Eigenvector Derivatives Research Articles

Comprehensive sensitivity analysis of repeated eigenvalues and eigenvectors for structures with viscoelastic elements

The paper discusses systems with viscoelastic elements that exhibit repeated eigenvalues in the eigenvalue problem. The mechanical behavior of viscoelastic elements can be described using classical rheological models as well as models that involve fractional derivatives. Formulas have been derived to calculate first- and second-order sensitivities of repeated eigenvalues and their corresponding eigenvectors. A specific case was also examined, where the first derivatives of eigenvalues are repeated. Calculating derivatives of eigenvectors associated with repeated eigenvalues is complex because they are not unique. To compute their derivatives, it is necessary to identify appropriate adjacent eigenvectors to ensure stable control of eigenvector changes. The derivatives of eigenvectors are obtained by dividing them into particular and homogeneous solutions. Additionally, in the paper, a special factor in the coefficient matrix has been introduced to reduce its condition number. The provided examples validate the correctness of the derived formulas and offer a more detailed analysis of structural behavior for structures with viscoelastic elements when altering a single design parameter or simultaneously changing multiple parameters.

Complex Standard Eigenvalue Problem Derivative Computation for Laminar–Turbulent Transition Prediction

As a high-fidelity approach to transition prediction, the coupled Reynolds-averaged Navier–Stokes (RANS) and linear stability theory (LST)-based method is widely used in engineering applications and is the preferred method for laminar flow optimization. However, the further development of gradient-based laminar flow wing optimization schemes is hindered by a lack of efficient and accurate derivative computation methods for LST-based eigenvalue problems with a large number of design variables. To address this deficiency and to compute the derivatives in the LST-based solution solver, we apply the adjoint method and analytical reverse algorithm differentiation (RAD), which scale well with the number of inputs. The core of this paper is the computation of the standard eigenvalue and eigenvector derivatives for the LST problem, which involves a complex matrix. We develop an adjoint method to compute these derivatives, and we couple this method with RAD to reduce computational costs. In addition, we incorporate the LST-based partial derivatives into the laminar–turbulent transition prediction framework for the computation of total derivatives. We verify our proposed method with reference to finite difference (FD) results for an infinite swept wing. Both the intermediate derivatives from the transition module and total derivatives agree with the FD reference results to at least three digits, demonstrating the accuracy of our proposed approach. The fully adjoint and the coupled adjoint–RAD methods both have considerable advantages in terms of computational efficiency compared with iterative RAD and FD methods. The LST-based transition method and the proposed method for efficient and accurate derivative computations have prospects for wide application to laminar flow optimization in aerodynamic design.

Enhanced algebraic substructuring for symmetric generalized eigenvalue problems

AbstractThis article proposes a new substructuring algorithm to approximate the algebraically smallest eigenvalues and corresponding eigenvectors of a symmetric positive‐definite matrix pencil . The proposed approach partitions the graph associated with into a number of algebraic substructures and builds a Rayleigh–Ritz projection subspace by combining spectral information associated with the interior and interface variables of the algebraic domain. The subspace associated with interior variables is built by computing substructural eigenvectors and truncated Neumann series expansions of resolvent matrices. The subspace associated with interface variables is built by computing eigenvectors and associated leading derivatives of linearized spectral Schur complements. The proposed algorithm can take advantage of multilevel partitionings when the size of the pencil. Experiments performed on problems stemming from discretizations of model problems showcase the efficiency of the proposed algorithm and verify that adding eigenvector derivatives can enhance the overall accuracy of the approximate eigenpairs, especially those associated with eigenvalues located near the origin.

