Eigenvector Derivatives Research Articles

Higher-order sensitivity analysis of periodic 3-D eigenvalue problems for electromagnetic field calculations

Abstract. An algorithm to perform a higher-order sensitivity analysis for electromagnetic eigenvalue problems is presented. By computing the eigenvalue and eigenvector derivatives, the Brillouin Diagram for periodic structures can be calculated. The discrete model is described using the Finite Integration Technique (FIT) with periodic boundaries, and the sensitivity analysis is performed with respect to the phase shift φ between the periodic boundaries. For validation, a reference solution is calculated by solving multiple eigenvalue problems (EVP). Furthermore, the eigenvalue derivatives are compared to reference values using finite difference (FD) formulas.

A new method for eigenvector derivatives of a quadratic eigenvalue problem

A new method is proposed to compute the eigenvector derivative of a quadratic eigenvalue problem (QEP) analytically dependent on a parameter. It avoids the linearization of the QEP. The proposed method can be seen as an improved incomplete modal method. Only a few eigenvectors of the QEP are required. The contributions of other eigenvectors to the desired eigenvector derivative are obtained by an iterative scheme. From this point of view, our method also can be seen as an iterative method. The convergence properties of the proposed method are analyzed. The techniques to accelerate the proposed method are provided. A strategy is developed for simultaneously computing several eigenvector derivatives by the proposed method. Finally some numerical examples are given to demonstrate the efficiency of our method.

Iterative Computation of Eigenvector Derivatives for Middle Eigenvalues

The computation of eigenvector derivatives for middle eigenvalues in the real symmetric eigensystems is discussed. Based on finite element modeling, the coefficient matrices of the governing equations for eigenvector derivatives are singular. This work concentrates on a new method for such computations. The method is based on an iteration technique and its convergence is proved. The number of iterations is adaptively determined based on some error estimation. Highly accurate approximations to eigenvector derivatives can be achieved with low computational cost. The proposed method can deal with both cases of simple and repeated eigenvalues in a unified manner. It is especially suitable for the large sparse matrices from finite element models. Finally, three examples including an aircraft wing structure are used to demonstrate the effectiveness of the proposed method.

Eigensensitivity of symmetric damped systems with repeated eigenvalues by generalized inverse

We consider computation of the derivatives of the semisimple eigenvalues and corresponding eigenvectors of a symmetric quadratic eigenvalue problem. Using the normalization condition, we can compute the derivatives of the differentiable eigenvalues of the quadratic eigenvalue problem. Using the constrained generalized inverse, we present an efficient algorithm to compute the particular solutions to the governing equation of the derivatives of eigenvectors. The proposed method is suitable for the computation of the eigenpair derivatives of a symmetric quadratic eigenvalue problem when the first-order derivatives of eigenvalues are distinct or repeated. A numerical example is included to illustrate the validity of the proposed method.

An eigenspace method for computing derivatives of semi-simple eigenvalues and corresponding eigenvectors of quadratic eigenvalue problems

This paper concerns computing derivatives of semi-simple eigenvalues and corresponding eigenvectors of the quadratic matrix polynomial Q(p,λ)=λ2M(p)+λC(p)+K(p) at p=p⁎. Computing derivatives of eigenvectors usually requires solving a certain singular linear system by transforming it into a nonsingular one. However, the coefficient matrix of the transformed linear system might be ill-conditioned. In this paper, we propose a new method for computing these derivatives, where the condition number of the coefficient matrix is the ratio of the maximum singular value to the minimum nonzero singular value of Q(p⁎,λ(p⁎)), which is generally smaller than those in current literature and hence leads to higher accuracy. Numerical examples show the feasibility and efficiency of our method.

Calculation of eigenpair derivatives for asymmetric damped systems with distinct and repeated eigenvalues

An algorithm is derived for the computation of eigenpair derivatives of asymmetric quadratic eigenvalue problem with distinct and repeated eigenvalues. In the proposed method, the eigenvector derivatives of the damped systems are divided into a particular solution and a homogeneous solution. By introducing an additional normalization condition, we construct two extended systems of linear equations with nonsingular coefficient matrices to calculate the particular solution. The method is numerically stable, and the homogeneous solutions are computed by the second-order derivatives of the eigenequations. Two numerical examples are used to illustrate the validity of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.

Calculation of eigenpair derivatives for symmetric quadratic eigenvalue problem with repeated eigenvalues

In this paper, we consider computing the derivatives of the semisimple eigenvalues and corresponding eigenvectors of symmetric quadratic eigenvalue problem. In the proposed method, the eigenvector derivatives of the symmetric quadratic eigenvalue problem are divided into a particular solution and a homogeneous solution; a simplified method is given to calculate the particular solution by solving a linear system with nonsingular coefficient matrix, the method is numerically stable and efficient. Two numerical examples are included to illustrate the validity of the proposed method.

