基于径向基神经网络的无网格法及其应用
|
摘要
无网格法的初衷是为了追踪物理量的局部高梯度变化,其显著特征是近似场函数构造不依赖于网格,并且数值积分网格与场函数构造方式无关。无网格法在弹塑性大变形、动态裂纹发展、冲击碰撞、流体力学等领域取得了很大的成功,受到众多研究人员重视,是计算力学的又一重要发展。
研究表明,即使在近似场函数有较高精度的情况下,近似场函数的导数也会产生局部数值震荡,极值点会发生相移。近似场函数导数的局部震荡和相移会对追踪局部物理量变化产生不利影响,导致数值计算结果的可靠性降低。另外,无网格法计算工作量大,计算效率低下,不利于大规模数值问题求解。
神经网络具有结构自适应确定、输出与初始权值无关及非线性收敛特征,径向基神经网络逼近原理实质是无网格法思想。将基于优化思想的神经网络算法用于构造近似场函数,可得到具有亚插值特性的近似场函数。
论文分析了产生近似导数数值局部震荡和相移的原因,讨论了影响无网格法计算量的因素。提出通过改进近似场函数的构造方式,提高近似场函数导数的精度,消除近似导数数值震荡和相移。利用基本解降低计算维数,赋予形函数Kroneckerδ函数性质,方便施加边界条件,减少数值计算工作量。
文中提出了两种消除近似导数数值局部震荡和相移的方法。
方法之一是将近似场函数和近似场函数的导数分别看作是独立变量,按照相同的方式分别构造近似场函数和近似场函数的导数,得到具有相同构造精度的函数。通过张量积的方式产生二维激励函数,构造直接径向基神经网络,依据混合变分原理建立系统控制方程,形成了基于张量积的径向基神经网络无网格法。适当选择激励函数,这种类型的无网格法不需要作数值积分,只要存储少量的计算基矩阵,就能在规则域上获得高精度的数值解。对不规则域,适当的数值处理也可求得满意的结果。
方法之二是利用误差在数学运算过程中的传播规律,构造积分形式的激励函数,得到高精度的近似导数。函数误差在微分过程中将进一步劣化,而在积分过程中光顺钝化。首先构造函数的高阶导数,通过不定积分得到带有权参数的新激励函数,用积分形式的激励函数构造间接径向基神经网络,显著地提高了场函数导数和场函数的近似精度,有效地消除了近似场函数导数出现的局部震荡和相移现象。
文中采取两种方法减少计算工作量。
第一种方法是利用间接边界型数值方法思想,提出了无网格虚边界法。边界型数值方法通过引入基本解,降低待求问题复杂程度,减少数值计算工作量。在无网格虚边界法中,虚边界始终不与真实边界相交,消除了其它边界型方法中存在的奇异积分和边界效应。虚边界几何形态简单灵活,与实边界的间距在较大范围内变化时,保持数值结果稳定收敛。无网格虚边界法计算不需要真实边界的外法线方向导数,不存在其它边界型方法中必须处理的角点问题。二维弹性力学的应用表明,在无论是简单域,还是复连通域,无网格虚边界法都能获得快速稳定的收敛结果。在物理量梯度变化较大的点,无网格虚边界法能高精度地给出结果,准确地反映出物理量的变化。
第二种方法是将边界节点邻近的激励函数进行线性组合,得到便于施加边界条件的扩展激励函数,消除了控制方程中的边界处理项,从而减少了计算工作量。
为满足无网格法场函数构造和数值积分计算的需要,文中还分析讨论了节点应满足的分布条件,编制了几何处理程序。程序能按几何变化特征,确定数值积分域内的数值积分点,给出较高精度稳定的数值结果。
数值计算结果证明了基于径向基神经网络无网格法的正确和有效。方法在计算精度、收敛速度、计算工作量上的优良表现,为其进一步研究发展奠定了基础。
The meshless methods are primarily motivated to trace large gradient variation of the variables in local domain. It is distinctive in that approached field functions can be constructed independently of the grids, and that the numerical integral cells are not concerned with the manner the approached field functions are constructed by. As a class of newly developed methods with great appeal, the meshless methods have made significant successes in such fields as large deformations in elasticity and plasticity, dynamic crack tracing, crash problems, flow mechanics, and etc.
However, it has been revealed that the derivative value may exhibit numerical oscillation in local domains and the peak values may be drifted even though the approached field functions are of high-level accuracy. Such local numerical oscillation and peak value drifts might have negative effects on tracing large gradient variation, making the calculation results less reliable. In addition, computation burden and low efficiency of the meshless methods is also an obstacle to solving large-scale numerical problems.
Neural networks have many excellent properties such as self-adapted determinant, output insensitive to the initial weighted values and non-linear convergence. The procedure of the radial basis neural networks (RBNN) approaches agrees with the idea of the meshless methods. It is expected that a quasi-interpolated function can be obtained by introducing radial basis neural network algorithms based on the optimum ideas into the meshless methods for constructing approached field functions.
The causes of local oscillations of the derivatives and drift of the peak values, as well as those factors that might have influence on the computation cost of the meshless methods, are analyzed in this dissertation. It is suggested that the local derivatives oscillations and the peak values drift should be eliminated by modifying the construction methods of the approached field functions with improved accuracy of the derivatives. On the other hand, in order to make less calculation, the fundamental solutions would be used for declining the dimensions by one, and the shape functions with Kroneckerδfunction properties be constructed so that the boundary conditions can be imposed easily.
Two methods for eliminating local oscillations of the derivatives and drift of the peak values are proposed.
One is to consider both approached field functions and their derivatives as independent variables respectively. Both are constructed with the same procedure to make the approached fields functions and their derivatives have the same accuracy. Applying two-dimensional prompted functions that are generated by tensor product method to the radial basis neural networks, a novel RBNN meshless method based on the tensor product is then developed, using the mixed variational principles to establish the governing equations of the system. In this meshless method, if the prompted functions are chosen properly, no more numerical integral is needed. Moreover, high accuracy results can be obtained in the normal region rapidly with only a few basis matrices required for completing numerical evaluation. A satisfactory result is also available if a moderate procedure is implemented in an irregular region.
Another method proposed here for eliminating the local oscillations of the derivatives and drift of the peak values is to create new prompted functions with the integral operations by taking advantages of the error propagation laws. Generally, the error generated in the approached functions would get worse during the differential process. Inversely, little difference occurs while the integral calculi are executed. By constructing high order derivatives first and establishing indirect radial basis neural networks with prompted functions generated by indeterminate integral operations, the accuracy of the field functions and their derivatives are improved significantly. The local oscillations of the derivatives and drift of the peak values vanish evidently.
So far as the calculation burden of meshless methods is concerned, two approaches are proposed to reduce the amount of computation.
Firstly, meshless virtual boundary method is developed by taking the profits of the ideas of the indirect boundary numerical procedures. It might be able to decline the complexity and reduce the calculations by introducing the fundamental solutions into the boundary methods. In this proposed method, singular integral and boundary effect associated with other boundary type methods disappear because no intersection between the virtual boundaries and true boundaries exists. The shapes of the virtual boundaries are simple and can be chosen easily. The numerical results are kept stable and convergent when the intervals between the virtual boundaries and true boundaries vary in large range. In addition, the corner problem associated with the other boundary methods is nonexistent as the directional cosines of the outer normal along the virtual boundaries are no more required in the meshless virtual boundary method. The results of the two-dimensional elastic problems illustrate that stabilized solutions are available either in the simple regions or in the complex regions, and the variable values varied in large gradient can be obtained exactly.
Secondly, a linear combination of the prompted functions neighboring the boundary neural units is made for generating new expanded prompted functions. Since the expanded prompted functions are subjected to the boundary conditions simply, the terms for imposing boundary conditions in system equations are eliminated. Consequently, the amount of calculation decreases.
To meet the requirement of the numerical integral and approached field functions construction in the methods proposed here, the conditions that scattered nodes must be subjected to have also been discussed. A code for determining the nodes in the regions with different geometric characteristics is developed. The code is capable of determining the integral points in each integral cell according to the shape of each integral domain. As a result, accurate numerical integral results can be obtained conveniently.
The numerical results have demonstrated the validity and effectiveness of the proposed RBNN meshless method. Such excellent performance the method exhibits as little calculation, high accuracy and rapid convergence, lays a foundation for the future progress.
研究表明,即使在近似场函数有较高精度的情况下,近似场函数的导数也会产生局部数值震荡,极值点会发生相移。近似场函数导数的局部震荡和相移会对追踪局部物理量变化产生不利影响,导致数值计算结果的可靠性降低。另外,无网格法计算工作量大,计算效率低下,不利于大规模数值问题求解。
神经网络具有结构自适应确定、输出与初始权值无关及非线性收敛特征,径向基神经网络逼近原理实质是无网格法思想。将基于优化思想的神经网络算法用于构造近似场函数,可得到具有亚插值特性的近似场函数。
论文分析了产生近似导数数值局部震荡和相移的原因,讨论了影响无网格法计算量的因素。提出通过改进近似场函数的构造方式,提高近似场函数导数的精度,消除近似导数数值震荡和相移。利用基本解降低计算维数,赋予形函数Kroneckerδ函数性质,方便施加边界条件,减少数值计算工作量。
文中提出了两种消除近似导数数值局部震荡和相移的方法。
方法之一是将近似场函数和近似场函数的导数分别看作是独立变量,按照相同的方式分别构造近似场函数和近似场函数的导数,得到具有相同构造精度的函数。通过张量积的方式产生二维激励函数,构造直接径向基神经网络,依据混合变分原理建立系统控制方程,形成了基于张量积的径向基神经网络无网格法。适当选择激励函数,这种类型的无网格法不需要作数值积分,只要存储少量的计算基矩阵,就能在规则域上获得高精度的数值解。对不规则域,适当的数值处理也可求得满意的结果。
方法之二是利用误差在数学运算过程中的传播规律,构造积分形式的激励函数,得到高精度的近似导数。函数误差在微分过程中将进一步劣化,而在积分过程中光顺钝化。首先构造函数的高阶导数,通过不定积分得到带有权参数的新激励函数,用积分形式的激励函数构造间接径向基神经网络,显著地提高了场函数导数和场函数的近似精度,有效地消除了近似场函数导数出现的局部震荡和相移现象。
文中采取两种方法减少计算工作量。
第一种方法是利用间接边界型数值方法思想,提出了无网格虚边界法。边界型数值方法通过引入基本解,降低待求问题复杂程度,减少数值计算工作量。在无网格虚边界法中,虚边界始终不与真实边界相交,消除了其它边界型方法中存在的奇异积分和边界效应。虚边界几何形态简单灵活,与实边界的间距在较大范围内变化时,保持数值结果稳定收敛。无网格虚边界法计算不需要真实边界的外法线方向导数,不存在其它边界型方法中必须处理的角点问题。二维弹性力学的应用表明,在无论是简单域,还是复连通域,无网格虚边界法都能获得快速稳定的收敛结果。在物理量梯度变化较大的点,无网格虚边界法能高精度地给出结果,准确地反映出物理量的变化。
第二种方法是将边界节点邻近的激励函数进行线性组合,得到便于施加边界条件的扩展激励函数,消除了控制方程中的边界处理项,从而减少了计算工作量。
为满足无网格法场函数构造和数值积分计算的需要,文中还分析讨论了节点应满足的分布条件,编制了几何处理程序。程序能按几何变化特征,确定数值积分域内的数值积分点,给出较高精度稳定的数值结果。
数值计算结果证明了基于径向基神经网络无网格法的正确和有效。方法在计算精度、收敛速度、计算工作量上的优良表现,为其进一步研究发展奠定了基础。
The meshless methods are primarily motivated to trace large gradient variation of the variables in local domain. It is distinctive in that approached field functions can be constructed independently of the grids, and that the numerical integral cells are not concerned with the manner the approached field functions are constructed by. As a class of newly developed methods with great appeal, the meshless methods have made significant successes in such fields as large deformations in elasticity and plasticity, dynamic crack tracing, crash problems, flow mechanics, and etc.
