结构模态重分析的预条件SRQCG方法
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摘要
本文针对自由度未发生改变的结构模态重分析问题,根据初始结构模态分析的刚度矩阵分解的信息构造了预条件矩阵,并选取修改前结构模态分析的特征向量作为开始迭代向量,然后采用预条件SRQCG方法求解修改后结构的固有频率和相应的振动模态.提出的预条件算子可以加快SRQCG方法的收敛速度.我们讨论了不同预条件算子的SRQCG方法,采用多个数值算例对各种方法的收敛速度做了比较.结果说明以结构修改前刚度矩阵的逆作为预条件算子的SRQCG方法对结构模态重分析问题是很有效的.
With the development of science and technology, many large-complicated structures need to be designed. In structural optimization, the solution is iterative and consists of repeated analyses followed by redesign steps. The high computational cost involved in repeated analyses of large-scale problems is one of the main obstacles in the solution process. In many problems the analysis part will require most of the computational effort, therefore only methods that do not involve numerous time-consuming implicit analyses might prove useful. Reanalysis methods, intended to reduce the computational cost, have been motivated by some typical difficulties involved in the solution process. The object of reanalysis is to evaluate the structural response for successive modifications efficiently and precisely without solving the set of the modified implicit equations so that the computational cost is significantly reduced.
In this thesis, the Simultaneous Rayleigh quotient modified conjugate gradient method which combine the preconditioned technique and iterate method is developed for structural modal reanalysis problems.
Suppose the stiffness matrix and mass matrix of initial design is and with dimension of . The frequencies and corresponding modesobtained from solution of the initial eigenproblem
Assume a change in the design, the corresponding stiffness and mass matrices can be expressed The modified analysis equations to be solved are
The object is to evaluate the modified structural responseλi, using the initial information from without solving the set of the modified implicit equations so that the computational cost is significantly reduced.
The SRQMCG2 scheme for structural modal reanalysis consists of the following steps:
Step 1. Use initial stiffness matrix as preconditioning matrix, that is K0
Step 2. Using the eigenvectors obtained from the modal analysis for initial structure as starting matrix X ( 0), a tolerance value TOLL, the allowed maximum number of iterations NITMAX and a“restart”value NREST. Compute the initial residual matrix as where is the diagonal matrix whose entries are the diagonal coefficients of the Rayleigh matrix , then compute the average relative residual (i teration index). ( k)As long as k is smaller than NITMAX and ra is greater than TOLL, execute Steps 3, 4, and 5; otherwise go to Step 6. 3.2.1. Programing a Ritz projection step, which is equivalent to solving the problem: Where y is a p-dimensional vector and . An orthogonal matrix is found such that Step 4. Evaluate the new matrix 4.1. The vector is evaluated by M-orthogonalizing p(j k) with respect to 4.2. The coefficientsαj are obtained by minimizing the Rayleigh quotient is evaluated by M-orthonormalizing xj with respect to Vj. Step 5. Compute the residual matrix M r X together with the value ra( k+1), Increment the iteration counter k and go to the loop start.
Step 6. If ra ( k) is smaller than TOLL, R x j and x(j k), ( j =1,,p),are the smallest p eigenvalues of (3), and the corresponding eigenvectors, respectively. Here, ra ( k) is the average relative residual, it is defined as follows:
Consider now the preconditioning matrix . In mathematics, two preconditioning matrices can be chosen. The cheapest selection for is provided by where D is the diagonal matrix formed by the diagonal entries of K ,named SRQMCG2(D). Another choice is
where L being the pointwise incomplete Cholesky factor of K, named SRQMCG2(LLT).
Numerical examples shows that, compared with the algorithms SRQMCG2(D) and SRQMGC2(LLT), the proposed algorithm SRQMCG2(K0) is an efficient method for solving structural modal reanalysis.
With the development of science and technology, many large-complicated structures need to be designed. In structural optimization, the solution is iterative and consists of repeated analyses followed by redesign steps. The high computational cost involved in repeated analyses of large-scale problems is one of the main obstacles in the solution process. In many problems the analysis part will require most of the computational effort, therefore only methods that do not involve numerous time-consuming implicit analyses might prove useful. Reanalysis methods, intended to reduce the computational cost, have been motivated by some typical difficulties involved in the solution process. The object of reanalysis is to evaluate the structural response for successive modifications efficiently and precisely without solving the set of the modified implicit equations so that the computational cost is significantly reduced.
