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.
Highlights
Most optimal design frameworks rely on a measure of the sensitivity of the objective function with respect to “small modifications” of shapes
One popular method in this direction is that of Hadamard, whereby variations of a shape are understood as perturbations of their boundaries; see e.g. [3, 12, 17, 18]. This information is sometimes combined with topological derivatives, as in [2]; these indicate where internal holes can be beneficially nucleated
Asymptotic expansions similar to those underlying topological derivatives would make it possible to account for the addition of small bubbles of material
Summary
Most optimal design frameworks rely on a measure of the sensitivity of the objective (and constraint) function with respect to “small modifications” of shapes. The sensitivity of the elastic displacement of the structure and that of a related quantity of interest (the pivotal ingredients of this viewpoint) can be calculated by borrowing techniques from the literature devoted to low-volume inhomogeneities. Such asymptotic problems have been quite extensively investigated; see [7], [4, 6] about thin tubular inclusions for the conductivity equation, and [5] in the 2d linearized elasticity case. Presentation of the structural optimization problem and relation with thin tubular inhomogeneities
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