Nonlinear eigenvalue topology optimization for structures with frequency-dependent material properties

Eigenvalue topology optimization problem has been a hot topic in recent years for its wide applications in many engineering areas. In the previous studies, the applied materials are usually assumed as elastic, and the resulting structural eigenfrequencies are obtained by solving a linear or quadratic eigenvalue problem. However, many engineering materials, such as viscoelastic materials, have frequency-dependent modulus, which results in a more complicated nonlinear eigenvalue problem. This paper presents a systematic study on the nonlinear eigenvalue topology optimization problem with frequency-dependent material properties. The nonlinear eigenvalue problem is solved by a continuous asymptotic numerical method based on the homotopy algorithm and perturbation expansion technique, which involves higher-order differentiation of the nonlinear term and shows a fast convergence. Several schemes are proposed to improve the computational accuracy, applicability, and robustness of the method for the application in topology optimization, including Faà di Bruno's theorem, bisection method, and inverse iteration based eigenvector modification method. Three optimization problems are solved to demonstrate the effectiveness of the developed methods, including the maximization of the fundamental frequency, the eigenfrequency separation interval between two adjacent eigenfrequencies of given orders, and the eigenfrequency separation interval at a given frequency. Numerical examples show the large influence of the frequency-dependent material properties on the optimized results and validate the effectiveness of the developed method.