Topology optimization of extruded beams modeled with the XFEM for maximizing their natural frequencies

In this paper, an efficient topology optimization approach is developed for maximizing the fundamental natural frequency of extruded beams. Mass fraction and static compliance bounds are defined using inequality-type constraints in the optimization problem. An XFEM approach, previously proposed by the authors for analyzing beam elements, is extended herein to compute the natural frequencies of the beam. The method allows for 3D modeling of beams with a significant reduction in the number of degrees of freedom and therefore also yields efficient optimization procedure. This reduction is made possible by incorporating global enrichment functions in the longitudinal direction, which enables a significant reduction in the number of elements in that direction without loss of accuracy. A nonlinear optimization problem is formulated using continuous density-based design variables that represent the material distribution in the beam’s cross-section. The optimization problem is then solved using a gradient-based approach with analytical sensitivities. The well-known Solid Isotropic Material with Penalization (SIMP) method is used to acquire discrete solutions. We study the optimal design of short and long beams. It is shown that for short beams, localized vibration modes appear within the cross-section, leading to a significant distortion deformation mode of the cross-section. The optimized design of the long beam shows global deformation modes with an increase of 15% in the fundamental frequency compared with a non-optimized design consisting of a hollow rectangular cross-section with the same mass.