Topology Optimization With Many Right-Hand Sides Using Mirror Descent Stochastic Approximation—Reduction From Many to a Single Sample
Topology Optimization With Many Right-Hand Sides Using Mirror Descent Stochastic Approximation—Reduction From Many to a Single Sample
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DOI：
10.1115/1.4045902
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发表时间：
2020-05
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影响因子：
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通讯作者：
X. Zhang;E. D. Sturler;A. Shapiro
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文献类型：
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作者：
X. Zhang;E. D. Sturler;A. Shapiro
实际工程设计通常涉及许多荷载工况。对于具有多个确定性荷载工况的拓扑优化问题，在每一步优化过程中都需要求解大量的线性方程组，计算量巨大。为了解决这一挑战，我们提出了一个镜像下降随机近似（MD-SA）框架与各种步长策略来解决拓扑优化问题与许多负载情况。我们重新制定的确定性目标函数和梯度到随机的，通过随机化，推导出MD-SA更新，并制定算法策略。所提出的MD-SA算法在随机梯度中仅需要低精度，因此每个优化步骤仅使用单个样本（即，样本量始终为1）。因此，我们将每步要解决的线性系统的数量从数百个减少到一个，这大大降低了总计算成本，同时保持了类似的设计质量。例如，对于其中一个设计问题，要解决的线性系统的总数和挂钟时间分别减少了223和22倍。
Practical engineering designs typically involve many load cases. For topology optimization with many deterministic load cases, a large number of linear systems of equations must be solved at each optimization step, leading to an enormous computational cost. To address this challenge, we propose a mirror descent stochastic approximation (MD-SA) framework with various step size strategies to solve topology optimization problems with many load cases. We reformulate the deterministic objective function and gradient into stochastic ones through randomization, derive the MD-SA update, and develop algorithmic strategies. The proposed MD-SA algorithm requires only low accuracy in the stochastic gradient and thus uses only a single sample per optimization step (i.e., the sample size is always one). As a result, we reduce the number of linear systems to solve per step from hundreds to one, which drastically reduces the total computational cost, while maintaining a similar design quality. For example, for one of the design problems, the total number of linear systems to solve and wall clock time are reduced by factors of 223 and 22, respectively.