Investigation of localized modes in topology optimization of dynamic structures

One of the main problems in the frequency maximization of dynamic structures by topology optimization is the appearance of localized modes in low-density areas. This paper aims at investigating the inherent cause by means of numerical examples. Based on the stiffness and mass matrices of the structurel the dynamic characteristic equation is solved to figure out how the density difference and penalty factor affect the natural frequencies and modal vectors. It is found that due to the penalty factor of the SIMP method) the ratio of the density to stiffness is greater in the solid region than in the void region. Such a difference consequently results in the appearance of localized modes in the void region. Furthermore, natural frequencies and modal vectors are studied to show their variations versus the density ratio between the solid and the void. In the meantime, the microstructure based on homogenization method is discussed. Based on the existing unit cell, a variety of new unit cells is proposed and numerically validated to avoid localized modes.