We propose an efficient probabilistic method to solve a deterministic problem -- we present a randomized optimization approach that drastically reduces the enormous computational cost of optimizing designs under many load cases for both continuum and truss topology optimization. Practical structural designs by topology optimization typically involve many load cases, possibly hundreds or more. The optimal design minimizes a, possibly weighted, average of the compliance under each load case (or some other objective). This means that in each optimization step a large finite element problem must be solved for each load case, leading to an enormous computational effort. On the contrary, the proposed randomized optimization method with stochastic sampling requires the solution of only a few (e.g., 5 or 6) finite element problems (large linear systems) per optimization step. Based on simulated annealing, we introduce a damping scheme for the randomized approach. Through numerical examples in two and three dimensions, we demonstrate that the stochastic algorithm drastically reduces computational cost to obtain similar final topologies and results (e.g., compliance) compared with the standard algorithms. The results indicate that the damping scheme is effective and leads to rapid convergence of the proposed algorithm.
An efficient and easy-to-extend Matlab code of the Moving Morphable Component (MMC) method for three-dimensional topology optimization
显式拓扑最优化方法近年来受到越来越多的关注。具体地，Zhang等人发布了基于二维（2D）移动可变形组件（MMC）的拓扑最优化方法的188行Matlab代码（Struct Multi Morphable Optim 53（6）：1243-1260，2016）。本工作的目的是提出一个有效的和易于扩展的256行Matlab代码的MMC方法的三维（3D）拓扑优化实施一些新的数值技术。具体地说，通过函数聚合技术，实现了精确的灵敏度分析，该方法也易于推广到其它问题。此外，基于一种有效的载荷传递路径识别算法，在有限元分析中去除了不属于载荷传递路径的自由度，大大加快了优化过程。结果表明，与相应的188行二维程序相比，优化结果的性能、有限元分析的计算效率、优化过程的收敛速度和鲁棒性都有很大提高。为了完整起见，还提供了实现2D-MMC方法的精炼的218行Matlab代码。