Eigenvalue problem derivatives computation for a complex matrix using the adjoint method

Derivatives of eigenvalues and eigenvectors with respect to design variables are required for gradient-based optimization in many engineering design problems. However, for the generalized and standard eigenvalue problems with general complex and non-Hermitian coefficient matrices, no method can accurately compute the eigenvalue and eigenvector derivatives while remaining efficient for large numbers of design variables. In this paper, we develop an adjoint method to compute complex eigenvalue and eigenvector derivatives with machine precision. For the special case when only the eigenvalue derivative is required, we propose a reverse algorithmic differentiation (RAD) formula using a newly developed dot product identity for complex functions. We verify the proposed method against the finite differences (FD) for a simple algebraic example with a 3-by-3 complex non-Hermitian matrix and a plane Poiseulle flow stability problem that is modeled as a generalized eigenvalue problem. The adjoint method is demonstrated to scale well with the number of design variables, matching the FD reference to about 5 to 7 digits.

Hyper-dual number-based numerical differentiation of eigensystems

Considering application to nonlinear material models, this study proposes a novel numerical method for computing arbitrary-order derivatives of eigenvalues and eigenvectors (eigensystems) for second-order tensors (matrices) based on hyper-dual numbers (HDNs). This study introduces two numerical differentiation approaches for eigensystems: one obtains derivatives of both eigenvalues and eigenvectors by inductively solving linear equations derived from an HDN formulation of eigensystem, and the other specializes in solving derivatives of a function represented by eigenvalues, such as the Helmholtz free energy function in the Ogden model, a representative example of the eigenvalue-based hyperelastic material models. Because these approaches differ considerably in applicability and computational cost, this study proposes a numerical differentiation method that uses both complementarily. The proposed method can accurately compute higher-order derivatives even if the second-order tensors have multiple solutions for eigenvalues. Moreover, applying the proposed method to an incremental variational formulation (IVF) leads to robust and accurate computations for the Ogden material model. These results imply that IVF facilitates the implementation of several constitutive models in a unified manner, and the proposed method broadens the scope of these models, including eigensystem expressions.

Seismic topology optimization based on spectral approaches

In this study, the structural design problem under earthquake excitations is handled using the proposed topology optimization scheme that successfully copes with the complexity of seismic loading. To achieve this success, the optimization scheme is formulated based on response spectrum method, adequately balancing the accuracy and computational cost in calculating seismic responses of structures. Also, an interpolation strategy, providing smooth curves to supersede the non-differentiable response spectra, is suggested to ensure that the derivatives of spectral values can be obtained for the sensitivity analysis. In addition, the terms including the derivatives of eigenvectors in the sensitivity expressions are annihilated through the derived adjoint approach, which is more accurate and convenient compared with directly solving the derivatives of eigenvectors. In the present numerical examples, the specified seismic responses of the optimized structures are significantly reduced, and these reductions are also observed in the time-domain measurement. Therefore, the efficacy of the proposed optimization scheme is validated.

New Sensitivity Analysis Methods for Complex Modal Parameters in Asymmetrical Damped System

Most of the sensitivity algorithms with respect to the design parameters adopted the normalization relationship with system property matrices weighted when analyzing an asymmetrical damping linear discrete dynamic system. In this paper, two new methods (an algebraic method and a direct method) based on the Euclidean norm as the normalization conditions are proposed. Both proposed methods are developed to calculate the first- and second-order sensitivities of the complex modal parameters for an asymmetric damped system. The precisions of the Taylor approximations of the second degree are also discussed in the proposed methods. Like other algebraic methods, this paper outlines the proof of numerical stability. Different from Nelson’s direct method (“Simplified Calculation of Eigenvector Derivatives,” AIAA Journal, Vol. 14, No. 9, 1976, pp. 1201–1205), this paper considers complexity of the solution space in complex vector space, and the partitioning scheme of the proposed direct algorithm is more practical. The usefulness and correctness are demonstrated by a one-parameter numerical experiment and a two-parameter numerical experiment.

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

A preconditioned conjugate projected gradient (PCPG) method is given to solve the repeated eigenvector derivatives with respect to design parameters in finite element models. The factored stiffness matrix is used as a preconditioner, the new method can provide the approximate solutions to repeated eigenvector derivatives. This method can be used as an eigenvector sensitivities solver which can be coupled as an eigensolver/derivative software. Moreover, the new method can be used for industrial-size models. Finally, a numerical example is provided to illustrate the superior efficiency of the present method.