However, it has been revealed that the derivative value may exhibit numerical oscillation in local domains and the peak values may be drifted even though the approached field functions are of high-level accuracy. Such local numerical oscillation and peak value drifts might have negative effects on tracing large gradient variation, making the calculation results less reliable. In addition, computation burden and low efficiency of the meshless methods is also an obstacle to solving large-scale numerical problems.
Neural networks have many excellent properties such as self-adapted determinant, output insensitive to the initial weighted values and non-linear convergence. The procedure of the radial basis neural networks (RBNN) approaches agrees with the idea of the meshless methods. It is expected that a quasi-interpolated function can be obtained by introducing radial basis neural network algorithms based on the optimum ideas into the meshless methods for constructing approached field functions.
The causes of local oscillations of the derivatives and drift of the peak values, as well as those factors that might have influence on the computation cost of the meshless methods, are analyzed in this dissertation. It is suggested that the local derivatives oscillations and the peak values drift should be eliminated by modifying the construction methods of the approached field functions with improved accuracy of the derivatives. On the other hand, in order to make less calculation, the fundamental solutions would be used for declining the dimensions by one, and the shape functions with Kroneckerδfunction properties be constructed so that the boundary conditions can be imposed easily.
Two methods for eliminating local oscillations of the derivatives and drift of the peak values are proposed.
One is to consider both approached field functions and their derivatives as independent variables respectively. Both are constructed with the same procedure to make the approached fields functions and their derivatives have the same accuracy. Applying two-dimensional prompted functions that are generated by tensor product method to the radial basis neural networks, a novel RBNN meshless method based on the tensor product is then developed, using the mixed variational principles to establish the governing equations of the system. In this meshless method, if the prompted functions are chosen properly, no more numerical integral is needed. Moreover, high accuracy results can be obtained in the normal region rapidly with only a few basis matrices required for completing numerical evaluation. A satisfactory result is also available if a moderate procedure is implemented in an irregular region.
Another method proposed here for eliminating the local oscillations of the derivatives and drift of the peak values is to create new prompted functions with the integral operations by taking advantages of the error propagation laws. Generally, the error generated in the approached functions would get worse during the differential process. Inversely, little difference occurs while the integral calculi are executed. By constructing high order derivatives first and establishing indirect radial basis neural networks with prompted functions generated by indeterminate integral operations, the accuracy of the field functions and their derivatives are improved significantly. The local oscillations of the derivatives and drift of the peak values vanish evidently.
So far as the calculation burden of meshless methods is concerned, two approaches are proposed to reduce the amount of computation.
Firstly, meshless virtual boundary method is developed by taking the profits of the ideas of the indirect boundary numerical procedures. It might be able to decline the complexity and reduce the calculations by introducing the fundamental solutions into the boundary methods. In this proposed method, singular integral and boundary effect associated with other boundary type methods disappear because no intersection between the virtual boundaries and true boundaries exists. The shapes of the virtual boundaries are simple and can be chosen easily. The numerical results are kept stable and convergent when the intervals between the virtual boundaries and true boundaries vary in large range. In addition, the corner problem associated with the other boundary methods is nonexistent as the directional cosines of the outer normal along the virtual boundaries are no more required in the meshless virtual boundary method. The results of the two-dimensional elastic problems illustrate that stabilized solutions are available either in the simple regions or in the complex regions, and the variable values varied in large gradient can be obtained exactly.
Secondly, a linear combination of the prompted functions neighboring the boundary neural units is made for generating new expanded prompted functions. Since the expanded prompted functions are subjected to the boundary conditions simply, the terms for imposing boundary conditions in system equations are eliminated. Consequently, the amount of calculation decreases.
To meet the requirement of the numerical integral and approached field functions construction in the methods proposed here, the conditions that scattered nodes must be subjected to have also been discussed. A code for determining the nodes in the regions with different geometric characteristics is developed. The code is capable of determining the integral points in each integral cell according to the shape of each integral domain. As a result, accurate numerical integral results can be obtained conveniently.
The numerical results have demonstrated the validity and effectiveness of the proposed RBNN meshless method. Such excellent performance the method exhibits as little calculation, high accuracy and rapid convergence, lays a foundation for the future progress.
引文
[1] Zienkiewicz O C. Finite Elements and Approximation. New York: John Wiley & Sons, 1982.
[2] 吴永礼. 计算固体力学方法. 北京: 科学出版社, 2003.
[3] 王烈衡. 有限元方法的数学基础. 北京: 科学出版社, 2004.
[4] C.A.布莱比亚. 工程师用边界元法.第一版. 武际可,傅子智译. 北京: 科学出版社, 1986.
[5] Rizzo F J. An integral equation approach to boundary value problems of classical elastostatics. Q. Appl. Math., 1966, xxv(1):83-95.
[6] Brebbia C A, Tells C F, Wrobel L C. Boundary Element Techniques: Theory and Applications in Engineering. Berlin Germany: Springer-Verlag, 1984.
[7] 杨德全. 边界元理论及应用. 北京: 北京理工大学出版社, 2003.
[8] Tells C F. The Boundary Element Method Applied to Inelastic Problems.
[9] 秦荣. 样条边界元方法. 南宁: 广西科学技术出版社, 1988.
[10] 王元淳. 边界元法基础. 上海: 上海交通大学出版社, 1988.
[11] 杜庆华. 边界积分方程方法-边界元. 北京: 高等教育出版社, 1989.
[12] 嵇醒. 边界元法进展及通用程序. 上海: 同济大学出版社, 1997.
[13] 马振华. 现代应用数学手册(计算与数值分析卷). 北京: 清华大学出版社, 2005.
[14] Benito J J, Urena F, Gavete L. Influence of several factors in the generalized finite difference method. Applied Mathematical Modelling, 2001, 25(12):1039-1053.
[15] 邓俊辉译. 计算几何-算法与应用. 北京: 清华大学出版社, 2005.
[16] Liu Wing Kam, Han Weimin, Lu Hongsheng, et al. Reproducing kernel element method. Part I: Theoretical formulation. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12-14):933-951.
[17] Li Shaofan, Lu Hongsheng, Han Weimin, et al. Reproducing kernel element method. Part II: Globally conforming ImCn hierarchies. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12-14):953-987.
[18] Lu Hongsheng, Li Shaofan, Simkins Jr, et al. Reproducing kernel element method.Part III: Generalized enrichment and applications. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12-14):989-1011.
[19] Simkins Jr, Daniel C, Li Shaofan, et al. Reproducing kernel element method. Part IV: Globally compatible Cn(n greater than or equal 1) triangular hierarchy. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12-14):1013-1034.
[20] Sukumar N, Moran B, Belytschko T. The natural element method in solid mechanics. International Journal for Numerical Methods in Engineering, 1998, 43(5):839-87.
[21] Melenk J M, Babuska I. The partition of unity finite element method: basic theory and applications. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4):289-314.
[22] Belytschko T, Krongauz Y, Organ D, et al. Meshless methods: an overview and recent developments. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4):3-47.
[23] Liu Wing-Kam, Li Shaofan, Belytschko Ted. Moving least-square reproducing kernel methods (I) methodology and convergence. Computer Methods in Applied Mechanics and Engineering, 1997, 143(1-2):113-154.
[24] 张雄, 刘岩. 无网格法. 北京: 清华大学出版社, 2004.
[25] Krysl P, Belytschko T, Element-free Galerkin method: convergence of the continuous and discontinuous shape functions. Computer Methods in Applied Mechanics and Engineering, 1997, 148(3-4):257-277.
[26] Shapiro V, Tsukanov I. Meshfree simulation of deforming domains. CAD Computer Aided Design, 1999, 31(7):459-471.
[27] Bruneel Herman C J, De Rycke Igor. Quick trace: A fast algorithm to detect contact. International Journal for Numerical Methods in Engineering, 2002, 54(2):299-316.
[28] Li S, Qian D, Liu W K, et al. A meshfree contact-detection algorithm. Computer Methods in Applied Mechanics and Engineering, 2001, 190(24-25):3271-3292.
[29] Krysl Petr, Belytschko Ted. Analysis of thin shells by the element-free Galerkin method. International Journal of Solids and Structures, 1996, 33(20-22):3057-3080.
[30] Guo Y M, Nakanishi K. A backward extrusion analysis by the rigid-plastic integralless-meshless method. Journal of Materials Processing Technology, 2003, 140(1-3 SPEC):19-24.
[31] Belytschko T, Lu Y Y, Gu L. Crack propagation by element-free Galerkin methods. Engineering Fracture Mechanics, 1995, 51(2):295-315.
[32] Basu P K, Jorge A B, Badri S, et al. Higher-Order Modeling of Continua by Finite-Element, Boundary-Element, Meshless, and Wavelet Methods. Computers and Mathematics with Applications, 2003, 46(1):15-33.
[33] Liew K M, Ng T Y, Wu Y C. Meshfree method for large deformation analysis-a reproducing kernel particle approach. Engineering Structures, 2002, 24(5):543-551.
[34] Lanson N, Vila J P. Meshless methods for conservation laws. Mathematics and Computers in Simulation, 2001, 55(4-6):493-501.
[35] Dolbow John, Moes Nicolas, Belytschko Ted. Modeling fracture in Mindlin-Reissner plates with the extended finite element method. International Journal of Solids and Structures, 2000, 37(48):7161-7183.
[36] Sukumar N, Chopp D L, Moes N, et al. Modeling holes and inclusions by level sets in the extended finite-element method. Computer Methods in Applied Mechanics and Engineering, 2001, 190(46-47):6183-6200.
[37] Belytschko Ted, Krongauz Yury, Fleming Mark, et al. Smoothing and accelerated computations in the element free Galerkin method. Journal of Computational and Applied Mathematics, 1996, 74(1-2):111-126.
[38] Xiao Shaoping, Belytschko T. Stability analysis of particle methods with corrected derivatives. Computers and Mathematics with Applications, 2002, 43(3-5):329-350.
[39] Liu Wing Kam, Chen Yijung, Belytschko Ted, et al. Three reliability methods for fatigue crack growth. Engineering Fracture Mechanics, 1996, 53(5):733-752.
[40] Liu Wing Kam, Jun Sukky. Multiple-scale Reproducing Kernel Particle Methods for large deformation problems. International Journal for Numerical Methods in Engineering, 1998, 41(7):1339-1362.
[41] Chen J S, Pan C, Wu C T. Large deformation analysis of rubber based on a reproducing kernel particle method. Computational Mechanics, 1997, 19(3):211-227.
[42] Liu Z S, Swaddiwudhipong S, Koh C G. High velocity impact dynamic response of structures using SPH method. International Journal of Computational Engineering Science, 2004, 5(2):315-326.
[43] Johnson G R, Stryk R A, Beissel S R. SPH for high velocity impact computations. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4):347-73.
[44] Lam K Y, Shen Y G, Gong S W. A study of axial impact of composite rods using SPH approach. Shock and Vibration, 2001, 8(5):303-312.
[45] Johnson G R, Stryk R A, Beissel S R. Interface effects for SPH impact computations.In Proceedings of the 1996 4th International Conference on Structures Under Shock and Impact, SUSI 96, Jul 1996; Udine, Italy: Computational Mechanics Inc, Billerica, MA, USA, 1996.
[46] Benz W, Asphaug E. Simulations of brittle solids using smooth particle hydrodynamics. Computer Physics Communications, 1995, 87(1-2):253-65.
[47] Ho S L, Yang S, Machado J M, et al., Application of a meshless method in electromagnetics. In Ninth Biennial Electromagnetic Field Computation (CEFC), Jun 4-7 2001; Milwaukee, WI: Institute of Electrical and Electronics Engineers Inc., 2001.
[48] Tran-Cong T, Mai-Duy N, Phan-Thien N. BEM-RBF approach for viscoelastic flow analysis. Engineering Analysis with Boundary Elements, 2002-, 26(9):757-62.
[49] Campbell J, Vignjevic R. Modelling hypervelocity impact in DYNA3D. in Proceedings of 3rd International Conference Dynamics and Control of Structures in Space (ISBN 185312415X), 27-31 May 1996. London, UK: Comput. Mech. Publications, 1996.
[50] Gray J P, Monaghan J J, Swift R P. SPH elastic dynamics. Computer Methods in Applied Mechanics and Engineering, 2001, 190(49-50):6641-62.
