Keywords: topology optimization, level set method, vibration problem, phase field method, structural optimization, optimum design, finite element method, structural analysis, Tikhonov regularization method.

Vibration characteristics such as eigenfrequencies and eigenmodes are important factors in structural design. The ability to design mechanical structures that have specified eigenfrequencies facilitates the development of high performance vibratory devices such as specialized sensors and vibromotors. However, trial and error approaches seldom provide optimal configurations for structures whose performance depends on sophisticated control of vibration. Topology optimization methods have been proposed for a variety of vibration problems, to solve particular design requirements. However, traditional topology optimization methods such as the homogenization design method and the SIMP method typically provide optimal configurations that include grayscales, areas with intermediate density values that do not belong to either the material or void domains. The issue of grayscales is particularly troublesome in specified eigenfrequency problems, when grayscales are usually widely distributed in the design domain.

To overcome the issue of grayscales, several structural optimization methods based on the level set method have been proposed during the last decade, as a new type of structural optimization method. Yamada et al. [1] proposed a topology optimization method using level set-based structural representations in which the Tikhonov regularization method is applied to regularize the topology optimization problem. As a result, the geometrical complexity of the obtained optimal configurations can be controlled by adjusting a regularization parameter. In addition, the dependency of several design parameters, such as the initial configuration and finite element mesh size, is extremely low in the Yamada design method.

This paper presents a topology optimization method, based on the level set method, for vibrating structures that have a number of specified eigenfrequencies. The proposed topology optimization method, which uses level set boundary expressions and the Tikhonov regularization method, is described first. An optimization problem is then formulated for vibrating structures that have a number of specified eigenfrequencies. Next, based on this formulation, the design's topological sensitivities are derived using the adjoint variable method. A new topology optimization algorithm is then constructed that uses the finite element method when solving the equilibrium equations and updating the level set function. Here, the level set function is updated based on a reaction-diffusion equation derived from a phase field concept. Finally, a numerical example of the formulated vibration problem is provided to confirm the usefulness of the proposed topology optimization method. The results allow us to confirm that an appropriate, clear, and smooth optimal configuration was obtained, and that topological changes occurred during the optimization procedure.

1

T. Yamada, K. Izui, S. Nishiwaki, A. Takezawa, "A Topology Optimization Method based on the Level Set Method Incorporating a Fictitious Interface Energy", Computer Methods in Applied Mechanics and Engineering, 199(45-48), 2876-2891, 2010. doi:10.1016/j.cma.2010.05.013

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