Year: 2026
Type: preprint
Abstract: In this paper we derive a framework for topological expansions of shape functionals with respect to multiple holes based on the expansion of one hole. Our strategy is similar to deriving a Taylor expansion of order two of functions in $\mathbf R^d$ by applying $d$-times a Taylor expansion in $\mathbf R$. For shape functionals defined on subsets of $\mathbf R^2$ we relate the topological expansion with multiple holes to a Taylor expansion with respect to the ball volumes where usual derivatives are replaced by topological state derivatives. The first topological state derivative was introduced in a previous paper and relates the first topological derivative of cost functionals. The second topological state derivative at two distinct points is introduced, similar to a second directional derivatives of functions defined on Banach spaces, as the topological state derivative of the topological state derivative. The second topological state derivative at the same point is defined as the "far-away" component of the compound layer expansion of the state variable. In dimension three the obtained topological expansion still involves first and second topological state derivatives, but it does not follow the same pattern of a Taylor expansion in the ball volumes. As we will show this dimension dependent behaviour is rather of geometrical nature and already appears in cost functionals without PDE constraints.
Source: arXiv (Cornell University)
Author: Sturm, Kevin
Subfield: Civil and Structural Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7164023968
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