A Simple Introduction to the SiMPL Method for Density-Based Topology Optimization

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.