A new algorithm for topology optimization using a level-set method

Abstract

The level-set method has been recently introduced in the field of shape optimization, enabling a smooth representation of the boundaries on a fixed mesh and therefore leading to fast numerical algorithms. However, most of these algorithms use a Hamilton-Jacobi equation to connect the evolution of the level-set function with the deformation of the contours, and consequently they cannot create any new holes in the domain (at least in 2D). In this work, we propose an evolution equation for the level-set function based on a generalization of the concept of topological gradient. This results in a new algorithm allowing for all kinds of topology changes.

Shape Optimization under Uncertainty from a Stochastic Programming Point of View

“…However, as T is defined only on O and we consider the case that we have no elastic material in D \ O, we have to be careful not to take away too much material at once, because it will never be added back again in the optimization procedure. This is different in the approach by Amstutz and Andrä [9] for example, where the void areas are simulated by a very soft elastic material, which also allows for the opposite operation, i.e. strengthening of the weak phase.…”

“…e.g. [3,9,52]). We also included the topological derivative in our algorithm which allows us to start the optimization process with a solid structure.…”

“…This procedure will be incorporated into the overall shape optimization algorithm, which is described in Section 4.4, in the following way: After a prespecified number of iterations using the shape derivative alone, a descent step based on the topological derivative is performed, according to the following algorithm (also cf. [9]). Moreover, set c k := s min x∈O k T k (x), where T k denotes the topological gradient computed in the domain O k .…”

Optimal damping of a membrane and topological shape optimization

“…In [3] a method was given where the topological gradient alone was used to write a level set equation, which was then made stable by using an orthogonal projection technique. This was the method we chose to compare against the traditional boundary variation methods.…”

Topological sensitivity analysis for time-dependent problems

“…It can also be utilized within a level-set-based algorithm (see e.g. [1,6,11]). From the theoretical point of view, most efforts for deriving the expansion (0.1) have been so far focused on problems associated with state equations of elliptic type, for which several generalizations of the above notion have been proposed (e.g.…”