Implementation of the Phase Field Method for Structural Topology Optimization in a FEM Framework Developed in C++

Resumo

Topological optimization, specifically with a focus on compliance minimization, seeks the best material distribution within a design domain given different loading and constraints configurations. The Phase Field Method (PFM) offers a continuous and well-posed formulation from a mathematical point of view, naturally regularizing the problem by controlling the interface thickness, without the need for artificial filters. This paper presents the implementation of the PFM for topological optimization of structures in a finite element (FEM) framework developed in C++. The main objectives are to incorporate the PFM directly into a FEM code and compare its results with other methods, such as Solid Isotropic Material with Penalization (SIMP). The theoretical foundation of the study is based on a variational formulation, combining the structural equilibrium equation with the Allen–Cahn evolution equation to govern the material distribution. The coupling of the equations is made through a staggered scheme, this implies that each equation is solved separately and the outcome of each equation is iteratively exchanged to update the other. The results are used to confirm the successful implementation of the method and to demonstrate the phase field method ability to produce smooth and physically consistent topologies.