结构特征灵敏度分析若干问题研究
摘要
本文系统地研究了对应于孤立和重特征值的实(复)特征向量的灵敏度计算方法。
对于对称无阻尼系统,提出了两种计算特解的新方法:扩展系统方法和改进的Nelson方法。在这两种方法中,我们通过调整相应的非奇异系数矩阵的元素降低了其条件数,从而提高了方法的数值稳定性和计算精度。对于一般的非对称非亏损系统,提出了一种新的规范化条件,该条件能使复特征向量及其导数唯一,并且利用该条件我们能计算任意可微的复特征向量一阶导数;此外,也提出一种计算特解的有效算法。提出的方法简单,应用范围广,而且易于在计算机上实现。最后,数值算例验证了方法的有效性。
The structural eigensensitivity is the information of derivatives of the eigenvalues and eigenvectors with respect to the design parameters, and it can be applied to the fields such as dynamical response analysis, model modification, structural optimization and damage identification. Supplying an efficient algorithm for the eigensensitivity of the large complicated structures is a challenging problem. Under such engineering background and requirement, we devote ourselves to the research of the computational methods of the eigensensitivity.
In this paper, we study the eigensensitivity of the real and complex eigenvectors of systems with distinct and repeated eigenvalues. For the symmetric undamped systems, we present two new methods to compute the particular solutions, i.e. the extending system method and the improved Nelson’s method. For the general asymmetric non-defective eigensystems, we propose a new normalization condition. In addition, we also put forward an efficient algorithm to compute the particular solutions. Numerical examples have demonstrated the validity of the proposed methods.
1. The eigensensitivity analysis of the symmetric undamped systems
Considering the following real and symmetric eigenvalue problem where K (p) and M (p) in R~(n×n) are the structural symmetric stiffness and mass matrices, respectively, whose elements depend continuously on the real parameter p .λ_j(p)is the eigenvalue, x_j(p)is the eigenvector corresponding toλ_j(p), n is the total degrees of freedom, andδ_(jk) is the Kronecker delta. The paper is concerned with the derivatives of eigenvalues and eigenvectors at p = p_0. For convenience, hereafter " (p_0)" is omitted for variable evaluated at p = p_0.
It is assumed that at p = p0 the eigenvalue problems (1)-(2) have m (1 < m≤n) repeated eigenvalues, without loss of generality, we denoteλ_1 ( p_0)=λ2(p0)=L =λ_m(p_0)=λ|~, and letΛ( p) =diag[λ_1 (p),λ_2(p),L,λ_m(p)], X ( p )= [x_1 (p),x_2(p),L,x_m(p)]. The repeated eigenvalue derivativesΛ′can be found by solving the subeigenproblem XT (K′-λ|~M′)XΓ=ΓΛ′, and assuming that theλ_j′be distinct, then the unique differentiable eigenvectors corresponding to the repeated eigenvalues are determined by Z = XΓ. We define F := K-λ|~M, G := (λ|~ M′-K′)Z+MZΛ′. So the governing equation of Z′can be written as F Z′=G, and let the solution be Z′=V+ZC, where the particular solution V satisfies FV = G. Once theV is obtained, the coefficient matrix C can be determined by the known algorithm. In the following, we will propose two new methods for the computation of the particular solutionV .
1) Extending system method
As V is a particular solution only, and noting the expression of Z′, we let the particular solution V be orthogonal to X with respect to M , i.e. X~TMV=0. Considering the following extended equations with unknowns V andμ∈C~(m×m)
We may prove the coefficient matrix is non-singular, and the V in Equation
(3) is just a particular solution. We suppose that the repeated eigenvalue be nonzero, i.e.λ|~≠0, and revise the Equation (3) as follows where . It is easy to show that Equations (3) and (4) have the same solution, but the later has much smaller condition number. Thus, we can obtain the particular solution V by solving the Equation (4). Note that the case of m =1 corresponds to a distinct eigenvalue and the proposed method is also applicable.