Highly efficient general method for sensitivity analysis of eigenvectors with repeated eigenvalues without passing through adjacent eigenvectors

SummaryIt is well known that the sensitivity analysis of the eigenvectors corresponding to multiple eigenvalues is a difficult problem. The main difficulty is that for given multiple eigenvalues, the eigenvector derivatives can be computed for a specific eigenvector basis, the so‐called adjacent eigenvector basis. These adjacent eigenvectors depend on individual variables, which makes the eigenvector derivative calculation elaborate and expensive from a computational perspective. This research presents a method that avoids passing through adjacent eigenvectors in the calculation of the partial derivatives of any prescribed eigenvector basis. As our method fits into the adjoint sensitivity analysis , it is efficient for computing the complete Jacobian matrix because the adjoint variables are independent of each variable. Thus our method clarifies and unifies existing theories on eigenvector sensitivity analysis. Moreover, it provides a highly efficient computational method with a significant saving of the computational cost. Additional benefits of our approach are that one does not have to solve a deficient linear system and that the method is independent of the existence of repeated eigenvalue derivatives of the multiple eigenvalues. Our method covers the case of eigenvectors associated to a single eigenvalue. Some examples are provided to validate the present approach.

An incomplete modal method for eigenvector derivatives of polynomial eigenvalue problems

Eigenvector derivatives of eigenproblems analytically dependent on parameters are required in diverse technical fields such as structural design, system identification. Modal methods, a kind of popular methods in engineering problems, calculate eigenvector derivatives by expressing them as linear combinations of eigenvectors of eigenproblems. In this paper, an incomplete modal method is presented to compute eigenvector derivatives of polynomial eigenproblems analytically dependent on parameters. The proposed method only uses a small part of eigenvectors of polynomial eigenproblems. The contributions of other eigenvectors to eigenvector derivatives are approximated by an iterative formula. Our method can simultaneously compute the eigenvector derivatives corresponding to several eigenvalues, and it is effective whether the differentiated eigenvectors correspond to simple or semisimple eigenvalues, and whether the leading coefficient matrix is singular or not. Numerical examples are given to test the efficiency of the proposed method.

Observations on the Computation of Eigenvalue and Eigenvector Jacobians

Many scientific and engineering problems benefit from analytic expressions for eigenvalue and eigenvector derivatives with respect to the elements of the parent matrix. While there exists extensive literature on the calculation of these derivatives, which take the form of Jacobian matrices, there are a variety of deficiencies that have yet to be addressed—including the need for both left and right eigenvectors, limitations on the matrix structure, and issues with complex eigenvalues and eigenvectors. This work addresses these deficiencies by proposing a new analytic solution for the eigenvalue and eigenvector derivatives. The resulting analytic Jacobian matrices are numerically efficient to compute and are valid for the general complex case. It is further shown that this new general result collapses to previously known relations for the special cases of real symmetric matrices and real diagonal matrices. Finally, the new Jacobian expressions are validated using forward finite differencing and performance is compared with another technique.

Calculating the minimal/maximal eigenvalue of symmetric parameterized matrices using projection

SummaryIn applications of linear algebra including nuclear physics and structural dynamics, there is a need to deal with uncertainty in the matrices. We focus on matrices that depend on a set of parameters ω and we are interested in the minimal eigenvalue of a matrix pencil (A,B) with A,B symmetric and B positive definite. If ω can be interpreted as the realization of random variables, one may be interested in statistical moments of the minimal eigenvalue. In order to obtain statistical moments, we need a fast evaluation of the eigenvalue as a function of ω. Because this is costly for large matrices, we are looking for a small parameterized eigenvalue problem whose minimal eigenvalue makes a small error with the minimal eigenvalue of the large eigenvalue problem. The advantage, in comparison with a global polynomial approximation (on which, e.g., the polynomial chaos approximation relies), is that we do not suffer from the possible nonsmoothness of the minimal eigenvalue. The small‐scale eigenvalue problem is obtained by projection of the large‐scale problem. Our main contribution is that, for constructing the subspace, we use multiple eigenvectors and derivatives of eigenvectors. We provide theoretical results and document numerical experiments regarding the beneficial effect of adding multiple eigenvectors and derivatives.

