Adjoint variable method for sensitivity analysis of performance metrics involving mode shapes

Abstract

An adjoint variable method is proposed for sensitivity analysis of performance metrics involving mode shapes. First, a functional is defined to render these performance metrics in scalar form so that their sensitivity analysis can be performed in a unified form. The adjoint variable method is then performed with the eigenproblem and normalization conditions as constraints on these performance metrics to form Lagrangian functional. After obtaining the Lagrangian multipliers, the sensitivity of these performance metrics can be directly calculated. In the framework of the adjoint variable method, three different methods are proposed to obtain these Lagrangian multipliers. Afterward, the performance of the direct differentiation method and the adjoint variable method are compared. Furthermore, modal assurance criteria (MAC) and elemental modal strain energy are used as examples of performance metrics, and two numerical examples are applied to verify the performance of the adjoint variable method and the direct differentiation method. The results show that the proposed method is more effective for sensitivity analysis of performance metrics with multiple design variables without loss of accuracy.