Topology optimization of bi-material structures with frequency-domain objectives using time-domain simulation and sensitivity analysis

In this paper, we propose to use time-domain transient analysis to compute the response of structures in a wide frequency band by means of Fourier transform. A time-domain adjoint variable method is then developed to carry out the sensitivity analysis of frequency-domain objective functions. By using the concept of frequency response function, it turns out that both the objective function and its sensitivity information at multiple frequencies can be obtained by one original simulation and at most one adjoint simulation, respectively. It is also demonstrated that some commonly used performance indices, e.g., dynamic compliance and input power, are indeed self-adjoint; thus, no extra adjoint simulations are needed, which makes the sensitivity analysis extremely efficient. An obvious distinction between the proposed method and the traditional frequency domain methods is that in our method, the frequency response curves in a wide band can be obtained in each iteration with no extra costs. It follows that it is easy to track the evolution of the frequency response curve in our method, which is essential in both computational and engineering sense. Several numerical examples are tested to show the effectiveness of the proposed method.