Abstract
The inverse acoustic scattering model for crack diagnosis is described by Helmholtz problem within mathematic framework and investigated for the sake of scientific computing. Minimizing the misfit from given measurements leads to an optimality conditionbased imaging function which is used for noniterative identification of the center of an unknown crack put in a test domain. The numerical tests are presented for the cracks of Tjunction shape and are carried out based on the PetrovGalerkin generalized FEM using wavelets basis and levelsets. This shows highprecision identification result and stability to noisy data of the diagnosis, which is illustrated for soundsoft as well as moderately soundhard cracks when varying the coefficient of surface impedance.
Highlights
 The inverse acoustic scattering problem is investigated for scientific computing of crack diagnosis
 Minimizing the misfit from given measurements leads to an optimality conditionbased imaging
 Imaging function is used for noniterative identification of the center of a crack put in test domain
 Numerical tests are based on PetrovGalerkin GFEM and wavelet basis
 Tjunction shaped cracks are soundsoft as well as soundhard with moderate surface impedance
 Highprecision identification result and stability to noisy data of the diagnosis are reported
1. Introduction
In this contribution, the Helmholtz problem describing scattering of homogeneous acoustic medium that contains the single defect described by a cluster of cracks like Tjunctions and characterized by surface impedance (thus, inhomogeneous medium) is considered. The inverse acoustic scattering problem is aimed at crack diagnosis motivated by applications to nondestructive testing with acoustic, electromagnetic, and elastic waves in engineering sciences.
The classic approach to inverse scattering problems in the context of mathematical modeling was established based on the operator theory, which results in direct (noniterative) solution methods such as factorization, sampling, enclosure, MUSICtype algorithms, and alike. The classic analysis utilizes fundamental solutions and Green’s functions for respective boundary value problems. In comparison, iterative methods are mostly used for the reason of numerical computing of the solution.
Recently, the shape optimization approach to identification of small geometric objects was developed based on the concept of topological derivative (when relating inhomogeneous medium to the background homogeneous one). The corresponding asymptotic analysis realizes the methods of singular perturbations. For the optimization approach adapted to inverse scattering problems, see the author’s works [1] related to soundsoft and soundhard obstacles, and [2] for diagnosis of inhomogeneities in the background medium. The firstorder topological derivative specified for geometric objects with realvalued refractive index was found in [3], and the highorder topological expansions were derived in [4].
Following the shape optimization, the imaging function is determined by the incident background field and the known measurement. In [1] it was applied for identification of the center of impedance obstacles from boundary measurements, and in [2] for identification of the center of an inhomogeneity from the known farfield pattern. The latter work used an equivalent formulation of the scattering problem in the form of a weakly singular integral equation of LippmannSchwinger. The numerical tests were carried out in 2d and 3d and demonstrated highprecision of the identification result and its stability with respect to the aposteriori errors due to discretization as well as noisy data.
In the current contribution we extend the optimality conditionbased imaging technique and successfully apply numerical algorithm for diagnosis of cracks. For the variational theory of cracks, see [5].
2. Theory
The theoretical result is established in the following manner. A least square cost functional of the misfit from a given measurement is minimized. To define the set of feasible parameters we describe the unknown crack $\omega $ with the help of geometric variables of its shape, center, size, and surface impedance coefficient $\alpha \in $ [0, $\infty $]. The weak variational formulation of the minimization problem with respect to these variables provides us with the zeroorder necessary optimality condition, which is derivativefree. Further applying asymptotic arguments as the size tends to zero, for the soundsoft crack as $\alpha \to \infty $ the optimality condition with respect to the center ${\mathbf{x}}^{*}$ was derived in [1, 2] as follows:
and the imaging function ${I}_{l}$ is defined for spatial points $\mathbf{x}$ by the following imaginary part:
In Eq. (2) the complexvalued function ${u}_{l}\left(\mathbf{x}\right)$ yields the incident field determined in the background homogeneous medium $\Omega $ with the wave number $k$ that satisfies the Helmholtz equation:
where $\nabla $ stands for the gradient vector and ${\nabla}^{2}$ for the Laplace operator. The other complexvalued function ${v}_{l}\left(x\right)$ solves the adjoint Helmholtz problem:
for the measurement ${u}_{l}^{*}\left(x\right)$ given at the observation boundary $\mathrm{\Gamma}$ with the normal vector $\mathbf{n}$, and $\overline{{v}_{l}}$ in Eq. (2) is its complex conjugate.
3. Methods
Motivated by the special need of highly accurate and stable interpolation for the variational solutions of the acoustic scattering problem, in [6] we developed the concept of PetrovGalerkin enrichment by a finite element method (FEM). To reduce the discretization error, we enriched the space of test functions based on necessary optimality conditions for interpolation properties. Using a PetrovGalerkin approach, we suggested low order interpolation polynomials for the trial space, and we enriched the test space with high order shape functions. Henceforth, the resulting PetrovGalerkin enrichment (PGE) has low computational costs, but it improves significantly the accuracy of interpolation which is of the seventh order. In [6] justification of PGE was provided by local wavelets with vanishing moments based on the Gegenbauer polynomial approximation. Practical formulas were derived for calculation of the system matrix for the reference Helmholtz equation given over uniform meshes in 2d and 3d. In the current work we apply the PGE algorithm for computation of the Helmholtz problems Eqs. (4) and (7).