A preconditioned conjugate gradient method for computing eigenvector derivatives with distinct and repeated eigenvalues

A preconditioned conjugate gradient method is proposed for computing eigenvector derivatives with distinct and repeated eigenvalues in the real symmetric eigensystems. In view of singular character of the coefficient matrices of the governing equations for particular solutions of eigenvector derivatives, a modified governing equation for the complementary part of the computed modal contribution excluding those of the repeated modes is introduced, and its coefficient matrix is symmetric and positive definite. The existing factored (shifted) stiffness matrix from an iterative eigensolution such as Lanczos or Subspace Iteration is then utilized as preconditioner. High accurate approximations to particular solutions of eigenvector derivatives can be provided with a few iterations. The present method can deal with both cases of simple and repeated eigenvalues in a unified manner, and can be integrated into a coupled eigensolver/derivative software module. It is especially suitable for the large sparse matrices that arise in industrial-size finite element models. Finally, two numerical examples are used to demonstrate the superior efficiency and fast convergence of the present method.

Improved modal truncation method for eigensensitivity analysis of asymmetric matrix with repeated eigenvalues

An improved modal truncation method with arbitrarily high order accuracy is developed for calculating the second- and third-order eigenvalue derivatives and the first- and second-order eigenvector derivatives of an asymmetric and non-defective matrix with repeated eigenvalues. If the different eigenvalues λ1, λ2, …, λr of the matrix satisfy |λ1| ⩽ … ⩽ |λr| and |λs| < |λs+1| (s ⩽ r−1), then associated with any eigenvalue λi (i ⩽ s), the errors of the eigenvalue and eigenvector derivatives obtained by the qth-order approximate method are proportional to |λi/λs+1|q+1, where the approximate method only uses the eigenpairs corresponding to λ1, λ2, …, λs. A numerical example shows the validity of the approximate method. The numerical example also shows that in order to get the approximate solutions with the same order accuracy, a higher order method should be used for higher order eigenvalue and eigenvector derivatives.

Free-Interface Modal Synthesis Based Substructural Damage Detection Method

Free-interface modal synthesis method is applied to civil structure, and a substructure method is proposed by introducing the method into global sensitivity method. The substructure expression of the derivatives of eigenvalues and eigenvectors with respect to elemental parameters is obtained. The accuracy of the application of free-interface modal synthesis method is evaluated with different retained modes in substructure, and then the effectiveness of the proposed substructure sensitivity method is illustrated through an 11-storey building under both single- and multidamage cases. Both the damage locations and the extent can be effectively identified. By comparing it with the identical results of global sensitivity method, the proposed method can be faster in detecting the damage location and more stable under multidamage cases. Since this substructure sensitivity method only needs to update sensitivity matrix in the substructure with relative small number of DOFs, it may save much computation effort and become more efficient.

Dynamic condensation approach to the calculation of eigensensitivity

Calculation of the eigensensitivity of a large and complex structural system requires considerable computational resources and is time-consuming. This paper derives the eigenvalue and eigenvector derivatives of a structure based on a dynamic condensation technique. The eigensensitivity of a structure are computed by iteratively updating the derivatives of the condensed system matrices and a transformation matrix. As the condensed model is much smaller than the original full model, the proposed method is quite efficient in the calculation of the eigensensitivity. The accuracy and efficiency of the proposed method are verified by the GARTEUR structure and a cantilever plate.

Eigensensitivity analysis of damped systems with distinct and repeated eigenvalues

This paper considers the computation of eigensolution sensitivity of viscously damped eigensystems with distinct and repeated eigenvalues. To simplify the computation, a combined normalization, which combines two traditional normalizations, is presented. Based on the combined normalization, a method for sensitivity analysis of eigenvalues and eigenvectors is studied. In the case of distinct eigenvalues, the proposed method can determine the eigenvector derivatives directly and is robust since the components of coefficient matrices are all of the same order of magnitude. The computational cost of the second-order sensitivities of eigenvectors can be reduced remarkably since the matrix decomposition of the coefficient matrix is available from the computation process of the first-order eigensensitivities. In the case of repeated eigenvalues, an algorithm is presented for computing the eigensolution sensitivities. The algorithm maintains N-space without using state-space equations such that the computational cost is reduced. The method is accurate, compact, numerically stable and easy to be implemented. Finally, three numerical examples have demonstrated the validity of the proposed method. The capacity of predicting the changes of eigensolutions with respect to the changes of design parameters in terms of the first- and second-order eigensensitivities is studied with application to the analysis of a two-stage floating raft isolation system.

Differentiable implicitly restarted Lanczos method for computing eigenpairs and their derivatives of large symmetric matrix-valued functions

A differentiable implicitly restarted Lanczos method is presented in this paper to compute the smallest eigenpairs and their derivatives of large matrix-valued function. The convergence of the proposed method is also established. This method improves the efficiency of computing the eigenpair derivatives greatly and there are three advantages of it. First, eigenpairs and their derivatives are computed simultaneously. Second, the equation system for eigenvector derivatives can be greatly reduced from the original matrix size. Third, the left eigenvector and its derivative are not required. Numerical experiments show that the method is efficient.

