http://web.mae.ufl.edu/nkim/Papers/paper19.pdf

Review of options for structural design sensitivity analysis. Part 1: Linear systems

Finite strain fracture of 2D problems with injected anisotropic softening elements

This paper presents and analyzes new algorithms for computing the numerical values of derivatives, of arbitrary order, and of eigenvalues and eigenvectors of ${\bf A}(\rho){\bf x}(\rho) = \lambda(\rho){\bf B}(\rho){\bf x}(\rho)$ at a point $\rho=\rho_0$ at which the eigenvalues considered are multiple. Here ${\bf A}(\rho)$ and ${\bf B}(\rho)$ are hermitian matrices which depend analytically on a single real variable $\rho,$ and ${\bf B}(\rho_0)$ is positive definite. The algorithms are valid under more general conditions than previous algorithms. Numerical results support the theoretical analysis and show that the algorithms are also useful when eigenvalues are merely very close rather than coincident.

Computation of Derivatives of Repeated Eigenvalues and the Corresponding Eigenvectors of Symmetric Matrix Pencils

Aniterativeprocedureispresentedforcomputingeigenvectorsensitivitiesdueto e niteelementmodelparameter variations. The present method is a preconditioned conjugate projected gradient-based technique and is intended to utilize the existing matrix factorizations developed for an iterative eigensolution such as Lanczos or Subspace Iteration. As such, this technique can be integrated into a coupled eigensolver/sensitivity software module and leverage the nonrecurring costs of the solver. The use of projection operators in the algorithm is dictated by the indee nite character of the governing coefe cient matrix of the eigenvector derivative. Two model examples are provided to demonstrate both the accuracy of the present technique and its superior efe ciency as compared to existing techniques with similar accuracy.

Efficient Computation of Eigenvector Sensitivities for Structural Dynamics

A state-of-the-art review on theory and engineering applications of eigenvalue and eigenvector derivatives

There are a number of applications where the sensitivity of eigenvectors with respect to physical parameters is desired. We develop an iterative solution scheme for calculating the eigenvector sensitivity in which only the lowest eigencharacteristics are required. It uses a least-squares formulation for the eigenvector sensitivity including the relation from the basic eigenvalue problem and the orthogonality and normality conditions with respect to the mass matrix. The iterative scheme uses the band structure of the stiffness matrix and an efficient use of the Householder transformation to reduce the number of calculations. Since only the lowest eigensolutions are used in the formulation, it is applicable to situations where only a partial eigensolution of the lowest eigenvectors and eigenvalues is available. The eigenvalues are assumed to be distinct and only the first-order variation is calculated. The stiffness matrix is assumed to be nonsingular and a Choleski decomposition of the stiffness matrix is required, but this is the only large matrix that needs to be decomposed. The least-squares solution to the eigenvector sensitivity is shown to reduce to the modal expansion method when appropriate weights are incorporated. From this expression, we show why the modal expansion is not always adequate for eigenvector sensitivity and give a criterion for evaluating this method in a given application.

Iterative least-squares calculation for modal eigenvector sensitivity

Non‐linear optimization technique for partial solution of generalized eigenvalue problems

The partial solution of damped structures involves the determination of only a few of the resonant frequencies and corresponding damping characteristics and mode shapes. The procedure outlined here uses a modification of the variable metric solution algorithm, in which the dicision variables are elements of the complex eigenvector and the eigenvalue. The quadratic eigenvalue problem is reformulated as a sequential unconstrained minimization problem. Both normality and orthogonality constraints are incorporated in the objective function. The method is compared to an adaptation of the subspace iteration technique, also applicable to the quadratic eigenvalue problem.

Partial eigensolution of damped structures by non-linear optimization

Advances in modal testing and in partial eigenvalue and eigenvector calculation of large matrix systems and in eigensensitivity make it feasible to estimate actual physical parameters of finite element or frame analysis models by comparing calculated modal information with measured modal characteristics. The relevent mathematics required for structural system identification is presented in this paper with regards to linear structural systems.

Structural System Identification from Modal Information

Numerical methods to solve the dynamic interaction of fluid/structure interaction fall into three general areas; boundary element techniques, element-by-element solvers, and reduction algorithms. Amongst the reduction methods there are two general approaches: subspace iteration involves the reduction to a smal Ier eigenvalue problem by assuming Ritz mode shapes whereas component mode synthesis techniques discretize the structure/fluid system into smaller substructures, such that a much smaller global system is solved at the end. The reduced number of eigenvalues and eigenvectors thus calculated are approximations to the lowest natural frequencies and mode shapes of vibration.

Jean-Guy Beliveau

Papers

Iterative least-squares calculation for modal eigenvector sensitivity

Non‐linear optimization technique for partial solution of generalized eigenvalue problems

Partial eigensolution of damped structures by non-linear optimization

Structural System Identification from Modal Information

Component Mode Iteration Technique Applied to Modal Calculation for Fluid/Structure Interaction