A new form of forbidden frequency band constraint for dynamic topology optimization

It is often required to avoid a specified natural frequency band when designing a structure in order to prevent vibration resonance under external excitations. However, the total number and the orders of free vibration modes falling within the forbidden frequency band are not known a priori and constantly change during the course of optimization iterations. Moreover, the mode switching issue also makes mode tracking a difficult task. Thus, it is not always realistic to fulfill the forbidden band requirement by restricting pre-specified orders of natural frequencies to be higher or lower than the forbidden band. In this paper, we propose a new form of constraint to keep the natural frequencies from falling within a given band. This constraint has a unified expression, which facilitates the design sensitivity analysis. Though this constraint form is also applicable for other types of structural design problems with forbidden frequency band constraints, we demonstrate its efficacy in topology optimization, in which imposing such constraints otherwise becomes even more challenging due to topological evolutions of the structural configuration during the optimization process. The velocity field level set method is used for formulating and solving such dynamic topology optimization problems. The design sensitivity analysis of the constraint is derived, for the cases of both distinct and repeated eigenfrequencies. Several numerical examples in 2D and 3D design domains are presented to illustrate effectiveness of the proposed method.