随机无网格伽辽金法在若干工程问题中的应用研究
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摘要
由于工程结构系统中存在着大量的不确定因素,这些不确定因素通常被描述为空间的随机参数,为了更好的处理这些参数的随机性,近年来出现了一种新兴的数值方法——随机无网格伽辽金法(SEFGM)。它采用移动最小二乘法构造形函数,从能量泛函的弱变分形式中得到控制方程,并用拉格朗日乘子满足位移边界条件,从而得到偏微分方程的数值解。这种方法可有效地处理结构分析中所涉及的有关参数的随机性和工程结构的可靠性。在处理数据时,不需要划分单元,剖分网格,简化了数据处理,提高了计算速度,并可解决某些随机有限元法所不能解决的问题。
论文首先阐述了弹性力学的一些相关理论,给出了弹性力学的基本控制方程(平衡方程,几何方程,物理方程)及应力、应变和位移的相关概念,为后面力学应用中求解具体问题,探究应力、应变和节点位移的分布规律提供了理论依据。
其次,论文运用无网格伽辽金法和小参数摄动技术构造摄动随机无网格伽辽金法,这种方法有效的避免了有限元法在处理复杂结构时需要重新划分网格或网格重构的缺陷。然后,论文运用摄动随机无网格伽辽金法分别对含裂纹的平面结构的可靠性和含节理的岩体滑坡的可靠性进行了分析,通过与有限元法结果的比较,证明了方法的有效性。
最后对全文做了一个总结并指明了以后的研究方向。
There are many uncertainties which are normally described as a random function of space in the engineering structure system. In recent years, a new numerical method, stochastic element-free Galerkin method (SFEGM), has bee- n found to be better to study the structure containing the response problem of random variables. The shape function is constructed by Moving Least Squares Method in EFGM, and control equations are produced from the weak form of variational equation. Lagrange multipliers are used to satisfy displacement bou- ndary conditions, and make out numerical solutions. SFEGM can effectively handle structure analysis of related parameters of the randomness and engineer- ing structure reliability. This method simplifies data processing and improves computing speed without grid. And it can solve the problems that can not solve by stochastic finite element method.
The paper first expounds some relevant theory of elastic mechanics, the basic control equation (the balance equation, the geometric equations, and the physical equations) of elastic mechanics and the related concept of the stress, strain and the displacement , provides theory basis for solving specific problems in the application of mechanics and exploring stress, strain and node displayce- ment distribution.
Secondly, the paper combining SEFGM with small parameter perturbation technology constructs perturbation stochastic element-free Galerkin method (P SEFGM), this method has effectively avoided flaw about the finite element me- thod dividing the grid or reconstructing grid when the processing multiple struc- ture needs. Then, with perturbation stochastic element-free Galerkin method, the paper analysis the reliability of the mix-mode crack with radom variable and rock slide with joints, respectively. From the comparisons of the outcomes of PSEFGM with stochastic finite element method results, this method is proved to be applicable.
Finally, a summary of the paper and the future research directions are forecasted.
论文首先阐述了弹性力学的一些相关理论,给出了弹性力学的基本控制方程(平衡方程,几何方程,物理方程)及应力、应变和位移的相关概念,为后面力学应用中求解具体问题,探究应力、应变和节点位移的分布规律提供了理论依据。
其次,论文运用无网格伽辽金法和小参数摄动技术构造摄动随机无网格伽辽金法,这种方法有效的避免了有限元法在处理复杂结构时需要重新划分网格或网格重构的缺陷。然后,论文运用摄动随机无网格伽辽金法分别对含裂纹的平面结构的可靠性和含节理的岩体滑坡的可靠性进行了分析,通过与有限元法结果的比较,证明了方法的有效性。
最后对全文做了一个总结并指明了以后的研究方向。
There are many uncertainties which are normally described as a random function of space in the engineering structure system. In recent years, a new numerical method, stochastic element-free Galerkin method (SFEGM), has bee- n found to be better to study the structure containing the response problem of random variables. The shape function is constructed by Moving Least Squares Method in EFGM, and control equations are produced from the weak form of variational equation. Lagrange multipliers are used to satisfy displacement bou- ndary conditions, and make out numerical solutions. SFEGM can effectively handle structure analysis of related parameters of the randomness and engineer- ing structure reliability. This method simplifies data processing and improves computing speed without grid. And it can solve the problems that can not solve by stochastic finite element method.
