We present a rigorous convergence analysis of a new method for density-based topology optimization: Sigmoidal Mirror descent with a Projected Latent variable. SiMPL, pronounced like "simple," provides point-wise bound preserving design updates and faster convergence than other popular first-order topology optimization methods. Due to its strong bound preservation, the method is exceptionally robust, as demonstrated in numerous examples here and in the companion article. Furthermore, it is easy to implement with clear structure and analytical expressions for the updates. Our analysis covers two versions of the method, characterized by the employed line search strategies. We consider a modified Armijo backtracking line search and a Bregman backtracking line search. For both line search algorithms, SiMPL delivers a strict monotone decrease in the objective function and further intuitive convergence properties, e.g., strong and pointwise convergence of the density variables on the active sets, norm convergence to zero of the increments, convergence of the Lagrange multipliers, and more. In addition, the numerical experiments demonstrate apparent mesh-independent convergence of the algorithm.

翻译：本文对一种新的基于密度的拓扑优化方法——带投影潜变量的Sigmoidal镜像下降法（SiMPL，发音同“simple”）——进行了严格的收敛性分析。SiMPL方法能提供逐点保界的设计更新，且收敛速度优于其他常用的一阶拓扑优化方法。得益于其强大的保界特性，该方法展现出卓越的鲁棒性，这一点在本文及其姊妹论文的众多算例中均得到验证。此外，该方法结构清晰、更新过程具有解析表达式，易于实现。我们的分析涵盖了该方法的两个版本，其区别在于所采用的线搜索策略：改进的Armijo回溯线搜索与Bregman回溯线搜索。对于这两种线搜索算法，SiMPL均能保证目标函数的严格单调递减，并具有更直观的收敛性质，例如：密度变量在有效集上的强收敛与逐点收敛、增量范数收敛至零、拉格朗日乘子的收敛性等。数值实验进一步表明，该算法具有明显的网格无关收敛特性。