Topology optimization with anisotropic materials, including a filter to smooth fiber pathways

In a recent publication, an approach to optimize the orientation of anisotropic materials was presented. This strategy was embedded into the thermodynamic topology optimization based on growth. In this paper, we show that the thermodynamic orientation optimization can also be used in more classical approaches to topology optimization. We furthermore enhance the approach by a novel filtering technique to provide control over the smoothness of the pathway of principal material directions, i.e., the curvature of fibers. The filter is based on a convolution operator and is applied to the material stiffness tensor, so that the filtering technique is not directly bounded to the actual parameterization for the design variables. To this end, the topology is defined by a continuous density approach with penalization of intermediate densities (SIMP) solved via the optimality criteria method (OCM). A set of three continuous Euler angles is used as additional design variables to describe the local material rotation of the anisotropic base material. The thermodynamic optimization of the material orientation is performed by evolution of the Euler angles to minimize the elastic energy. The related evolution equations are derived by means of the Hamilton principle, well-known from material modeling.