Multi material stress-based topology optimization using the phase field method

The structural topology optimization for stress-related objective and multi-material design have been two focal points in the advanced manufacturing. The stress minimization issues become particularly challenging due to the stress singularities, its local sensitivity, and the complex nonlinear nature of stress. This study establishes a topology optimization framework with a primary focus on stress minimization, which is also capable of achieving multi-material design. The energy function contains the original stress minimization objective, the Ginzburg-Landau free energy for different materials, the volume constraints for each individual phase and the total fraction constraint for multi phases. The detailed sensitivity analysis for the established energy is provided. We use the finite element method for the linear elasticity problem and the Method of Moving Asymptotes for the minimization problem. Several structural benchmarks with multi-materials in both two- and three-dimension are performed to demonstrate the effectiveness of the present framework. Both single material and multi material designs can be achieved in two- and three-dimensional situations, while the corresponding constraint conditions can be satisfied. The numerical results demonstrate that the proposed phase field topology optimization framework effectively addresses the three challenges of stress minimization problems and exhibits strong design flexibility.