The paper first expounds some relevant theory of elastic mechanics, the basic control equation (the balance equation, the geometric equations, and the physical equations) of elastic mechanics and the related concept of the stress, strain and the displacement , provides theory basis for solving specific problems in the application of mechanics and exploring stress, strain and node displayce- ment distribution.
Secondly, the paper combining SEFGM with small parameter perturbation technology constructs perturbation stochastic element-free Galerkin method (P SEFGM), this method has effectively avoided flaw about the finite element me- thod dividing the grid or reconstructing grid when the processing multiple struc- ture needs. Then, with perturbation stochastic element-free Galerkin method, the paper analysis the reliability of the mix-mode crack with radom variable and rock slide with joints, respectively. From the comparisons of the outcomes of PSEFGM with stochastic finite element method results, this method is proved to be applicable.
Finally, a summary of the paper and the future research directions are forecasted.
引文
1王勖成,邵敏.有限单元法基本原理和数值方法(第2版).北京:清华大学出版社, 1997:1-2
2李景湧.有限元法.北京:邮电大学出版社, 2004:51-87
3谢贻权,何福保.弹性和塑性力学中的有限单元法.北京:机械工业出版社, 1981:1-5
4薛守义.有限单元法.北京:中国建材工业出版社, 2005.2
5顾元通,丁桦.无网格法及其最新进展.力学进展, 2005,35(3):323-332
6曹国金,姜弘道.无单元法研究和应用现状及动态.力学进展, 2002,32(4):526-534
7 L. B. Lucy. A Numerical Approach to the Testing of the Fission Hypothesis. The Astr- on Astron J, l977,8(12):1013-1020
8 C. T. Dyka. Addressing Tension Instability in SPH Method. Technical Report NRL/ MR6384, NRL, 1994
9 J. W. Swegle, D. L. Hicks, S. W. Attaway. Smoothed Partical Hydrodynamics Stability Analysis. J Compout Phys, 1995,116:123-134
10 G. R. Johnson, R. A. Stryk, S. R. Beisssel. SPH for High Velocity Impact Computatio- ns Computer Methods in Applied Mechanics and Engineering, 1996,139:347-373
11 J. J. Monaghan. Smoothed Particle Hydrodynamics. Annual Review Astronomics and Astrophysics, 1992,30:543-574
12 J. W. Swegle, Hicks, S. W. Attaway. Smoothed Particle Hydrodynamics Stability Ana- lysis. J Comput Phys, 1995,116:123-134
13 J. K. Chen, J. E. Beraun, C. J. Jih. An Improvement for Tensile Instability in Smooth- ed Particke Hydrodynamics. Comput Mech, 1999,23:279-287
14 C. T. Dyka. Addressing Tension Insrability in SPH Methods. Technical Report NRL/ MR/6384, NRL, 1994
15 J. W. Swegle, S. W. Attaway. On the Feasibility of Using Smoothed Particle Hydrody- namics for under Water Explosion Calculations. Comput Mech, 1995,17:151- 168
16 G. R. Johnson, R. A. Stryk, S. R. Beissel. SPH for High Velocity Impact Computations. Comput Methods Appl Mech Engrg, 1996,139:347-373
17 L. D. Libersky, A. G. Petschek, etal. High Strain Lagrangian Hydrodynamics: A Three-dimensional SPH Code for Dynamic Mmaterizl Response. Comput Phys, 1993, 109:67-75
18 G. R. Johnson, S. R. Beussel. Normalized Smoothing Functions for SPH Impact Com- putations. Int J Numer Methods Engrg, 1996,39:2725-2741
19张锁春.光滑质点流体动力学(SPH)方法(综述).计算物理, 1996,13(4):385-397
20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
21 B. Nayroles, G. Touzot, P. Villon. Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements. Comput Mech, 1992,10:307-318
22刘更,刘天祥,谢琴.无网格法及其应用.西安:西北工业大学出版社,2005.9
