Quasi-Newton Methods for Topology Optimization Using a Level-Set Method

The ability to efficiently solve topology optimization problems is of great importance for many practical applications. Hence, there is a demand for efficient solution algorithms. In this paper, we propose novel quasi-Newton methods for solving PDE-constrained topology optimization problems. Our approach is based on and extends the popular solution algorithm of Amstutz and Andr\"a (A new algorithm for topology optimization using a level-set method, Journal of Computational Physics, 216, 2006). To do so, we introduce a new perspective on the commonly used evolution equation for the level-set method, which allows us to derive our quasi-Newton methods for topology optimization. We investigate the performance of the proposed methods numerically for the following examples: Inverse topology optimization problems constrained by linear and semilinear elliptic Poisson problems, compliance minimization in linear elasticity, and the optimization of fluids in Navier-Stokes flow, where we compare them to current state-of-the-art methods. Our results show that the proposed solution algorithms significantly outperform the other considered methods: They require substantially less iterations to find a optimizer while demanding only slightly more resources per iteration. This shows that our proposed methods are highly attractive solution methods in the field of topology optimization

ZASTOSOWANIE ELEKTRYCZNEJ TOMOGRAFII IMPEDANCYJNEJ W UKŁADACH POMIARU LINIOWEGO

The article presents an application to the topology optimization in electrical impedance tomography using the level set method. The level set function is based on shape and topology optimization for areas with partly continuous conductivities. The finite element method has been used to solve the forward problem. The proposed algorithm is initialized using topological sensitivity analysis. Shape derivative and material derivative have been incorporated with the level set method to investigate shape optimization problems. The coupled algorithm is a relatively new procedure to overcome this problem. Using the line measurement model is very useful to solve the inverse problem in the copper-mine ceiling and the flood embankment.W artykule przedstawiono aplikację do optymalizacji topologicznej w elektrycznej tomografii impedancyjnej przy użyciu metody zbiorów poziomicowych. Funkcja poziomicowa oparta jest optymalizacji topologii i kształtu dla obszarów z częściowo ciągłymi konduktywnościami. Metoda elementów skończonych została wykorzystana do rozwiązania tego problemu. Proponowany algorytm jest inicjalizowany przy użyciu topologicznej analizy wrażliwościowej. Pochodna kształtu i pochodna topologiczna zostały zaimplementowane z metodą zbiorów poziomicowych do rozwiązania problemu optymalizacji. Sprzężony algorytm jest stosunkowo nową procedurą do rozwiązania tego zadania. Zastosowanie modelu pomiaru tablicowego jest bardzo użyteczne w celu rozwiązania problemu odwrotnego m.in. w chodniku kopalni miedzi i wałach przeciwpowodziowych

Variational approach to relaxed topological optimization: closed form solutions for structural problems in a sequential pseudo-time framework

The work explores a specific scenario for structural computational optimization based on the following elements: (a) a relaxed optimization setting considering the ersatz (bi-material) approximation, (b) a treatment based on a non-smoothed characteristic function field as a topological design variable, (c) the consistent derivation of a relaxed topological derivative whose determination is simple, general and efficient, (d) formulation of the overall increasing cost function topological sensitivity as a suitable optimality criterion, and (e) consideration of a pseudo-time framework for the problem solution, ruled by the problem constraint evolution. In this setting, it is shown that the optimization problem can be analytically solved in a variational framework, leading to, nonlinear, closed-form algebraic solutions for the characteristic function, which are then solved, in every time-step, via fixed point methods based on a pseudo-energy cutting algorithm combined with the exact fulfillment of the constraint, at every iteration of the non-linear algorithm, via a bisection method. The issue of the ill-posedness (mesh dependency) of the topological solution, is then easily solved via a Laplacian smoothing of that pseudo-energy. In the aforementioned context, a number of (3D) topological structural optimization benchmarks are solved, and the solutions obtained with the explored closed-form solution method, are analyzed, and compared, with their solution through an alternative level set method. Although the obtained results, in terms of the cost function and topology designs, are very similar in both methods, the associated computational cost is about five times smaller in the closed-form solution method this possibly being one of its advantages. Some comments, about the possible application of the method to other topological optimization problems, as well as envisaged modifications of the explored method to improve its performance close the workPeer ReviewedPostprint (author's final draft

Layout optimization for multi-bi-modulus materials system under multiple load cases

Financial support from the National Natural Science Foundation of China (Grant No. 51179164) and the Australian Research Council (Grant No. DP140103137) is acknowledged