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## FEM-BASED APPROACH FOR CAVITY IDENTIFICATION

**Introduction**

Defect identification using TS under transient dynamical conditions have so far been the subject of only a few investigations, notably Dominguez et al. (2005) where the connection with time-reversal is ex-plored, Bonnet (2006) in which an adjoint-based form of the TS is derived for 3D elastodynamics and acoustics, Malcolm and Guzina (2008) and Chikichev and Guzina (2008) where the case of penetrable inclusions in acoustic and elastic media (respectively) is considered, and Bellis and Bonnet (2009) which is devoted to a specialized formulation for crack identification problems. This chapter addresses defect identification in elastic solids by means of the TS function defined for small-cavity nucleation in the con-text of 3D time-domain elastodynamics. In a previous publication (Bonnet, 2006), the TS function was obtained as a bilinear expression featuring the (time-forward) free field and the (time-backward) adjoint solution by considering the asymptotic behavior of a system of governing integral equations based on the transient full-space elastodynamic Green’s tensor, the corresponding (analogous and simpler) formu-lation for scalar waves was derived as a by-product, and a semi-analytical example based on transient 3D acoustic data was presented. As in many other derivations of TS formulations published thus far, the integral-equation setting is convenient for performing the mathematical asymptotic analysis but is then just one of several possible approaches for doing numerical computations once the necessary formulae are established.

The intended contributions of this chapter are twofold. Firstly, on the theoretical side, the derivation of the TS field proposed in Bonnet (2006) is clarified and extended as follows: (a) the validity of the previously-established asymptotic behavior of the time-domain governing integral equation (and hence of the resulting TS formulation) is shown to depend on smoothness assumptions on the free field, an issue not touched upon in Bonnet (2006); (b) a simpler and more compact version of the derivation, using Green’s tensors rather than full-space fundamental solutions, is presented; (c) proofs are also given for two-dimensional problems. Secondly, a comprehensive set of numerical experiments, including 3D elastodynamic examples, is reported and discussed. Unlike previous publications where the time-domain TS is computed by means of specialized techniques based on Green’s tensors, this study emphasizes the implementation and exploitation of TS fields using the standard displacement-based FEM, and indeed the ease of doing so once the correct sensitivity formulation is available. To the authors’ best knowledge, this chapter presents the first comprehensive numerical study of TS-based defect identification methodology in time-dependent 3D settings and implemented within general-purpose computational environments.

This chapter is organized as follows. The forward and inverse problems of interest are reviewed in Section 1.2. Topological sensitivity is defined and established, in both direct and adjoint-based forms, in Section 1.3, the more technical parts of the derivations being deferred to 1.A for ease of reading. Sec-tion 1.4 then discusses some important features of the methodology and introduces additional concepts and notations pertaining to the FEM-based implementation and its exploitation in subsequently presented numerical results. Then, the results of FEM-based numerical experiments are presented and discussed in Sections 1.5 (2D scalar wave equation) and 1.6 (3D and 2D elastodynamics).

### Cavity identification model problem

Let Ω denote a finite elastic body in Rd (d = 3 or d = 2), bounded by the external surface S and characterized by the shear modulus µ, Poisson’s ratio ν and mass density ρ, and referred in the following as the reference body. A cavity (or a set thereof) B bounded by the closed traction-free surface(s) is embedded in Ω. The external surface S, which is identical for the reference domain Ω and the cavitated domain Ω(B) = Ω \B, is split into a Neumann part SN and a Dirichlet part SD, respectively associated with prescribed time-varying tractions tN and displacements uD. Under this dynamical loading, an elastodynamic state uB arises in Ω(B), which satisfies the following set of field equations,

**Topological sensitivity**

**Small-cavity asymptotics**

The topological sensitivity of the cost functional (1.2.5) is defined as its sensitivity with respect to the creation of an infinitesimal object of characteristic size ε at a given location z in Ω. Here, such infinites-imal object is taken to be a trial cavity Bε(z), defined by Bε(z) = z + εB in terms of its center z, its shape specified by the unit bounded set B ⊂ Rd (with boundary S and volume |B|) containing the origin, and its radius ε > 0. The corresponding trial cavitated solid is denoted Ωε(z). Following Sokolowski and Zochowski (1999) or Garreau et al. (2001), one seeks the asymptotic behavior of J(Ωε(z), T ) as ε → 0 through the expansion: J(Ωε(z), T ) = J(Ω, T ) + η(ε)|B|T(z, T ) + o(η(ε)) (ε → 0) (1.3.1)

where the function η(ε), to be determined, vanishes in the limit ε → 0 and the topological sensitivity T(z, T ) is a function of the sampling point z and duration T .

