The topological ligament in shape optimization: a connection with thin tubular inhomogeneities

Abstract

In this article, we propose a formal method for evaluating the asymptotic behavior of a shape functional when a thin tubular ligament is added between two distant regions of the boundary of the considered domain. In the contexts of the conductivity equation and the linear elasticity system, we relate this issue to a perhaps more classical problem of thin tubular inhomogeneities: we analyze the solutions to versions of the physical partial differential equations which are posed inside a fixed "background" medium, and whose material coefficients are altered inside a tube with vanishing thickness. Our main contribution from the theoretical point of view is to propose a heuristic energy argument to calculate the limiting behavior of these solutions with a minimum amount of effort. We retrieve known formulas when they are available, and we manage to treat situations which are, to the best of our knowledge, not reported in the literature (including the setting of the 3d linear elasticity system). From the numerical point of view, we propose three different applications of the formal "topological ligament" approach derived from these expansions. At first, it is an original way to account for variations of a domain, and it thereby provides a new type of sensitivity for a shape functional, to be used concurrently with more classical shape and topological derivatives in optimal design frameworks. Besides, it suggests new, interesting algorithms for the design of the scaffold structure sustaining a shape during its fabrication by a 3d printing technique, and for the design of truss-like structures. Several numerical examples are presented in two and three space dimensions to appraise the efficiency of these methods.

Small perturbations in the type of boundary conditions for an elliptic operator

“…As we have illustrated at in Remark 3.2, asymptotic formulas for the solution to "small" perturbations of a "background" boundary value problem allow to appraise the sensitivity of a quantity of interest (or a performance criterion) with respect to such perturbations. This idea plays into the concepts of topological derivative [55] and "topological ligaments" [51,24] in optimal design. In our context it would allow us to appraise the sensitivity of a performance criterion with respect to the introduction of a new, "small" region supporting Dirichlet or Neumann boundary condition in the physical boundary value problem.…”

An asymptotic representation formula for scattering by thin tubular structures and an application in inverse scattering

“…In the next section we extend the result from [13] to thin neighborhoods of smooth curves of (D ρn ) n as in (2.6), and we derive a spectral representation of the polarization tensor in terms of the center curve K and the two-dimensional polarization tensor of the cross-sections (D ρn ) n . In [13,23,29], the authors expressed interest such a characterization of the polarisation tensor for thin tubular objects for various applications. Therewith, the leading order terms in the asymptotic representation formulas (3.6) and (3.7) can be evaluated very efficiently.…”

“…A similar method for electrical impedance tomography has been considered in [29] (see also [13] for a related inverse problem with thin straight cylinders). Further applications of asymptotic representation formulas for electrostatic potentials as well as elastic and electromagnetic fields with thin objects in inverse problems, image processing, or shape optimization can, e.g., be found in [2,3,16,23,27,38].…”

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

“…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. That work uses the material ersatz approximation to replace the linear elasticity system on the frame with an approximate problem on a design domain where the voids in the design domain are filled with a smooth inhomogeneous material that coincides with the frame properties at the frame domain, while extending smoothly to a soft, artificial material at the voids.…”

“…That work uses the material ersatz approximation to replace the linear elasticity system on the frame with an approximate problem on a design domain where the voids in the design domain are filled with a smooth inhomogeneous material that coincides with the frame properties at the frame domain, while extending smoothly to a soft, artificial material at the voids. The two formulations, namely, the formulation proposed in Dapogny (2020Dapogny ( , 2021 and the one introduced in this work follow fundamentally different approaches and entail different computations for the calculation of the topological derivatives.…”