The Error Estimation of Approximating the Crack to Identify the Interfaces Crack in the Mobile's Wire

This paper is devoted to estimate the error resulting from approximating crack, i.e. approximating the support of the jump of crack in mobile's wire. This estimation is important to identify the interfaces crack which have been completed by using the derivation of Reciprocity Gap of this model. This derivation is based on the principle of Galerkin finite element method.


1-Introduction:
This work has focused on the reconstruction of line segment cracks (in 2D situations ) or planar cracks (in 3D situations) of a mobile's wire . In this area, there are many theoretical works, and almost all of them deal with 2D cases [8]: a uniqueness result for a buried crack has been investigated by Friedman and Vogelius [6]. For the case of emerging cracks at an a priori known point of the boundary , a uniqueness result and a local Lipschitz stability one have been proved in 1996 by Abda ,Andrieux and Jaoua [1] .In the case of a family of emerging cracks, a uniqueness result has been proved in 2001 by Elcrrat,Isakov and Necoloin [3] . As for the 3D situations , a few uniqueness results exist, and they all assume the knowledge of all the possible measurements , namely the full Neumann-to-Dirichlet operator (see [4] and [5]) . In this work we will identify the 2D or 3D crack interfaces of mobile's wire by deriving the concept of reciprocity gap of this model which is based on finite element method ,and then we will estimate the error producted by this identification.

2-The Mathematical Model of mobile's wire and Uniqueness results for 3D planar cracks:
Let denote the 3D bounded domain occupied by the body, and its external boundary which we shall assume to be C 2 . Let be the interface between the two materials in this domain which divided it , into two parts, .
The body is supposed to contain one co-planar crack where is the affine plane in R 3 containing the crack .Crack is propagated inside the interface as shown in figure (1) .

Figure (1):the Representation of part of wire containing crack
The affine space is equipped with a direct orthonormal frame (0,e 1 ,e 2 ,e 3 ). Denoting by (x 1 ,x 2 ,x 3 ) the corresponding Cartesian coordinates system ,the plane equation of the interface will be given by: where N=(n 1 ,n 2 ,n 3 ) is a unit normal vector to on the boundary and on the interface . Let us denote by a given heat flux on satisfying and ,(in practice , will be chosen to be piecewise continuous). To solve the problem of conduction (Dirichlet -Neumann boundary value problem (BVP)) : We suppose to ensure the existene s of the solution and we Assume that to ensure of the uniqueness and let f* be the trace of The conductivity k(x) is picewise constant with the int. inside The discontinuity in k(x) necessitates the consideration of the weak solution of (2a) which satisfies: . In our case , is the interface between two material and it is sufficiently smooth .
In the classical formulation of (2a) , the solution satisfies the Laplace equation and so-called transmission conditions across the interface : .
i.e. continuity of the potential and of the flux. So that :(2a) is equivalent to the two bellow problems ( 2b ), i =1,2.
i.e.The Mathematical model of the Interface of the planar crack in mobile's wire is : 3-The Derivation of the Reciprocity Gap (the inversion process) for the planar cracks in wire of mobile: .
In this section, we first will derive the numerical principle of the reciprocity gap (inversion process) notion and the functional associated to it using the principles of Galerkin FEM. Then, we use this function to

3-1-The Numerical Principle of Reciprocity gap concept : [7]
In fact, this principle is general and is valid in the case of symmetric operators For the sake of simplicity, the principle is presented in the case of elliptic operators. The variation formulation associated with this kind of problem can be phrased as follows: . .
Where H is a Hilbert space, a bilinear , symmetric and coercive form , continuous form on H .L 1 and L 2 being two different linear forms defined on H, let us consider the two corresponding problems (i=1,2): . .
Then choosing v=u 2 as a test function for the first problem, and v=u 1 for the second one, we derive that L 1 (v 2 )=L 2 (v 1 ). This is the explicit reciprocity principle , due to the symmetry of a. .

Lemma(1):
Suppose is empty and let u be the solution of the problem in the safe wire of mobile : which is the solution of the equation of conduction: . .
Satisfying (3) in and by using the principles of FEM,we have By the union of the region we have Now let be un flux that verify (3),so we have : subtracting (6) from (5) we get: φ appears as a linear form defined on the set: The following theorem is the key to the use of the numerical explicit reciprocity gap functional method for our model: .

Theorem(1):
For which is the solution of the equation of conduction: .
By union the region we have : (8)

Proof:
One has and is known by equation (11) Where the polynomial harmonic function satisfies the previous conditions. This means that one has explicit inversion formulae that give the Cartesian equation of the interface plane containing the crack .
4-The complete identification of the interfaces plane: In this section , a constructive method is now proposed to achieve the interfaces that contain cracks identification . Once again , the numerical explicit reciprocity gap that we derived in the previous section is a basic tool . Based on its two expressions (7) and (11) , the identification of that contains is performed by interpreting as a linear form of L 2 (S) , S being some square domain of the plane containing the interfaces and the crack . .
Consider now a new frame ( O,T,V,N) obtained by a simple translation, such that the new origin O belongs to . Let , and w be some open "big box" containing . setting d=2 does not reduce the generality , and then for example: .

Remark(2): [2]
Equation (17) gives, in fact the Fourier coefficinets of on the square S and by using a truncated fourier expansion,that is the quadratic partial sum at order n . we can reconstructing its fourier expansion .
Now in order to complete the identification of the interfaces crack , the error resulted from approximating the support of this jump should be estimated, i.e. approximating of the crack . .

5-The Error Estimation of approximating the Cracks:
In order to provide an approximation of the cracks, we need to define , for a given positive real number , and a given integer n , the following sets. The first one is expected to be an approximation of , and the second one an approximation of . Let us denote by d the Hausdorff distance in the plane ( ) . The following results then hold : . Figure

Lemma(3): [7]
For a prescribed real positive number , we have: . Under the assumption that the distance to the boundary does not vanish on , there exists some constant c, and some real positive number depending on and s.t. for any we have : We are now able to give the error estimate in the following theorem:    so that it is sufficient to choose to get Now let , the condition on becomes then: Now according to Zhizhiashvili, the uniform convergence of g n to g . for any given ,there exists So that choosing insures that : and according to the proof of lemma (3) and (4) [ 7] we conclude that:

6-Conclusion:
The Rreciprocity Gap concept derived in this paper seems to be quite efficient, both from the theoretical and the numerical viewpoints. It leads to uniqueness results for the planar crack inverse problem as well as explicit inversion formulae for the interface containing the cracks. We have introduced and derived this concept to identify the interfaces of crack in the mobile's wire and complete this identification by estimating the error of approximating these cracks .