New theoretical developments on eigenvector derivatives with repeated eigenvalues

Abstract Many different methods have been developed since the pioneering works of Mills-Curran (1988) and Dailey (1989) on eigenvector derivatives with repeated eigenvalues. In spite of the increasing mathematical complexities witnessed in many of the newly emerged methods, some underlying fundamental theories governing the eigenvector derivatives have neither been much discussed, nor fully established to date. The present approach seeks to fill such an outstanding theoretical gap and to lay down the necessary theoretical foundation on which existing methods can be mathematically unified and further improved in numerical accuracy and computational efficiency. The particular solutions of eigenvector derivatives generally required have been derived in terms of modal properties, thereby avoiding the computationally expensive and potentially erroneous procedure of solving a set of algebraic equations of system dimension. The contributions of higher unavailable modes have been theoretically derived, enhancing the practical applicability of the proposed method to the general case where only partial eigensolutions are made. To avoid degeneration of eigenvector space in the case of repeated eigenvalues, a concept of global design variable is developed in which all intended multivariate design modifications are grouped into a single global variable to which eigenvector derivatives are derived, rendering real major applications of the proposed method to the predictions of structural design modifications. A discrete parameter model of a turbine bladed disk assembly, which is known to have many pairs of repeated eigenvalues due to its cyclic symmetry, as well as a finite element model of a cantilevered beam with large DOFs have been employed. Numerical results have demonstrated the accuracy and the practical applicability of the proposed new theoretical developments, as well as the proposed new method.

Eigenvalue and eigenvector derivatives of fractional vibration systems

Dynamic characterizations of fractional vibration systems have recently attracted significant research interest. Increasingly, successful applications of fractional derivatives have been found to the modeling of mechanical damping, vibration transmissions, improved fractional vibration controls and nonlinear vibration analyses. To facilitate further development, the eigenvalue problem including its derivatives, which are the central issues of vibration analysis, have to be fully established. This paper examines how eigenvalue and eigenvector derivatives of fractional systems can be derived when system matrices become functions of physical design parameters. First, new important orthonormal constraints are proposed since the modes are no longer orthonormal to the mass matrix, in this case due to its complex and frequency dependent nature. Next, new methods of eigenvector derivatives are developed for distinct eigenvalues for the cases of complete, incomplete and single mode modal data. Realistic and practical FE models incorporating fractional derivatives in the form of viscoelastic supports are employed to demonstrate the numerical accuracy and computational efficiency of the proposed methods. However, when repeated eigenvalues are considered due to structural spatial symmetries, the eigenvector space degenerates and further differentiation of system matrices are required in order to uniquely determine the eigenvector derivatives. Consequently, a new and effective general method is developed which can be applied to compute eigenvector derivatives of repeated eigenvalues with any multiplicity m. A simplified turbine bladed disk vibration model which is known to have repeated eigenvalues due to its cyclic symmetry, is then used to demonstrate the accuracy and salient features of the proposed method.

Dynamic and Sensitivity Analysis General Non-Conservative Asymmetric Mechanical Systems

Abstract In this paper the concept of generalized form of proportional damping is proposed. Classical modal analysis of non-conservative continua is extended to multi DOF linear dynamic systems with asymmetric matrices. Mode orthogonality relationships have been generalized to non-conservative systems. Several discretization methods of continua are presented. Finally, an expression for derivatives of eigenvalues and eigenvectors of non-conservative system is presented. Examples are provided to illustrate the proposed methods.

