Franck Delvare
email: franck.delvare@ensi-bourges.fr

Alain Cimetiere

A first order method for the Cauchy problem for the Laplace equation using BEM

Keywords: Cauchy problem, inverse problem, boundary values, boundary element method, regularization

The purpose is to propose an improved method for inverse boundary value problems. This method is presented on a model problem. It introduces a higher order problem. BEM numerical simulations highlight the efficiency, the improved accuracy, the robustness to noisy data of this new approach, as well as its ability to deblur noisy data.

Introduction

The purpose is to propose an improved method using boundary element method (BEM) to solve Cauchy inverse problems. These problems are called data completion problems. They consist in recovering data on a part of the boundary given the partial differential equation within the domain and overspecified data available on the remaining part of the boundary. This kind of problems arises in many engineering fields like thermal problems, electrostatic or elastostatic ones. For convenience, we will present the new method in the framework of a model problem: the Cauchy problem for the Laplace equation. Other methods exist to solve this problem. The main methods are Tikhonov like methods, [START_REF] Engl | Regularization of Inverse problems[END_REF], [START_REF] Hayashi | Direct method of solution for general boundary value problem of the Laplace equation[END_REF], [START_REF] Pasquetti | Inverse diffusion by boundary elements[END_REF], [START_REF] Tikhonov | Solutions of Ill-posed problems[END_REF], quasi reversibility methods [3], [START_REF] Fasino | An inverse Robin problem for the Laplace's equation: Theorical results and numerical methods[END_REF], [START_REF] Klibanov | A computational quasi-reversibility method for Cauchy problems for Laplace's equation[END_REF], [START_REF] Lattès | An iterative boundary element method for solving the Cauchy problem for the Laplace equation[END_REF], and iterative methods [START_REF] Andrieux | Solving Cauchy problems by minimizing an energy-like functional[END_REF], [START_REF] Jourhmane | Relaxation procedures for an iterative algorithm for solving the Cauchy problem for the Laplace equation[END_REF], [START_REF] Kozlov | An iterative method for solving the Cauchy problem for elliptic equations[END_REF], [16]. Quasi reversibility methods and Tikhonov regularization methods present the avantage to lead to well-posed problems after modifying the partial derivative operator. The iterative methods, where the number of iterations acts as a regularization parameter, are based on the use of a sequence of well-posed problems. In previous works [START_REF] Cimetière | Solution of the Cauchy problem using iterated Tikhonov regularization[END_REF], [START_REF] Delvare | An iterative boundary element method for Cauchy inverse problems[END_REF] we have presented a somewhat different resolution approach for the Cauchy problem for the Laplace equation. This approach was introduced in [START_REF] Cimetière | Solution of the Cauchy problem using iterated Tikhonov regularization[END_REF] and reduces to the resolution of a sequence of optimization problems under equality constraints. The functional is composed of two terms. The first one represents the gap between the optimal element and the overspecified data, the second one the gap between the optimal element and the previous optimal element (regularization term). The equality constraints characterize the equilibrium. So, we obtain at each step an optimal element which is an exact equilibrium and is near to the overprescribed data. In case of compatible data, we have also proved that the sequence limit is the solution of the Cauchy problem. This method was numerically implemented using both the finite element method [START_REF] Cimetière | Solution of the Cauchy problem using iterated Tikhonov regularization[END_REF] and the boundary element method [START_REF] Delvare | An iterative boundary element method for Cauchy inverse problems[END_REF]. We obtained the same results in both situations, accuracy and robutness features especially for the function. In order to improve the reconstruction of the normal derivatives, reduce the extension of the boundary part where overspecified data are given and deblur noisy normal derivatives data, we have introduced [START_REF] Cimetière | Une methode inverse d'ordre un pour les problèmes de complétion de données[END_REF] a higher order method which will be called first order method. This paper is devoted to the numerical implementation of the first order method using BEM. The efficiency of the proposed method, as well as its robustness and ability to deblur noisy data will be highlighted.

