Inverse design of structure and fiber orientation by means of topology optimization with tensor field variables

This paper presents a topology optimization method which is capable of designing both topology and orientation distribution of anisotropic composite material simultaneously. Topology optimization is a well established structural optimization framework which optimizes the material distribution, i.e. density, in a given design domain for maximum performance. However, a method for structures made of inhomogeneously distributed anisotropic composite material is still under research. In this paper, a topology optimization method is extended to handle orientation distribution together with density distribution. A tensor field design variable is used for modeling orientation based on the idea of the orientation tensor. All tensor components are updated in similar manner to the Free Material Optimization technique while maintaining the physical feasibility by using the existing material tensor for interpolation. Thanks to the tensor representation, the method is free from complications derived from three-dimensional rotation. At the same time, the method works well with common non-linear programming algorithms because the tensor invariants are kept constant by multi-variable projection without point-wise constraints. The proposed method is built upon a modern topology optimization technique, thus, it is versatile and flexible enough to solve multi-load problems. Single loaded and multiply loaded stiffness maximization problems are provided as numerical examples, and characteristics of concurrent density and fiber orientation optimization are investigated.