A connection between topological ligaments in shape optimization and thin tubular inhomogeneities

TL;DR

This paper develops a formal framework to analyze how adding thin ligaments or tubular inhomogeneities affects the properties of elastic structures, with practical formulas and applications in shape optimization.

Contribution

It introduces a novel sensitivity analysis method for thin ligaments in shape optimization, bridging topological ligaments and tubular inhomogeneities in elastic structures.

Findings

Derived a practical sensitivity formula for thin tubular inhomogeneities.

Applied the framework to structural optimization problems.

Demonstrated numerical implementation of the sensitivity analysis.

Abstract

In this note, we propose a formal framework accounting for the sensitivity of a function of the domain with respect to the addition of a thin ligament. To set ideas, we consider the model setting of elastic structures, and we approximate this question by a thin tubular inhomogeneity problem: we look for the sensitivity of the solution to a partial differential equation posed inside a background medium, and that of a related quantity of interest, with respect to the inclusion of a thin tube filled with a different material. A practical formula for this sensitivity is derived, which lends itself to numerical implementation. Two applications of this idea in structural optimization are presented.