Efficient limitation of resonant peaks by topology optimization including modal truncation augmentation

In many engineering applications, the dynamic frequency response of systems is of high importance. In this paper, we focus on limiting the extreme values in frequency response functions, which occur at the eigenfrequencies of the system, better known as resonant peaks. Within an optimization, merely sampling the frequency range and limiting the maximum values result in high computational effort. Additionally, the sensitivities of this method are not complete, since only information about the resonance peak amplitude is included. The design dependence with respect to the frequency of the extreme value is missed, thus hampering the convergence. To overcome these difficulties, we propose a constraint which can efficiently and accurately limit resonant peaks in a frequency response function. It has a close relation with eigenfrequency maximization; however, in that case, the amplitudes of the frequency response are unconstrained. In order to keep the computational time low, efficient implementation of this constraint is studied using reduced-order models based on modal truncation and modal truncation augmentation. Furthermore, approximated sensitivities are investigated, resulting in a large computational gain, while still yielding very accurate sensitivities and designs with almost equivalent performance compared with the non-approximated case. Conditions are established for the accuracy and computational efficiency of the proposed methods, depending on the number of peaks to be limited, numbers of inputs and outputs, and whether or not the system input and output are collocated.