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We present a two-step method for recovering an unknown sound-hard crack in

In this paper we consider an acoustic scattering problem from a sound-hard crack. This problem is modeled by an exterior boundary value problem governed by Helmholtz equation for an open arc with Neumann boundary conditions on both sides of the arc. Our major concern is the inverse problem which aim is to reconstruct the crack from some measurements. This kind of problem is of fundamental importance, for example, in material investigation, nondestructive testing, or in seismic exploration.

The inverse scattering problem for an open arc was first investigated by Kress [

The nonlinear integral equations method, proposed by Kress and Rundell [

Another group of method which is not iterative is the decomposition method (see [

In [

Given a regular nonintersecting

The direct scattering problem for a sound-hard crack that we are considering is as follows.

Given an incident plane wave

In (

Note that the boundary conditions (

Using boundary integral equations, this direct problem can be solved via the layer approach. We refer to the monograph [

The direct Neumann Problem

At this place, we introduce the far-field pattern

It is convenient for our further treatment to rewrite our integral equation (

After introducing the notations in the last section, we consider the following inverse problem.

Determine the crack

For the uniqueness of this inverse problem, that is, for the identifiability of the arc, we refer to [

Motivated by the method of nonlinear integral equations in [

In [

Start the inverse problem with the same system (

As in Section

The system of integral equation (

Now we want to introduce our two-step algorithm. The parametrized systems (

The two-step scheme reads

At this point, we want to discuss the uniqueness and the solvability of the system (

This reflects the fact that different parametrizations of the arc leading to the same set of points turn out to have the same far-field pattern. We can avoid this ambiguity by limiting our solution space to the set of arcs representable as the graph of a function as suggested in [

If the pairs

Numerically, we solve the systems (

The algorithm for our method can be formulated as follows:

given an initial guess

iterative steps, for

Step

Step

In this section we will demonstrate the applicability of our method via some examples. We reconstruct the unknown crack from the knowledge of the far-field pattern at a number of points resulted from just one incident wave. For the direct problem, the forward solver is applied with 63 collocation points. In order to avoid committing an inverse crime, the number of collocation points used in the inverse problem is chosen to be different from that of the forward solver. The point is, in the inverse problem, (

The basis functions for the parametrization are taken from the space:

For the first example, we take the arc

From the numerical results, we see that the reconstructions for exact data are very good. The reconstructions in the case where random are present, the reconstructions are not bad.

Exact data,

3% error,

Exact data,

3% error,

To demonstrate the benefit of our method, we choose a curve which does not belong to our solution space. To this end, we take the arc

The results are shown in Figures

Exact data,

3% error,

Exact data,

3% error,

We see that the reconstructions are very good, even in the case of erroneous data.

In this final example, we choose the same curve as in the last example for the case of limited aperture. We assume that the data are only measurable within a certain range apart from the incident angle. Figure

Exact data,

3% error,

Exact data,

3% error,

We conclude this paper with some remarks. First of all, our examples above show the feasibility of the proposed numerical method. On one hand, being a Newton-type method, our method is conceptually simple and numerically more accurate than the traditional decomposition method. On the other hand, being a variant of the nonlinear integral equations method, the derivative is directly computable from the algorithm itself which makes the method more easily accessible than the classical Newton’s method. The splitting of the problem into two smaller parts makes our method more competitive as the computational cost concerns. Finally we want to point out that our method can be carried over to other type of boundary conditions and also to other type of scattering problems.

This work is partially supported by the NSC Grant NSC-100-2115-M-006-003-MY2.