To evaluate the expansion (1.3.1) and find the value of T(z, T ), it is necessary to consider the asymp-totic behavior of the displacement uε governed by problem P(Bε(z)). Towards that aim, it is convenient to decompose uε as uε(ξ, t) = u(ξ, t) + vε(ξ, t) (1.3.2)

where the free field u is the response of the cavity-free domain Ω to the prescribed excitation, i.e.

#### Leading contribution of vε as ε → 0

To address this issue, it is convenient to reformulate the governing boundary-initial problem (1.3.4) in terms of an integral equation. Let U (x, t, ξ) and T (x, t, ξ; n) denote the time-impulsive elastodynamic Green’s tensors, defined such that ek •U (x, t, ξ) and ek •T (x, t, ξ) are the displacement and traction vectors at ξ ∈ Ω resulting from a unit time-impulsive point force acting at x in the k-th direction at time t = 0 and satisfying the boundary conditions

U (x, t, ξ) = 0 (ξ ∈ SD, t > 0), T (x, t, ξ; n) = 0 (ξ ∈ SN, t > 0), (1.3.6)

One also defines the elastodynamic full-space fundamental tensors U ∞(x, t, ξ) and T ∞(x, t, ξ; n) in a similar way, replacing boundary conditions (1.3.6) with decay and radiation conditions at infinity (Erin-gen and Suhubi, 1975, see Section 1.A.2). The governing integral equation for the scattered field vε then reads (see Section 1.A.1)

Expression (1.3.22) provides a useful basis for discussing some of the features of the time-domain TS, see Section 1.4.1. It can also conceivably be used for the purpose of computing the field T(z, T ), and is indeed so used in Chikichev and Guzina (2008) wherein Ω is an elastic half-space with a traction-free surface, a configuration for which the Green’s tensor is known. For arbitrary reference bodies Ω, an implementation of (1.3.22) would require a numerical evaluation of the Green’s tensor for source points located on Sobs (typically taken as Gauss quadrature points associated with the evaluation of the integral over Sobs) and field points taken as sampling points z.

However, a computationally more efficient approach for evaluating the field T(z, T ), based on an adjoint solution, is usually preferable and was used for all numerical examples presented thereafter.

**Adjoint field formulation**

The adjoint formulation, previously presented in Bonnet (2006) and now summarized for completeness, stems from treating the integral in (1.3.5) as one of the terms arising in the elastodynamic reciprocity identity. For any generic domain O and pair of elastodynamic states u1, u2 satisfying the homogeneous elastodynamic field equations in O as well as homogenous initial conditions

u1(ξ, 0) = u˙1(ξ, 0) = 0 and u2(ξ, 0) = u˙2(ξ, 0) = 0 (ξ ∈ O),

**Remark 1.** The O(εd) asymptotic behavior (1.3.26) of J(Ωε(z), T ) relies on vε approaching (up to a scaling factor) a static solution as ε → 0. This requires the free-field to be sufficiently regular at (z, t), e.g. according to the sufficient condition given in Lemmas 1 and 2. To put this another way, the TS (1.3.26) may (invoking the Fourier convolution theorem) be formulated as the inverse Fourier transform of the (previously established in Bonnet and Guzina, 2004) frequency-domain expression

The Fourier integral then converges if ω T(z, ω) ∈ L1(R), i.e. provided the high-frequency content of the excitation is limited. Related considerations are developed in Ammari et al. (2009), where the order in ε of the leading perturbation by a small inclusion of the fundamental solution of the transient wave equation is shown to depend on the high-frequency content of the time-modulated point source.