23 T. Belytschko, Y. Y. Lu, L. Gu. Element-free Galerkin Methods. International Journal for Numerical Method in Engineering, 1994,37:229-256
24 M. Fleming, Y. A. Chu, B. Moran, et al. Enriched Element-free Galerkin Methods for Crack Tip Fields. Int J Numer Methods Engrg, 1997,40:1483-1504
25 P. Kryal, T. Belytschko. The Element Free Galerkin Method for Dynamic Propagation of Arbitrary 3-D Cracks. Int J Numer Methods Engrg, 1999,44(6):767-800
26 W. Barry, S. Saigal. A Three-dimensional Element-free Galerkin Elastic and Elastopl- astic Formulation. Int J Numer Methods Engrg, 1999,46:671-693
27 Y. Krongauz, T. Belytschko. EFG Approximation with Discontinuous Derivatives. Int J Numer Methods Engrg, 1998,41:1215-1233
28 H. Modaressi, P. Aubert. Element-free Galerkin Method for Deforming Multiphase Porous Media. International Journal for Numerial Method in Engineering, 1998,42: 313-340
29庞作会,葛修润,郑宏等.一种新的数值计算方法无网格伽辽金法.计算力学学报, 1999,16(3):320-329
30 T. E. Voth, M. A. Christon. Discretization Errors Associated with Reproducing KernelMethods: One-dimensional Domains. Computer Methods in Applied Mechanics and Engineering, 2001,190:2429-2446
31 E. Onarte. A Finite Point Method in Computational Mechanics. International Journal for Numerical Method in Engineering, 1996,39,3839-3866
32 Y. X. Mukherjee, S. Mukherjee. The Boundary Node Method for Potential Problems. Int J Numer Methods Engrg, 1997,40(6):797-815
33 J. T. Oden, C. A. M. Duarte, O. C. Zienkiewicz. A New Cloud-based Hp Finite Eleme- nt Method. Compute Methods App1 Mech Engrg, 1998,153(1-2):117-126
34 T. Zhu, J. Zhang, S. N. Arluri. A Meshless Local Boundary Integral Equation Method for Solving Nonlinear Problems. Computational Mechanics, 1998,22(3):174-186
35 S. N. Arluri, T. Zhu. A New Meshless Local Petrov-Galerkin Approach in Computatio- nal Mechanics. Computational Mechanics, 1998,22(2):117-127
36 T. Belytschko, Y. Krongauz, J. Dolbow, C. Gerlach. On the Completeness of Meshfree Particle Methods. Int J Numer Methods Engrg, 1998,43(5):785-819
37 J. Dolbow, T. Belytschko. Volumetric Locking in the Element Free Galerkin Method. Int J Numer Methods Engrg, 1999,46(6):925-942
38 P. Kryal, T. Belytschko. The Element Free Galerkin Method for Dynamic Propagation of Arbitrary 3-D Cracks. Int J Numer Methods Engrg, 1999,44(6):767-800
39 N. Sukumar, B. Moran, T. Belytschko. The Natural Element Method in Solid Mechan- ics. Int J Numer Method. Engrg, 1998,43(5):839-887
40 T. Belytschko, D. Organ, C. Gerlach. Element-free Galerkin Methods for Dynamic Fracture in Concrete. Comput Methods Appl Mech Engrg, 2000,187(3):385-399
41 P. Venini, P. Morana. An Adaptive Wavelet-Galerkin Methods for an Elastic-plastic Damage Constitutive Model: 1D Problems. Comput Methods App1 Mech Engrg, 2001,190(42):5619-5638
42 G. R. Liu, Y. T. Gu. A Point Interpolation Mmethod for Two-dimensional Solids. Inter- national Journal for Numerical Mmethods in Engineering, 2001,50(4):937-951
43 N. Perrone and R. A. Kao. A General Finite Difference Method for Arbitrary Meshes. Comput Struct, 1975,5:45-58
44 T. Liszka and J. Orkisa. The Finite Difference Method at Arbitrary Irregular Grids and Its Application in Application in Applied Mechanics. Comput Struct, 1980,11:83-95
45 G. R. Liu. Mesh free methods: Moving Beyond the Finite Element Method. Boca Raton: CRC Press LLC, 2003
