摘要
随着科学技术的快速进步和经济条件的迅速改善，世界各地对大型结构的需求越来越多。对这些大型结构进行地震反应分析时，存在两个科学技术上的挑战：（1）结构有限元模型的单元数和自由度数巨大；（2）该类结构受高阶振型影响较大，需要建立高阶的求解方程。并行计算技术的出现，为顺利的求解该类型结构，确保计算精度，提供了很好的方法和手段。本研究利用并行计算技术，研究了大型结构的地震反应分析方法。研究内容主要包括以下几个方面：
1．深入研究了区域剖分方法的国内外研究现状，对部分方法编制了C/C++程序并进行区域剖分效果的对比，分析了各个方法的优缺点及适用范围。
2．对各种区域分解算法进行了深入的研究，分析总结了各种常用算法的优缺点。
3．专门对子结构分析方法进行了研究，编制了相应的C/C++程序，并利用ANSYS作前处理器，得到相应的结构刚度矩阵和荷载矩阵后，对二个单元和四个单元的简单悬臂梁进行了试验性并行计算算法的研究，此方法的详细研究为后续方法的研究奠定了基础。
4．详细研究了BDD（Balancing Domain Decomposition）方法，BDD方法是一种子结构方法，其主要是把结构分成若干个子区域，每个子区域分成区域与区域间的边界节点和剩余的内部节点，每个子区域首先进行自由度静力凝聚，形成界面节点自由度方程。然后应用BDD预处理子的预处理共轭梯度法求解各个界面方程，在求解过程中要交换界面之间的信息，最终得到各个节点的位移和应力情况。
5．基于区域分解算法对结构的静力和动力问题进行了并行计算方法研究，在进行大型结构的动力并行计算分析时，利用Newmark-β法对时间离散化积分。对于动力分析，基于ADVENTURE编制了并行计算程序，实例证明，编译的程序可以用于大型结构的动力并行计算分析。
6．基于区域分解算法对静力非线性和动力非线性有限元进行了并行计算方法的研究，在非线性方程求解中，在每个荷增量步内采用了牛顿-拉夫逊迭代法，在牛顿-拉夫逊的每个迭代步内使用了共轭梯度法进行相应的迭代计算，对于动力非线性分析，也基于ADVENTURE编制了并行计算程序，实例证明，编译的程序可以用来求解大型结构的动力非线性反应。
7．应用上述方法对意大利的万神庙进行了三维地震反应分析，该结构总共划分为1,329,027个四结点的四面体单元，结果表明，当采用96个计算节点时，只需要短短的十一分钟左右的计算时间。
With the development of science and technology and the rapid expansionof economies, a large number of large-scale structures are built around theworld. There are two technical challenges in the earthquake response analysisof these large-scale structures:(1) the number of elements of the finite elementmodels of these structures and the number of degrees of freedom are so huge;(2) these structures are affected by higher order mode and a large number ofequations are needed to better analyze the structural response. In order to solvethe problems of these structures accurately, parallel computing tools can beused. In this dissertation, seismic response analysis of large-scale structure isstudied based on the application of parallel computing techniques. The researchmainly includes the following aspects:
1. Current research of all kinds of domain decomposition methods arestudied, and C/C++programs of some methods are compiled. The merits anddemerits of each method are summarized, and the applicable scope of eachmethod is also studied.
2. A variety of domain decomposition algorithms are analyzed, and thenthe advantages and disadvantages of the commonly used algorithms aresummarized.
3. Sub-structure analysis method is studied carefully, and correspondingC/C++program is compiled where ANSYS is used for the preprocessor to getthe structural stiffness matrix and load matrix. Then static analysis of thesimple cantilever beam which is divided into two substructures is solved usingsubstructural parallel computing algorithms. Detailed study of this methodprovides the basis for subsequent study.
4. BDD (Balancing Domain Decomposition) method is studied detailedly.BDD method is a substructure method, and the structure is split into severalnon-overlapping sub-structures. The node of each sub-structure is divided intointerface boundary nodes and the remaining internal nodes. Firstly, Staticcondensation of the degree of freedom in each sub-structure is carried out, and then equations of degree of freedom of the interface nodes are formulated.Secondly preconditioning conjugate gradient method are applied using BDDpreconditioner to solve the equations. Information between interface nodesmust be exchanged in the solving process. Finally displacement and stress ofeach node can be acquired.