In this thesis, the Simultaneous Rayleigh quotient modified conjugate gradient method which combine the preconditioned technique and iterate method is developed for structural modal reanalysis problems.
Suppose the stiffness matrix and mass matrix of initial design is and with dimension of . The frequencies and corresponding modesobtained from solution of the initial eigenproblem
Assume a change in the design, the corresponding stiffness and mass matrices can be expressed The modified analysis equations to be solved are
The object is to evaluate the modified structural responseλi, using the initial information from without solving the set of the modified implicit equations so that the computational cost is significantly reduced.
The SRQMCG2 scheme for structural modal reanalysis consists of the following steps:
Step 1. Use initial stiffness matrix as preconditioning matrix, that is K0
Step 2. Using the eigenvectors obtained from the modal analysis for initial structure as starting matrix X ( 0), a tolerance value TOLL, the allowed maximum number of iterations NITMAX and a“restart”value NREST. Compute the initial residual matrix as where is the diagonal matrix whose entries are the diagonal coefficients of the Rayleigh matrix , then compute the average relative residual (i teration index). ( k)As long as k is smaller than NITMAX and ra is greater than TOLL, execute Steps 3, 4, and 5; otherwise go to Step 6. 3.2.1. Programing a Ritz projection step, which is equivalent to solving the problem: Where y is a p-dimensional vector and . An orthogonal matrix is found such that Step 4. Evaluate the new matrix 4.1. The vector is evaluated by M-orthogonalizing p(j k) with respect to 4.2. The coefficientsαj are obtained by minimizing the Rayleigh quotient is evaluated by M-orthonormalizing xj with respect to Vj. Step 5. Compute the residual matrix M r X together with the value ra( k+1), Increment the iteration counter k and go to the loop start.
Step 6. If ra ( k) is smaller than TOLL, R x j and x(j k), ( j =1,,p),are the smallest p eigenvalues of (3), and the corresponding eigenvectors, respectively. Here, ra ( k) is the average relative residual, it is defined as follows:
Consider now the preconditioning matrix . In mathematics, two preconditioning matrices can be chosen. The cheapest selection for is provided by where D is the diagonal matrix formed by the diagonal entries of K ,named SRQMCG2(D). Another choice is
where L being the pointwise incomplete Cholesky factor of K, named SRQMCG2(LLT).
Numerical examples shows that, compared with the algorithms SRQMCG2(D) and SRQMGC2(LLT), the proposed algorithm SRQMCG2(K0) is an efficient method for solving structural modal reanalysis.
引文
[1]薛明德.力学与工程技术的进步.北京:高等教育出版社,2001.
[2]林家浩.结构动力优化中的灵敏度分析.震动与冲击,1985,4(1):1-6 .
[3]吕振华.结构动力学修改重分析方法的发展.计算结构力学及其应用,1994:85-91.
[4] Kirsch U, Bogomolni M. Procedures for approximate eigenproblem reanalysis of structures. International Journal for Numerical Methods in Engineering, 60(2004):1969-1986.
[5] Kirsch U, Bogomolni M, Sheinman I. Efficient dynamic reanalysis of structures. Journal of Structural Engineering, ASCE 133(2007): 440-448.
[6] Zhang G, Mourelatos Z P. An efficient re-analysis methodology for vibration of large-scale structures. SAE 2007 Noise and Vibration Conference and Exhibition. 2007-01-2326.
[7] Benzi M. Preconditioning techniques for large linear systems: a survey. Journal of Computational Physics, 182 (2002): 418-477.
[8] Neymeyr K. On preconditioned eigensolvers and Invert–Lanczos processes. Linear Algebra and Its Applications 430 (2009): 1039–1056.
[9] Bai Z, Demmel J, Dongarra J, Ruhe A, Van der Vorst H. Templates for the solution of algebraic eigenvalue problems: a practical guide. Society for Industrial and Applied Mathematics, Philadelphia, 2000.