46张雄,刘岩.无网格法.北京:清华大学出版社, 2004
47 T. Zhu. A New Meshless Regular Local Boundary Integral Equation (MRLBIE) Appr- oach. Int J Numer Methods Engrg, 1999,46:1237-1252
48 T. Belytschko, D. Organ, Y. Krongauz. A Couoled Finite Element-free Galerkin Meth- od. Comput Mech, 1995,17:186-195
49刘天祥,刘更,徐华.二维无网格法伽辽金-有限元耦合方法.西北工业大学学报, 2002,24(4):602-607
50刘天祥,刘更.二维接触问题的无网格伽辽金-有限元耦合方法.西北工业大学学报, 2003,4:449-503
51 T. Liu, G. Liu, Q. Wang. An Element-free Galerkin-finite Element Coupling Method for Elasto-plastic Contact Problem, ASME Journal of Tribology, 2005
52 H. Karutz, R. Chudoba, W. B. Kratzig. Automatic Adaptive Gengration of A Coupled Finite Element/element-free Galerkin Discretization. Finite Element Analysis and Design, 2002,38:1075-1091
53 S. W. Attaway, M. W. Heinstein, J. W. Swegle. Coupling of Smooth Particle Hydrody- namics with the Finite Element Method. Nucl Engrg Des, 1994,150:199-205
54 G. R. Johnson. Linking Lagrangian Particle Methods to Standard Finite Element Met- hods for High Velocity Impact Computations. Nucl Engrg Des, 1994,150:265-274
55 D. Hegen. Numerical Techniques for the Simulation of Crack Growth Technical Repo- rt Eindhoven University of Technology Final Report of the Postgraduate Programme Mathematics for Industry, 1994
56 Y. T. Gu, G. R. Liu. A Coupled Element Free Galerkin/boundary Element Method for Stress Analysis of Two-dimensional Solids. Comput Methods Appl Mech Engrg, 2001, 190:4405-4419
57 T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl. Meshless Methods: AnOverview and Recent Developments. Comput Methods Appl Mech Engrg, 1996,139:3 -47
58秦权,林道锦,梅刚.结构可靠度随机有限元:理论及工程应用.北京:清华大学出版社, 2006
59 R. G. Ghanem, P. D. Spanos. Stochastic Finite Elements: A Spectral Approach. New York: Springer, 1990
60游世辉,钟志华,李光耀,邹传平.基于随机无网格伽辽金法的结构可靠性分析.机械强度, 2005,27(6):770-773
61 S. Li, W. K. Liu. Meshfree and Particle Methods and Their Applications. Applied Mechanics Review, 2002,55:1-34
62 S. Rahman, B. N. Rao. An Element-free Galerkin Method for Probabilistic Mechanics and Reliability. International Journal of Solids and Structures, 2001,38:9313-9330
63 S. Rahman, B. N. Rao. A Perturbation Method for Stochastic Meshless Analysis in Elastostatics. International Journal for Numerical Methods in Engineering, 2001,50:19 69-1991
64 P. Lancaster, K. Salkauskas. Surfaces Generated by Moving Least Square Methods. Mathematics of Computation, 1981,35(155):141-158
65张汝清,吕恩琳,蹇开林.现代计算力学.重庆大学出版社, 2004
66 R. S. Barsoum. Triangular Quarter-point Elements as Elastic and Perfectly-plastic Crack Tip Elements. Int J Mech Eng, 1997(11):85-98
67毕忠宇,丁德馨,毕忠伟.随机有限元在采矿工程中的应用.矿业工程, 2006,2(4): 52-53
68 E. Vanmarcke, M. Shinozuka, S. Nakagiri, G. I. Schueller, M. Geigoriu. Random Fields and Stochastic Finite Element. Structural Safety, 1986,(3):143-166
69谢康和,周健.岩土工程有限元分析理论与应用.科学出版社, 2002
2李景湧.有限元法.北京:邮电大学出版社, 2004:51-87
3谢贻权,何福保.弹性和塑性力学中的有限单元法.北京:机械工业出版社, 1981:1-5
4薛守义.有限单元法.北京:中国建材工业出版社, 2005.2
5顾元通,丁桦.无网格法及其最新进展.力学进展, 2005,35(3):323-332
6曹国金,姜弘道.无单元法研究和应用现状及动态.力学进展, 2002,32(4):526-534
7 L. B. Lucy. A Numerical Approach to the Testing of the Fission Hypothesis. The Astr- on Astron J, l977,8(12):1013-1020
8 C. T. Dyka. Addressing Tension Instability in SPH Method. Technical Report NRL/ MR6384, NRL, 1994
