A strategy based on the strain‐to‐kinetic energy ratio to ensure stability and convergence in topology optimization of globally resonating one‐material structures

The authors propose a new formulation of SIMP-based topology optimization problems aiming to obtain resonating one-material structures through a stable maximization process. The new formulation is capable of achieving globally resonant lightly damped "0-1" structures for frequencies close to those of interest. The proposed strategy successfully deals with known issues like design "degeneration" and instability of the gradient-based optimization process around the resonance frequency (which is a local maximum when maximizing vibration responses) at which harmonic excitation forces shall be applied. In this work, the authors use the concept of complex input power to overcome the mentioned issues. It is proposed a topology optimization procedure where a weighted sum between the active input power (real part of the complex input power) and the static compliance is minimized with a constraint on the reactive input power (imaginary part of the complex input power), which is converted to a ratio between the time-averaged potential and kinetic energies of the system named quotient R. By this way, it is possible to ensure viability to the procedure by preventing the resonance frequency from reaching exactly the excitation frequency throughout the process, which otherwise causes difficulties on convergence. The process achieves resonance frequencies very close to given values of interest by keeping the "side" of RR>1. Several examples are presented to illustrate the potential of the proposed method.