5. Based on domain decomposition algorithm, parallel computing methodsof static and dynamic analysis of large-scale structure are researched. Thisdomain decomposition has been implemented within a finite element code forlinear implicit transient dynamic analysis, and time integration is performedusing Newmark-β constant average acceleration method. The example showsthat the compiled program can be used for parallel computing analysis oflarge-scale structure.
6. Based on domain decomposition algorithm, parallel computing methodsof static and dynamic nonlinear analysis of large-scale structure are alsoresearched. In the nonlinear equations, Newton-Raphson iteration method isused within each load increment or each time step, and conjugate gradientmethod is used for the corresponding iterative calculation in each iteration stepof the Newton–Raphson. Based on ADVENTURE the corresponding dynamicnonlinear parallel computing program is also compiled, and the example showsthat it can be used for large-scale dynamic nonlinear analysis of large-structure.
7. Parallel computing analysis of Pantheon in Itali is implemented usingthe above methods on massive parallel processors. The finite element model ofthis structure is divided into about1,329,027four-node tetrahedral elements.The results show that about eleven minutes are needed when96processors areused.
引文
[1]张汝清.并行计算结构力学[M].重庆大学出版社，1993.
[2]郭迅.地震工程试验联网最新进展[J].地震工程与工程振动，2006，26(4):37-41.
[3]田石柱，蔡新江.远程协同结构试验方法研究与发展[J].地震工程与工程振动，2006，26(5):47-54.
[4]龚强.地震工程远程协同试验支持系统—NEES Grid CN的研究[J].信息技术，2006，(3):1-6.
[5]胡宁，张汝清.一种改进的有限元子结构并行算法[J].重庆交通学院学报，1990，9(4):73-78.
[6]胡宁，张汝清.一种改进的有限元子结构静凝聚并行算法[J].陕西机械学院学报，1992，8(1):62-67.
[7]胡宁，张汝清.一种多波前并行处理的有限元算法[J].重庆大学学报，1991，14(4):16-19.
[8]金先龙，李渊印.结构动力学并行计算方法及应用[M].国防工业出版社，2008.
[9]钟万勰.结构动力方程的精细时程积分法[J].大连理工大学学报，1994，34(2):131-136.
[10]李红云，宋华茂.任意荷载作用下结构动力响应的并行算法[J].振动工程学报，1996，9(4):333-339.
[11]王元丰，储德文.精细时程积分法在地震响应分析中的应用[J].土木工程学报，2004，37(1):21-23.
[12]李渊印，金先龙，李丽君，等.基于精细时程法的结构动力响应并行分析系统[J].岩石力学与工程学报，2005，24(1):18-22.
[13]李丽君，金先龙，李渊印，等.有限元软件结构分析模块的并行开发及应用[J].上海交通大学学报，2004，38(8):1354-1357.
[14]余天堂，姜弘道.一种求解结构动力响应的并行解法[J].河海大学学报，1999，27(3):75-78.
[15]胡宁，张汝清.弹性接触问题参数变分原理的有限元并行算法[J].应用数学与力学，1992，13(7):583-591.
[16]张林波，迟学斌，莫则尧等.并行计算导论[M].清华大学出版社，2006.
[17]李晓梅，吴建平.数值并行算法与软件[M].科学出版社，2007.
[18]吕涛，石济民，林振宝.区域分解算法-偏微分方程数值解新技术[M].科学技术出版社，1999.
[19]茹忠亮，叶亚齐，赵洪波.区域分解并行有限元计算方法研究[J].河南理工大学学报，2007，26(2):198-201.
[20]王勖成.有限单元法[M].清华大学出版社，2003.
[21]李强，邹经湘.独立子结构并行计算方法[J].哈尔滨工业大学学报，1996，28(2):144-148.
[22]现代应用数学手册编委会.现代应用数学手册计算与数值分析卷[M].清华大学出版社，2005.
[23] Anil K. Chopra.结构动力学-理论及其在地震工程中的应用（第3版）[M].清华大学出版社，2009.
[24] R.W.克拉夫，J.彭津著，王光远等译.结构动力学[M].科学出版社，1983.
[25] A.Noor, R.E.Fulton.Impact of the CDC-STAR-100Computer on Finite ElementSystems[J].Jounuary of the Structure Division,1975, ASCE,101, No. ST4,731-750.
[26] R.J.Melosh,Senol Utku. Direct Finite Element Equation Solving Algorithm[J].Computers&Sturctures,1985,20(1-3):99-105.
[27] D.Goehlich, L.Komzisk, R.E.Fulton. Application of a Parallel Equation Solver toStatic FEM Problems[J]. Computers&Structures,1989,13(2):121-129.
[28] Charbel Farhat, Edward Wilson. A Parallel Active Column Equation Solver[J].Computers&Structures,1988,28(2):289-304.
[29] David R.Kincaid, Thomas C.Oppe. A Parallel Algorithm for the General LUFactorization[J]. Communication in Applied Numerical Method,1988,4:349-359.
[30] Charbel Farhat. A Simple and Efficient Automatic FEM Domain Decomposer[J].Computers&Structures,1988,28(5):579-602.
[31] Zhang Ruqing. Parallel Computational Algorithm of Substructure Method ofLarge-Scale Structure Analysis[J].Applied Methematics and Mechanics,1991,12(1):93-100.
[32] G.F.Carey, E.Barragy, R.Mclay, etc.. Element-by-Element Vector and ParallelComputation[J]. Communications in Applied Numerical Methods,1988,4:299-307.
[33] E.Barragy, G.F.Carey. A Parallel Element-by-Element Solution Scheme[J].International Journal for Numerical Method in Engineering,1988,26:2367-2382.
[34] L.Adams, J.Ortega. A Multi-color SOR Method for Parallel Computation[J].Proceedings of the1982International Conference on Parallel Processing, IEEEComputer Society.
[35] M.P.Pattrick, T.W.Prat. Communications Oriented Programming of ParallelIterative Solutions for Sparse Linear Systems[J]. Communications in AppliedNumerical Methods,1986,2:255-261.
[36] Charbel Farhat. A Method of Finite Element Tearing and Interconnecting and itsParallel Solution Algorithm[J]. International Journal for Numerical Methods inEngineering,1991,32:1205-1227.
[37] A.Noor, J.Lambiotte. Finite Element Dynamic Analysis on CDC STAR-100Computer[J]. Computers&Structures,1978,8:621-632
[38] Olaf Storaasli, Jonathan Ransom. Structural Dynamic Analysis on a ParallelComputer: the Finite Element Machine[J]. Computers&Structures,1987,26(4):551-559.
[39] Ronfu Ou, Robert E. Fulton. An Investigation of Parallel Numerical IntegrationMethods for Nonlinear Dynamics[J]. Computers&Structures,1988,30(12):403-409.
[40] M.Ortiz, Pinsky P.M.,Taylor R M. Unconditionally stable element-by-elementalgorithms for dynamic problems[J]. Computer Methods Applied MechanicsEngineering,1983,36:223–239.
[41] M.Oritz, Bahram Nour-omid. Unconditionally stable concurrent procedures fortransient finite element analysis[J]. Computer Methods Applied MechanicsEngineering,1986,58:157–174.
[42] M.Oritz, Bahram Nour-omid, Elisa D.Sotelino. Accuracy of a Class of ConcurrentAlgorithms for Transient Finite Element Analysis[J]. International Journal forNumerical Methods in Engineering,1988,26:379-391.
[43] M.Oritz, Bahram Nour-omid. Efficiency of Group Implicit Concurrent algorithmsfor Transient Finite Element Analysis[J]. International Journal for NumericalMethods in Engineering,1989,28:2761-2776.
[44] Modak S, Sotelino E.D.. The iterative group implicit algorithm for nonlinearstructural analysis. International Journal for Numerical Methods in Engineering,2000,47(4):869-885.
[45] Susan W. Bostic, R.E.Fulton. Implementation of the Lanczos Method for StructuralVibration Analysis on a Parallel Computer[J]. Computers&Structures,1987,25(3):395-403.
[46] Yuji Aoyama, Genki Yagawa. Component Mode Synthesis for Large-scaleStructural Eigenanalysis[J]. Computers&Structures,2001,79:605-615.
[47] Paresh Murthy, Petra Poschmann, Mike Reymond, ect. Automated ComponentModal Synthesis with Parallel Processing[J].
[48] Geng Zhang, Matthew P.Castanier, and Christophe Pierre. Efficient ComponentMode Synthesis with a New Interface Reduction Method[J].
[49] Chao Yang, Weiguo Gao, Zhaojun Bai, etc.. An Algebraic Substructuring Methodfor Large-scale Eigenvalue Calculation[J]. SIAM J. SCI. COMPUT.,2005,27(3):873-892.
[50] Pierre Guérin, Anne-Marie Baudron, Jean-Jacques Lautard, etc.. Component ModeSynthesis Methods for3D Heterogeneous core Calculations Applied to the MixedDual Finite Element Solver Minos[J].
[51] Jerome F. Hajjar, John F.Abel. Parallel Processing for Transient NonlinearStructural Dynamics of Three-dimensional Framed Structures using DomainDecomposition[J]. Computers&Structures,1988,30(6):1237-1254.
[52] Jerome F. Hajjar, John F.Abel. Parallel Processing of central difference transientanalysis for three-dimensional nonlinear framed Structures[J]. Communication andApplied Numerical Methods,1989,5:39-46.
[53] Charbel Farhat, Edward Wilson. A New Finite Element Concurrent ComputerProgram Architecture[J]. International Journal for Numerical Methods inEngineering,1987,24(9):1771-1792.
[54] Edward Wilson, Charbel Farhat. Linear and Nonlinear Finite Element Analysis onMultiprocessor Computer Systems[J]. Communications in Applied NumericalMethods,1988,4:425-434.
[55] Charbel Farhat, Luis Crivell. A General Approach to Nonlinear FE Computationson Shared-memory Multiprocessors[J]. Computer Methods in Applied Mechanicsand Engineering,1989,72:153-171.
[56] S.Kacou and I.D.Parsons. A parallel multigrid method for history-dependent elasto-plasticity computations[J]. Computer Methods Applied Mechanics andEngineering,1993,108(1-2):1-21.
[57] I.D.Parsons. Parallel adaptive multigrid methods for elasticity, plasticity andeigenvalue problems[J]. Parallel solution methods in computational mechanics,Edited by Papadrakakis M, Wiley,1997,Chapter4:143-180.
[58] J. Fish, M. Pandheeradi and V. Belshy. An efficient multilevel solution scheme forlarge scale non-linear systems[J]. International Journal for Numerical Methods inEngineering,1995,38:1597-1610.
[59] Johan Z, Mathur KK, etc.. Scalability of finite element applications ondistributed-memory parallel computers[J]. Computer Methods Applied Mechanicsand Engineering,1994,119:61-72.
[60] Fahmy M W, Namini A H. A survey of parallel nonlinear dynamic analysismethodologies[J]. Computers and Structures,1994,53(4):1033~1043.
[61] Charbel Farhat, Luis Crivelli. A Transient FETI Methodology for Large-ScaleParallel Implicit Computations in Structural Mechanics[J]. International Journal forNumerical Methods in Engineering,1994,37:1945-1975.
[62] Farhat C and Roux FX. Implicit Parallel Processing in Structural Mechanics[M].Computational mechanics advances,1994,2:1-124.
[63] Charbel Farhat，Michel Geradin. On the General Solution by a Direct Method of aLarge-scale Singular System of Linear Equations: Application to the Analysis ofFloating Structures[J]. International Journal for Numerical Methods in Engineering,1998,41:675-696.
[64] Charbel Farhat, Jan Mandel, Francois Xavier Roux. Optimal ConvergenceProperties of the FETI Domain Decomposition Method[J]. Computer MethodsApplied Mechanics and Engineering,1994,115:365-385.
[65] Charbel Farhat, Po-shu CHEN. A Scalable Lagrange Multiplier Based DomainDecomposition Method for Time-Dependent Problems[J]. International Journal forNumerical Methods in Engineering,1995,38:3831-3853.
[66] Charbel Farhat, Jan Mandel. Scalable Substructuring by Lagrange Multipliers inTheory and Practice[J]. Ninth International Conference on Domain DecompositionMethods,1998.
[67] Charbel Farhat, Kendall Pierson, Michel Lesoinne. The Second Generation FETIMethods and Their Application to the Parallel Solution of Large-scale Linear andGeometrically Non-linear Structural Analysis Problems[J]. Computer MethodsApplied Mechanics and Engineering,2000,184:333-374.
[68] Charbel Farhat, Jan Mandel. The Two-level FETI Method for Static and DynamicPlate Problems Part I: An Optimal Iterative Solver for Biharmonic Systems[J].Computer Methods Applied Mechanics and Engineering,1998,155:129-151.
[69] Charbel Farhat, P.S. Chen, J.Mandel and F.X. Roux. The two-level FETI methodPart II: Extension to shell problems, parallel implementation and performanceresults[J]. Computer Methods Applied Mechanics and Engineering,1998,155:153-179.
[70] Charbel Farhat, Michel Lesoinne, etc.. FETI-DP: a Dual-primal Unified FETImethod-part I: a Faster Alternative to the Two-level FETI Method[J]. InternationalJournal for Numerical Methods in Engineering,2001,50:1523-1544.
[71] JAN MANDEL. Balancing Domain Decomposition[J]. Communications onNumericalMethods in Engineering,1993,9:223-241.
[72] Hiroyuki TAKUBO and Shinobu YOSHIMURA. Parallel Decomposition of100-million DOF Meshes into Hierarchical subdomains[R]. Technical Report ofADVENTURE Project ADV-99-1,1999.
[73] Tomoshi MIYAMURA, Hirohisa NOGUCHI. Static Elastic-Plastic Stress Analysesof Ten-million DOFs Problems using HDDM[R]. Annual Report of ADVENTUREProject,1999.
[74] Atsuya Oishi. Large-scale Dynamic Analysis with Contact-impact using theHierarchical Domain Decomposition Method[R]. Annual Report of ADVENTUREProject,1999.
[75] Takashi Furumura, Kazuki Koketsu. Parallel3-D Simulation of Groud Motion forthe1995Kobe Earthquake: The Component Decomposition Approach[J]. Pure andApplied Geophysics,2000,157:2047-2062.
[76] Takashi Furumura, Kazuki Koketsu, Kuo-Liang Wen. Parallel PSM/FDM HybridSimulation of Ground Motions from the1999Chi-Chi, Taiwan, Earthquake[J]. Pureand Applied Geophysics,2002,159:2133-2146.
[77] T.Furumura, L.Chen. Parallel Simulation of Strong Ground Motions during Recentand Historical Damaging Earthquakes in Tokyo, Japan[J]. Parallel Computing,2005,31:149-165.
[78] Takashi Furumura, Tatsuhiko Saito. Integrated Ground Motion and TsunamiSimulation for the1944Tonankai Earthquake Using High-PerformanceSupercomputers[J]. Journal of Disaster Research,2009,4(2).
[79] Dimitri Komatitsch, Jean-Pierre Vilotte. The Spectral Element Method: AnEfficient Tool to Simulate the Seismic Response of2D and3D GeologicalStructures[J]. Bulletin of the Seismological Society of America, April1998,88(2):368-392.
[80] Dimitri Komatitsch, Jeroen Tromp. Introduction to the Spectral Element Methodfor Three-dimensional Seismic Wave Propagation[J]. Geophys. J. Int.1999,139:806-822.
[81] Dimitri Komatitsch, Qinya Liu, etc. Simulations of Ground Motion in the LosAngeles Basin Based upon the Spectral-Element Method[J]. Bulletin of theSeismological Society of America, February2004,94(1):187-206.
[82] http://www.geodynamics.org/cig/software/specfem3d （2012年2月）
[83] Muneo HORI, Tsuyoshi Ichimura and Kenji Oguni. Integrated EarthquakeSimulation-Estimation of Strong Ground Motions and Structural Response. Bull.Earthq. Res. Inst. Univ. Tokyo.2006,81:331-339.
[84] Muneo HORI, Teruyuki Kato etc.. On Development of Crustal Deformation/stressstate monitoring system.
[85] H. Zolghadr Jahromi, B.A. Izzuddin, L. Zdravkovic. Partitioned analysis ofnonlinear soil-structure interaction using iterative coupling[J]. Interaction andMultiscale Mechanics,2007,1(1):33-51.
[86] H. Zolghadr Jahromi, B.A. Izzuddin, L. Zdravkovic. A domain decompositionapproach for coupled modelling of nonlinear soil–structure interaction[J].Comput. Methods Appl. Mech. Engrg,2009,198:2738–2749.
[87] Boris Jeremic, Guanzhou Jie, etc. Time Domain Simulation of Soil-Foundation-Structure Interaction in non-Uniform Soils[J]. EARTHQUAKE ENGINEERINGAND STRUCTURAL DYNAMICS,2008:1-23.
[88] Boris Jeremic and Guanzhou Jie. Parallel Finite Element Computations forSoil-Foundation-Structure Interaction Problems, Report UCD-Comp GeoMech-02-07,2008.
[89] David Michea, Dimitri Komatitsch. Accelerating a three-dimensionalfinite-difference wave propagation code using GPU graphics cards[J]. GeophysicalJournal International,2010,182:389-402.
[90] Dimitri Komatitsch, Gordon Erlebacher, etc. High-order Finite-element SeismicWave Propagation Modeling with MPI on a Large GPU Cluster[J]. Journal ofComputational Physics,2010,229:7692-7714.
[91] D imitri Komatitsch. Fluid-solid Coupling on a Cluster of GPU Graphics Cards forSeismic Wave Propagation[J]. Comptes Rendus Mecanique,2011,339:125-135.
[92] Flynn M J. Very High-speed Computing Systems[J]. Proc.of the IEEE,1966,54(12):1901-1909.
[93] Handler W. The Impact of Classification Schemes on Computer Architecture[J].Proc. Int’1Conf. On Parallel Processing, New York,,Aug.1977,7-15.
[94] David J. Kuck. The Structure of Computers and Computations[M]. New York,1978.
[95] E.J.Plaskacz, M.R.Ramirez and S.Gupta. Nonlinear explicit finite element analysison the Intel Delta[J]. Computing Systems in Engineering,1994,5(1):1-17.
[96] G.M. Amdahl. Validity of the Single Processor Approach to Achieving Large-ScaleComputing Capabilities[J]. Proc. Am. Federation of Information Processing Soc.Spring Joint Computer Conf.(AFIPS07), AFIPS Press,1967,483-485.
[97] SHANG-HSIEN HSIEH. Evaluation of Automatic Domain Partitioning Algorithmsfor Parallel Finite Element Analysis[J]. International Journal for NumericalMethods in Engineering,1997,40:1025-1051.
[98] J.G.Malone. Automatic mesh decomposition and concurrent finite element analysisfor hypercube multiprocessor computers[J]. Comput. Methods Appl. Mech. Eng.,1988,70:27-58.
[99] H.D.Simon. Partitioning of Unstructured Problems for Parallel Processing[J].Comput. Systems Eng.,1991,2:135-148.
[100] S.H.Hsieh, G.H.Paulino and J.F.Abel. Recursive Spectral Algorithms for AutomaticDomain Partitioning in Parallel Finite Element Analysis[J]. Comput. Methods Appl.Mech. Eng.,1995,121:137-162.
[101] M.Al-Nasra and D.T.Nguyen. An Algorithm for domain decomposition in finiteelement analysis[J]. Comput. Struct.,1991,39:227-289.
[102] Boris Jeremi, Guanzhou Jie. Plastic Domain Decomposition Method for ParallelElastic-Plastic Finite Element Computations in Geomechanics[R]. Report UCD-CompGeoMech-03-07,2008.
[103] George Karypis and Vipin Kumar. A fast and high quality multilevel scheme forpartitioning irregular graphs[R]. Technical report TR95-035, Department ofComputer Science, University of Minnesota, USA,1995.
[104] George Karypis and Vipin Kumar. Multilevel k-way Partitioning Scheme forIrregular Graphs[J]. Journal of Parallel and Distributed Computing,199848:96-129.
[105] B.W.Kernighan and S.Lin. An Efficient Heruistic Procedure for PartitioningGraphs[J]. The Bell Technical Journal,1970,49(2):291-307.
[106] C. M. Fiduccia and R. M. Mattheyses. A linear time heuristic for improvingnetwork partitions[C]. in Proc.19th IEEE Design Automation Conference,1982,175–181.
[107] George Karypis and Vipin Kumar. Parallel Multilevel k-way Partitioning Schemefor Irregular Graphs[J]. Society for Industrial and Applied Mathematics,1999,41(2):278-300.
[108] G. Yagawa, N. Soneda and S. Yoshimura. A Large Scale Finite Element AnalysisUsing Domain Decomposition Method on a Parallel Computer[J]. Comuters&Structures,1991,38(5/6):615-625.
[109] G. Yagawa, A.Yoshioka, S. Yoshimura and N. Soneda. A Parallel Finite ElementMethod with a Supercomputer Network[J]. Comuters&Structures,1993,47(3):407-418.
[110] G.Yagawa and R.Shioya. Parallel Finite Elements on a Massively Parallel Computerwith Domain Decomposition[J]. Computing Systems in Engineering,1993,4(4-6):495-503.
[111] A.Toselli, O.Widlund.Domain Decomposition Methods-Algorithms and Theory[M].科学出版社,2006.
[112] Masao OGINO, Ryuji SHIOYA, Hiroshi KANAYAMA. an Inexact BalancingPreconditioner for Large-scale Structural Analysis[J]. Journal of ComputationalScience and Technology,2008,2(1).
[113] Mandel, J. and Brezina, M.. Balancing Domain Decomposition: Theory andPerformance in Two and Three Dimensions[R]. UCD/CCM Report,1993,2.
[114] LeTallec, P., Mandel, J. and Vidrascu,M.. Balancing Domain Decomposition forPlates[C]. Proceedings of Seventh International Symposium on DomainDecomposition Methods,1993,515-524.
[115] Ryuji SHIOYA. an Implementation of Balancing Domain Decomposition Methodon Static Elastic Stress Analyses[R].Annual Report of ADVENTURE ProjectADV-99-1,1999.
[116] Ryuji SHIOYA,Masao OGINO, etc.. Large Scale Finite Element Analysis with aBalancing Domain Decomposition Method.
[117] LeTallec, P., Mandel, J. and Vidrascu, M.. a Neumann-Neumann DomainDecompostion Algorithm for Solving Plate and Shell Problems[J]. SIAM Journalon Numerical Analysis,1998,35:836-867.
[118] Dohrmann, C.R.. a Preconditioner for Substructuring based on Constrained EnergyMinimization[J]. SIAM Journal on Scientific Computing,2003,25:246-258.
[119] Masao Ogino, Ryuji Shioya, etc.. Seimic Response Analysis of Nuclear PressureVessel Model with ADVENTURE System on the Earth Simulator[J]. Journal of theEarth Simulator,2005,2:41-54.
[120] http://adventure.sys.t.u-tokyo.ac.jp/（2012年3月）
[121] ADVENTURE_Solid User Manual, http://adventure.sys.t.u-tokyo.ac.jp/
[122] Robert D.Cook, David S.Malkus, etc.. Concepts and Applications of Finite ElementAnslysis[M].Third Edition,1988.
[123] Klaus-Jürgen Bathe. Finite Element Procedures[M]. Prentice Hall, Upper SaddleRiver, New Jersey07458,1982.
[124] Newmark N M. A Method of Computation for Structural Dynamics[J]. J. Eng.Mech. Div. ASCE,1959,85(EM3),67-94.
[125] A RAMA MOHANRAO, TVSR APPARAO, etc.. Comparative Efficiencies of threeParallel Algorithms for Nonlinear Implicit Transient Dynamic Analysis[J].Sadhana,2004,29,Part1.
[126] Soneda N, Yagawa G and Yoshimura S. Elastic-Plastic Finite Element Analysisusing Domain Decomposition Method[J]. Proc. SMIRT-11, Tokyo, Japan B,1991:243-248.
[127] Riqing LAN, Biao FENG, Xuqian CHEN. Study on Seismic Response Analysis of3-D Structure[C]. International Conference on Advances in ComputationalModeling and Simulation,2011.