9 J. W. Swegle, D. L. Hicks, S. W. Attaway. Smoothed Partical Hydrodynamics Stability Analysis. J Compout Phys, 1995,116:123-134
10 G. R. Johnson, R. A. Stryk, S. R. Beisssel. SPH for High Velocity Impact Computatio- ns Computer Methods in Applied Mechanics and Engineering, 1996,139:347-373
11 J. J. Monaghan. Smoothed Particle Hydrodynamics. Annual Review Astronomics and Astrophysics, 1992,30:543-574
12 J. W. Swegle, Hicks, S. W. Attaway. Smoothed Particle Hydrodynamics Stability Ana- lysis. J Comput Phys, 1995,116:123-134
13 J. K. Chen, J. E. Beraun, C. J. Jih. An Improvement for Tensile Instability in Smooth- ed Particke Hydrodynamics. Comput Mech, 1999,23:279-287
14 C. T. Dyka. Addressing Tension Insrability in SPH Methods. Technical Report NRL/ MR/6384, NRL, 1994
15 J. W. Swegle, S. W. Attaway. On the Feasibility of Using Smoothed Particle Hydrody- namics for under Water Explosion Calculations. Comput Mech, 1995,17:151- 168
16 G. R. Johnson, R. A. Stryk, S. R. Beissel. SPH for High Velocity Impact Computations. Comput Methods Appl Mech Engrg, 1996,139:347-373
17 L. D. Libersky, A. G. Petschek, etal. High Strain Lagrangian Hydrodynamics: A Three-dimensional SPH Code for Dynamic Mmaterizl Response. Comput Phys, 1993, 109:67-75
18 G. R. Johnson, S. R. Beussel. Normalized Smoothing Functions for SPH Impact Com- putations. Int J Numer Methods Engrg, 1996,39:2725-2741
19张锁春.光滑质点流体动力学(SPH)方法(综述).计算物理, 1996,13(4):385-397
20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
21 B. Nayroles, G. Touzot, P. Villon. Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements. Comput Mech, 1992,10:307-318
22刘更,刘天祥,谢琴.无网格法及其应用.西安:西北工业大学出版社,2005.9
23 T. Belytschko, Y. Y. Lu, L. Gu. Element-free Galerkin Methods. International Journal for Numerical Method in Engineering, 1994,37:229-256
24 M. Fleming, Y. A. Chu, B. Moran, et al. Enriched Element-free Galerkin Methods for Crack Tip Fields. Int J Numer Methods Engrg, 1997,40:1483-1504
25 P. Kryal, T. Belytschko. The Element Free Galerkin Method for Dynamic Propagation of Arbitrary 3-D Cracks. Int J Numer Methods Engrg, 1999,44(6):767-800
26 W. Barry, S. Saigal. A Three-dimensional Element-free Galerkin Elastic and Elastopl- astic Formulation. Int J Numer Methods Engrg, 1999,46:671-693
27 Y. Krongauz, T. Belytschko. EFG Approximation with Discontinuous Derivatives. Int J Numer Methods Engrg, 1998,41:1215-1233
28 H. Modaressi, P. Aubert. Element-free Galerkin Method for Deforming Multiphase Porous Media. International Journal for Numerial Method in Engineering, 1998,42: 313-340
29庞作会,葛修润,郑宏等.一种新的数值计算方法无网格伽辽金法.计算力学学报, 1999,16(3):320-329
30 T. E. Voth, M. A. Christon. Discretization Errors Associated with Reproducing KernelMethods: One-dimensional Domains. Computer Methods in Applied Mechanics and Engineering, 2001,190:2429-2446
31 E. Onarte. A Finite Point Method in Computational Mechanics. International Journal for Numerical Method in Engineering, 1996,39,3839-3866
32 Y. X. Mukherjee, S. Mukherjee. The Boundary Node Method for Potential Problems. Int J Numer Methods Engrg, 1997,40(6):797-815
33 J. T. Oden, C. A. M. Duarte, O. C. Zienkiewicz. A New Cloud-based Hp Finite Eleme- nt Method. Compute Methods App1 Mech Engrg, 1998,153(1-2):117-126
34 T. Zhu, J. Zhang, S. N. Arluri. A Meshless Local Boundary Integral Equation Method for Solving Nonlinear Problems. Computational Mechanics, 1998,22(3):174-186
