels. Singular integrals in boundary element methods,1998:85-124
[84]Aliabadi M H. Boundary element formulations in fracture mechanics. Applied Mechanics Reviews,1997,50(2):83-96
[85]Mi Y M, Aliabadi M H. Dual boundary element method for three-dimensional fracture mechanics analysis. Engineering Analysis with Boundary Elements, 1992,10(2):161-171
[86]Wang Q, Noda N A, Honda M A. Variation of stress intensity factor along the front of a 3D rectangular crack by using a singular integral equation method. International journal of fracture,2001,108(2):119-131
[87]Murakami Y. Analysis of stress intensity factors of modes Ⅰ, Ⅱ and Ⅲ for inclined surface cracks of arbitrary shape. Engineering Fracture Mechanics, 1985,22(1):101-114
[88]Guidera J T, Lardner R W. Penny-shaped cracks. Journal of Elasticity,1975, 5(1):59-73
[89]Cruse T A. Boundary element analysis in computational fracture mechanics. Berlin:Springer,1988
[90]Weaver J. Three-dimensional crack analysis. International Journal of Solids and Structures,1977,13(4):321-330
[91]Balas J, Sladek J. Method of boundary integral equations for analysis of three dimensional crack problems. In:Brebbia CA ed. Boundary Element Methods. Berlin:Springer Berlin Heidelberg,1981,183-205
[92]Ioakimidis N I. Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity. Acta mechanica,1982,45(1-2):31-47
[93]赵明皞,刘元杰.三维裂纹问题的第二类边界积分方程.机械强度,1992,14(1)：17-21
[94]秦太验.三维断裂力学的有限部积分边界元法研究：[上海交通大学博士学位论文].上海：上海交通大学工程力学系,1993
[95]乐金朝,冯新,韩连元.双材料平面裂纹问题的超奇异积分方程方法.固体力学学报,1999,20：34-37
[96]Cruse T A, Vanburen W. Three-dimensional elastic stress analysis of a fracture specimen with an edge crack. International Journal of Fracture Mechanics,1971, 7(1):1-15
[97]Luchi M L, Poggialini A. Computation of 3-dimensional stress intensity factors using special boundary element. In:Proc 5th Int Conf BEM. Berlin,1983: 461-470
[98]Luchi M L, Rizzuti S. Boundary elements for three-dimensional elastic crack analysis. International journal for numerical methods in engineering,1987, 24(12):2253-2271
[99]Chen J T, Hong H K. Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series. Applied Mechanics Reviews, 1999,52(1):17-33
[100]Chen J T, Hong H K, Chyuan S W. Boundary element analysis and design in seepage problems using dual integral formulation. Finite Elements in Analysis and Design,1994,17(1):1-20
[101]Hong H K, Chen J T. Derivations of integral equations of elasticity. Journal of Engineering Mechanics,1988,114(6):1028-1044
[102]Portela A, Aliabadi MH, Rooke DP. Dual boundary element incremental analysis of crack propagation. Computers and Structures,1993,46(2):237-247
[103]Portela A, Aliabadi MH, Rooke DP. The dual boundary element method: effective implementation for crack problems. International Journal for Numerical Methods in Engineering,1992,33(6):1269-1287
[104]Mi Y, Aliabadi M H. Discontinuous crack-tip elements:application to 3D boundary element method. International journal of fracture,1994,67(3): R67-R71
[105]王朋波,姚振汉.快速多极对偶边界元法在大规模裂纹分析中的应用.见：北京力学会第11届学术年会论文摘要集.北京：北京力学会,2005,116-117
[106]孙玉周,王银邦.反平面裂纹问题的边界元解法.兰州大学学报：自然科学版,2001,37(4)：25-30
[107]Sun Y Z, Yang S S, Wang Y B. A new formulation of boundary element method for cracked anisotropic bodies under anti-plane shear. Computer methods in applied mechanics and engineering,2003,192(22):2633-2648
[108]Wang Y B, Sun Y Z. A new boundary integral equation method for cracked 2-D anisotropic bodies. Engineering fracture mechanics,2005,72(13): 2128-2143
