A nodal finite element approximation of a phase field model for shape and topology optimization

We propose a nodal finite element method to the problem of finding optimal structural shapes based on a phase field model motivated by the work of Takezawa et al. (2010). Compared to finite differences used in the original study, the proposed method better characterizes optimal configurations and is not sensitive to initial guesses or element shapes. Using nodal finite elements as a basis, we also investigate the application of two semi-implicit time-stepping schemes, the first-order and second-order semi-implicit backward Euler time-stepping schemes (1-SBEM and 2-SBDF), to the optimization problem. We then discuss the stability of these schemes and a classic finite-difference based upwind scheme using benchmark problems of compliance minimization with volume constraints. Numerical evidences show that the nodal FEM approach alleviates the initial dependency problem of structural optimization, and the 1-SBEM scheme is more stable than the other two schemes in tracking the moving boundary.