An inverse problem for Voronoi diagrams: A simplified model of non‐destructive testing with ultrasonic arrays

Abstract

In this paper, we study the inverse problem of recovering the spatially varying material properties of a solid polycrystalline object from ultrasonic travel time measurements taken between pairs of points lying on the domain boundary. We consider a medium of constant density in which the orientation of the material's lattice structure varies in a piecewise constant manner, generating locally anisotropic regions in which the wave speed varies according to the incident wave direction and the material's known slowness curve. This particular problem is inspired by current challenges faced by the ultrasonic non‐destructive testing of polycrystalline solids. We model the geometry of the material using Voronoi tessellations and study two simplified inverse problems where we ignore wave refraction. In the first problem, the Voronoi geometry itself and the orientations associated to each region are unknowns. We solve this nonsmooth, nonconvex optimisation problem using a multistart non‐linear least squares method. Good reconstructions are achieved, but the method is shown to be sensitive to the addition of noise. The second problem considers the reconstruction of the orientations on a fixed square mesh. This is a smooth optimisation problem but with a much larger number of degrees of freedom. We prove that the orientations can be determined uniquely given enough boundary measurements and provide a numerical method that is more stable with respect to the addition of noise.

Sensitivity analysis and tailored design of minimization diagrams

“…The other inclusion is clear. Now we prove the second equality in (13). Let x ∈ γ k1 (t) ∩ γ k2 (t), then x ∈ M {k1,k2} (t) in view of (12).…”

“…We start with the first equality in (13). Suppose x ∈ k∈K γ k (t) \ k∈K γ k (t), then, in view of (12), φ k1 (x, t) = 0 and φ k2 (x, t) = 0 for some {k 1 , k 2 } ⊂ K, which proves k∈K γ k (t) \ k∈K γ k (t) ⊂ I={k1,k2}∈I 2 γ k1 (t) ∩ γ k2 (t).…”

“…In this context, the optimization usually consists in obtaining centroidal Voronoi tessellations, see the reviews [19,20] and the references therein; see also [34] for alternative approaches. Other applications include land-use optimization [37] and inverse problems [13]. In some cases, the optimization of Voronoi diagrams is based on a sensitivity analysis, which has been performed in the literature for specific classes of energies and minimization diagrams such as centroidal Voronoi tessellation functions [19], centroidal power diagrams [14] and for an inverse problem for Voronoi diagrams in [13].…”

“…Other applications include land-use optimization [37] and inverse problems [13]. In some cases, the optimization of Voronoi diagrams is based on a sensitivity analysis, which has been performed in the literature for specific classes of energies and minimization diagrams such as centroidal Voronoi tessellation functions [19], centroidal power diagrams [14] and for an inverse problem for Voronoi diagrams in [13]. The sensitivity analysis developed in the present paper widely generalizes these approaches and provides a rigorous mathematical construction of the bi-Lipschitz mappings required for integration by substitution.…”

A phase-field approach for detecting cavities via a Kohn-Vogelius type functional

“…We work in the framework of linear elasticity, representing the medium by a bounded domain Ω ⊂ R d , where d = 2, 3. These kinds of inverse problems appear in nondestructive testing techniques used by industry to detect defects, voids, cracks in a medium, which can appear during manufacturing processes, and to evaluate properties of materials and structures without causing damage to the medium ( [6,22,35,53,60]). For instance, non-destructive testing methods are particularly important in the fields where the new techniques of additive manufacturing are replacing traditional methods of metal manufacture [39,58,70,73].…”