Discrete variable topology optimization for maximizing single/multiple natural frequencies and frequency gaps considering the topological constraint

Finding optimized structural topology design for maximizing natural frequencies and frequency gaps of continuum structures is crucial for engineering applications. However, two significant numerical issues must be addressed: non-smoothness caused by multiple frequencies and Artificial Localized Rigid Motion (ALRM) modes due to the violation of the topological constraint related to isolated islands and point-connections. The above two issues are solved by employing the discrete variable topology optimization method based on Sequential Approximate Integer Programming (SAIP). First, the directional differentiability and multiple frequencies preservation constraints are formulated as the linear integer constraints. And then, the integer programming with these linear integer constraints is established and solved by the discrete variable linear or quadratic programming solver with multi-constraint Canonical Relaxation Algorithm (CRA). We also prove that the popular Average Modal Frequencies (AMF) strategy, like the Kreisselmeier-Steinhauser (KS) function, cannot rigorously tackle this non-smoothness caused by multiple frequencies. Furthermore, to eliminate the ALRM modes and concentrate on the real structural global modes, the burning method is employed to impose the topological constraint of the first Betti number that represents the number of isolated islands and point-connections. Numerical examples, including 2D and 3D, two-fold and three-fold multiple frequencies, natural frequencies and frequency gaps, are presented to show the effectiveness of the proposed method.