From Eqs. (1) and (2) it follows that the true center ${\mathbf{x}}^{*}$ belongs to zerolevel set ${Z}_{l}$ of ${I}_{l}$ that is:
hence forces the numerical algorithm finding approximate center ${\mathbf{x}}_{h}^{*}$ from multiple measurements ${u}_{1}^{*}$, ${u}_{2}^{*}$, … and zerolevel sets ${Z}_{1}^{h}$, ${Z}_{2}^{h}$, … by the mean of intersection:
where $h$ associates the mesh size appeared in numerical computation of problem Eq. (4).
4. Results
We present our numerical tests over the uniform quadrilateral grid of size $h=$ 2^{7} in 2d implying the degrees of freedom DOF = 16641 in the computational domain $\Omega =$ (0, 1)$\times $(0, 1). The crack of Tjunction shape and of size 0.0194 posed at the center ${\mathbf{x}}^{*}=$ (0.1467, 0.3750) is drawn in Fig. 1 and Fig. 2. The measurements${u}_{1}^{*}$ and ${u}_{2}^{*}$ are synthesized by solving numerically the forward scattering problem for $l=$ 1, 2:
$\mathbf{n}\bullet \nabla {u}_{l}^{*}\left(\mathbf{x}\right)+{\alpha u}_{l}^{*}\left(\mathbf{x}\right)=0,\mathbf{x}\in \omega ,$
where the incident field is prescribed by the plane wave:
in the scattering directions ${\theta}_{1}=\pi $/8 and ${\theta}_{2}=\pi $/4, and the wave number $k=$$\pi $/2 fixed.
We distinguish the crack scattering by varying the surface impedance coefficient $\alpha \in $ [0, $\infty $] in Eq. (7) thus approaching the soundhard $\alpha =$ 0 and the soundsoft $\alpha =\infty $ cases. The typical identification result of ${\mathbf{x}}_{h}^{*}$ due to Eq. (6) for large $\alpha $ is depicted in Fig. 1 as $\alpha =$ 1.
Fig. 1Identification of the center of crack when impedance α= 1 (approaching soundsoft case)
a) Wave direction $\pi $/8
b) Wave direction $\pi $/4
c) Zero level sets
In Fig. 2(a) and 2(b) respectively there are drawn the numerically computed imaging functions ${I}_{1}^{h}$ and ${I}_{2}^{h}$. Its zerolevel sets ${Z}_{1}^{h}$ and ${Z}_{2}^{h}$ are utilized numerically by the narrow band technique and form the straight lines crossing the crack. Both the zerolevel sets are depicted together in the computational test domain $\Omega $ in Fig. 2(c). Their intersection point ${\mathbf{x}}_{h}^{*}$ is very close to the true center ${\mathbf{x}}^{*}$ as can be verified in Table 1.
For comparison, the crosspoint ${\mathbf{x}}_{h}^{*}$ for small $\alpha $ is presented in Fig. 2 as $\alpha =$ 0.01. This picture shows the error of approximation of ${\mathbf{x}}^{*}$ caused by curving the zerolevel sets.
All the results of numerical tests are gathered together in Table 1 also quantifying the absolute error ${\mathbf{x}}^{*}{\mathbf{x}}_{h}^{*}$ of the identification algorithm. In particular, in the last column we observe the relative error ${\mathbf{x}}^{*}{\mathbf{x}}_{h}^{*}/h$ given with respect to the meshsize $h$. This fact clearly demonstrates that for moderate $\alpha \ge $ 1 the Euclidean distance between the centers ${\mathbf{x}}^{*}$ and ${\mathbf{x}}_{h}^{*}$ is significantly less than the meshsize (in the range of 130 % of $h$), which definitively justifies the highprecision property of our algorithm.
Fig. 2Approximation of the center of crack when impedance α= 0.01 (approaching soundhard case)
a) Wave direction $\pi $/8
b) Wave direction $\pi $/4
c) Zero level sets
Table 1Identification result of center x*= (0.1467, 0.3750)
Surface impedance $\alpha $  Approximate center ${\mathbf{x}}_{h}^{*}$  Absolute error  Error related to $h$ 
$\infty $  (0.1490, 0.3750)  0.0023  0.2984 
1000  (0.1489, 0.3750)  0.0022  0.2798 
100  (0.1479, 0.3750)  0.0012  0.1486 
10  (0.1459, 0.3750)  0.0008  0.0975 
1  (0.1479, 0.3752)  0.0012  0.1530 
0.1  (0.1592, 0.3906)  0.0200  2.5589 
0.01  (0.2031, 0.4280)  0.0774  9.9084 
0.0001  (0.2422, 0.4438)  0.0977  12.5074 
The error ${\mathbf{x}}^{*}{\mathbf{x}}_{h}^{*}$ versus $\alpha $ from Table 1 is also illustrated for convenience in the loglog plot in Fig. 1. This curve exhibits the bounded limit behavior as $\alpha \to $ 0 and $\alpha \to \infty $.
Fig. 3Identification error when varying impedance (soundhard α= 0 and soundsoft α=∞)
5. Conclusions
Although the inverse acoustic scattering theory of Section 2 is derived rigorously when getting to the limit of the size and the surface impedance coefficient, the computer simulation ensures that the imaging function in Eq. (2) is still acceptable for moderate values of the parameters. This conclusion holds true also for the diagnosis of cracks treated here.
Acknowledgements
The author is supported by the Austrian Science Fund (FWF) Project P26147N26: ‘Object Identification Problems: Numerical Analysis’ (PION) and the Austrian Academy of Sciences (OeAW), the RFBR and JSPS research Project 195150004.
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