A domain evolution method to suppress the eigenfrequency and eigenmode errors caused by low-density elements in dynamic topology optimization

In dynamic topology optimization involving eigenfrequencies, the solution of the direct problem is expected to reasonably reflect the mechanical performance of the real structure. However, due to the presence of low-density elements in the fixed design domain, there are always some errors compared to the real structure obtained through post-processing. These errors include errors in eigenfrequencies and eigenmodes, which may adversely affect the optimization process. This issue becomes especially pronounced when optimizing higher-order eigenfrequencies, where the errors can lead to discontinuities in the solution space and hinder convergence. To overcome this issue, this paper proposes a domain evolution method (DEM). In this method, the fixed design domain is divided into three domains: the solid domain, the narrow-band, and the low-density domain. The direct problem analysis is solved within the computational domain, which consists of the solid domain and the narrow-band, while the low-density domain remains inactive. Several examples are used to validate the proposed method. Numerical results indicate that the errors primarily arise during the form-finding process and become more significant with increasing order of eigenfrequency. The proposed method effectively mitigates these errors, ensuring stable convergence of the optimization process. Furthermore, a comparative analysis between the proposed method and the traditional approach shows that, in higher-order problems, low-density elements are closely related to classical issues in dynamic topology optimization, including localized modes, repeated eigenfrequencies, and mode switching phenomena. This provides further insight into the intrinsic difficulties of high-order eigenfrequency topology optimization.