Exact treatment of volume constraint for RDE-based topology optimization of elastoplastic structures

Abstract

For the reaction–diffusion equation (RDE) based topology optimization of elastoplastic structure, exactness in volume constraint can be crucial. As a non-traditional numerical method, the recently proposed exact volume constraint requires iterations to determine the precise Lagrangian multiplier. Conversely, conventional inexact volume constraint methods resemble a time-forward scheme, potentially leading to convergence issues. An approximate topological derivative for the 2D elastoplastic problem is derived and utilized to investigate the difference between employing exact and inexact volume constraint methods. A comprehensive examination is conducted by varying parameters such as mesh density, design domain aspect ratio, applied load, constrained volume ratio, and the diffusion coefficient $\tau$. Results indicate that the inexactness of volume constraint can lead to more severe issues in elastoplasticity compared to elasticity. The exact volume constraint method not only yields significantly improved convergence in structural optimization but also reduces structural compliance and computational runtime. There might be speculation that the fluctuation caused by the traditional inexact treatment of volume constraints could prevent the optimization process from being trapped in a local minimum. However, contrary to this assumption, in elastoplastic cases, it often has the opposite effect, frequently driving the structure away from a global optimum. Particularly noteworthy is the observation that inexact volume constraint quite often results in very poor structures with exceedingly high compliance. On the other hand, increasing the normalization parameter can lead to substantial improvements in results. These findings underscore the necessity of exact volume constraint for nonlinear topology optimization problems.