The first order method

The Cauchy problem for the Laplace equation

The presentation of the method is carried out for the model problem [START_REF] Andrieux | Solving Cauchy problems by minimizing an energy-like functional[END_REF], where Ω is a two-dimensional bounded domain and ∆ denotes the Laplacian operator. On the part Γ d of the boundary Γ = ∂ Ω , overspecified boundary conditions are given. The aim is to calculate u and its normal derivative denoted by u on

Γ u = Γ \ Γ d .    ∆ u = 0 in Ω u = φ d on Γ d u = ψ d on Γ d (1)
The problem (1) admits a unique solution when the data φ d and ψ d are compatible, but its solution is very sensitive to small data perturbations, since Cauchy inverse problems are ill-posed problems [START_REF] Hadamard | Lectures on Cauchy's problem in Linear Partial Differential Equation[END_REF].

The first order problem

In order to improve the reconstruction of u , we introduce the following first order method:

             ∆ u = 0, ∆ u 1 = 0, ∆ u 2 = 0 in Ω u = u 1 n 1 + u 2 n 2 on Γ Γ (u 1 n 1 + u 2 n 2 )ds = 0 u = φ d , u 1 = φ 1d , u 2 = φ 2d on Γ d u = ψ d , u 1 = ψ 1d , u 2 = ψ 2d on Γ d (2)
where n 1 and n 2 are the outward normal components. The first order method is associated with the problem [START_REF] Bonnet | Boundary Integral Equations Methods in Solids and Fluids[END_REF]. The expressions of the additional data (φ 1d ,φ 2d , ψ 1d and ψ 2d ) are related to φ d and ψ d by (3):

     φ 1d = ψ d n 1 - dφ d ds n 2 , φ 2d = ψ d n 2 + dφ d ds n 1 ψ 1d = - dφ 2d ds , ψ 2d = dφ 1d ds (3)
When the data φ d and ψ d are compatible and the solution u is sufficiently regular, all the data of the problem (2) are also compatible. Then u is the solution of the problem (1) and u 1 and u 2 are respectively the two partial derivatives of u.

Equivalent formulation for the first order problem

We rewrite the problem (2) distinguishing the equilibrium equations ( 4)

∆ u = 0, ∆ u 1 = 0, ∆ u 2 = 0 in Ω Γ (u 1 n 1 + u 2 n 2 )ds = 0 (4)
and the boundary conditions ( 5)

   u = φ d , u 1 = φ 1d , u 2 = φ 2d on Γ d u = ψ d , u 1 = ψ 1d , u 2 = ψ 2d on Γ d u = u 1 n 1 + u 2 n 2 on Γ (5)
Given

Φ d = (φ d , φ 1d , φ 2d , ψ d , ψ 1d , ψ 2d
), an equivalent for- mulation for (2) is given by [START_REF] Delvare | An iterative boundary element method for Cauchy inverse problems[END_REF], where H(Γ ) is a space of traces of functions satisfying (4).

   Find U = (u, u 1 , u 2 , u , u 1 , u 2 ) ∈ H(Γ ) such that : U = Φ d on Γ d u = u 1 n 1 + u 2 n 2 on Γ (6)
The problem (2) and its reformulation (6) are ill-posed problems in Hadamard's sense.

The iterative regularizing algorithm

In order to solve the ill-posed problem (6), we introduce the iterative regularizing algorithm [START_REF] Eilers | Flexible smoothing with B-splines and penalties[END_REF]:

             Let us consider c > 0 and U 0 ∈ H(Γ ) Find U k+1 ∈ H(Γ ) such that : J k c (U k+1 ) ≤ J k c (V ) ∀V ∈ H(Γ ) with : J k c (V ) = ||v -(v 1 n 1 + v 2 n 2 )|| 2 -1 2 ,Γ +||V -Φ d || 2 Γ d + c||V -U k || 2 Γ (7)
It is based on the same idea as the algorithm proposed in [START_REF] Delvare | An iterative boundary element method for Cauchy inverse problems[END_REF].