[51] Jubelgas M, Springel V, Dolag K. Thermal conduction in cosmological SPH simulations. Monthly Notices of the Royal Astronomical Society, 2004, 351(2):423-35.
[52] Strouboulis T, Copps K, Babuska I. The generalized finite element method. Computer Methods in Applied Mechanics and Engineering, 2001, 190(32-33):4081-193.
[53] 贝新源, 岳宗五. 三维SPH程序及其在斜高速碰撞问题的应用. 计算物理, 1997, 14(2):155-166.
[54] 张锁春. 光滑质点流体动力学(SPH)方法. 计算物理, 1996, 13(4):385-397.
[55] Lucy L B. A numerical approach to the testing of the fission hypothesis. The Astron. Journal, 1977, 8(12):1013-1024.
[56] Monaghan J J. An introduction to SPH. Comput. Phys. Comm., 1988, (48):89-96.
[57] Monaghan J J. Smoothed partic hydrodynamics. Annual Review Astronomics and Astrophysics, 1992,(30):543-574.
[58] Monaghan J J. A shock simulation by the Partical Method SPH. J. Comput. Phys., 1983, (52):374-389.
[59] Randles P W, Smoothed partical hydrodynamics: some recent improvments and applications. Comput. Methods Appl. Mech. Engrg., 1996, (139):375-408.
[60] Nitsche Ludwig C, Zhang Weidong. Atomistic SPH and a Link between Diffusion and Interfacial Tension. AIChE Journal, 2002, 48(2):201-211.
[61] Belytschko T, Fleming M. Smoothing enrichment and contact in the element-free Galerkin method. Computers and Structures, 1999, 71(2):173-195.
[62] Bonet J, Kulasegaram S. A simplified approach to enhance the performance of smooth particle hydrodynamics methods. Applied Mathematics and Computation (New York), 2002, 126(2-3):133-155.
[63] Liew K M, Ng T Y, Zhao X, et al. Harmonic reproducing kernel particle method for free vibration analysis of rotating cylindrical shells. Computer Methods in Applied Mechanics and Engineering, 2002, 191(37-38):4141-4157.
[64] Chen W. Meshfree boundary particle method applied to Helmholtz problems. Engineering Analysis with Boundary Elements, 2002, 26(7):577-581.
[65] Li Shaofan, Liu Wing Kam. Meshfree and particle methods and their applications. Applied Mechanics Reviews, 2002, 55(1):1-34.
[66] Aluru N R. Point collocation method based on reproducing kernel approximations. International Journal for Numerical Methods in Engineering, 2000, 47(6):1083-1121.
[67] Liew K M, Zou G P, Rajendran S. A spline strip kernel particle method and its application to two-dimensional elasticity problems. International Journal for Numerical Methods in Engineering, 2003, 57(5):599-616.
[68] Cueto-Felgueroso L, Colominas I, Mosqueira G, et al. On the Galerkin formulation of the smoothed particle hydrodynamics method. International Journal for Numerical Methods in Engineering, 2004, 60(9):1475-1512.
[69] Kim Do Wan, Kim Yongsik. Point collocation methods using the fast moving least-square reproducing kernel approximation. International Journal for Numerical Methods in Engineering, 2003, 56(10):1445-1464.
[70] Han Young Yoon, Koshizuka S, Oka Y. A particle-gridless hybrid method for incompressible flows. International Journal for Numerical Methods in Fluids, 1999, 30(4):407-24.
[71] Belytschko T, Krongauz Y, Dolbow J, et al. On the completeness of meshfree particle methods. International Journal for Numerical Methods in Engineering, 1998, 43(5):785-819.
[72] Chen Jiun-Shyan, Pan Chunhui, Wu Cheng-Tang, et al. Reproducing kernel particle methods for large deformation analysis of non-linear structures. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4):195-227.
[73] Schoenberg I J. Cardinal Spline Interpolation. Bristol England: JW Arrowsmith Ltd., 1973.
[74] Carl de Boor. A Practical Guide to Splines. New York: Springer-Verlag Inc., 1978.
[75] Kouibia A, Pasadas M. Approximation of surfaces by fairness bicubic splines. Advances in Computational Mathematics, 2004, 20(1-3):87-103.
[76] Dierckx P. Algorithm for fitting data over a circle use tensor product splines. Journal of Computational and Applied Mathematics, 1986, 15(2):161-173.
[77] Hollig Klaus, Reif Ulrich. Nonuniform web-splines. Computer Aided Geometric Design, 2003, 20(5):277-294.
[78] Onah E S. On direct methods for the discretization of a heat-conduction equation using spline functions. Applied Mathematics and Computation, 1997, 85(1):87-96.
[79] Hongzhi Zhong. Spline-based differential quadrature for fourth order differential equations and its application to Kirchhoff plates. Applied Mathematical Modelling, 2004, 28(4):353-66.
[80] 石钟慈. 样条有限元法. 计算数学, 1976,(1): 50-72.
[81] 刘效尧. 样条函数与结构力学. 北京: 人民交通出版社, 1990.
[82] Hollig K. Finite Element Methods with B-splines. Stuttgart Germany: SIAM, Philadelphia, 2003.
[83] Shen Pengcheng, He Peixiang. The multivariable spline finite element method. Acta Mechanica Solida Sinica, 1994, 15(3):234-43.
[84] Mizusawa T, Kajita T, Naruoka M. Vibration of skew plates by using B-spline functions. 1979, 62(2):301-308.
[85] Antes H. Bicubic fundamental splines in plate bending. 1974, 8(3):503-511.
[86] Wray Jonathan, Green Gary G. R. Neural networks, approximation theory, and finite precision computation. Neural Networks, 1995, 8(1):31-37.
[87] 飞思科技产品研发中心. 神经网络理论与Matlab7.0实现. 北京: 电子工业出版社, 2005.
[88] Braun J. A numerical method for solving partial differential equations on highly irregular grids. Nature, 1995, 376(24):655-660.
[89] Niu Zhongrong, Zhou Huanlin. The natural boundary integral equation in potential problems and regularization of the hypersingular integral. Computers and Structures, 2004, 82(2-3):315-323.
[90] Martinez M A, Cueto E, Doblare M, et al. Natural element meshless simulation of flows involving short fiber suspensions. Journal of Non-Newtonian Fluid Mechanics, 2003, 115(1):51-78.
[91] Farwig R. Multivariate interpolation of arbitrarily spaced data by moving least squares methods. Journal of Computational and Applied Mathematics, 1986, 16(1):79-93.
[92] Gu Lei. Moving kriging interpolation and element-free Galerkin method. International Journal for Numerical Methods in Engineering, 2003, 56(1):1-11.
[93] Hao Su, Park Harold S, Liu Wing Kam. Moving particle finite element method. International Journal for Numerical Methods in Engineering, 2002, 53(8):1937-1958.
[94] Dilts Gary A. Moving-least-squares-particle hydrodynamics - I. Consistency and stability. International Journal for Numerical Methods in Engineering, 1999, 44(8):1115-1155.
[95] Hao Su, Liu Wing Kam, Belytschko Ted. Moving particle finite element method with global smoothness. International Journal for Numerical Methods in Engineering, 2004, 59(7):1007-1020.
[96] Liew K M, Huang Y Q, Reddy, et al. Moving least squares differential quadrature method and its application to the analysis of shear deformable plates. International Journal for Numerical Methods in Engineering, 2003, 56(15):2331-2351.
[97] Dilts GaryA. Moving least-squares particle hydrodynamics II: Conservation and boundaries. International Journal for Numerical Methods in Engineering, 2000, 48(10):1503-1524.
[98] Gingold R A. Smoothed partical hydrodynamics: theory and applications to non-spherical stars. Mon. Not. Roy. Astrou. Soc., 1977, (18):375-389.
[99] Rao B N, Rahman S. A coupled meshless-finite element method for fracture analysis of cracks. International Journal of Pressure Vessels and Piping, 2001, 78(9):647-657.
[100] Chen Jiun-Shyan, Wang Hui-Ping. New boundary condition treatments in meshfree computation of contact problems. Computer Methods in Applied Mechanics and Engineering, 2000, 187(3): 441-468.
[101] Bulatovic R M. On the heavily damped response in viscously damped dynamicsystems. Journal of Applied Mechanics, Transactions ASME, 2004, 71(1):131-134.
[102] Bhushan Bharat, Peng Wei. Contact mechanics of multilayered rough surfaces. Applied Mechanics Reviews, 2002, 55(5):435-479.
[103] Campo Antonio, Morrone Biagio. Meshless approach for computing the heat liberation from annular fins of tapered cross section. Applied Mathematics and Computation (New York), 2004, 156(1):137-144.
[104] Carpinteri A, Ferro G, Ventura G. The partition of unity quadrature in element-free crack modelling. Computers and Structures, 2003, 81(18-19):1783-1794.
[105] Zhao X., Liew K M, Ng T Y. Vibration analysis of laminated composite cylindrical panels via a meshfree approach. International Journal of Solids and Structures, 2003, 40(1):161-180.
[106] Tsukanov I, Shapiro V, Zhang S. A meshfree method for incompressible fluid dynamics problems. International Journal for Numerical Methods in Engineering, 2003, 58(1):127-158.
[107] Charoenphan S, Plesha M E, Bank L C. Simulation of crack growth in composite material shell structures. International Journal for Numerical Methods in Engineering, 2004, 60(14): 2399-417.
[108] Zavarise G, Wriggers P. Contact with friction between beams in 3-D space. International Journal for Numerical Methods in Engineering, 2000, 49(8):977-1006.
[109] Sladek J, Sladek V. A meshless method for large deflection of plates. Computational Mechanics, 2003, 30(2):155-163.
[110] Belytschko T, Organ D, Gerlach C. Element-free Galerkin methods for dynamic fracture in concrete. Computer Methods in Applied Mechanics and Engineering, 2000, 187(3):385-399.
[111] Belytschko Ted, Babuska Ivo, Tinsley, et al. Research directions in computational mechanics. Computer Methods in Applied Mechanics and Engineering, 2003, 192(7-8):913-922.
[112] Zhou X, Hon Y C, Cheung K F. A grid-free, nonlinear shallow-water model with moving boundary. BEM in China: Engineering Analysis with Boundary Elements, 2004, 28(8):967-973.
[113] Belytschko T, Lu Y Y, Gu L, Crack propagation by element free Galerkin methods. Proceedings of the 1993 ASME Winter Annual Meeting, Nov 28-Dec 3 1993. New Orleans, LA, USA: Publ by ASME, New York, NY, USA, 1993.
[114] Xu Yonglin, Moran Brian, Belytschko Ted. Uncoupled characteristics of three-dimensional planar cracks. International Journal of Engineering Science, 1998, 36(1):33-48.
[115] Lancaster P. "Moving weighted least-squares method" in Polynomial and Spline Approximation. NATO Advanced Study InstituteSeries C., 1979, 103-120.
[116] Frank G N R. Smooth interpolation os Large Sets of Scattered Data. Technical Report. 1979.
[117] Lancaster K S P. Surface Generated by Moving Least Squares Method. Mathematics of Computation, 1981, 37(155):141-158.
[118] Nayroles G T B. Generalizing the finite element method: diffuse approximation and diffuse elements. Computational Mechanics, 1992, 10:307-318.
[119] Lu Y Y, Gu L, Belytschko T. Element-free Galerkin Methods. International Journal for numerical methods in engineering, 1994, 37:229-256.
[120] Belytschko T, Lu YY. A new implementation of the element free Galerkin method. Comput. Methods Appl. Mech. Engrg., 1994, 113:397-414.
[121] Beissel S, Belytschko Ted. Nodal integration of the element-free Galerkin method. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4):49-74.
[122] Duflot M, Nguyen-Dang Hung. A truly meshless Galerkin method based on a moving least squares quadrature. Communications in Numerical Methods in Engineering, 2002, 18(6):441-449.
[123] Krysl P, Belytschko T. The element free Galerkin method for dynamic propagation of arbitrary 3-D cracks. International Journal for Numerical Methods in Engineering, 1999, 44(6):767-800.