2) Improved Nelson’s method
Let Z_(m,m) be a submatrix composed of the l_j th ( j= 1, 2, L, m) rows of Z such that Then, we have det( Z m, m)≠0. Setting the l_j th rows and columns of F equal to zeros, while the l_j th diagonal elements of F equal to the corresponding diagonal elements k ljljof the stiffness matrix K , respectively, to form F? which is non-singular; setting l_j th rows of G equal to zeros to form G|~ . Then we may get the particular solution V by solving the equation FV= G|~. Note that the case of m =1 corresponds to a distinct eigenvalue and the proposed method degenerate into the Nelson’s method.
2. The eigensensitivity analysis of the general asymmetric non-defective systems
Let A (p) and B (p) in R~(n×n) be the general asymmetric matrices, whose elements depend continuously on the real parameter p , and B (p) is non-singular. Considering the following right and left eigenvalue problem whereλ_j(p) is the complex eigenvalue, x_j(p) and y j(p) are the right and left eigenvectors corresponding toλ_j(p), n is the total degrees of freedom, andδjk is the Kronecker delta.
Here, we still letλ_1 ( p 0)=λ_2(p0)=L =λm(p_0)=λ|~,Λ(p) and X (p) are the same as the expression above, and Y ( p )= [y1 (p),y2(p),L,ym(p)]. The derivativesΛ′can be found by solving a left-right subeigenproblem where D := YT (A′-λ|~B′)X. Assuming that theλ′j ( j=1, 2, L, m) be distinct, the differentiable right and left eigenvectors are determined by Z~ R = [z|~_(R1) ,z|~_(R2),L ,z|~_(Rm)]=Xαand Z|~ _L = [z|~_(L1) ,z|~_(L2),L ,z|~_(Lm]=Yβ. Sinceαj is uncertain, z|~ Rj is also uncertain. For each j ( j= 1,2,L,m), we may choose an arbitrary z|~ Rj , denoted as z_(Rj) , to compute its derivative. Then, the unique differentiable left eigenvector z_(Lj) can be determined. Thus, we obtain the differentiable right and left eigenvector matrices Z R and Z L.
Let z_(Rj) = a_j+ib_j be a differentiable right eigenvector corresponding toλ|~. We define which satisfies l Tj zRj=1( j = 1, 2, L,m). To render the derivatives unique, we propose the following normalization condition for p∈U0 ( p0) We will utilize the condition (13) and governing equations to derive the right and left eigenvector derivatives Z′R and Z′L .
For convenience, we define . Then the governing equations can be written as F Z′R =G and F~T Z′L=H, respectively, and let the solutions be Z′R =VR+ZRCR and Z′L =VL+Z_LC_L, where the particular solutions V_R and V_L satisfy the equations FV_R =G and F~T VL=H, respectively.
We first compute the particular solutions V_R and V_L. Similarly, we let the particular solution V_R satisfy the equation Y~T BVR=0. Considering the following extended equations with unknowns V_R andμ∈C m×m We may prove the coefficient matrix is non-singular, and the V_R in Equation (14) is just a particular solution. Thus, we can obtain the particular solution V_R by solving the Equation (14).
In a similar way, we only need to solve the following equation to obtain the particular solution V_L
It is noted that the coefficient matrix in Equation (15) is the transpose of that in Equation (14). Hence, the computational cost may be reduced.
Next, we can compute C_R and C L using the particular solutions. Differentiating the equation A ( p )Z_R (p)= B(p)Z_R(p)Λ(p) with respect to the parameter p twice at p = p_0, premultiplying it by Z_L~T and noting the equation Y T BVR=0, we have where Let R =[ r_(jk)], C_R =[ c_(Rjk)]. Using Equation (16), we may obtain the off-diagonal elements of matrix C_R
The diagonal elements will be solved by using the normalization condition (13). Differentiating Equation (13) with respect to the parameter p at p = p_0 and rearranging it yield Differentiating the equation Z_L~T ( p )B(p)Z_R(p)=Im with respect to the parameter p at p = p_0, we get
Substituting the expressions of Z_R′and Z_L′into Equation (20), and utilizing the choices of the particular solutions V R and V L, yield
So far, we have obtained the V_R, V_L, C_R and C_L. Thus, the right and left eigenvector derivatives Z_R′and Z_L′can be computed, respectively. From the above discussion, we know that the proposed algorithm is feasible, and the corresponding calculations are also simple. Note that the case of m =1 corresponds to a distinct eigenvalue and the proposed method is also applicable with simple modification.