The ill-posed problem ( 2) is replaced by a sequence of wellposed optimization problems in H(Γ ). In order to remove instabilities, the last term is introduced. It is a control term on the whole boundary Γ . The optimal elements obtained at each step are solutions of the equilibrium equations ( 4).

When the data (φ d , ψ d ) are compatible, it is also proved that this iterative process converges for all U 0 in H(Γ ) and for all c > 0. The first component of the limit is associated with the solution of problem (1). The proof is similar to the one carried out in [START_REF] Delvare | An iterative boundary element method for Cauchy inverse problems[END_REF].

BEM implementation and numerical simulations

3.1 BEM discretization of the iterative algorithm

The boundary element method

The boundary element method [START_REF] Bonnet | Boundary Integral Equations Methods in Solids and Fluids[END_REF] is used to discretize the iterative algorithm [START_REF] Eilers | Flexible smoothing with B-splines and penalties[END_REF] and specially to obtain a discrete representation of the H(Γ ) space. By using the usual fundamental solution of the Laplace equation and Green's identities then the governing partial differential equation ∆ u = 0 is transformed into the following integral equation [START_REF] Bonnet | Boundary Integral Equations Methods in Solids and Fluids[END_REF]:

η(x)u(x) = Γ (G(x, y) ∂ u(y) ∂ n -u(y) ∂ G(x, y) ∂ n )dΓ y ( 8
)
where

η(x) = 1 if x ∈ Ω and η(x) = 1 2 if x ∈ Γ (smooth)
, and G is the fundamental solution of the Laplace equation which in two dimensions is given by

G(x, y) = - 1 2π ln(r), r 2 = (x 1 -y 1 ) 2 + (x 2 -y 2 ) 2 (9)
In practice the boundary integral equation cannot be solved analytically and thus some form of numerical approximation is needed. Generically, if the boundary Γ is discretized into N boundary elements and the boundary integral equation is applied at the midpoint of each element, then a system of N linear algebraic equations is obtained in the form

AU + BU = 0 ( 10
)
where A and B are matrices which only depend on the geometry of the boundary and the vectors U and U are respectively the discretized values of the function and its normal derivative, which are assumed to be constant over each boundary element and take their values at the midpoint of each element. Expression (10) represents a system of N linear algebraic equations with 2N unknowns.

Discrete representation of the space H(Γ )

The boundary element method leads to the definition of H h (Γ ), the H(Γ ) discretization. H h (Γ ) reads as follows:

H h (Γ ) =          Ψ = (U,U 1 ,U 2 ,U ,U 1 ,U 2 ) ∈ R 6N : AU + BU = 0 AU 1 + BU 1 = 0 AU 2 + BU 2 = 0 N T 1 U 1 + N T 2 U 2 = 0          (11)
where U 1 ,U 2 , U 1 and U 2 are respectively the discretized values of u 1 and u 2 and their normal derivatives and where N 1 ,N 2 are the discretized values of the unit normal components over each boundary element. Expression (11) represents a system of 3N + 1 linear algebraic equations with 6N unknowns.

Discretization of the iterative algorithm

In the discrete case, the equivalence of the norms in finite dimension makes it possible to replace the H -1 2 (Γ ) norm by the L 2 (Γ ) norm. We also take in all situations U 0 = 0 as initialization. The parameter c value influences only the convergence rate. For non-noisy data, one takes a value c = 10 -4 and for noisy data, a value c = 1. In all cases, the algorithm converges in less than 100 iterations.

Description of one step

A discrete formulation of the continuous iterative process (7) reads as follows:

Find Ψ k+1 ∈ H h (Γ ) such that : J k hc (Ψ k+1 ) ≤ J k hc (Ψ ) ∀Ψ ∈ H h (Γ ) (12)
This optimization problem is equivalent to the following optimization process under equality constraints:

                 Find Ψ k+1 ∈ R 6N such that : J k hc (Ψ k+1 ) ≤ J k hc (Ψ ) ∀Ψ ∈ R 6N under the equality constraints : AU + BU = 0 AU 1 + BU 1 = 0 AU 2 + BU 2 = 0 N T 1 U 1 + N T 2 U 2 = 0 (13)
We introduce a 3N +1 vector of Lagrange multipliers to take into account the 3N + 1 equality constraints. Each iteration in the algorithm needs to solve a system of 9N + 1 linear algebraic equations with 9N + 1 unknowns.