A new method for computation of eigenvector derivatives with distinct and repeated eigenvalues in structural dynamic analysis

Abstract Determining eigenvector derivatives is a challenging task due to the singularity of the coefficient matrices of the governing equations, especially for those structural dynamic systems with repeated eigenvalues. An effective strategy is proposed to construct a non-singular coefficient matrix, which can be directly used to obtain the eigenvector derivatives with distinct and repeated eigenvalues. This approach also has an advantage that only requires eigenvalues and eigenvectors of interest, without solving the particular solutions of eigenvector derivatives. The Symmetric Quasi-Minimal Residual (SQMR) method is then adopted to solve the governing equations, only the existing factored (shifted) stiffness matrix from an iterative eigensolution such as the subspace iteration method or the Lanczos algorithm is utilized. The present method can deal with both cases of simple and repeated eigenvalues in a unified manner. Three numerical examples are given to illustrate the accuracy and validity of the proposed algorithm. Highly accurate approximations to the eigenvector derivatives are obtained within a few iteration steps, making a significant reduction of the computational effort. This method can be incorporated into a coupled eigensolver/derivative software module. In particular, it is applicable for finite element models with large sparse matrices.

An Improved Method for Computing Eigenpair Derivatives of Damped System

The calculation of eigenpair derivatives plays an important role in vibroengineering. This paper presents an improved algorithm for the eigenvector derivative of the damped systems by dividing it into a particular solution and general solution of the corresponding homogeneous equation. Compared with the existing methods, the proposed algorithm can significantly reduce the condition number of the equation for particular solution. Therefore, the relative errors of the calculated solutions are notably cut down. The results on two numerical examples show that such strategy is effective in reducing the condition numbers for both distinct and repeated eigenvalues.

Computing eigenpair derivatives of asymmetric damped system by generalized inverse

Many existing approaches for asymmetric damped system are based on the assumption that the eigenvalues are simple or semisimple with separated derivatives. This paper presents a new algorithm for computing the derivatives of the semisimple eigenvalues and corresponding eigenvectors of asymmetric damped system. Compared with the existing methods, the algorithm can be applicable to problems whether the repeated eigenvalues have well separated derivatives. In the proposed method, the derivatives of eigenvectors are divided into a particular solution and a homogeneous solution, where the particular solution is constructed by using generalized inverse matrix. The effectiveness of the proposed algorithm is illustrated by one numerical example.

Methods for solving underdetermined systems

SummaryWe compare some alternative methods for computing solutions of underdetermined linear systems, Ax=b. Each method involves solving an associated system with a different nonsingular coefficient matrix, . We obtain bounds on the condition numbers of these nonsingular matrices and test the methods on numerical examples. We discuss implications for computing eigenvector derivatives and make some recommendations.

Modification of natural frequencies of an automotive belt drive system based on eigen-sensitivity analysis of its configuration parameters

Abstract A methodology is presented to simultaneously modify natural frequencies of an automotive belt drive system via configuration modification to match natural frequencies of the system with its working conditions. In this methodology, the configuration modification is regarded as an inverse eigenvalue problem in which necessary changes of a set of configuration parameters are determined for achieving a desired natural frequency variation relative to the original design value. Natural frequencies and mode shapes of the system are obtained by solving an eigenvalue problem that is directly derived from established governing equations of motion of a typical automotive belt drive system. Explicit expressions of the first and second order derivatives of natural frequencies and eigenvectors with respect to the configuration parameters are derived by differentiating the eigenvalue equation. Several configuration parameters are selected as design variables and their corresponding modification regions and directions are given based on frequency sensitivity analysis. In the inverse eigenvalue problem, variations of natural frequencies caused by changes of multiple design variables are approximated using the first and second order Taylor series expansions, which are based on explicit expressions of the first and second order derivatives of natural frequencies and eigenvectors with respect to the configuration parameters. By solving the inverse eigenvalue problem, necessary changes of design variables are determined and the desired natural frequencies are achieved so that resonance conditions are successfully avoided.