Second-order design sensitivities to assess the applicability of sensitivity analysis

Abstract

This paper examines critically the practice of using first-order sensitivities for extrapolation to predict the effect of structural changes. A quality criterion, based on the Cauchy ratio test for convergence of infinite series, is suggested to examine the liability of systems to perform poorly under extrapolation. The efficacy of this approach is demonstrated oh a simple numerical example.

Iterative computation of second‐order derivatives of eigenvalues and eigenvectors

“…Differentiation of (1) with respect to Pj and rearranging yields and premultiplying (5) by X i , using (3), yields, after further rearrangement,…”

“…. , n (1) where the (real or complex) n x n matrix A (and consequently the eigenvalues X i and eigenvectors x i ) are functions of a number of real design parameters P I , Pz, ..., P,,,. We are concerned here with numerical computation of Xi,;[ and x;,;r for a small number of X i and x i , where the subscript 'J' denotes partial differentiation with respect to Pj and the subscript ', j/' denotes the second-order partial derivative with respect to P; and PI.…”

Computation of mixed partial derivatives of eigenvalues and eigenvectors by simultaneous iteration

“…These second-order derivatives are especially useful in re-analysis. 4 Second-order derivatives of eigenvalues, which are relatively easy to compute, were the subject of several earlier papers. More recently attention has been given to methods for the numerical computation of second-order derivatives of eigenvectors.…”

“…More recently attention has been given to methods for the numerical computation of second-order derivatives of eigenvectors. Previous methods include modal expansion methods 4 (which, unlike the method proposed here, require accurate knowledge of all eigenvalues and right and left eigenvectors for the accurate calculation of a single x i,jl ), and various direct methods, 5,6 one of which 7 is valid for problems with non-linear dependence on the eigenvalue parameter. For ®rst derivatives, several iterative methods have been proposed 2,8±12 which oer a number of advantages, 2,13,14 at least for the large sparse matrices that arise in ®nite element computations, but the solitary iterative method 14 that has been proposed for computing l i,jl and x i,jl is really suitable only for the dominant eigenvalue, or one which can be made dominant by a suitable origin shift 14,15 or for a dominant complex conjugate pair.…”

A new method for finding the first‐ and second‐order eigenderivatives of asymmetric non‐conservative systems with application to an FGM plate actively controlled by piezoelectric sensor/actuators

“…Later, many other researchers have modified the modal method to account for conservative or non-conservative asymmetric systems. Brandon [12] extended the modal method to asymmetric non-proportional damped systems using state-space equations based on 2N -space. Zhao et al [13] improved the modal method for damped systems by giving an accurate expression of contribution due to the higher truncated modes in terms of the available modes.…”