[10]王勖成,邵敏.有限单元法基本原理和数值方法.北京:清华大学出版社,2006:443-448.
[11]刘延柱,陈文良,陈立群,振动力学.北京:高等教育出版社,2002:105-107.
[12] Longsine D E, McCormick S F. Simultaneous Rayleigh-quotient minimization methods for Ax =λBx. Linear Algebra and Its Applications, 1980, 34: 195-233.
[13] Schwarz H R. Simultaneous Rayleigh-quoient iteration methods for large sparse generalized eigenvalue problems. Lecture Notes in Mathematics, 1982, Vol.968: 384-398.
[14] Perdon A, Gambolati G.. Extreme eigenvalues of large sparse matrices by Rayleigh quotient and modified conjugate gradients. Computer Methods inApplied Mechanics and Engineering,1986, 56: 251-264.
[15] Sartoretto F, Pini G, Gambolati G.. Accelerated simultaneous iterations for large finite element eigenproblems. Journal of Computational Physics, 1989, 81:53-69.
[16] Ajiz M A , Jennings A. A robust incomplete Cholesky-conjugate gradient algorithm. Intermational Journal for Numerical Methods in Engineering, 1984, 20: 949-966.
[17] David S K. The incomplete Cholesky-conjugate gradient method for the iterative solution of systems of linear equations. Journal of Computational Physics, 1978, 26: 43-65.
[18]张志涌.精通MATLAB 6.5版.北京:北京航空航天大学出版社,2005.
[2]林家浩.结构动力优化中的灵敏度分析.震动与冲击,1985,4(1):1-6 .
[3]吕振华.结构动力学修改重分析方法的发展.计算结构力学及其应用,1994:85-91.
[4] Kirsch U, Bogomolni M. Procedures for approximate eigenproblem reanalysis of structures. International Journal for Numerical Methods in Engineering, 60(2004):1969-1986.
[5] Kirsch U, Bogomolni M, Sheinman I. Efficient dynamic reanalysis of structures. Journal of Structural Engineering, ASCE 133(2007): 440-448.
[6] Zhang G, Mourelatos Z P. An efficient re-analysis methodology for vibration of large-scale structures. SAE 2007 Noise and Vibration Conference and Exhibition. 2007-01-2326.
[7] Benzi M. Preconditioning techniques for large linear systems: a survey. Journal of Computational Physics, 182 (2002): 418-477.
[8] Neymeyr K. On preconditioned eigensolvers and Invert–Lanczos processes. Linear Algebra and Its Applications 430 (2009): 1039–1056.
[9] Bai Z, Demmel J, Dongarra J, Ruhe A, Van der Vorst H. Templates for the solution of algebraic eigenvalue problems: a practical guide. Society for Industrial and Applied Mathematics, Philadelphia, 2000.
[10]王勖成,邵敏.有限单元法基本原理和数值方法.北京:清华大学出版社,2006:443-448.
[11]刘延柱,陈文良,陈立群,振动力学.北京:高等教育出版社,2002:105-107.
[12] Longsine D E, McCormick S F. Simultaneous Rayleigh-quotient minimization methods for Ax =λBx. Linear Algebra and Its Applications, 1980, 34: 195-233.
[13] Schwarz H R. Simultaneous Rayleigh-quoient iteration methods for large sparse generalized eigenvalue problems. Lecture Notes in Mathematics, 1982, Vol.968: 384-398.
[14] Perdon A, Gambolati G.. Extreme eigenvalues of large sparse matrices by Rayleigh quotient and modified conjugate gradients. Computer Methods inApplied Mechanics and Engineering,1986, 56: 251-264.
[15] Sartoretto F, Pini G, Gambolati G.. Accelerated simultaneous iterations for large finite element eigenproblems. Journal of Computational Physics, 1989, 81:53-69.
[16] Ajiz M A , Jennings A. A robust incomplete Cholesky-conjugate gradient algorithm. Intermational Journal for Numerical Methods in Engineering, 1984, 20: 949-966.
[17] David S K. The incomplete Cholesky-conjugate gradient method for the iterative solution of systems of linear equations. Journal of Computational Physics, 1978, 26: 43-65.
[18]张志涌.精通MATLAB 6.5版.北京:北京航空航天大学出版社,2005.