对于对称无阻尼系统,提出了两种计算特解的新方法:扩展系统方法和改进的Nelson方法。在这两种方法中,我们通过调整相应的非奇异系数矩阵的元素降低了其条件数,从而提高了方法的数值稳定性和计算精度。对于一般的非对称非亏损系统,提出了一种新的规范化条件,该条件能使复特征向量及其导数唯一,并且利用该条件我们能计算任意可微的复特征向量一阶导数;此外,也提出一种计算特解的有效算法。提出的方法简单,应用范围广,而且易于在计算机上实现。最后,数值算例验证了方法的有效性。
The structural eigensensitivity is the information of derivatives of the eigenvalues and eigenvectors with respect to the design parameters, and it can be applied to the fields such as dynamical response analysis, model modification, structural optimization and damage identification. Supplying an efficient algorithm for the eigensensitivity of the large complicated structures is a challenging problem. Under such engineering background and requirement, we devote ourselves to the research of the computational methods of the eigensensitivity.
In this paper, we study the eigensensitivity of the real and complex eigenvectors of systems with distinct and repeated eigenvalues. For the symmetric undamped systems, we present two new methods to compute the particular solutions, i.e. the extending system method and the improved Nelson’s method. For the general asymmetric non-defective eigensystems, we propose a new normalization condition. In addition, we also put forward an efficient algorithm to compute the particular solutions. Numerical examples have demonstrated the validity of the proposed methods.
1. The eigensensitivity analysis of the symmetric undamped systems
Considering the following real and symmetric eigenvalue problem where K (p) and M (p) in R~(n×n) are the structural symmetric stiffness and mass matrices, respectively, whose elements depend continuously on the real parameter p .λ_j(p)is the eigenvalue, x_j(p)is the eigenvector corresponding toλ_j(p), n is the total degrees of freedom, andδ_(jk) is the Kronecker delta. The paper is concerned with the derivatives of eigenvalues and eigenvectors at p = p_0. For convenience, hereafter " (p_0)" is omitted for variable evaluated at p = p_0.
It is assumed that at p = p0 the eigenvalue problems (1)-(2) have m (1 < m≤n) repeated eigenvalues, without loss of generality, we denoteλ_1 ( p_0)=λ2(p0)=L =λ_m(p_0)=λ|~, and letΛ( p) =diag[λ_1 (p),λ_2(p),L,λ_m(p)], X ( p )= [x_1 (p),x_2(p),L,x_m(p)]. The repeated eigenvalue derivativesΛ′can be found by solving the subeigenproblem XT (K′-λ|~M′)XΓ=ΓΛ′, and assuming that theλ_j′be distinct, then the unique differentiable eigenvectors corresponding to the repeated eigenvalues are determined by Z = XΓ. We define F := K-λ|~M, G := (λ|~ M′-K′)Z+MZΛ′. So the governing equation of Z′can be written as F Z′=G, and let the solution be Z′=V+ZC, where the particular solution V satisfies FV = G. Once theV is obtained, the coefficient matrix C can be determined by the known algorithm. In the following, we will propose two new methods for the computation of the particular solutionV .
1) Extending system method
As V is a particular solution only, and noting the expression of Z′, we let the particular solution V be orthogonal to X with respect to M , i.e. X~TMV=0. Considering the following extended equations with unknowns V andμ∈C~(m×m)
We may prove the coefficient matrix is non-singular, and the V in Equation
(3) is just a particular solution. We suppose that the repeated eigenvalue be nonzero, i.e.λ|~≠0, and revise the Equation (3) as follows where . It is easy to show that Equations (3) and (4) have the same solution, but the later has much smaller condition number. Thus, we can obtain the particular solution V by solving the Equation (4). Note that the case of m =1 corresponds to a distinct eigenvalue and the proposed method is also applicable.
2) Improved Nelson’s method
Let Z_(m,m) be a submatrix composed of the l_j th ( j= 1, 2, L, m) rows of Z such that Then, we have det( Z m, m)≠0. Setting the l_j th rows and columns of F equal to zeros, while the l_j th diagonal elements of F equal to the corresponding diagonal elements k ljljof the stiffness matrix K , respectively, to form F? which is non-singular; setting l_j th rows of G equal to zeros to form G|~ . Then we may get the particular solution V by solving the equation FV= G|~. Note that the case of m =1 corresponds to a distinct eigenvalue and the proposed method degenerate into the Nelson’s method.
