This paper deals with the use of the topological derivative in detection problems involving waves. In the first part, a framework to carry out the topological sensitivity analysis in this context is proposed. Arbitrarily shaped holes and cracks with Neumann boundary condition in 2 and 3 space dimensions are considered. In the second part, a numerical example concerning the treatment of ultrasonic probing data in metallic plates is presented. With moderate noise in the measurements, the defects (air bubbles) are detected and satisfactorily localized by means of a single sensitivity computation.
Inspection problems can generally be seen as shape inversion problems. If techniques borrowed from shape optimization are now commonly accepted as good theoretical candidates to address shape inversion problems, their applications to inspection problems such as non-destructive testing or medical imaging are today relatively restricted. The main reason is that, in such problems, the possibility to handle topology changes is crucial. Therefore the use of the topological derivative concept, which directly deals with the variable “topology”, seems to be particularly well-suited. We recall the basic principles of this approach, introduced by Schumacher [1], Sokolowski and Zochowski [2] in structural optimization. Consider a cost function J(Ω)=JΩ(uΩ)J(Ω)=JΩ(uΩ) where uΩuΩ is the solution of a system of partial differential equations defined in the domain Ω⊂RNΩ⊂RN, View the MathML sourceN=2or3, a point x0∈Ωx0∈Ω and a fixed open and bounded subset ωω of RNRN containing the origin. The “topological asymptotic expansion” is an expression of the form
equation(1)
View the MathML sourceJ(Ω⧹(x0+ρω¯))-J(Ω)=f(ρ)g(x0)+o(f(ρ)),
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where f(ρ)f(ρ) is a positive function tending to zero with ρρ. Therefore, to minimize J(Ω)J(Ω), we have interest to remove matter where the “topological gradient” (also called “topological derivative”, or “topological sensitivity”) g is negative.
A general framework enabling to calculate the topological asymptotic expansion for a large class of shape functionals has been worked out by Masmoudi [3]. It is based on an adaptation of the adjoint method and a domain truncation technique providing an equivalent formulation of the PDE in a fixed function space. Using this framework, Garreau, Guillaume, Masmoudi and Sididris [4], [5] and [6] have obtained the topological asymptotic expansions for several problems associated with linear and homogeneous differential operators. For such operators, but with a different approach, more general shape functionals are considered in [7]. The link between the shape and the topological derivatives has been established by Feijóo et al. [8] and [9]. This gives rise to a generic method for deriving the latter. However, it seems rather restricted to circular or spherical holes. For the first time a topological sensitivity analysis for a non-homogeneous operator was performed in [10]. The case of a circular hole with a Dirichlet condition imposed on its boundary was considered. For completeness, we point out that extensive research efforts using related techniques have been done in the context of reconstruction problems from boundary measurements (see e.g. [11], [12], [13], [14] and [15] among others). In contrast to our approach, they do not deal with an explicit cost function. Instead, the sensitivity of the PDE solution uΩuΩ at the location of the measurements (or its integral against special test functions) is computed. Then the data are interpreted by signal processing methods. In addition, those works are focused on the detection of inhomogeneities, which means that the material density inside the inclusions is non-zero.
The problem of interest in this paper is related to non-destructive testing by means of ultrasounds in the context of elastodynamics. The governing equations at a fixed frequency involve a non-homogeneous differential operator of the form
equation(2)
u↦div(A∇u)+k2u,u↦div(A∇u)+k2u,
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where A is a symmetric positive definite tensor. For such a problem, the topological asymptotic expansion is determined in dimensions 2 and 3 with respect to the creation of an arbitrarily shaped hole and an arbitrarily shaped crack on which a Neumann boundary condition is prescribed. For the sake of simplicity, the analysis is presented for the Helmholtz operator (A=IA=I), but it applies similarly to any operator of the form (2). We introduce an adjoint method that takes into account the variation of the function space, so that a domain truncation is not needed. This formalism brings several technical simplifications, notably for the study of criteria depending explicitly on ΩΩ, for which the truncation necessitates to transport the cost function in the fixed domain (see [5]). Similar results have been obtained in [16]. However, the analysis is not done there in a rigorous mathematical framework. Furthermore, the crack problem is not addressed and less general cost functionals are considered.
The rest of the present paper is organized as follows. The adjoint method is presented in Section 2. The framework of the study is described in Section 3. The topological asymptotic analysis for a hole and a crack is carried out in 4 and 5, respectively, the intermediate proofs being reported in Section 8. The case of some particular cost functions is examined in Section 6. Section 7 is devoted to numerical experiments that highlight the relevance of the topological sensitivity approach for non-destructive testing applications.