Computing Derivatives of Repeated Eigenvalues and Corresponding Eigenvectors of Quadratic Eigenvalue Problems

We consider quadratic eigenvalue problems in which the coefficient matrices, and hence the eigenvalues and eigenvectors, are functions of a real parameter. Our interest is in cases in which these functions remain differentiable when eigenvalues coincide. Many papers have been devoted to numerical methods for computing derivatives of eigenvalues and eigenvectors, but most require the eigenvalues to be well separated. The few that consider close or repeated eigenvalues place severe restrictions on the eigenvalue derivatives. We propose, analyze, and test new algorithms for computing first and higher order derivatives of eigenvalues and eigenvectors that are valid much more generally. Numerical results confirm the effectiveness of our methods for tightly clustered eigenvalues.

About eigen sensitivity analysis of mechanical structures

Several methods for a calculation of derivatives of eigenvectors with respect to design parameters are described here. These are the finite-difference method, the modal method, a modified modal method, Nelson's method, an improved first-order approximation of eigenvalues and eigenvectors and an iterative method. By combining the other structural reanalysis techniques and one of these sensitivity methods, it is possible to enhance the efficiency and the accuracy of structural optimization techniques for determining the optimum condition of mechanical structure specified by an analyst. The sensitivity approach is based on the prior selection of updating parameters (design variables) in the initial FE model.

Efficient Procedures of Sensitivity Analysis for Structural Vibration Systems with Repeated Frequencies

Derivatives of eigenvectors with respect to structural parameters play an important role in structural design, identification, and optimization. Particularly, calculation of eigenvector sensitivity is considered when the eigenvalues are repeated. A relaxation factor embedded in the combined approximations (CA) method makes it effective to the structural response at various modified designs. The proposed method is feasible after overcoming the defection of irreversibility of the characteristic matrix. Numerical examples show that it is easy to implement the computational procedure, and the method presented in this paper is efficient for the general linear vibration damped systems with repeated frequencies.

A new method for calculating derivatives of eigenvalues and eigenvectors for discrete structural systems

A new method is presented for calculating the derivatives of eigenvalues and eigenvectors for discrete structural systems. In the cases of distinctive eigenvalues, proportional damping is assumed and the exact eigen-pair derivatives of arbitrary order are calculated. For a repeated eigenvalue, its first-order derivatives are calculated by solving a reduced eigenvalue problem of order equal to the eigenvalue's geometric multiplicity. The new method is computationally efficient and numerically robust. Examples are presented to illustrate the use and effectiveness of the new method.

A parallel way for computing eigenvector sensitivity of asymmetric damped systems with distinct and repeated eigenvalues

A high efficient method is proposed for computing the eigensolution derivatives of asymmetric damped systems with distinct and repeated eigenvalues. A new normalization for the left eigenvectors is proposed, from which a new form of the constraints of the left eigenvector sensitivities can be derived. The aim of this paper is to show how to compute the left and right eigenvector derivatives separately and independently such that those can be computed in a parallel way. In addition, the calculation of eigenvalue sensitivity of N damped asymmetric systems with repeated eigenvalues is derived. The proposed method is well-conditioned since the components of coefficient matrices are all of the same order of magnitude. The method is simple, compact and easy to be implemented on computers. Three numerical examples are used to illustrate the application, accuracy and efficiency of the proposed method.

An iterative method for partial derivatives of eigenvectors of quadratic eigenvalue problems

An iterative method is proposed to compute partial derivatives of eigenvectors of quadratic eigenvalue problems with respect to system parameters. Convergence theory of the proposed method is established. Numerical experiments demonstrate that the proposed method can be used efficiently for partial derivatives of eigenvectors corresponding to dominant eigenvalues.

Approximate method for eigensensitivity analysis of a defective matrix

Based on the exact modal expansion method, an arbitrary high-order approximate method is developed for calculating the second-order eigenvalue derivatives and the first-order eigenvector derivatives of a defective matrix. The numerical example shows the validity of the method. If the different eigenvalues μ ( 1 ) , … , μ ( q ) of the matrix are arranged so that | μ ( 1 ) | ≤ ⋯ ≤ | μ ( q ) | and satisfy the condition that | μ ( q 1 ) | < | μ ( q 1 + 1 ) | for some q 1 < q , and if the approximate method only uses the left and right principal eigenvectors associated with μ ( 1 ) , … , μ ( q 1 ) , then associated with μ ( h ) ( h ≤ q 1 ) the errors of the eigenvalue and eigenvector derivatives by the p th-order approximate method are nearly proportional to | μ ( h ) / μ ( q 1 + 1 ) | p + 1 .