On the robustness of the topological derivative for Helmholtz problems and applications

Abstract

We consider Helmholtz problems in two and three dimensions. The topological sensitivity of a given cost function J(u ∈) with respect to a small hole B ∈ around a given point x 0 ∈ B ∈ ⊂ Ω depends on various parameters, like the frequency k chosen or certain material parameters or even the shape parameters of the hole B ∈. These parameters are either deliberately chosen in a certain range, as, e.g., the frequencies, or are known only up to some bounds. The problem arises as to whether one can obtain a uniform design using the topological gradient. We show that for 2-d and 3-d Helmholtz problems such a robust design is achievable.

Limits of stabilization of a networked hyperbolic system with a circle

“…Moreover, it is interesting to consider more general graphs, for example a cycle made from three edges and three attached single links. This topic is linked to the analysis in Leugering and Sokolowski (2008), where the elliptic case has been considered, see also Gugat, Qian and Sokolowski (2023) for the topological derivative method for control of wave equation on networks. A disadvantage of the spectral approach is that for more complex graphs also the spectral equations become more complicated.…”

On a cellular developmental method for layout optimization via the two-point topological derivative

“…Additive topological-derivatives are few, and the existing ones have not yet been used for topology optimization in the way that subtractive topological derivative has in approaches such level set methods. Indeed, Nazarov et al (2004Nazarov et al ( , 2005 developed a topological derivative for the energy functional due to the addition of thin ligaments when the state equation is a Poisson equation, Gangl (2020) developed a topological derivative for multi-material problems, and for graphs, such as trusses, Leugering and Sokolowski (2008) performed sensitivity analysis with respect to changing its topology-see also Lee (2008), which considers an asymptotic expansion of the topological gradient for a hole to decide whether an existing hole must be removed or not in the design domain. Recently, an independent work by Dapogny (2020Dapogny ( , 2021 considered an additive approach in a different context, where the bar can be a curve in twodimensional or three-dimensional space.…”