2. The eigensensitivity analysis of the general asymmetric non-defective systems
Let A (p) and B (p) in R~(n×n) be the general asymmetric matrices, whose elements depend continuously on the real parameter p , and B (p) is non-singular. Considering the following right and left eigenvalue problem whereλ_j(p) is the complex eigenvalue, x_j(p) and y j(p) are the right and left eigenvectors corresponding toλ_j(p), n is the total degrees of freedom, andδjk is the Kronecker delta.
Here, we still letλ_1 ( p 0)=λ_2(p0)=L =λm(p_0)=λ|~,Λ(p) and X (p) are the same as the expression above, and Y ( p )= [y1 (p),y2(p),L,ym(p)]. The derivativesΛ′can be found by solving a left-right subeigenproblem where D := YT (A′-λ|~B′)X. Assuming that theλ′j ( j=1, 2, L, m) be distinct, the differentiable right and left eigenvectors are determined by Z~ R = [z|~_(R1) ,z|~_(R2),L ,z|~_(Rm)]=Xαand Z|~ _L = [z|~_(L1) ,z|~_(L2),L ,z|~_(Lm]=Yβ. Sinceαj is uncertain, z|~ Rj is also uncertain. For each j ( j= 1,2,L,m), we may choose an arbitrary z|~ Rj , denoted as z_(Rj) , to compute its derivative. Then, the unique differentiable left eigenvector z_(Lj) can be determined. Thus, we obtain the differentiable right and left eigenvector matrices Z R and Z L.
Let z_(Rj) = a_j+ib_j be a differentiable right eigenvector corresponding toλ|~. We define which satisfies l Tj zRj=1( j = 1, 2, L,m). To render the derivatives unique, we propose the following normalization condition for p∈U0 ( p0) We will utilize the condition (13) and governing equations to derive the right and left eigenvector derivatives Z′R and Z′L .
For convenience, we define . Then the governing equations can be written as F Z′R =G and F~T Z′L=H, respectively, and let the solutions be Z′R =VR+ZRCR and Z′L =VL+Z_LC_L, where the particular solutions V_R and V_L satisfy the equations FV_R =G and F~T VL=H, respectively.
We first compute the particular solutions V_R and V_L. Similarly, we let the particular solution V_R satisfy the equation Y~T BVR=0. Considering the following extended equations with unknowns V_R andμ∈C m×m We may prove the coefficient matrix is non-singular, and the V_R in Equation (14) is just a particular solution. Thus, we can obtain the particular solution V_R by solving the Equation (14).
In a similar way, we only need to solve the following equation to obtain the particular solution V_L
It is noted that the coefficient matrix in Equation (15) is the transpose of that in Equation (14). Hence, the computational cost may be reduced.
Next, we can compute C_R and C L using the particular solutions. Differentiating the equation A ( p )Z_R (p)= B(p)Z_R(p)Λ(p) with respect to the parameter p twice at p = p_0, premultiplying it by Z_L~T and noting the equation Y T BVR=0, we have where Let R =[ r_(jk)], C_R =[ c_(Rjk)]. Using Equation (16), we may obtain the off-diagonal elements of matrix C_R
The diagonal elements will be solved by using the normalization condition (13). Differentiating Equation (13) with respect to the parameter p at p = p_0 and rearranging it yield Differentiating the equation Z_L~T ( p )B(p)Z_R(p)=Im with respect to the parameter p at p = p_0, we get
Substituting the expressions of Z_R′and Z_L′into Equation (20), and utilizing the choices of the particular solutions V R and V L, yield
So far, we have obtained the V_R, V_L, C_R and C_L. Thus, the right and left eigenvector derivatives Z_R′and Z_L′can be computed, respectively. From the above discussion, we know that the proposed algorithm is feasible, and the corresponding calculations are also simple. Note that the case of m =1 corresponds to a distinct eigenvalue and the proposed method is also applicable with simple modification.
引文
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