Some comments and remarks

The obtained matrix is symmetric. It is the same for each step of the iterative process and so needs to be computed and decomposed by a Crout factorization only once. From the second step of the algorithm, only actualization of the second member is needed. The computational time for the first step of the algorithm (computation of different matrices, Crout decomposition) is equivalent to the time needed for three hundred additional iterations.

The stopping test is based on a comparison between two successive optimal elements.

Evaluation of additional data

The evaluation of additional data by the use of the relations (3) needs differentiations of the data φ d and ψ d . First it is proceeded to an interpolation by B-splines of these data [START_REF] Eilers | Flexible smoothing with B-splines and penalties[END_REF].

Then the differentiations are obtained by the analytical differentiation of the B-splines. For further details, the reader can see the description of the computational algorithm (figure 1). This is a way to obtain additional data but we have already used quadratic regression to compute differentiations and obtained similar results [START_REF] Cimetière | Une methode inverse d'ordre un pour les problèmes de complétion de données[END_REF].

The input data for the inversion algorithm are on one hand the data φ d and ψ d and on the other hand the generated ad- ditional data φ 1d , φ 2d , ψ 1d and ψ 2d . It can be noticed that the interpolated functions φ d and ψ d of the data are only used to evaluate differentiations of the given data.

Numerical simulations

This section is devoted to compare numerical results obtained by the proposed method (first order method) to whose obtained by the inverse method introduced (so called previous method) in [START_REF] Delvare | An iterative boundary element method for Cauchy inverse problems[END_REF].

The function u to be recovered is defined by: u(x, y) = cos(x)cosh(y) + sin(x)sinh(y).

Smooth boundary

In this section, numerical simulations are performed on the domain Ω which is the disk with radius 1 2 centered at point ( 12 , 1 2 ). In the whole section, the boundary mesh is a regular one.

Behaviour with respect to boundary mesh

In this section, half of the boundary carries prescribed data. Figure 2 (respectively Figure 3) displays L 2 errors on u (respectively on its normal derivative u ) obtained by our previous method [START_REF] Delvare | An iterative boundary element method for Cauchy inverse problems[END_REF] and by the first order method, with respect to the number of boundary elements. We can notice that the first order method is stable with respect to boundary mesh as the previous method. But the first order method

Behaviour with respect to Γ d boundary extension

In this section, we introduce the ratio ρ between the measures of Γ d and Γ . The number of boundary elements is fixed to 180. Figure 4 (respectively Figure 5) displays L 2 errors on u (respectively on its normal derivative u ) obtained by our previous method [START_REF] Delvare | An iterative boundary element method for Cauchy inverse problems[END_REF] and by the first order method with respect to the ratio ρ. We can notice that the first order method is again more accurate than the previous method. As a consequence, the Γ d extension can be reduced by the use of the first order method. Figure 6 (respectively Figure 7) gives the function (respectively its normal derivative) reconstructions for ρ = 1 4 using the previous method and the first order one. We can notice that the normal derivative reconstruction (Figure 7) is improved by the first order method. This reconstruction is as accurate as the function reconstruction (Figure 6) obtained by the previous method. Behaviour with respect to noisy data In this section, we discuss the numerical stability of the algorithm to noisy data. Half of the boundary carries prescribed data, the mesh is a 180 elements regular one. For each case, the noise is a random one whose amplitude is equal to 10% of the data maximum value on the boundary. Firstly, we look at the algorithm stability when the data φ d is noisy. Figure 8 gives the noisy data used by the algorithm. Figures 9 and10 give the function and its normal derivative reconstruction. We can notice that the algorithm by recomputing informations on Γ d is able to deblur the noisy data φ d . It also gives accurate reconstructions and is stable. Secondly, we look at the algorithm stability when the data ψ d is noisy. Figure 11 gives the noisy data used by the al- gorithm. Figures 12 and13 give the function and its normal derivative reconstruction. We can notice that the first order method, by opposition to the previous one [START_REF] Delvare | An iterative boundary element method for Cauchy inverse problems[END_REF], is also able to deblur the noisy data ψ d . The deblurring effect of the first order method does not rely on the use of the interpolation by B-Splines for generating the additional data. The justification comes from the following points:

-The zero order method (previous method) already deblurred the field data φ d but not the flux data ψ d , -The first order method behaves roughly like three zero order methods for the evaluation of u and its two partial derivatives u 1 and u 2 . So it deblurs φ d , φ 1d and φ 2d data and therefore the flux data ψ d .

Non smooth boundary

Numerical simulations are now performed on a square domain (Ω =]0, 1[×]0, 1[). The Γ d boundary part is composed of two sides (y = 0 and x = 1) (Figure 14). The boundary mesh is a regular one with 120 elements on each side. Comparison with other methods This example was already studied by our previous method [START_REF] Delvare | An iterative boundary element method for Cauchy inverse problems[END_REF], a Tikhonov like method [START_REF] Hayashi | Direct method of solution for general boundary value problem of the Laplace equation[END_REF] and an iterative method [16] which all gave similar results. Figure 15 (respectively Figure 16) compares function (respectively normal derivatives) reconstructions obtained with one of these methods [START_REF] Delvare | An iterative boundary element method for Cauchy inverse problems[END_REF] and with the first order method. We can notice improved reconstructions by the first order method.

Behaviour with respect to noisy data φ d and ψ d We look at the algorithm stability when the φ d and ψ d data are both noisy. Figure 17 and Figure 18 give the function and its normal derivative reconstruction in this case and show the efficiency and the stability of this new approach. In particular, it is stable with respect to perturbations on the data (1% noise level).

Conclusion

A new method has been introduced for Cauchy inverse problems. This method has been carried out for a model problem.

It has introduced a higher order problem which has been numerically implemented using BEM.

Although this new method requires a computational time approximately nine times greater than that of the previous method, the presented numerical simulations prove the efficiency, the accuracy and the robustness of the method as well as the following properties:

-It reconstructs the function like its normal derivative with an increased precision, -It allows us to reduce the extension of the boundary part Γ d where prescribed data are given, -It deblurs the two boundary data φ d and ψ d when they are noisy.

This method can be extended without difficulty to 3D problems as well as to other problems, like elasticity problems.

Fig. 1

 1 Fig. 1 Computational algorithm for additionnal data evaluations

Fig. 2 Fig. 3

 23 Fig.2Error L 2 (Γ ) (in %) on the function with respect to the boundary mesh

Fig. 4 Fig. 5 Fig. 6 4 -Fig. 7 4 Fig. 8 φ

 4564748 Fig.[START_REF] Cimetière | Solution of the Cauchy problem using iterated Tikhonov regularization[END_REF] Error L 2 (Γ ) (in %) on the function with respect to the ratio ρ of given data

Fig. 9 uFig. 10 uFig. 11 ψFig. 12 uFig. 13 uFig. 15 uFig. 16 u

 9101112131516 Fig. 9 u first order method reconstruction (φ d noisy data (10%))

Fig. 17 uFig. 18 u

 1718 Fig. 17 u first method reconstruction (noisy data φ d and ψ d (1%))

Acknowledgements

The authors would like to thank reviewers for their relevant remarks and comments.


