Adjoint method for design sensitivity analysis of multiple eigenvalues and associated eigenvectors

Abstract

The adjoint method for design sensitivity analysis of multiple eigenvalues and their associated eigenvectors is proposed for real symmetric structural eigenvalue problems, where we restrict that the design sensitivity coefficients of the multiple eigenvalues are distinct. The developed adjoint method is required to compute adjoint variables from the simultaneous linear system equation, the so-called adjoint equation, without the linear combination of eigenvectors or the second-order derivatives of the eigenvalue problem. Once the adjoint variables are evaluated, the sensitivity analysis of the multiple eigenvalues and their associated eigenvectors can be computed directly. In this way, the design sensitivity analysis of eigenvectors can be obtained by using the eigenvalues and their eigenvectors of the mode being differentiated only. To verify the proposed method, analytical and numerical examples are demonstrated. This can have considerable impact on computer implementation of the method in the design sensitivity analysis of the eigenvalue problem needed for practical application.