2025-07-02 ブラウン大学
Topology optimization is an iterative process in which computers test small design tweaks to converge on optimal usage of materials. A new algorithm helps optimizers arrive at solutions in fewer iterations, saving valuable computing time.
＜関連情報＞
- https://www.brown.edu/news/2025-07-02/topology-optimization
- https://epubs.siam.org/doi/10.1137/24M1708863
- https://link.springer.com/article/10.1007/s00158-025-04008-9
密度に基づくトポロジー最適化のためのSiMPL法の分析 Analysis of the SiMPL Method for Density-Based Topology Optimization
Brendan Keith, Dohyun Kim, Boyan S. Lazarov, and Thomas M. Surowiec
SIAM Journal on Optimization Published:22 May 2025
DOI:https://doi.org/10.1137/24M1708863
Abstract
We present a rigorous convergence analysis of a new method for density-based topology optimization that provides pointwise 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 [D. Kim et al., Struct. Multidiscip. Optim., 68 (2025), 74]. 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, our algorithm 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. We refer to the new algorithm as the SiMPL method (pronounced “simple”), which stands for Sigmoidal Mirror descent with a Projected Latent variable.
密度に基づくトポロジー最適化のためのSiMPL法の簡単な紹介 A simple introduction to the SiMPL method for density-based topology optimization
Dohyun Kim,Boyan S. Lazarov,Thomas M. Surowiec & Brendan Keith
Structural and Multidisciplinary Optimization Published:25 April 2025
DOI:https://doi.org/10.1007/s00158-025-04008-9
Abstract
We introduce a novel method for solving density-based topology optimization problems: Sigmoidal Mirror descent with a Projected Latent 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. 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.
