On the use of active and reactive input power in topology optimization of one-material structures considering steady-state forced vibration problems

As studied by the authors in a previous work, applying topology optimization to minimize overall dynamic response of structures under time-harmonic loads may present serious drawbacks. It happens e.g. for an excitation frequency above the first resonance of the initial design, especially when the objective is to obtain one-material structures. Essential part of these drawbacks is due to the use of dynamic quantities that present strong influence of local responses, with antiresonances in their frequency spectrum, as is the case of the dynamic compliance. When minimizing for dynamic compliance, density-based topology optimization may present early convergence to designs filled predominantly with intermediate artificial densities. In this article, the authors present their studies on how to overcome these drawbacks by using a better objective function for the problem. The active input power (real part of complex input power) is proportional to time-averaged strain energy and/or to kinetic energy, being a global measure of vibration. Furthermore, it does not have antiresonances in its spectrum. It is illustrated that its minimization produces in most of the cases well defined structures consisting of practically only one material, with reduction of overall vibration at frequencies of interest, even above the first resonance of the initial design. In this work, the authors also show that the application of a constraint on the reactive input power (imaginary part of complex input power) allows to introduce displacement antiresonances for the excitation region at frequencies of interest. Several examples are presented to illustrate the physical issues and the effectiveness of the use of complex input power in topological optimization procedures for overall and local vibration reduction considering harmonic problems.