Adjoint Method for Design Sensitivity Analysis of Multiple Eigenvalues and Associated Eigenvectors

Abstract

IGENPROBLEMS are commonly considered in structural stability, noise, and vibration analyses. Design sensitivity analysiscomputestherateofchangeofresponse-dependentfunction with respect to the change of design variables. Design sensitivity analysis is an essential step to improve systematically the existing design and to optimize a system with the aid of gradient-based optimization. Therefore an efficient and accurate method for design sensitivity analysis is necessary for diverse design optimizations. Fox and Kapoor have developed a general technique to compute design sensitivity of eigenvalues for symmetric matrices [1]. Plaut andHusseyin [2],Rudisill[3],andGodoyetal.[4]havedevelopeda formula for second-order design sensitivity analysis of eigenvalues. Forthedesignsensitivityanalysisofeigenvectors,themodalmethod [1,5], the modified modal method [6], and Nelson’s method [7] are oftenusedtocalculatethederivativesofeigenvectors.Acomparison of the above methods for calculating eigenvector derivatives has beencarriedout[8].Themodalmethodapproximatesthederivatives of eigenvectors as a linear combination of all eigenvectors. The modified modal method is developed to reduce the number of eigenvectors needed to represent the design derivative by including an additional term in the linear combination of eigenvectors. The modal and modified modal methods can be computationally expensive if large numbers of eigenvectors are needed to represent accurately the derivatives of eigenvectors. Nelson’s method is a direct differentiation method for calculating eigenvector derivatives, where it requires only the eigenvalue and eigenvectors for the mode being differentiated. Recently an adjoint method for calculating derivativesofdistincteigenvalueandtheirassociatedeigenvectorsis suggested, where an adjoint equation of simultaneous linear system equations requires only the eigenvalues and eigenvectors for the mode being differentiated [9]. Formanytypicalstructures,thereexistmultipleornearlyidentical eigenvalues due to structural symmetry. In the case of multiple eigenvalues, their associated eigenvectors are generally not differentiable. Therefore the methods for the design sensitivity analysis of distinct eigenvalues will no longer be valid. The eigenvectorscorrespondingtomultipleeigenvalueshaveagreatdeal of uncertainty compared with those associated with distinct eigenvalues because anylinear combination of eigenvectorsis also a valid eigenvector. For design sensitivity analysis of multiple eigenvalues, Ojalvo [10]hasprovided anequation tocalculate thederivatives of multiple eigenvaluesandtheirassociatedeigenvectorsbyextendingNelson’s method when the derivatives of the eigenvalues are distinct. MillsCurran [11] and Dailey [12] have independently modified and correctedOjalvo’smethodbydifferentiatingtheeigenvalueequation twice, that is, the second order design sensitivity analysis of eigenvalue. Friswell [13] has also extended the Nelson’s method to handle the multiple eigenvalue and their associated eigenvectors. Juang et al. [14], Bernard and Bronowicki [15], and Lee and Jung [16] have improved the modal method to evaluate the multiple eigenvalue and eigenvector sensitivity. Calculation of sensitivities for multiple eigenvalues has been obtained without using associated eigenvectors for the structural problem [17,18]. The proposed new method for design sensitivity analysis of multiple eigenvalues and their associated eigenvectors is an adjoint method that requires evaluation of the adjoint variables from the simultaneous system equation, the so-called adjoint equation with added side constraints. First we define the augmented function that consists ofthe response function andeigenvalue equations. Then the design derivatives of the response function are represented as an explicit design variation of the augmented function and an adjoint equation is obtained by requiring the implicit design variation of the augmented function to vanish. It is important to note that the adjoint equation is composed of eigenvalues and their associated eigenvectors of the mode being differentiated. Once we evaluate the adjoint variables, design sensitivity analysis of multiple eigenvalues and their associated eigenvectors can be computed directly. To verify the proposed method, an analytical example of a twodegree-of-freedom spring-mass system, planar grillage structure, anda finiteelementexampleofcantileverbeamareincluded.Further the developed method can be easily implemented into a commercial finite element program to carry out the design sensitivity analysis of eigenproblems needed for practical applications.