Study on the design sensitivity analysis based on complex variable in eigenvalue problem

In the sensitivity techniques, the adjoint variable method is quite popular because it reduces computation time and save computer resources. Commonly, the adjoint variable method employs exact analytical differentiation with respect to the design variables. But, it can be cumbersome to precisely differentiate every given type of finite element. For improving this trouble, the numerical differentiation scheme can replace this exact manner of differentiation. Even though the numerical differentiation has some advantages, it suffers from severe inaccuracy due to the perturbation size dilemma. This paper employs a complex variable which is not much influenced by the perturbation size. Then, the adjoint variable method combined with complex variables is applied to obtain the shape and size sensitivity for structural optimization. Numerical examples demonstrate that the proposed method can predict stable sensitivity results and that its accuracy is remarkably superior to traditional sensitivity evaluation methods.