**Table of contents :**

Introduction

Context

Overview of the thesis

**I Topological Sensitivity Method **

Introduction and Overview

**1 FEM-based approach for cavity identification **

1.1 Introduction

1.2 Cavity identification model problem

1.3 Topological sensitivity

1.3.1 Small-cavity asymptotics

1.3.2 Leading contribution of vε as ε → 0

1.3.3 Adjoint field formulation

1.4 Discussion and implementation

1.4.1 Discussion

1.4.2 Implementation and numerical experiments

1.5 Defect imaging using acoustic time-domain data

1.6 Defect imaging using elastodynamic time-domain data

1.6.1 Methodology

1.6.2 Single or dual cavity identification

1.6.3 Influence of experiment duration

1.6.4 Influence of observation surface configuration

1.6.5 Influence of data noise

1.6.6 Identification of non-cavity defects

1.7 Conclusion

1.A Asymptotic behavior of elastodynamic integral operators

1.A.1 Elastodynamic governing BIE

1.A.2 Elastodynamic fundamental solutions and proof of Lemmas 1 to 3

1.A.3 3D Scalar wave equation

1.A.4 Two-dimensional case

1.B Summary of explicit formulae for polarization tensors

**2 Qualitative identification of cracks **

2.1 Introduction

2.2 Elastic topological sensitivity

2.2.1 Preliminaries

2.2.2 Asymptotic analysis

2.3 Acoustic topological sensitivity

2.4 TS-based crack identification: heuristics and implementation

2.5 Numerical examples

2.5.1 Cubic domain

2.5.2 Cylindrical shell

2.5.3 Discussion

2.6 Extension to interface cracks

2.6.1 Polarization tensor for a penny-shaped crack

2.6.2 Numerical examples

2.7 Conclusion

2.A Polarization tensors

2.A.1 Matrix representation of fourth-order tensors

2.A.2 Major symmetry of polarization tensor

2.A.3 Elliptical crack

2.A.4 Elliptical sound-hard screen

2.B Elastodynamic fundamental solutions and proof of Lemmas 4,5

2.C Radon transform

**II Linear Sampling Method **

Introduction and Overview

**3 Multi-frequency obstacle reconstruction **

3.1 Introduction

3.2 Preliminaries

3.3 Inverse scattering via the linear sampling method

3.3.1 Relationship with the solution to the interior problem

3.3.2 Regularized solution

3.4 Multi-frequency reconstruction

3.4.1 “Serial” indicator function

3.4.2 “Parallel” indicator function

3.4.3 Behavior of the solution in a neighborhood of an eigenvalue

3.5 Results

3.5.1 Analytical study: spherical scatterer in R3

3.5.2 Numerical study: square obstacle in R2

3.6 Conclusions

**4 Well-posedness of the interior transmission problem **

4.1 Introduction

4.2 Preliminaries

4.3 Interior transmission problem

4.3.1 Weak formulation of the modified ITP

4.4 Existence and uniqueness of a solution to the modified ITP

4.5 Well-posedness of the ITP

4.5.1 Relaxed solvability criterion

4.6 Can the set of transmission eigenvalues be empty?

4.6.1 Energy balance

4.7 Results and discussion

4.7.1 Analytical examples

4.8 Conclusions

**5 Elastic interior transmission eigenvalue problem **

5.1 Introduction

5.2 Preliminaries

5.2.1 Interior transmission eigenvalue problem

5.2.2 Analytical example

5.3 Configurations with material similitude

5.3.1 Equal elastic tensors

5.3.2 Equal mass densities

5.4 Configurations without material similitude

5.4.1 Elasticity and mass density contrasts of opposite sign

5.4.2 Elasticity and mass density contrasts of the same sign

5.5 Conclusions

**III A Glimpse at the Relationship between TSM and LSM **

**6 Analytical comparative study in acoustics **

6.1 Introduction

6.2 Preliminaries

6.2.1 Forward problem

6.2.2 Inverse problem

6.3 Analytical formulation for a spherical scatterer

6.3.1 Scattered field

6.3.2 Analytical topological sensitivity method

6.3.3 Analytical linear sampling method

6.4 Effect of noisy data

6.4.1 Topological sensitivity method

6.4.2 Linear sampling method

6.5 Conclusion

6.A Partial and discrete observations

6.A.1 Topological sensitivity method

6.A.2 Linear sampling method

6.B Spherical harmonics and their properties

6.C Asymptotic behavior of special functions and their derivatives

6.C.1 Spherical Bessel and Hankel functions of the first kind

6.C.2 Derivatives of spherical Bessel and Hankel functions

6.C.3 Legendre polynomials

6.C.4 Derivatives of Legendre polynomials

Conclusion

**Conclusions **

Possible directions for future work

**Bibliography **