35 S. N. Arluri, T. Zhu. A New Meshless Local Petrov-Galerkin Approach in Computatio- nal Mechanics. Computational Mechanics, 1998,22(2):117-127
36 T. Belytschko, Y. Krongauz, J. Dolbow, C. Gerlach. On the Completeness of Meshfree Particle Methods. Int J Numer Methods Engrg, 1998,43(5):785-819
37 J. Dolbow, T. Belytschko. Volumetric Locking in the Element Free Galerkin Method. Int J Numer Methods Engrg, 1999,46(6):925-942
38 P. Kryal, T. Belytschko. The Element Free Galerkin Method for Dynamic Propagation of Arbitrary 3-D Cracks. Int J Numer Methods Engrg, 1999,44(6):767-800
39 N. Sukumar, B. Moran, T. Belytschko. The Natural Element Method in Solid Mechan- ics. Int J Numer Method. Engrg, 1998,43(5):839-887
40 T. Belytschko, D. Organ, C. Gerlach. Element-free Galerkin Methods for Dynamic Fracture in Concrete. Comput Methods Appl Mech Engrg, 2000,187(3):385-399
41 P. Venini, P. Morana. An Adaptive Wavelet-Galerkin Methods for an Elastic-plastic Damage Constitutive Model: 1D Problems. Comput Methods App1 Mech Engrg, 2001,190(42):5619-5638
42 G. R. Liu, Y. T. Gu. A Point Interpolation Mmethod for Two-dimensional Solids. Inter- national Journal for Numerical Mmethods in Engineering, 2001,50(4):937-951
43 N. Perrone and R. A. Kao. A General Finite Difference Method for Arbitrary Meshes. Comput Struct, 1975,5:45-58
44 T. Liszka and J. Orkisa. The Finite Difference Method at Arbitrary Irregular Grids and Its Application in Application in Applied Mechanics. Comput Struct, 1980,11:83-95
45 G. R. Liu. Mesh free methods: Moving Beyond the Finite Element Method. Boca Raton: CRC Press LLC, 2003
46张雄,刘岩.无网格法.北京:清华大学出版社, 2004
47 T. Zhu. A New Meshless Regular Local Boundary Integral Equation (MRLBIE) Appr- oach. Int J Numer Methods Engrg, 1999,46:1237-1252
48 T. Belytschko, D. Organ, Y. Krongauz. A Couoled Finite Element-free Galerkin Meth- od. Comput Mech, 1995,17:186-195
49刘天祥,刘更,徐华.二维无网格法伽辽金-有限元耦合方法.西北工业大学学报, 2002,24(4):602-607
50刘天祥,刘更.二维接触问题的无网格伽辽金-有限元耦合方法.西北工业大学学报, 2003,4:449-503
51 T. Liu, G. Liu, Q. Wang. An Element-free Galerkin-finite Element Coupling Method for Elasto-plastic Contact Problem, ASME Journal of Tribology, 2005
52 H. Karutz, R. Chudoba, W. B. Kratzig. Automatic Adaptive Gengration of A Coupled Finite Element/element-free Galerkin Discretization. Finite Element Analysis and Design, 2002,38:1075-1091
53 S. W. Attaway, M. W. Heinstein, J. W. Swegle. Coupling of Smooth Particle Hydrody- namics with the Finite Element Method. Nucl Engrg Des, 1994,150:199-205
54 G. R. Johnson. Linking Lagrangian Particle Methods to Standard Finite Element Met- hods for High Velocity Impact Computations. Nucl Engrg Des, 1994,150:265-274
55 D. Hegen. Numerical Techniques for the Simulation of Crack Growth Technical Repo- rt Eindhoven University of Technology Final Report of the Postgraduate Programme Mathematics for Industry, 1994
56 Y. T. Gu, G. R. Liu. A Coupled Element Free Galerkin/boundary Element Method for Stress Analysis of Two-dimensional Solids. Comput Methods Appl Mech Engrg, 2001, 190:4405-4419
57 T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl. Meshless Methods: AnOverview and Recent Developments. Comput Methods Appl Mech Engrg, 1996,139:3 -47
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