We introduce a novel method for solving density-based topology optimization problems: \underline{Si}gmoidal \underline{M}irror descent with a \underline{P}rojected \underline{L}atent variable (SiMPL). The SiMPL method (pronounced as "the simple method") optimizes a design using only first-order derivative information of the objective function. The bound constraints on the density field are enforced with the help of the (negative) Fermi--Dirac entropy, which is also used to define a non-symmetric distance function called a Bregman divergence on the set of admissible designs. This Bregman divergence leads to a simple update rule that is further simplified with the help of a so-called latent variable. %Introducing a generalized Barzilai-Borwein step size rule accelerates the convergence of SiMPL. Because the SiMPL method involves discretizing the latent variable, it produces a sequence of pointwise-feasible iterates, even when high-order finite elements are used in the discretization. Numerical experiments demonstrate that the method outperforms other popular first-order optimization algorithms. To outline the general applicability of the technique, we include examples with (self-load) compliance minimization and compliant mechanism optimization problems.

翻译：本文提出一种求解基于密度的拓扑优化问题的新方法：\underline{Si}gmoidal \underline{M}irror descent with a \underline{P}rojected \underline{L}atent variable（简称SiMPL）。SiMPL方法（发音同“the simple method”）仅利用目标函数的一阶导数信息进行设计优化。密度场的边界约束通过（负）费米-狄拉克熵实现，该熵同时用于在可行设计集上定义一种称为Bregman散度的非对称距离函数。该Bregman散度导出一个简洁的更新规则，并借助所谓潜变量进一步简化。由于SiMPL方法涉及潜变量的离散化，即使采用高阶有限元进行离散，该方法仍能生成逐点可行的迭代序列。数值实验表明，该方法优于其他常用的一阶优化算法。为说明该技术的普适性，我们给出了（自载荷）柔度最小化和柔顺机构优化问题的算例。