[124] Onate E, Perazzo F, Miquel J. A finite point method for elasticity problems. Computers and Structures, 2001, 79(22-25):2151-2163.
[125] Lohner Rainald, Sacco Carlos, Onate Eugenio, et al. A finite point method for compressible flow. International Journal for Numerical Methods in Engineering, 2002, 53(8):1765-1779.
[126] Duarte C A, Oden J T. An h-p adaptive method using clouds. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4):237-62.
[127] Liszka T J, Duarte C A, Tworzydlo W W. hp-meshless cloud method. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4):263-288.
[128] Li Hua, Ng T Y, Cheng J Q, et al. Hermite-Cloud: A novel true meshless method.Computational Mechanics, 2003, 33(1):30-41.
[129] Aluru N R, Li G. Finite cloud method: A true meshless technique based on fixed reproducing kernel approximation. International Journal for Numerical Methods in Engineering, 2001, 50(10):2373-2410.
[130] Meschke G. Finite Element Modelling of Cracks Based on the Partition of Unity Method. Proc. Appl. Math. Mech., 2003, 226-227.
[131] Fan S C, Liu X, Lee C K. Enriched partition-of-unity finite element method for stress intensity factors at crack tips. Computers and Structures, 2004, 82(4-5):445-461.
[132] Simone A. Partition of unity-based discontinuous elements for interface phenomena: Computational issues. Communications in Numerical Methods in Engineering, 2004, 20(6):465-478.
[133] Alves Marcelo Krajnc, Rossi Rodrigo. A modified element-free Galerkin method with essential boundary conditions enforced by an extended partition of unity finite element weight function. International Journal for Numerical Methods in Engineering, 2003, 57(11): 1523-1552.
[134] Carpinteri A, Ferro G, Ventura G. The partition of unity quadrature in meshless methods. International Journal for Numerical Methods in Engineering, 2002, 54(7):987-1006.
[135] Munts E A, Hulshoff S J, de Borst R. The partition-of-unity method for linear diffusion and convection problems: Accuracy, stabilization and multiscale interpretation. International Journal for Numerical Methods in Fluids, 2003, 43(2):199-213.
[136] Dolbow John, Moes Nicolas, Belytschko Ted. Discontinuous enrichment in finite elements with a partition of unity method. Finite Elements in Analysis and Design, 2000, 36(3):235-260.
[137] Zu T, zhang J. A local boundary integral equation (LIBE) method in computational mechanics, a meshless discretization approach. Comp. Mech., 1998, 21:223-235.
[138] Atluri S N, Zhu T. A new meshless local Petrov-Galerkin (MLPG) approach to nonlinear problems in computer modeling and simulation. Computer Modeling and Simulation in Engineering, 1998, 3(3):187-96.
[139] Atluri S N, Zhu T. New Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics. Computational Mechanics, 1998, 22(2):117-127.
[140] De S, Bathe K J. The method of finite spheres with improved numerical integration.Computers and Structures, 2001, 79(22-25):2183-2196.
[141] Liu G R. A point assembly method for stress analysis for two-dimensional solids. International Journal of Solids and Structures, 2002, 39(1):261-276.
[142] Gu Y T, Liu G R. Hybrid boundary point interpolation methods and their coupling with the element free Galerkin method. Engineering Analysis with Boundary Elements, 2003, 27(9):905-917.
[143] Cheng M, Liu G R. A novel finite point method for flow simulation. International Journal for Numerical Methods in Fluids, 2002, 39(12):1161-1178.
[144] Wang J G, Liu G R. A point interpolation meshless method based on radial basis functions. International Journal for Numerical Methods in Engineering, 2002, 54(11):1623-1648.
[145] Liu G R, Gu Y T. A point interpolation method for two-dimensional solids. International Journal for Numerical Methods in Engineering, 2001, 50(4):937-951.
[146] Sibson R. A brief description of nature neighbor interpolation. 1981,21-36.
[147] Sibson R, A vector identity for a Dirichlet tesselation. Math. Proc., 1980, 87:151-155.
[148] Watson D F. Neighborhood-based interpolation. Geobyte, 1987, 2(2):12-16.
[149] Traversoni L. An algorithm for nature spline interpolation. Numer. Algorithms, 1993, (5):63-70.
[150] Traversoni L. Nature neighbor finite elements. Int.Con.on Hydroaulic Engineering Software, Hydrosoft Proc., 1994.
[151] Wang Kai, Zhou Shenjie, Shan Guojun. The natural neighbour Petrov-Galerkin method for elasto-statics. International Journal for Numerical Methods in Engineering, 2005, 63(8):1126-1145.
[152] Mai-Duy Nam, Tran-Cong Thanh. Approximation of function and its derivatives using radial basis function networks. Applied Mathematical Modelling, 2003, 27(3):197-220.
[153] Mai-Duy N, Tran-Cong T. Solving biharmonic problems with scattered-point discretization using indirect radial-basis-function networks. Engineering Analysis with Boundary Elements, 2006, 30(2):77-87.
[154] Mai-Duy Nam. An effective spectral collocation method for the direct solution of high-order ODEs. Communications in Numerical Methods in Engineering, 2006, 22(6):627-642.
[155] Nam Mai-Duy, Thanh Tran-Cong. Approximation of function and its derivativesusing radial basis function networks. Applied Mathematical Modelling, 2003, 27(3):197-220.
[156] Nam Mai-Duy, Thanh Tran-Cong. Numerical solution of differential equations using multiquadric radial basis function networks. Neural Networks, 2001-, 14(2):185-99.
[157] Mai-Duy Nam, Tran-Cong Thanh. RBF interpolation of boundary values in the BEM for heat transfer problems. International Journal of Numerical Methods for Heat and Fluid Flow, 2003, 13(5-6):611-632.
[158] Mai-Duy Nam, Tran-Cong Thanh. An effective RBFN-boundary integral approach for the analysis of natural convection flow. International Journal for Numerical Methods in Fluids, 2004, 46(5):545-568.
[159] Mai-Duy Nam, Tanner R I. Computing non-Newtonian fluid flow with radial basis function networks. International Journal for Numerical Methods in Fluids, 2005, 48(12):1309-1336.
[160] Mai-Duy N, Tran-Cong T. Boundary integral-based domain decomposition technique for solution of Navier Stokes equations. Computer Modeling in Engineering & Sciences, 2004, 6(1):59-75.
[161] Mai-Duy Nam, Tran-Cong Thanh. Neural networks for BEM analysis of steady viscous flows. International Journal for Numerical Methods in Fluids, 2003, 41(7):743-763.
[162] Mai-Duy N, Tran-Gong T. Boundary integral-based domain decomposition technique for solution of Navier Stokes equations. CMES - Computer Modeling in Engineering and Sciences, 2004, 6(1):59-75.
[163] Mai-Duy N, Tran-Cong T. Numerical solution of Navier-Stokes equations using multiquadric radial basis function networks. International Journal for Numerical Methods in Fluids, 2001, 37(1):65-86.
[164] Mai-Duy N, Tran-Cong T. Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson's equations. Engineering Analysis with Boundary Elements, 2002, 26(2):133-156.
[165] Nam Mai-Duy, Thanh Tran-Cong. Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson's equations. Engineering Analysis with Boundary Elements, 2002, 26(2):133-56.
[166] Prime M B, Sebring R J, Edwards J M, et al. Laser surface-contouring and spline data-smoothing for residual stress measurement. Experimental Mechanics, 2004, 44(2):176-184.
[167] Hughes Matt C, Westwick David T. Identification of IIR Wiener systems with spline nonlinearities that have variable knots. IEEE Transactions on Automatic Control, 2005, 50(10): 1617-1622.
[168] Hardy R L. Theory and applications of the multiquadric-biharmonic method. 20 years of discovery 1968-88. Computers & Mathematics with Applications, 1990, 19 (8-9):163-208.
[169] Atluri S N, Sladek J, Sladek V, et al. Local boundary integral equation (LBIE) and it's meshless implementation for linear elasticity. Computational Mechanics, 2000, 25(2-3):180-198.
[170] Zhu T, Zhang J D, Atluri S N. Local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach. Computational Mechanics, 1998, 21(3):223-235.
[171] Zhu T, Zhang J, Atluri S N. Meshless local boundary integral equation (LBIE) method for solving nonlinear problems. Computational Mechanics, 1998, 22 (2):174-186.
[172] Gowrishankar Ramesh, Mukherjee Subrata. A 'pure' boundary node method for potential theory. Communications in Numerical Methods in Engineering, 2002, 18(6):411-427.
[173] Chati Mandar K, Mukherjee S. Boundary node method for three-dimensional problems in potential theory. International Journal for Numerical Methods in Engineering, 2000, 47(9):1523-1547.
[174] Miao Yu, Wang Yuan-Han. Meshless analysis for three-dimensional elasticity with singular hybrid boundary node method. Applied Mathematics and Mechanics (English Edition), 2006, 27(5):673-681.
[175] Miao Yu, Mao Feng, Wang Yuan-Han, Zhang Jun, et al.. Hybrid boundary node method in geotechnical engineering. Yantu Lixue/Rock and Soil Mechanics, 2005, 26(9):1452-1455.
[176] Chen W, Tanaka M. A meshless, integration-free, and boundary-only RBF technique. Computers and Mathematics with Applications, 2002, 43(3-5):379-391.
[177] Liu G R, Gu Y T. Boundary meshfree methods based on the boundary pointinterpolation methods. Twenty-Fourth International Conference on the Boundary Element Method Incorporating Meshless Solution Seminar, BEM XXIV, Jun 17-19 2002; Sintra, Portugal: WITPress, Southampton, United Kingdom, 2002.
[178] Wu C-K C, Plesha M E. Essential boundary condition enforcement in meshless methods: Boundary flux collocation method. International Journal for Numerical Methods in Engineering, 2002, 53(3):499-514.
[179] Yagawa G. Node-by-node parallel finite elements: A virtually meshless method. International Journal for Numerical Methods in Engineering, 2004, 60(1):69-102.
[180] 孙海涛, 王元汉. 无网格法在我国的起源. 见戴念祖主编, 力学史与方法论, 第二届力学史与方法论会议, 上海. 上海: 上海大学出版社, 2005.
[181] Gordon W J, Wixom J A. Shepard's method of `metric interpolation' to bivariate and multivariate interpolation. Mathematics of Computation, 1978, 32(141):253-64 .
[182] Chen J K, Beraun J E. A generalized smoothed particle hydrodynamics method for nonlinear dynamic problems. Computer Methods in Applied Mechanics and Engineering, 2000, 190(1-2):225-39.
[183] Chen Jiun-Shyan, Pan Chunhui, Wu Cheng-Tang. Application of Reproducing Kernel Particle Method to large deformation contact analysis of elastomers. Rubber Chemistry and Technology, 1998, 71(2):191-213.
[184] Yoon Sangpil, Chen Jiun-Shyan. Accelerated meshfree method for metal forming simulation. Finite Elements in Analysis and Design, 2002, 38(10):937-948.
[185] Liu W K, Lua Y J, Chen Y, et al. Study of three reliability methods for fatigue crack growth. Proceedings of the IUTAM Symposium, Jun 7-10 1993; San Antonio, TX, USA: Springer-Verlag New York Inc., 1993.
[186] Lu Y Y, Belytschko T, Tabbara M. Element-free Galerkin method for wave propagation and dynamic fracture. Computer Methods in Applied Mechanics and Engineering, 1995, 126(1-2): 131-153.
[187] Liu W K. Reproducing kernel partical methods. International journal for numerical methods in fluids, 1995, 20:1081-1106.
[188] Johnson G R, Beissel S R. Normalized smoothing functions for SPH impact computations. International Journal for Numerical Methods in Engineering, 1996, 39(16):2725-41.
[189] Johnson G R. Artificial viscosity effects for SPH impact computations. InternationalJournal of Impact Engineering, 1996, 18(5):477-88.
[190] Johnson G R, Beissel S R. Normalized smoothing functions for SPH impact computations. International Journal for Numerical Methods in Engineering, 1996, 39(16):2725-2741.
[191] Johnson Gordon R, Stryk Robert A, Beissel Stephen R. SPH for high velocity impact computations. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4):347-373.
[192] Mao Yi-ming, Wu Wen-yuan, Chen Guang-lin, et al. Numerical simulation of high velocity impact problems with SPH method. Journal of PLA University of Science and Technology (Natural Science Edition), 2003, 4(5):84-7.
[193] Kikuchi Masanori, Miyamoto Masayuki. Numerical simulations of impact crush/buckling of circular tube using SPH method. Fracture and Strength of Solids VI: Key Engineering Materials, 2006, 306-308 I:697-702.
[194] Yan Xiaojun, Zhang Yuzhu, Nie Jingxu. Numerical simulation of space debris hypervelocity impact using SPH method. Beijing Hangkong Hangtian Daxue Xuebao/Journal of Beijing University of Aeronautics and Astronautics, 2005, 31(3):351-354.
[195] Li Jin-Zhu, Zhang Qing-Ming, Long Ren-Rong. SPH simulation of hypervelocity impacts. Journal of Beijing Institute of Technology, 2004, 13(3):266-9.
[196] Libersky L D, Randles P W, Carney T C, et al. Recent improvements in SPH modeling of hypervelocity impact. Proceedings of the 1996 Symposium on Hypervelocity Impact. Part 2 (of 2), Oct 8-10 1996: International Journal of Impact Engineering, 1997, 20(6-10):525-532.
[197] Li Jin-Zhu, Zhang Qing-Ming, Long Ren-Rong. SPH simulation of hypervelocity impacts. Journal of Beijing Institute of Technology (English Edition), 2004, 13(3):266-269.
[198] Lukyanov A A, Reveles J R, Vignjevic R, et al. Simulation of hypervelocity debris impact and spacecraft shielding performance. 4th European Conference on Space Debris, Apr 18-20 2005. Darmstadt, Germany: European Space Agency, Noordwijk, 2200 AG, Netherlands, 2005.
[199] Brunger M J, Campbell L, Cartwright D C. Studying the effects of electron-impact excitation in the upper atmosphere. Trends in Applied Spectroscopy, 2004, 5:39-54.
[200] Campbell R. Vignjevic. Lagrangian hydrocode modelling of hypervelocity impact onspacecraft. Third International Conference in Computational Structures Technology. Advances in Computational Methods for Simulation, 21-23 Aug. 1996; Budapest, Hungary: Civil-Comp Press, 1996.
[201] Vandemark D, Chapron B, Elfouhaily T, et al. Impact of high-frequency waves on the ocean altimeter range bias. Journal of Geophsical Research-Part C-Oceans, 2005, 110(C11):12 pp.
[202] Kanel G I, Ivanov M F, Parshikov A N. Computer simulation of the heterogeneous materials response to the impact loading. Hypervelocity Impact. 1994 Symposium, 17-19 Oct. 1994: International Journal of Impact Engineering; Sante Fe, NM, USA : Elsevier, 1995.
[203] Parshikov Anatoly N, Medin Stanislav A, Loukashenko Igor I, et al. Improvements in SPH method by means of interparticle contact algorithm and analysis of perforation tests at moderate projectile velocities. International Journal of Impact Engineering, 2000, 24(8):779-796.
[204] Dolbow J, Moes N, Belytschko T. An extended finite element method for modeling crack growth with frictional contact. Computer Methods in Applied Mechanics and Engineering, 2001, 190(51-52):6825-6846.
[205] Kremmer M, Favier J F. A method for representing boundaries in discrete element modelling - Part I: Geometry and contact detection. International Journal for Numerical Methods in Engineering, 2001, 51(12):1407-1421.
[206] Pandolfi A, Kane C, Marsden J E, et al. Time-discretized variational formulation of non-smooth frictional contact. International Journal for Numerical Methods in Engineering, 2002, 53(8):1801-1829.
[207] Wang F, Cheng J, Yao Z. FFS contact searching algorithm for dynamic finite element analysis. International Journal for Numerical Methods in Engineering, 2001, 52(7):655-672.
[208] Jiang F, Oliveira, Sousa A C M. SPH simulation of low reynolds number planar shear flow and heat convection. Materialwissenschaft und Werkstofftechnik, 2005, 36(10):613-619.
[209] Sakai Yuzuru, Yang Zong Yi, Jung Young Guan. Incompressible viscous flow analysis by SPH. Nippon Kikai Gakkai Ronbunshu, B Hen/Transactions of the Japan Society of Mechanical Engineers, Part B, 2004, 70(696):1949-1956.
[210] Monaghan J J. SPH compressible turbulence. Monthly Notices of the RoyalAstronomical Society, 2002, 335(3):843-52.
[211] Cummins S J, Rudman M J. Truly incompressible SPH. Proceedings of the 1998 ASME Fluids Engineering Division Summer Meeting, Jun 21-25 1998; Washington, DC, USA: ASME, Fairfield, NJ, USA, 1998.
[212] Fujisawa Toshimitsu, Ito Satoshi, Inaba Masakazu, et al. Node-based parallel computing of three-dimensional incompressible flows using the free mesh method. Engineering Analysis with Boundary Elements, 2004, 28(5):425-441.
[213] Idelsohn S R, Onate E, Del Pin F. A Lagrangian meshless finite element method applied to fluid-structure interaction problems. Computers and Structures, 2003, 81(8-11):655-671.
[214] Nam Mai-Duy, Thanh Tran-Cong, Numerical solution of Navier-Stokes equations using multiquadric radial basis function networks. International Journal for Numerical Methods in Fluids, 2001, 37(1):65-86.
[215] Tran-Cong T, Mai-Duy N, Phan-Thien N. BEM-RBF approach for viscoelastic flow analysis. Engineering Analysis with Boundary Elements, 2002, 26(9):757-762.
[216] Lee Kok-Meng, Li Qiang, Sun Hungson. Effects of numerical formulation on magnetic field computation using meshless methods. IEEE Transactions on Magnetics, 2006, 42(9):2164-2171.
[217] Li Qiang, Lee Kok-Meng. An adaptive meshless method for magnetic field computation. IEEE Transactions on Magnetics, 2006, 42(8):1996-2003.
[218] Yong Zhang, Shao K R, Xie D X, Lavers J D. Meshless method based on orthogonal basis for computational electromagnetics. IEEE Transactions on Magnetics, 2005, 41(5):1432-5.
[219] Mai-Duy N. Indirect RBFN method with scattered points for numerical solution of PDEs. Computer Modeling in Engineering & Sciences, 2004, 6(2):209-26.
[220] Mai-Duy Nam. Solving high order ordinary differential equations with radial basis function networks. International Journal for Numerical Methods in Engineering, 2005, 62(6):824-852.
[221] Powell M J D. Radial basis functions for multivariable interpolation: A review. IMA Conference on Algorithms for the Approximation of Founctions and Data; Shrivenham England. England: RMCS, 1985.
[222] Powell M J D. Radial basis function approximations to polynomials. Numerical Analysis 1987, 23-26 June 1987. Dundee, UK: Longman Sci. &, Tech, 1988.
[223] Powell M J D. Some convergence properties of the conjugate gradient method. Mathematical Programming, 1976, 11(1):42-9.
[224] 胡海昌. 弹性力学的变分原理及其应用. 北京: 科学出版社, 1981.
[225] Atluri S N, Zhu Tulong. New concepts in meshless methods. International Journal for Numerical Methods in Engineering, 2000, 47(1):537-556.
[226] Burgess Gary, Mahajerin E. Comparison of the boundary element and superposition methods. Computers and Structures, 1984, 19(5-6):697-705.
[227] Xu Qiang, Sun Huanchun. Virtual boundary element-least square method for solving three dimensional problems of thick shell. Dalian Ligong Daxue Xuebao/Journal of Dalian University of Technology, 1996, 36(4):413-418.
[228] 余德浩. 自然边界元法的数学理论. 北京: 科学出版社, 1993.
[229] Guzelbey Ibrahim H, Tonuc Galip. Boundary element analysis using artificial boundary node approach. Communications in Numerical Methods in Engineering, 2000, 16(11):769-776.
[230] G. H. 戈卢布, C F 范洛恩. 矩阵计算. 袁亚湘译. 北京: 科学出版社, 2001.
[231] Timoshenko S P. Theory of Elasticity. New York: McGraw-Hill, 1973.
[2] 吴永礼. 计算固体力学方法. 北京: 科学出版社, 2003.
[3] 王烈衡. 有限元方法的数学基础. 北京: 科学出版社, 2004.
[4] C.A.布莱比亚. 工程师用边界元法.第一版. 武际可,傅子智译. 北京: 科学出版社, 1986.
[5] Rizzo F J. An integral equation approach to boundary value problems of classical elastostatics. Q. Appl. Math., 1966, xxv(1):83-95.
[6] Brebbia C A, Tells C F, Wrobel L C. Boundary Element Techniques: Theory and Applications in Engineering. Berlin Germany: Springer-Verlag, 1984.
[7] 杨德全. 边界元理论及应用. 北京: 北京理工大学出版社, 2003.
[8] Tells C F. The Boundary Element Method Applied to Inelastic Problems.
[9] 秦荣. 样条边界元方法. 南宁: 广西科学技术出版社, 1988.
[10] 王元淳. 边界元法基础. 上海: 上海交通大学出版社, 1988.
[11] 杜庆华. 边界积分方程方法-边界元. 北京: 高等教育出版社, 1989.
[12] 嵇醒. 边界元法进展及通用程序. 上海: 同济大学出版社, 1997.
[13] 马振华. 现代应用数学手册(计算与数值分析卷). 北京: 清华大学出版社, 2005.
[14] Benito J J, Urena F, Gavete L. Influence of several factors in the generalized finite difference method. Applied Mathematical Modelling, 2001, 25(12):1039-1053.
[15] 邓俊辉译. 计算几何-算法与应用. 北京: 清华大学出版社, 2005.
[16] Liu Wing Kam, Han Weimin, Lu Hongsheng, et al. Reproducing kernel element method. Part I: Theoretical formulation. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12-14):933-951.
[17] Li Shaofan, Lu Hongsheng, Han Weimin, et al. Reproducing kernel element method. Part II: Globally conforming ImCn hierarchies. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12-14):953-987.
[18] Lu Hongsheng, Li Shaofan, Simkins Jr, et al. Reproducing kernel element method.Part III: Generalized enrichment and applications. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12-14):989-1011.
[19] Simkins Jr, Daniel C, Li Shaofan, et al. Reproducing kernel element method. Part IV: Globally compatible Cn(n greater than or equal 1) triangular hierarchy. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12-14):1013-1034.
[20] Sukumar N, Moran B, Belytschko T. The natural element method in solid mechanics. International Journal for Numerical Methods in Engineering, 1998, 43(5):839-87.
[21] Melenk J M, Babuska I. The partition of unity finite element method: basic theory and applications. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4):289-314.
[22] Belytschko T, Krongauz Y, Organ D, et al. Meshless methods: an overview and recent developments. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4):3-47.
[23] Liu Wing-Kam, Li Shaofan, Belytschko Ted. Moving least-square reproducing kernel methods (I) methodology and convergence. Computer Methods in Applied Mechanics and Engineering, 1997, 143(1-2):113-154.
[24] 张雄, 刘岩. 无网格法. 北京: 清华大学出版社, 2004.
[25] Krysl P, Belytschko T, Element-free Galerkin method: convergence of the continuous and discontinuous shape functions. Computer Methods in Applied Mechanics and Engineering, 1997, 148(3-4):257-277.
[26] Shapiro V, Tsukanov I. Meshfree simulation of deforming domains. CAD Computer Aided Design, 1999, 31(7):459-471.
[27] Bruneel Herman C J, De Rycke Igor. Quick trace: A fast algorithm to detect contact. International Journal for Numerical Methods in Engineering, 2002, 54(2):299-316.
[28] Li S, Qian D, Liu W K, et al. A meshfree contact-detection algorithm. Computer Methods in Applied Mechanics and Engineering, 2001, 190(24-25):3271-3292.
[29] Krysl Petr, Belytschko Ted. Analysis of thin shells by the element-free Galerkin method. International Journal of Solids and Structures, 1996, 33(20-22):3057-3080.
[30] Guo Y M, Nakanishi K. A backward extrusion analysis by the rigid-plastic integralless-meshless method. Journal of Materials Processing Technology, 2003, 140(1-3 SPEC):19-24.
[31] Belytschko T, Lu Y Y, Gu L. Crack propagation by element-free Galerkin methods. Engineering Fracture Mechanics, 1995, 51(2):295-315.
[32] Basu P K, Jorge A B, Badri S, et al. Higher-Order Modeling of Continua by Finite-Element, Boundary-Element, Meshless, and Wavelet Methods. Computers and Mathematics with Applications, 2003, 46(1):15-33.
[33] Liew K M, Ng T Y, Wu Y C. Meshfree method for large deformation analysis-a reproducing kernel particle approach. Engineering Structures, 2002, 24(5):543-551.
[34] Lanson N, Vila J P. Meshless methods for conservation laws. Mathematics and Computers in Simulation, 2001, 55(4-6):493-501.
[35] Dolbow John, Moes Nicolas, Belytschko Ted. Modeling fracture in Mindlin-Reissner plates with the extended finite element method. International Journal of Solids and Structures, 2000, 37(48):7161-7183.
[36] Sukumar N, Chopp D L, Moes N, et al. Modeling holes and inclusions by level sets in the extended finite-element method. Computer Methods in Applied Mechanics and Engineering, 2001, 190(46-47):6183-6200.
[37] Belytschko Ted, Krongauz Yury, Fleming Mark, et al. Smoothing and accelerated computations in the element free Galerkin method. Journal of Computational and Applied Mathematics, 1996, 74(1-2):111-126.
[38] Xiao Shaoping, Belytschko T. Stability analysis of particle methods with corrected derivatives. Computers and Mathematics with Applications, 2002, 43(3-5):329-350.
[39] Liu Wing Kam, Chen Yijung, Belytschko Ted, et al. Three reliability methods for fatigue crack growth. Engineering Fracture Mechanics, 1996, 53(5):733-752.
[40] Liu Wing Kam, Jun Sukky. Multiple-scale Reproducing Kernel Particle Methods for large deformation problems. International Journal for Numerical Methods in Engineering, 1998, 41(7):1339-1362.
[41] Chen J S, Pan C, Wu C T. Large deformation analysis of rubber based on a reproducing kernel particle method. Computational Mechanics, 1997, 19(3):211-227.
[42] Liu Z S, Swaddiwudhipong S, Koh C G. High velocity impact dynamic response of structures using SPH method. International Journal of Computational Engineering Science, 2004, 5(2):315-326.
[43] Johnson G R, Stryk R A, Beissel S R. SPH for high velocity impact computations. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4):347-73.
[44] Lam K Y, Shen Y G, Gong S W. A study of axial impact of composite rods using SPH approach. Shock and Vibration, 2001, 8(5):303-312.
[45] Johnson G R, Stryk R A, Beissel S R. Interface effects for SPH impact computations.In Proceedings of the 1996 4th International Conference on Structures Under Shock and Impact, SUSI 96, Jul 1996; Udine, Italy: Computational Mechanics Inc, Billerica, MA, USA, 1996.
[46] Benz W, Asphaug E. Simulations of brittle solids using smooth particle hydrodynamics. Computer Physics Communications, 1995, 87(1-2):253-65.
[47] Ho S L, Yang S, Machado J M, et al., Application of a meshless method in electromagnetics. In Ninth Biennial Electromagnetic Field Computation (CEFC), Jun 4-7 2001; Milwaukee, WI: Institute of Electrical and Electronics Engineers Inc., 2001.
[48] Tran-Cong T, Mai-Duy N, Phan-Thien N. BEM-RBF approach for viscoelastic flow analysis. Engineering Analysis with Boundary Elements, 2002-, 26(9):757-62.
[49] Campbell J, Vignjevic R. Modelling hypervelocity impact in DYNA3D. in Proceedings of 3rd International Conference Dynamics and Control of Structures in Space (ISBN 185312415X), 27-31 May 1996. London, UK: Comput. Mech. Publications, 1996.
[50] Gray J P, Monaghan J J, Swift R P. SPH elastic dynamics. Computer Methods in Applied Mechanics and Engineering, 2001, 190(49-50):6641-62.
[51] Jubelgas M, Springel V, Dolag K. Thermal conduction in cosmological SPH simulations. Monthly Notices of the Royal Astronomical Society, 2004, 351(2):423-35.
[52] Strouboulis T, Copps K, Babuska I. The generalized finite element method. Computer Methods in Applied Mechanics and Engineering, 2001, 190(32-33):4081-193.
[53] 贝新源, 岳宗五. 三维SPH程序及其在斜高速碰撞问题的应用. 计算物理, 1997, 14(2):155-166.
[54] 张锁春. 光滑质点流体动力学(SPH)方法. 计算物理, 1996, 13(4):385-397.
[55] Lucy L B. A numerical approach to the testing of the fission hypothesis. The Astron. Journal, 1977, 8(12):1013-1024.
[56] Monaghan J J. An introduction to SPH. Comput. Phys. Comm., 1988, (48):89-96.
[57] Monaghan J J. Smoothed partic hydrodynamics. Annual Review Astronomics and Astrophysics, 1992,(30):543-574.
[58] Monaghan J J. A shock simulation by the Partical Method SPH. J. Comput. Phys., 1983, (52):374-389.
[59] Randles P W, Smoothed partical hydrodynamics: some recent improvments and applications. Comput. Methods Appl. Mech. Engrg., 1996, (139):375-408.
[60] Nitsche Ludwig C, Zhang Weidong. Atomistic SPH and a Link between Diffusion and Interfacial Tension. AIChE Journal, 2002, 48(2):201-211.
[61] Belytschko T, Fleming M. Smoothing enrichment and contact in the element-free Galerkin method. Computers and Structures, 1999, 71(2):173-195.
[62] Bonet J, Kulasegaram S. A simplified approach to enhance the performance of smooth particle hydrodynamics methods. Applied Mathematics and Computation (New York), 2002, 126(2-3):133-155.
[63] Liew K M, Ng T Y, Zhao X, et al. Harmonic reproducing kernel particle method for free vibration analysis of rotating cylindrical shells. Computer Methods in Applied Mechanics and Engineering, 2002, 191(37-38):4141-4157.
[64] Chen W. Meshfree boundary particle method applied to Helmholtz problems. Engineering Analysis with Boundary Elements, 2002, 26(7):577-581.
[65] Li Shaofan, Liu Wing Kam. Meshfree and particle methods and their applications. Applied Mechanics Reviews, 2002, 55(1):1-34.
[66] Aluru N R. Point collocation method based on reproducing kernel approximations. International Journal for Numerical Methods in Engineering, 2000, 47(6):1083-1121.
[67] Liew K M, Zou G P, Rajendran S. A spline strip kernel particle method and its application to two-dimensional elasticity problems. International Journal for Numerical Methods in Engineering, 2003, 57(5):599-616.
[68] Cueto-Felgueroso L, Colominas I, Mosqueira G, et al. On the Galerkin formulation of the smoothed particle hydrodynamics method. International Journal for Numerical Methods in Engineering, 2004, 60(9):1475-1512.
[69] Kim Do Wan, Kim Yongsik. Point collocation methods using the fast moving least-square reproducing kernel approximation. International Journal for Numerical Methods in Engineering, 2003, 56(10):1445-1464.
[70] Han Young Yoon, Koshizuka S, Oka Y. A particle-gridless hybrid method for incompressible flows. International Journal for Numerical Methods in Fluids, 1999, 30(4):407-24.
[71] Belytschko T, Krongauz Y, Dolbow J, et al. On the completeness of meshfree particle methods. International Journal for Numerical Methods in Engineering, 1998, 43(5):785-819.
[72] Chen Jiun-Shyan, Pan Chunhui, Wu Cheng-Tang, et al. Reproducing kernel particle methods for large deformation analysis of non-linear structures. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4):195-227.
[73] Schoenberg I J. Cardinal Spline Interpolation. Bristol England: JW Arrowsmith Ltd., 1973.
[74] Carl de Boor. A Practical Guide to Splines. New York: Springer-Verlag Inc., 1978.
[75] Kouibia A, Pasadas M. Approximation of surfaces by fairness bicubic splines. Advances in Computational Mathematics, 2004, 20(1-3):87-103.
[76] Dierckx P. Algorithm for fitting data over a circle use tensor product splines. Journal of Computational and Applied Mathematics, 1986, 15(2):161-173.
[77] Hollig Klaus, Reif Ulrich. Nonuniform web-splines. Computer Aided Geometric Design, 2003, 20(5):277-294.
[78] Onah E S. On direct methods for the discretization of a heat-conduction equation using spline functions. Applied Mathematics and Computation, 1997, 85(1):87-96.
[79] Hongzhi Zhong. Spline-based differential quadrature for fourth order differential equations and its application to Kirchhoff plates. Applied Mathematical Modelling, 2004, 28(4):353-66.
[80] 石钟慈. 样条有限元法. 计算数学, 1976,(1): 50-72.
[81] 刘效尧. 样条函数与结构力学. 北京: 人民交通出版社, 1990.
[82] Hollig K. Finite Element Methods with B-splines. Stuttgart Germany: SIAM, Philadelphia, 2003.
[83] Shen Pengcheng, He Peixiang. The multivariable spline finite element method. Acta Mechanica Solida Sinica, 1994, 15(3):234-43.
[84] Mizusawa T, Kajita T, Naruoka M. Vibration of skew plates by using B-spline functions. 1979, 62(2):301-308.
[85] Antes H. Bicubic fundamental splines in plate bending. 1974, 8(3):503-511.
[86] Wray Jonathan, Green Gary G. R. Neural networks, approximation theory, and finite precision computation. Neural Networks, 1995, 8(1):31-37.
[87] 飞思科技产品研发中心. 神经网络理论与Matlab7.0实现. 北京: 电子工业出版社, 2005.
[88] Braun J. A numerical method for solving partial differential equations on highly irregular grids. Nature, 1995, 376(24):655-660.
[89] Niu Zhongrong, Zhou Huanlin. The natural boundary integral equation in potential problems and regularization of the hypersingular integral. Computers and Structures, 2004, 82(2-3):315-323.
[90] Martinez M A, Cueto E, Doblare M, et al. Natural element meshless simulation of flows involving short fiber suspensions. Journal of Non-Newtonian Fluid Mechanics, 2003, 115(1):51-78.
[91] Farwig R. Multivariate interpolation of arbitrarily spaced data by moving least squares methods. Journal of Computational and Applied Mathematics, 1986, 16(1):79-93.
[92] Gu Lei. Moving kriging interpolation and element-free Galerkin method. International Journal for Numerical Methods in Engineering, 2003, 56(1):1-11.
[93] Hao Su, Park Harold S, Liu Wing Kam. Moving particle finite element method. International Journal for Numerical Methods in Engineering, 2002, 53(8):1937-1958.
[94] Dilts Gary A. Moving-least-squares-particle hydrodynamics - I. Consistency and stability. International Journal for Numerical Methods in Engineering, 1999, 44(8):1115-1155.
[95] Hao Su, Liu Wing Kam, Belytschko Ted. Moving particle finite element method with global smoothness. International Journal for Numerical Methods in Engineering, 2004, 59(7):1007-1020.
[96] Liew K M, Huang Y Q, Reddy, et al. Moving least squares differential quadrature method and its application to the analysis of shear deformable plates. International Journal for Numerical Methods in Engineering, 2003, 56(15):2331-2351.
[97] Dilts GaryA. Moving least-squares particle hydrodynamics II: Conservation and boundaries. International Journal for Numerical Methods in Engineering, 2000, 48(10):1503-1524.
[98] Gingold R A. Smoothed partical hydrodynamics: theory and applications to non-spherical stars. Mon. Not. Roy. Astrou. Soc., 1977, (18):375-389.
[99] Rao B N, Rahman S. A coupled meshless-finite element method for fracture analysis of cracks. International Journal of Pressure Vessels and Piping, 2001, 78(9):647-657.
[100] Chen Jiun-Shyan, Wang Hui-Ping. New boundary condition treatments in meshfree computation of contact problems. Computer Methods in Applied Mechanics and Engineering, 2000, 187(3): 441-468.
[101] Bulatovic R M. On the heavily damped response in viscously damped dynamicsystems. Journal of Applied Mechanics, Transactions ASME, 2004, 71(1):131-134.
[102] Bhushan Bharat, Peng Wei. Contact mechanics of multilayered rough surfaces. Applied Mechanics Reviews, 2002, 55(5):435-479.
[103] Campo Antonio, Morrone Biagio. Meshless approach for computing the heat liberation from annular fins of tapered cross section. Applied Mathematics and Computation (New York), 2004, 156(1):137-144.
[104] Carpinteri A, Ferro G, Ventura G. The partition of unity quadrature in element-free crack modelling. Computers and Structures, 2003, 81(18-19):1783-1794.
[105] Zhao X., Liew K M, Ng T Y. Vibration analysis of laminated composite cylindrical panels via a meshfree approach. International Journal of Solids and Structures, 2003, 40(1):161-180.
[106] Tsukanov I, Shapiro V, Zhang S. A meshfree method for incompressible fluid dynamics problems. International Journal for Numerical Methods in Engineering, 2003, 58(1):127-158.
[107] Charoenphan S, Plesha M E, Bank L C. Simulation of crack growth in composite material shell structures. International Journal for Numerical Methods in Engineering, 2004, 60(14): 2399-417.
[108] Zavarise G, Wriggers P. Contact with friction between beams in 3-D space. International Journal for Numerical Methods in Engineering, 2000, 49(8):977-1006.
[109] Sladek J, Sladek V. A meshless method for large deflection of plates. Computational Mechanics, 2003, 30(2):155-163.
[110] Belytschko T, Organ D, Gerlach C. Element-free Galerkin methods for dynamic fracture in concrete. Computer Methods in Applied Mechanics and Engineering, 2000, 187(3):385-399.
[111] Belytschko Ted, Babuska Ivo, Tinsley, et al. Research directions in computational mechanics. Computer Methods in Applied Mechanics and Engineering, 2003, 192(7-8):913-922.
[112] Zhou X, Hon Y C, Cheung K F. A grid-free, nonlinear shallow-water model with moving boundary. BEM in China: Engineering Analysis with Boundary Elements, 2004, 28(8):967-973.
[113] Belytschko T, Lu Y Y, Gu L, Crack propagation by element free Galerkin methods. Proceedings of the 1993 ASME Winter Annual Meeting, Nov 28-Dec 3 1993. New Orleans, LA, USA: Publ by ASME, New York, NY, USA, 1993.
[114] Xu Yonglin, Moran Brian, Belytschko Ted. Uncoupled characteristics of three-dimensional planar cracks. International Journal of Engineering Science, 1998, 36(1):33-48.
[115] Lancaster P. "Moving weighted least-squares method" in Polynomial and Spline Approximation. NATO Advanced Study InstituteSeries C., 1979, 103-120.
[116] Frank G N R. Smooth interpolation os Large Sets of Scattered Data. Technical Report. 1979.
[117] Lancaster K S P. Surface Generated by Moving Least Squares Method. Mathematics of Computation, 1981, 37(155):141-158.
[118] Nayroles G T B. Generalizing the finite element method: diffuse approximation and diffuse elements. Computational Mechanics, 1992, 10:307-318.
[119] Lu Y Y, Gu L, Belytschko T. Element-free Galerkin Methods. International Journal for numerical methods in engineering, 1994, 37:229-256.
[120] Belytschko T, Lu YY. A new implementation of the element free Galerkin method. Comput. Methods Appl. Mech. Engrg., 1994, 113:397-414.
[121] Beissel S, Belytschko Ted. Nodal integration of the element-free Galerkin method. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4):49-74.
[122] Duflot M, Nguyen-Dang Hung. A truly meshless Galerkin method based on a moving least squares quadrature. Communications in Numerical Methods in Engineering, 2002, 18(6):441-449.
[123] Krysl P, Belytschko T. The element free Galerkin method for dynamic propagation of arbitrary 3-D cracks. International Journal for Numerical Methods in Engineering, 1999, 44(6):767-800.
[124] Onate E, Perazzo F, Miquel J. A finite point method for elasticity problems. Computers and Structures, 2001, 79(22-25):2151-2163.
[125] Lohner Rainald, Sacco Carlos, Onate Eugenio, et al. A finite point method for compressible flow. International Journal for Numerical Methods in Engineering, 2002, 53(8):1765-1779.
[126] Duarte C A, Oden J T. An h-p adaptive method using clouds. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4):237-62.
[127] Liszka T J, Duarte C A, Tworzydlo W W. hp-meshless cloud method. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4):263-288.
[128] Li Hua, Ng T Y, Cheng J Q, et al. Hermite-Cloud: A novel true meshless method.Computational Mechanics, 2003, 33(1):30-41.
[129] Aluru N R, Li G. Finite cloud method: A true meshless technique based on fixed reproducing kernel approximation. International Journal for Numerical Methods in Engineering, 2001, 50(10):2373-2410.
[130] Meschke G. Finite Element Modelling of Cracks Based on the Partition of Unity Method. Proc. Appl. Math. Mech., 2003, 226-227.
[131] Fan S C, Liu X, Lee C K. Enriched partition-of-unity finite element method for stress intensity factors at crack tips. Computers and Structures, 2004, 82(4-5):445-461.
[132] Simone A. Partition of unity-based discontinuous elements for interface phenomena: Computational issues. Communications in Numerical Methods in Engineering, 2004, 20(6):465-478.
[133] Alves Marcelo Krajnc, Rossi Rodrigo. A modified element-free Galerkin method with essential boundary conditions enforced by an extended partition of unity finite element weight function. International Journal for Numerical Methods in Engineering, 2003, 57(11): 1523-1552.
[134] Carpinteri A, Ferro G, Ventura G. The partition of unity quadrature in meshless methods. International Journal for Numerical Methods in Engineering, 2002, 54(7):987-1006.
[135] Munts E A, Hulshoff S J, de Borst R. The partition-of-unity method for linear diffusion and convection problems: Accuracy, stabilization and multiscale interpretation. International Journal for Numerical Methods in Fluids, 2003, 43(2):199-213.
[136] Dolbow John, Moes Nicolas, Belytschko Ted. Discontinuous enrichment in finite elements with a partition of unity method. Finite Elements in Analysis and Design, 2000, 36(3):235-260.
[137] Zu T, zhang J. A local boundary integral equation (LIBE) method in computational mechanics, a meshless discretization approach. Comp. Mech., 1998, 21:223-235.
[138] Atluri S N, Zhu T. A new meshless local Petrov-Galerkin (MLPG) approach to nonlinear problems in computer modeling and simulation. Computer Modeling and Simulation in Engineering, 1998, 3(3):187-96.
[139] Atluri S N, Zhu T. New Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics. Computational Mechanics, 1998, 22(2):117-127.
[140] De S, Bathe K J. The method of finite spheres with improved numerical integration.Computers and Structures, 2001, 79(22-25):2183-2196.
[141] Liu G R. A point assembly method for stress analysis for two-dimensional solids. International Journal of Solids and Structures, 2002, 39(1):261-276.
[142] Gu Y T, Liu G R. Hybrid boundary point interpolation methods and their coupling with the element free Galerkin method. Engineering Analysis with Boundary Elements, 2003, 27(9):905-917.
[143] Cheng M, Liu G R. A novel finite point method for flow simulation. International Journal for Numerical Methods in Fluids, 2002, 39(12):1161-1178.
[144] Wang J G, Liu G R. A point interpolation meshless method based on radial basis functions. International Journal for Numerical Methods in Engineering, 2002, 54(11):1623-1648.
[145] Liu G R, Gu Y T. A point interpolation method for two-dimensional solids. International Journal for Numerical Methods in Engineering, 2001, 50(4):937-951.
[146] Sibson R. A brief description of nature neighbor interpolation. 1981,21-36.
[147] Sibson R, A vector identity for a Dirichlet tesselation. Math. Proc., 1980, 87:151-155.
[148] Watson D F. Neighborhood-based interpolation. Geobyte, 1987, 2(2):12-16.
[149] Traversoni L. An algorithm for nature spline interpolation. Numer. Algorithms, 1993, (5):63-70.
[150] Traversoni L. Nature neighbor finite elements. Int.Con.on Hydroaulic Engineering Software, Hydrosoft Proc., 1994.
[151] Wang Kai, Zhou Shenjie, Shan Guojun. The natural neighbour Petrov-Galerkin method for elasto-statics. International Journal for Numerical Methods in Engineering, 2005, 63(8):1126-1145.
[152] Mai-Duy Nam, Tran-Cong Thanh. Approximation of function and its derivatives using radial basis function networks. Applied Mathematical Modelling, 2003, 27(3):197-220.
[153] Mai-Duy N, Tran-Cong T. Solving biharmonic problems with scattered-point discretization using indirect radial-basis-function networks. Engineering Analysis with Boundary Elements, 2006, 30(2):77-87.
[154] Mai-Duy Nam. An effective spectral collocation method for the direct solution of high-order ODEs. Communications in Numerical Methods in Engineering, 2006, 22(6):627-642.
[155] Nam Mai-Duy, Thanh Tran-Cong. Approximation of function and its derivativesusing radial basis function networks. Applied Mathematical Modelling, 2003, 27(3):197-220.
[156] Nam Mai-Duy, Thanh Tran-Cong. Numerical solution of differential equations using multiquadric radial basis function networks. Neural Networks, 2001-, 14(2):185-99.
[157] Mai-Duy Nam, Tran-Cong Thanh. RBF interpolation of boundary values in the BEM for heat transfer problems. International Journal of Numerical Methods for Heat and Fluid Flow, 2003, 13(5-6):611-632.
[158] Mai-Duy Nam, Tran-Cong Thanh. An effective RBFN-boundary integral approach for the analysis of natural convection flow. International Journal for Numerical Methods in Fluids, 2004, 46(5):545-568.
[159] Mai-Duy Nam, Tanner R I. Computing non-Newtonian fluid flow with radial basis function networks. International Journal for Numerical Methods in Fluids, 2005, 48(12):1309-1336.
[160] Mai-Duy N, Tran-Cong T. Boundary integral-based domain decomposition technique for solution of Navier Stokes equations. Computer Modeling in Engineering & Sciences, 2004, 6(1):59-75.
[161] Mai-Duy Nam, Tran-Cong Thanh. Neural networks for BEM analysis of steady viscous flows. International Journal for Numerical Methods in Fluids, 2003, 41(7):743-763.
[162] Mai-Duy N, Tran-Gong T. Boundary integral-based domain decomposition technique for solution of Navier Stokes equations. CMES - Computer Modeling in Engineering and Sciences, 2004, 6(1):59-75.
[163] Mai-Duy N, Tran-Cong T. Numerical solution of Navier-Stokes equations using multiquadric radial basis function networks. International Journal for Numerical Methods in Fluids, 2001, 37(1):65-86.
[164] Mai-Duy N, Tran-Cong T. Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson's equations. Engineering Analysis with Boundary Elements, 2002, 26(2):133-156.
[165] Nam Mai-Duy, Thanh Tran-Cong. Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson's equations. Engineering Analysis with Boundary Elements, 2002, 26(2):133-56.
[166] Prime M B, Sebring R J, Edwards J M, et al. Laser surface-contouring and spline data-smoothing for residual stress measurement. Experimental Mechanics, 2004, 44(2):176-184.
[167] Hughes Matt C, Westwick David T. Identification of IIR Wiener systems with spline nonlinearities that have variable knots. IEEE Transactions on Automatic Control, 2005, 50(10): 1617-1622.
[168] Hardy R L. Theory and applications of the multiquadric-biharmonic method. 20 years of discovery 1968-88. Computers & Mathematics with Applications, 1990, 19 (8-9):163-208.
[169] Atluri S N, Sladek J, Sladek V, et al. Local boundary integral equation (LBIE) and it's meshless implementation for linear elasticity. Computational Mechanics, 2000, 25(2-3):180-198.
[170] Zhu T, Zhang J D, Atluri S N. Local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach. Computational Mechanics, 1998, 21(3):223-235.
[171] Zhu T, Zhang J, Atluri S N. Meshless local boundary integral equation (LBIE) method for solving nonlinear problems. Computational Mechanics, 1998, 22 (2):174-186.
[172] Gowrishankar Ramesh, Mukherjee Subrata. A 'pure' boundary node method for potential theory. Communications in Numerical Methods in Engineering, 2002, 18(6):411-427.
[173] Chati Mandar K, Mukherjee S. Boundary node method for three-dimensional problems in potential theory. International Journal for Numerical Methods in Engineering, 2000, 47(9):1523-1547.
[174] Miao Yu, Wang Yuan-Han. Meshless analysis for three-dimensional elasticity with singular hybrid boundary node method. Applied Mathematics and Mechanics (English Edition), 2006, 27(5):673-681.
[175] Miao Yu, Mao Feng, Wang Yuan-Han, Zhang Jun, et al.. Hybrid boundary node method in geotechnical engineering. Yantu Lixue/Rock and Soil Mechanics, 2005, 26(9):1452-1455.
[176] Chen W, Tanaka M. A meshless, integration-free, and boundary-only RBF technique. Computers and Mathematics with Applications, 2002, 43(3-5):379-391.
[177] Liu G R, Gu Y T. Boundary meshfree methods based on the boundary pointinterpolation methods. Twenty-Fourth International Conference on the Boundary Element Method Incorporating Meshless Solution Seminar, BEM XXIV, Jun 17-19 2002; Sintra, Portugal: WITPress, Southampton, United Kingdom, 2002.
[178] Wu C-K C, Plesha M E. Essential boundary condition enforcement in meshless methods: Boundary flux collocation method. International Journal for Numerical Methods in Engineering, 2002, 53(3):499-514.
[179] Yagawa G. Node-by-node parallel finite elements: A virtually meshless method. International Journal for Numerical Methods in Engineering, 2004, 60(1):69-102.
[180] 孙海涛, 王元汉. 无网格法在我国的起源. 见戴念祖主编, 力学史与方法论, 第二届力学史与方法论会议, 上海. 上海: 上海大学出版社, 2005.
[181] Gordon W J, Wixom J A. Shepard's method of `metric interpolation' to bivariate and multivariate interpolation. Mathematics of Computation, 1978, 32(141):253-64 .
[182] Chen J K, Beraun J E. A generalized smoothed particle hydrodynamics method for nonlinear dynamic problems. Computer Methods in Applied Mechanics and Engineering, 2000, 190(1-2):225-39.
[183] Chen Jiun-Shyan, Pan Chunhui, Wu Cheng-Tang. Application of Reproducing Kernel Particle Method to large deformation contact analysis of elastomers. Rubber Chemistry and Technology, 1998, 71(2):191-213.
[184] Yoon Sangpil, Chen Jiun-Shyan. Accelerated meshfree method for metal forming simulation. Finite Elements in Analysis and Design, 2002, 38(10):937-948.
[185] Liu W K, Lua Y J, Chen Y, et al. Study of three reliability methods for fatigue crack growth. Proceedings of the IUTAM Symposium, Jun 7-10 1993; San Antonio, TX, USA: Springer-Verlag New York Inc., 1993.
[186] Lu Y Y, Belytschko T, Tabbara M. Element-free Galerkin method for wave propagation and dynamic fracture. Computer Methods in Applied Mechanics and Engineering, 1995, 126(1-2): 131-153.
[187] Liu W K. Reproducing kernel partical methods. International journal for numerical methods in fluids, 1995, 20:1081-1106.
[188] Johnson G R, Beissel S R. Normalized smoothing functions for SPH impact computations. International Journal for Numerical Methods in Engineering, 1996, 39(16):2725-41.
[189] Johnson G R. Artificial viscosity effects for SPH impact computations. InternationalJournal of Impact Engineering, 1996, 18(5):477-88.
[190] Johnson G R, Beissel S R. Normalized smoothing functions for SPH impact computations. International Journal for Numerical Methods in Engineering, 1996, 39(16):2725-2741.
[191] Johnson Gordon R, Stryk Robert A, Beissel Stephen R. SPH for high velocity impact computations. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4):347-373.
[192] Mao Yi-ming, Wu Wen-yuan, Chen Guang-lin, et al. Numerical simulation of high velocity impact problems with SPH method. Journal of PLA University of Science and Technology (Natural Science Edition), 2003, 4(5):84-7.
[193] Kikuchi Masanori, Miyamoto Masayuki. Numerical simulations of impact crush/buckling of circular tube using SPH method. Fracture and Strength of Solids VI: Key Engineering Materials, 2006, 306-308 I:697-702.
[194] Yan Xiaojun, Zhang Yuzhu, Nie Jingxu. Numerical simulation of space debris hypervelocity impact using SPH method. Beijing Hangkong Hangtian Daxue Xuebao/Journal of Beijing University of Aeronautics and Astronautics, 2005, 31(3):351-354.
[195] Li Jin-Zhu, Zhang Qing-Ming, Long Ren-Rong. SPH simulation of hypervelocity impacts. Journal of Beijing Institute of Technology, 2004, 13(3):266-9.
[196] Libersky L D, Randles P W, Carney T C, et al. Recent improvements in SPH modeling of hypervelocity impact. Proceedings of the 1996 Symposium on Hypervelocity Impact. Part 2 (of 2), Oct 8-10 1996: International Journal of Impact Engineering, 1997, 20(6-10):525-532.
[197] Li Jin-Zhu, Zhang Qing-Ming, Long Ren-Rong. SPH simulation of hypervelocity impacts. Journal of Beijing Institute of Technology (English Edition), 2004, 13(3):266-269.
[198] Lukyanov A A, Reveles J R, Vignjevic R, et al. Simulation of hypervelocity debris impact and spacecraft shielding performance. 4th European Conference on Space Debris, Apr 18-20 2005. Darmstadt, Germany: European Space Agency, Noordwijk, 2200 AG, Netherlands, 2005.
[199] Brunger M J, Campbell L, Cartwright D C. Studying the effects of electron-impact excitation in the upper atmosphere. Trends in Applied Spectroscopy, 2004, 5:39-54.
[200] Campbell R. Vignjevic. Lagrangian hydrocode modelling of hypervelocity impact onspacecraft. Third International Conference in Computational Structures Technology. Advances in Computational Methods for Simulation, 21-23 Aug. 1996; Budapest, Hungary: Civil-Comp Press, 1996.
[201] Vandemark D, Chapron B, Elfouhaily T, et al. Impact of high-frequency waves on the ocean altimeter range bias. Journal of Geophsical Research-Part C-Oceans, 2005, 110(C11):12 pp.
[202] Kanel G I, Ivanov M F, Parshikov A N. Computer simulation of the heterogeneous materials response to the impact loading. Hypervelocity Impact. 1994 Symposium, 17-19 Oct. 1994: International Journal of Impact Engineering; Sante Fe, NM, USA : Elsevier, 1995.
[203] Parshikov Anatoly N, Medin Stanislav A, Loukashenko Igor I, et al. Improvements in SPH method by means of interparticle contact algorithm and analysis of perforation tests at moderate projectile velocities. International Journal of Impact Engineering, 2000, 24(8):779-796.
[204] Dolbow J, Moes N, Belytschko T. An extended finite element method for modeling crack growth with frictional contact. Computer Methods in Applied Mechanics and Engineering, 2001, 190(51-52):6825-6846.
[205] Kremmer M, Favier J F. A method for representing boundaries in discrete element modelling - Part I: Geometry and contact detection. International Journal for Numerical Methods in Engineering, 2001, 51(12):1407-1421.
[206] Pandolfi A, Kane C, Marsden J E, et al. Time-discretized variational formulation of non-smooth frictional contact. International Journal for Numerical Methods in Engineering, 2002, 53(8):1801-1829.
[207] Wang F, Cheng J, Yao Z. FFS contact searching algorithm for dynamic finite element analysis. International Journal for Numerical Methods in Engineering, 2001, 52(7):655-672.
[208] Jiang F, Oliveira, Sousa A C M. SPH simulation of low reynolds number planar shear flow and heat convection. Materialwissenschaft und Werkstofftechnik, 2005, 36(10):613-619.
[209] Sakai Yuzuru, Yang Zong Yi, Jung Young Guan. Incompressible viscous flow analysis by SPH. Nippon Kikai Gakkai Ronbunshu, B Hen/Transactions of the Japan Society of Mechanical Engineers, Part B, 2004, 70(696):1949-1956.
[210] Monaghan J J. SPH compressible turbulence. Monthly Notices of the RoyalAstronomical Society, 2002, 335(3):843-52.
[211] Cummins S J, Rudman M J. Truly incompressible SPH. Proceedings of the 1998 ASME Fluids Engineering Division Summer Meeting, Jun 21-25 1998; Washington, DC, USA: ASME, Fairfield, NJ, USA, 1998.
[212] Fujisawa Toshimitsu, Ito Satoshi, Inaba Masakazu, et al. Node-based parallel computing of three-dimensional incompressible flows using the free mesh method. Engineering Analysis with Boundary Elements, 2004, 28(5):425-441.
[213] Idelsohn S R, Onate E, Del Pin F. A Lagrangian meshless finite element method applied to fluid-structure interaction problems. Computers and Structures, 2003, 81(8-11):655-671.
[214] Nam Mai-Duy, Thanh Tran-Cong, Numerical solution of Navier-Stokes equations using multiquadric radial basis function networks. International Journal for Numerical Methods in Fluids, 2001, 37(1):65-86.
[215] Tran-Cong T, Mai-Duy N, Phan-Thien N. BEM-RBF approach for viscoelastic flow analysis. Engineering Analysis with Boundary Elements, 2002, 26(9):757-762.
[216] Lee Kok-Meng, Li Qiang, Sun Hungson. Effects of numerical formulation on magnetic field computation using meshless methods. IEEE Transactions on Magnetics, 2006, 42(9):2164-2171.
[217] Li Qiang, Lee Kok-Meng. An adaptive meshless method for magnetic field computation. IEEE Transactions on Magnetics, 2006, 42(8):1996-2003.
[218] Yong Zhang, Shao K R, Xie D X, Lavers J D. Meshless method based on orthogonal basis for computational electromagnetics. IEEE Transactions on Magnetics, 2005, 41(5):1432-5.
[219] Mai-Duy N. Indirect RBFN method with scattered points for numerical solution of PDEs. Computer Modeling in Engineering & Sciences, 2004, 6(2):209-26.
[220] Mai-Duy Nam. Solving high order ordinary differential equations with radial basis function networks. International Journal for Numerical Methods in Engineering, 2005, 62(6):824-852.
[221] Powell M J D. Radial basis functions for multivariable interpolation: A review. IMA Conference on Algorithms for the Approximation of Founctions and Data; Shrivenham England. England: RMCS, 1985.
[222] Powell M J D. Radial basis function approximations to polynomials. Numerical Analysis 1987, 23-26 June 1987. Dundee, UK: Longman Sci. &, Tech, 1988.
[223] Powell M J D. Some convergence properties of the conjugate gradient method. Mathematical Programming, 1976, 11(1):42-9.
[224] 胡海昌. 弹性力学的变分原理及其应用. 北京: 科学出版社, 1981.
[225] Atluri S N, Zhu Tulong. New concepts in meshless methods. International Journal for Numerical Methods in Engineering, 2000, 47(1):537-556.
[226] Burgess Gary, Mahajerin E. Comparison of the boundary element and superposition methods. Computers and Structures, 1984, 19(5-6):697-705.
[227] Xu Qiang, Sun Huanchun. Virtual boundary element-least square method for solving three dimensional problems of thick shell. Dalian Ligong Daxue Xuebao/Journal of Dalian University of Technology, 1996, 36(4):413-418.
[228] 余德浩. 自然边界元法的数学理论. 北京: 科学出版社, 1993.
[229] Guzelbey Ibrahim H, Tonuc Galip. Boundary element analysis using artificial boundary node approach. Communications in Numerical Methods in Engineering, 2000, 16(11):769-776.
[230] G. H. 戈卢布, C F 范洛恩. 矩阵计算. 袁亚湘译. 北京: 科学出版社, 2001.
[231] Timoshenko S P. Theory of Elasticity. New York: McGraw-Hill, 1973.