[109]Pan E N. A general boundary element analysis of 2-D linear elastic fracture mechanics. International Journal of Fracture,1997,88(1):41-59
[110]Yue Z Q, Xiao H T, Tham L G. Boundary element analysis of crack problems in functionally graded materials. International journal of solids and structures,2003,40(13):3273-3291
[111]肖洪天.梯度材料断裂力学的新型边界元法分析.北京：高等教育出版社,2007
[112]方志民,柴国钟.三维线弹性断裂分析的混合边界元法.浙江工业大学学报,1994,3：8-17
[113]柴国钟.线弹性断裂力学的混合边界元法：[华东理工大学博士学位论文].上海：华东理工大学化工过程机械系,1993
[114]Nowak A J, Neves A C. The multiple reciprocity boundary element method. Southampton and Boston:Computational Mechanics Publications,1994
[115]Gao X W. The radial integration method for evaluation of domain integrals with boundary-only discretization. Engineering Analysis with Boundary Elements,2002,26:905-916
[116]Yang K, Gao X W. Radial integration BEM for transient heat conduction problems. Engineering analysis with boundary elements,2010,34(6):557-563
[117]Yang K, Gao X W, Liu Y F. Using analytical expressions in radial integration BEM for variable coefficient heat conduction problems. Engineering Analysis with Boundary Elements,2011,35(10):1085-1089
[118]Partridge P W, Brebbia C A, Wrobel L C. The dual reciprocity boundary element method. Southampton and Springer-Verlag:Computational Mechanics Publications,1992
[119]Scuderi, Letizia. On the computation of nearly singular integrals in 3D BEM collocation. International journal for numerical methods in engineering,2008, 74(11):1733-1770
[120]Hayami K, Brebbia CA. A new coordinate transformation method for singular and nearly singular integrals over general curved boundary elements. In:Proc of 9th Int Con BEM. Berlin,1987,375-399
[121]Kane J H. Boundary element analysis in engineering continuum mechanics. Englewood Cliffs:Prentice-Hall Inc Press,1994
[122]Hughes T J, Cottrell J A, Bazilevs Y. Isogeometric analysis:CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer methods in applied mechanics and engineering,2005,194(39):4135-4195
[123]Hadamard J. Lectures on Cauchy's problem in linear partial differential equations. Courier Dover Publications,2003
[124]黄玲珍,聂毓琴.计算断裂力学研究的现状与进展.吉林工业大学学报,1995,25(1)：120-124
[125]范天佑.断裂力学基础.南京：江苏科学技术出版社,1978
[126]Murakami Y. Stress intensity factors handbook. Oxford:Pergarnon Press, 1987
[127]Gupta G D, Erdogan F. The problem of edge cracks in an infinite strip. Journal of Applied Mechanics,1974,41(4):1001-1006
[128]黎在良,王乘.高等边界元法.北京：科学出版社,2008
[129]Wen P H, Aliabadi M H, Rooke D P. Mixed-mode weight functions in three-dimensional fracture mechanics static. Engineering fracture mechanics, 1998,59(5):563-575
[130]Tada H, Paris P C, Irwin G R. The Analysis of Cracks Handbook. New York: ASME Press,2000
[131]Raju I S, Newman J C. Three dimensional finite-element analysis of finite-thickness fracture specimens. New York:National Aeronautics and Space Administration,1977
[132]Raju I S, Newman J C. Stress-intensity factors for a wide range of semi-elliptical surface cracks in finite-thickness plates. Engineering Fracture Mechanics,1979,11(4):817-829
[133]李亚莎.高精度快速边界元法及其在绝缘子电场计算中应用研究：[华北电力大学博士学位论文].北京：华北电力大学电气与电子工程学院,2007
[134]Zhang L P, Cui T, Liu H. A set of symmetric quadrarure rules on triangles and tetrahedra. Journal of Computational Mathematics,2009,27(1):89-96
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |