基于Woodbury非线性方法的迭代算法对比分析
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摘要
在结构局部非线性求解过程中,刚度矩阵仅部分元素发生改变,此时切线刚度矩阵可写成初始刚度矩阵与其低秩修正矩阵和的形式,每个增量步的位移响应可用数学中快速求矩阵逆的Woodbury公式高效求解,但通常情况下迭代计算在结构非线性分析中是不可避免的,因此迭代算法的计算性能也对分析效率有重要影响。该文以基于Woodbury非线性方法为基础,分别采用Newton-Raphson(N-R)法、修正牛顿法、3阶两点法、4阶两点法及三点法求解其非线性平衡方程,并对比分析5种迭代算法的计算性能。利用算法时间复杂度理论,得到了5种迭代算法求解基于Woodbury非线性方法平衡方程的时间复杂度分析模型,定量对比了5种迭代算法的计算效率。通过2个数值算例,从收敛速度、时间复杂度和误差等方面对比了各迭代算法的计算性能,分析了各算法适用的非线性问题。最后,计算了5种算法求解基于Woodbury非线性方法平衡方程的综合性能指标。
The elements of the stiffness matrix are often partially changed in the solution of local nonlinear problems, in which the tangent stiffness matrix can be written as the sum of the initial stiffness matrix and its low rank perturbation matrix so that the displacement response in each incremental step can be efficiently solved by the Woodbury formula that is used to calculate the inverse matrix in mathematics. However, the iterative calculation is often unavoidable in the structural nonlinear analysis, and the performance of the nonlinear iterative algorithm also has a great impact on the efficiency of the structural nonlinear analysis. This paper studies the iterative solution of the nonlinear method based on the Woodbury formula. The Newton-Raphson(N-R) method,the modified Newton method, the two-point method with three convergence order, the two-point method with four convergence order, and the three-point method, are chosen to solve the equilibrium equations of the nonlinear method based on the Woodbury formula, and the performance of these five iterative algorithms is compared. The time complexity analysis models of the five iterative algorithms solving the equilibrium equations of the nonlinear method based on the Woodbury formula are obtained, and the efficiency of the five algorithms is quantitatively compared. The calculation performance of the five iterative algorithms is compared through two cases from the perspective of convergence rate, time complexity and error; then the applicable nonlinear problems of the five algorithms are analyzed. Finally, the comprehensive performance index of the five iterative algorithms solving the equilibrium equations of the nonlinear method based on the Woodbury formula is calculated.
The elements of the stiffness matrix are often partially changed in the solution of local nonlinear problems, in which the tangent stiffness matrix can be written as the sum of the initial stiffness matrix and its low rank perturbation matrix so that the displacement response in each incremental step can be efficiently solved by the Woodbury formula that is used to calculate the inverse matrix in mathematics. However, the iterative calculation is often unavoidable in the structural nonlinear analysis, and the performance of the nonlinear iterative algorithm also has a great impact on the efficiency of the structural nonlinear analysis. This paper studies the iterative solution of the nonlinear method based on the Woodbury formula. The Newton-Raphson(N-R) method,the modified Newton method, the two-point method with three convergence order, the two-point method with four convergence order, and the three-point method, are chosen to solve the equilibrium equations of the nonlinear method based on the Woodbury formula, and the performance of these five iterative algorithms is compared. The time complexity analysis models of the five iterative algorithms solving the equilibrium equations of the nonlinear method based on the Woodbury formula are obtained, and the efficiency of the five algorithms is quantitatively compared. The calculation performance of the five iterative algorithms is compared through two cases from the perspective of convergence rate, time complexity and error; then the applicable nonlinear problems of the five algorithms are analyzed. Finally, the comprehensive performance index of the five iterative algorithms solving the equilibrium equations of the nonlinear method based on the Woodbury formula is calculated.
引文
[1]孙宝印,古泉,张沛洲,等.钢筋混凝土框架结构弹塑性数值子结构分析方法[J].工程力学,2016,33(5):44-49.Sun Baoyin,Gu Quan,Zhang Peizhou,et al.Elastoplastic numerical substructures method of reinforced concrete frame structures[J].Engineering Mechanics,2016,33(5):44-49.(in Chinese)
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[7]高国强.车身结构设计中快速重分析方法研究[D].长沙:湖南大学,2015:16-30.Gao Guoqiang.The research of fast reanalysis algorithm for structural design of automobile body[D].Changsha:Hunan University,2015:16-30.(in Chinese)
[8]王琥,种浩,高国强,等.重分析方法研究进展及展望[J].工程力学,2017,34(5):1-16.Wang Hu,Chong Hao,Gao Guoqiang,et al.Review of advances and outlook in reanalysis methods[J].Engineering Mechanics,2017,34(5):1-16.(in Chinese)
[9]宋琦,杨任,陈璞.一种新的结构修改算法及其在工程设计中的应用[J].工程力学,2016,33(7):1-6.Song Qi,Yang Ren,Chen Pu.A new algorithm for structural modifications and its applications[J].Engineering Mechanics,2016,33(7):1-6.(in Chinese)
[10]Huang C,Verchery G.An exact structural static reanalysis method[J].Communications in numerical methods in engineering,1997,13(2):103-112.
[11]Deng L,Ghosn M.Pseudoforce method for nonlinear analysis and reanalysis of structural systems[J].Journal of Structural Engineering,2001,127(5):570-578.
[12]Kirsch U.Reanalysis and sensitivity reanalysis by combined approximations[J].Structural and Multidisciplinary Optimization,2009,40(1):1-15.
[13]李书,冯太华,范绪箕.结构静力问题的重特征值灵敏度分析[J].工程力学,1996,13(4):97-104.Li Shu,Feng Taihua,Fan Xuji.Sensitivity analysis with repeated eigenvalues in structural static problem[J].Engineering Mechanics,1996,13(4):97-104.(in Chinese)
[14]Kirsch U,Bogomolni M.Nonlinear and dynamic structural analysis using combined approximations[J].Computers&structures,2007,85(10):566-578.
[15]Kirsch U,Bogomolni M,Sheinman I.Nonlinear dynamic reanalysis of structures by combined approximations[J].Computer Methods in Applied Mechanics&Engineering,2006,195:4420-4432.
[16]Hurtado J E.Reanalysis of linear and nonlinear structures using iterated shanks transformation[J].Computer Methods in Applied Mechanics&Engineering,2002,191:4215-4229.
[17]Huang G,Chen X,Yang Z,et al.Exact analysis and reanalysis methods for structures with nonlinear boundary conditions[J].Computers&Structures,2018,198:12-22.
[18]Wong K K F,Yang R.Inelastic dynamic response of structures using force analogy method[J].Journal of Engineering Mechanics,1999,125(10):1190-1199.
[19]靳永强,李钢,李宏男.基于拟力法的钢支撑非线性滞回行为模拟[J].工程力学,2017,34(10):139-148.Jin Yongqiang,Li Gang,Li Hongnan.Numerical simulation of steel brace hysteretic behavior based on the force analogy method[J].Engineering Mechanics,2017,34(10):139-148.(in Chinese)
[20]李钢,余丁浩,李宏男.基于拟力法的纤维梁有限元非线性分析方法[J].建筑结构学报,2016,37(9):61-68.Li Gang,Yu Dinghao,Li Hongnan.Nonlinear fiber beam element analysis based on force analogy method[J].Journal of Building Structures,2016,37(9):61-68.(in Chinese)
[21]Li G,Yu D,Li H N.Seismic response analysis of reinforced concrete frames using inelasticity-separated fibre beam-column model[J].Earthquake Engineering&Structural Dynamics,2018,47(5):1291-1308.
[22]Yu D H,Li G,Li H N.Improved woodbury solution method for nonlinear analysis with high-rank modifications based on a sparse approximation approach[J].Journal of Engineering Mechanics,2018,144(11):04018103-1-04018103-13.
[23]李钢,贾硕,余丁浩.基于算法复杂度理论的拟力法计算效率评价[J].计算力学学报,2018,35(2):129-137.Li Gang,Jia Shuo,Yu Dinghao.The efficiency evaluation of force analogy method based on the algorithm complexity theory[J].Chinese Journal of Computational Mechanics,2018,35(2):129-137.(in Chinese)
[24]何政,欧进萍.钢筋混凝土结构非线性分析[M].哈尔滨:哈尔滨工业大学出版社,2007:168-177.He Zheng,Ou Jinping.Nonlinear analysis of reinforced concrete structures[M].Harbin:Harbin Institute of Technology Press,2007:168-177.(in Chinese)
[25]Scott M H,Fenves G L.Krylov subspace accelerated newton algorithm:Application to dynamic progressive collapse simulation of frames[J].Journal of Structural Engineering,2010,136(5):473-480.
[26]Bathe K J,Cimento A P.Some practical procedures for the solution of nonlinear finite element equations[J].Computer Methods in Applied Mechanics&Engineering,1980,22(1):59-85.
[27]Kim J H,Kim Y H.A predictor-corrector method for structural nonlinear analysis[J].Computer Methods in Applied Mechanics&Engineering,2001,191(8-10):959-974.
[28]Saffari H,Mansouri I,Bagheripour M H,et al.Elastoplastic analysis of steel plane frames using homotopy perturbation method[J].Journal of Constructional Steel Research,2012,70(2):350-357.
[29]Mansouri I,Saffari H.A fast hybrid algorithm for nonlinear analysis of structures[J].Asian Journal of Civil Engineering,2014,15(2):213-230.
[30]Zhang L X,Hu M Y,Wu P C.Application analysis of an improved N-R iterative method about finite element numerical calculation[J].Key Engineering Materials Trans Tech Publications,2011,450:518-521.
[31]王勖成.有限单元法[M].北京:清华大学出版社,2003:547-555.Wang Xucheng.Finite element method[M].Beijing:Tsinghua University Press,2003:547-555.(in Chinese)
[32]Suarjana M.Conjugate gradient method for linear and nonlinear structural analysis on sequential and parallel computers[D].USA:Stanford University,1994:1-30.
[33]De Borst R,Crisfield M A,Remmers J J C,et al.Nonlinear finite element analysis of solids and structures[M].New York:John Wiley&Sons,2012:30-46.
[34]Bathe K J.Finite element method[M].New Jersey:Prentice Hall,1996:755-759.
[35]RuizáA M,Argyros I K.Two-step newton methods[J].Journal of Complexity,2014,30(4):533-553.
[36]Chun C.Some fourth-order iterative methods for solving nonlinear equations[J].Applied Mathematics&Computation,2008,195(2):454-459.
[37]D?uni?J,Petkovi?M S,Petkovi?L D.A family of optimal three-point methods for solving nonlinear equations using two parametric functions[J].Applied Mathematics&Computation,2011,217(19):7612-7619.
[38]Petkovi M S,Petkovi L D.Families of optimal multipoint methods for solving nonlinear equations:a survey[J].Applicable Analysis&Discrete Mathematics,2010,4(1):1-22.
[39]Ralston A,Rabinowitz P.A first course in numerical analysis[M].Courier Corporation,2001:25-29.
[40]王晓东.计算机算法设计与分析[M].第3版.北京:电子工业出版社,2007:1-3.Wang Xiaodong.The design and analysis of computer algorithms[M].3rd ed.Beijing:Publishing House of Electronics Industry,2007:1-3.(in Chinese)
[41]Rezaiee-Pajand M,Estiri H.Mixing dynamic relaxation method with load factor and displacement increments[J].Computers&Structures,2016,168:78-91.
[2]Kim D J,Duarte C A,Proenca S P.A generalized finite element method with global-local enrichment functions for confined plasticity problems[J].Computational Mechanics,2012,50(5):563-578.
[3]Lin T H.Theory of inelastic structures[M].New York:John Wiley&Sons,1968:43-55.
[4]Li G,Zhang Y,Li H N.Nonlinear seismic analysis of reinforced concrete frames using the force analogy method[J].Earthquake Engineering&Structural Dynamics,2014,43(14):2115-2134.
[5]Akgün M A,Garcelon J H,Haftka R T.Fast exact linear and non-linear structural reanalysis and the ShermanMorrison-Woodbury formulas[J].International Journal for Numerical Methods in Engineering,2001,50(7):1587-1606.
[6]Li G,Yu D.Efficient inelasticity-separated finite element method for material nonlinearity analysis[J].Journal of Engineering Mechanics,2018,144(4):04018008-1-04018008-11.
[7]高国强.车身结构设计中快速重分析方法研究[D].长沙:湖南大学,2015:16-30.Gao Guoqiang.The research of fast reanalysis algorithm for structural design of automobile body[D].Changsha:Hunan University,2015:16-30.(in Chinese)
[8]王琥,种浩,高国强,等.重分析方法研究进展及展望[J].工程力学,2017,34(5):1-16.Wang Hu,Chong Hao,Gao Guoqiang,et al.Review of advances and outlook in reanalysis methods[J].Engineering Mechanics,2017,34(5):1-16.(in Chinese)
[9]宋琦,杨任,陈璞.一种新的结构修改算法及其在工程设计中的应用[J].工程力学,2016,33(7):1-6.Song Qi,Yang Ren,Chen Pu.A new algorithm for structural modifications and its applications[J].Engineering Mechanics,2016,33(7):1-6.(in Chinese)
[10]Huang C,Verchery G.An exact structural static reanalysis method[J].Communications in numerical methods in engineering,1997,13(2):103-112.
[11]Deng L,Ghosn M.Pseudoforce method for nonlinear analysis and reanalysis of structural systems[J].Journal of Structural Engineering,2001,127(5):570-578.
[12]Kirsch U.Reanalysis and sensitivity reanalysis by combined approximations[J].Structural and Multidisciplinary Optimization,2009,40(1):1-15.
[13]李书,冯太华,范绪箕.结构静力问题的重特征值灵敏度分析[J].工程力学,1996,13(4):97-104.Li Shu,Feng Taihua,Fan Xuji.Sensitivity analysis with repeated eigenvalues in structural static problem[J].Engineering Mechanics,1996,13(4):97-104.(in Chinese)
[14]Kirsch U,Bogomolni M.Nonlinear and dynamic structural analysis using combined approximations[J].Computers&structures,2007,85(10):566-578.
[15]Kirsch U,Bogomolni M,Sheinman I.Nonlinear dynamic reanalysis of structures by combined approximations[J].Computer Methods in Applied Mechanics&Engineering,2006,195:4420-4432.
[16]Hurtado J E.Reanalysis of linear and nonlinear structures using iterated shanks transformation[J].Computer Methods in Applied Mechanics&Engineering,2002,191:4215-4229.
[17]Huang G,Chen X,Yang Z,et al.Exact analysis and reanalysis methods for structures with nonlinear boundary conditions[J].Computers&Structures,2018,198:12-22.
[18]Wong K K F,Yang R.Inelastic dynamic response of structures using force analogy method[J].Journal of Engineering Mechanics,1999,125(10):1190-1199.
[19]靳永强,李钢,李宏男.基于拟力法的钢支撑非线性滞回行为模拟[J].工程力学,2017,34(10):139-148.Jin Yongqiang,Li Gang,Li Hongnan.Numerical simulation of steel brace hysteretic behavior based on the force analogy method[J].Engineering Mechanics,2017,34(10):139-148.(in Chinese)
[20]李钢,余丁浩,李宏男.基于拟力法的纤维梁有限元非线性分析方法[J].建筑结构学报,2016,37(9):61-68.Li Gang,Yu Dinghao,Li Hongnan.Nonlinear fiber beam element analysis based on force analogy method[J].Journal of Building Structures,2016,37(9):61-68.(in Chinese)
[21]Li G,Yu D,Li H N.Seismic response analysis of reinforced concrete frames using inelasticity-separated fibre beam-column model[J].Earthquake Engineering&Structural Dynamics,2018,47(5):1291-1308.
[22]Yu D H,Li G,Li H N.Improved woodbury solution method for nonlinear analysis with high-rank modifications based on a sparse approximation approach[J].Journal of Engineering Mechanics,2018,144(11):04018103-1-04018103-13.
[23]李钢,贾硕,余丁浩.基于算法复杂度理论的拟力法计算效率评价[J].计算力学学报,2018,35(2):129-137.Li Gang,Jia Shuo,Yu Dinghao.The efficiency evaluation of force analogy method based on the algorithm complexity theory[J].Chinese Journal of Computational Mechanics,2018,35(2):129-137.(in Chinese)
[24]何政,欧进萍.钢筋混凝土结构非线性分析[M].哈尔滨:哈尔滨工业大学出版社,2007:168-177.He Zheng,Ou Jinping.Nonlinear analysis of reinforced concrete structures[M].Harbin:Harbin Institute of Technology Press,2007:168-177.(in Chinese)
[25]Scott M H,Fenves G L.Krylov subspace accelerated newton algorithm:Application to dynamic progressive collapse simulation of frames[J].Journal of Structural Engineering,2010,136(5):473-480.
[26]Bathe K J,Cimento A P.Some practical procedures for the solution of nonlinear finite element equations[J].Computer Methods in Applied Mechanics&Engineering,1980,22(1):59-85.
[27]Kim J H,Kim Y H.A predictor-corrector method for structural nonlinear analysis[J].Computer Methods in Applied Mechanics&Engineering,2001,191(8-10):959-974.
[28]Saffari H,Mansouri I,Bagheripour M H,et al.Elastoplastic analysis of steel plane frames using homotopy perturbation method[J].Journal of Constructional Steel Research,2012,70(2):350-357.
[29]Mansouri I,Saffari H.A fast hybrid algorithm for nonlinear analysis of structures[J].Asian Journal of Civil Engineering,2014,15(2):213-230.
[30]Zhang L X,Hu M Y,Wu P C.Application analysis of an improved N-R iterative method about finite element numerical calculation[J].Key Engineering Materials Trans Tech Publications,2011,450:518-521.
[31]王勖成.有限单元法[M].北京:清华大学出版社,2003:547-555.Wang Xucheng.Finite element method[M].Beijing:Tsinghua University Press,2003:547-555.(in Chinese)
[32]Suarjana M.Conjugate gradient method for linear and nonlinear structural analysis on sequential and parallel computers[D].USA:Stanford University,1994:1-30.
[33]De Borst R,Crisfield M A,Remmers J J C,et al.Nonlinear finite element analysis of solids and structures[M].New York:John Wiley&Sons,2012:30-46.
[34]Bathe K J.Finite element method[M].New Jersey:Prentice Hall,1996:755-759.
[35]RuizáA M,Argyros I K.Two-step newton methods[J].Journal of Complexity,2014,30(4):533-553.
[36]Chun C.Some fourth-order iterative methods for solving nonlinear equations[J].Applied Mathematics&Computation,2008,195(2):454-459.
[37]D?uni?J,Petkovi?M S,Petkovi?L D.A family of optimal three-point methods for solving nonlinear equations using two parametric functions[J].Applied Mathematics&Computation,2011,217(19):7612-7619.
[38]Petkovi M S,Petkovi L D.Families of optimal multipoint methods for solving nonlinear equations:a survey[J].Applicable Analysis&Discrete Mathematics,2010,4(1):1-22.
[39]Ralston A,Rabinowitz P.A first course in numerical analysis[M].Courier Corporation,2001:25-29.
[40]王晓东.计算机算法设计与分析[M].第3版.北京:电子工业出版社,2007:1-3.Wang Xiaodong.The design and analysis of computer algorithms[M].3rd ed.Beijing:Publishing House of Electronics Industry,2007:1-3.(in Chinese)
[41]Rezaiee-Pajand M,Estiri H.Mixing dynamic relaxation method with load factor and displacement increments[J].Computers&Structures,2016,168:78-91.