Nonlinear diffusions in topology optimization

Filtering has been a major technique used in homogenization-based methods for topology optimization of structures. It plays a key role in regularizing the basic problem into a well-behaved setting, but it has the drawback of a smoothing effect around the boundary of the material domain. In this paper, a diffusion technique is presented as a variational approach to the regularization of the topology optimization problem. A nonlinear or anisotropic diffusion process not only leads to a suitable problem regularization but also exhibits strong "edge"-preserving characteristics. Thus, we show that the use of nonlinear diffusions brings the desirable effects of boundary preservation and even enhancement of lower-dimensional features such as flow-like structures. The proposed diffusion techniques have a close relationship with the diffusion methods and the phase-field methods from the fields of materials and digital image processing. The proposed method is described and illustrated with 2D examples of minimum compliance that have been extensively studied in recent literature of topology optimization.