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 . 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.

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基于RDE的弹塑性结构拓扑优化体积约束的精确处理


对于基于反应扩散方程 (RDE) 的弹塑性结构拓扑优化，体积约束的精确性至关重要。作为一种非传统的数值方法，最近提出的精确体积约束需要迭代来确定精确的拉格朗日乘数。相反，传统的不精确体积约束方法类似于时间前移方案，可能导致收敛问题。推导了二维弹塑性问题的近似拓扑导数，并用于研究采用精确和非精确体积约束方法之间的差异。通过改变网格密度、设计域纵横比、施加载荷、约束体积比和扩散系数等参数进行全面检查。结果表明，与弹性相比，体积约束的不精确可能会导致弹塑性问题更严重。精确的体积约束方法不仅可以显着提高结构优化的收敛性，而且可以减少结构合规性和计算运行时间。可能有人猜测，传统的体积约束处理不精确所引起的波动可能会阻止优化过程陷入局部最小值。然而，与这一假设相反，在弹塑性情况下，它通常会产生相反的效果，经常使结构偏离全局最优值。特别值得注意的是，不精确的体积约束常常会导致结构非常差且顺应性极高。另一方面，增加归一化参数可以导致结果的显着改善。 这些发现强调了非线性拓扑优化问题精确体积约束的必要性。