Franck Delvare

Alain Cimetière

Jean-Luc Hanus

Patrice Bailly

Alain Cimeti Ère

An iterative method for the Cauchy problem in linear elasticity with fading regularization effect

Keywords: Cauchy problem, Inverse problem, Data completion, Linear elasticity, Regularization, Boundary conditions

Introduction

Inverse problems arise in many engineering fields such as for instance in thermal sciences, electrostatics, solid and fluid mechanics. Kubo [START_REF] Kubo | Inverse problems related to the mechanics and fracture of solids and structures[END_REF] has defined inverse problems in opposition to direct problems. Direct problems are problems for which the geometry of the domain, the equilibrium equations, the constitutive equations, the boundary conditions on the whole boundary of the domain and the initial conditions are all given. For Kubo, inverse problems can be characterized by the lack of at least one of these pieces of information. According to this definition, many mechanical problems, for example, identification of material parameters, identification of unknown boundaries (such as contact zones, cavities or cracks), identification of residual stresses, identification of initial boundary conditions, identification of inaccessible boundary conditions can be considered as inverse problems. Some examples arising from fracture mechanics can be found in [START_REF] Bui | Fracture mechanics. Inverse problems and Solutions[END_REF] and some relating to elasticity problems in [START_REF] Bonnet | Inverse problems in elasticity[END_REF]. In a mathematical sense, direct problems can be considered as well-posed problems. In linear cases, these problems have a unique solution which is stable (continuously dependent on the data). Conversely, inverse problems are generally ill-posed problems in the Hadamard sense [START_REF] Hadamard | Lectures on Cauchy's problem in linear partial differential equations[END_REF], since existence or uniqueness or continuous dependence on the data of their solutions may not be ensured. This paper examines an inverse boundary value problem in linear elasticity. It consists in recovering missing data (both displacement and traction vectors are unknown) on some part of the boundary of a domain from overspecified data (both displacement and traction vectors are given) on the other part. In this case, the equilibrium equations, the constitutive equations, the domain and its boundary are known. This problem is namely known as a Cauchy problem and is a data completion problem. In order to solve Cauchy problems for elliptic equations, many regularization methods have been introduced. The references [START_REF] Andrieux | Solving Cauchy problems by minimizing an energy-like functional[END_REF][START_REF] Bourgeois | A mixed formulation of quasi-reversibility to solve the Cauchy problem for the Laplace equation[END_REF][START_REF] Cimetiere | Une methode inverse avec regularisation evanescente[END_REF][START_REF] Cimetiere | Solution of the Cauchy problem using iterated Tikhonov regularisation[END_REF][START_REF] Cimetiere | Une methode inverse d'ordre un pour les problèmes de complétion de données[END_REF][START_REF] Delvare | An iterative boundary element method for Cauchy inverse problems[END_REF][START_REF] Delvare | A first order method for the Cauchy problem for the Laplace equation using BEM[END_REF][START_REF] Jourhmane | Relaxation procedures for an iterative algorithm for solving the cauchy problem for the laplace equation[END_REF][START_REF] Klibanov | A computational quasi-reversibility method for Cauchy problems for Laplace's equation[END_REF][START_REF] Kozlov | An iterative method for solving the Cauchy problem for elliptic equations[END_REF][START_REF] Lattes | Methode de quasi-reversibilite et applications[END_REF][START_REF] Lesnic | An iterative boundary element method for solving the Cauchy problem for the Laplace equations[END_REF] propose different methods of solving the Cauchy problem for the Laplace equation. References [START_REF] Andrieux | An energy error-based method for the resolution of the Cauchy problem in 3D linear elasticity[END_REF][START_REF] Baranger | An optimization approach for the Cauchy problem in linear elasticity[END_REF][START_REF] Delvare | Completion de donnees par methode inverse en elasticite lineaire lineaire[END_REF][START_REF] Koya | An inverse problem in elasticity with partially overspecified boundary conditions. II. Numerical details[END_REF][START_REF] Kozlov | An iterative method for solving the Cauchy problem for elliptic equations[END_REF][START_REF] Maniatty | Finite element analysis of some elasticity problems[END_REF][START_REF] Marin | Boundary element method for the Cauchy problem in linear elasticity[END_REF][START_REF] Marin | Regularized boundary element solution for an inverse boundary value problem in linear elasticity[END_REF][START_REF] Marin | Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition[END_REF][START_REF] Marin | Boundary element regularization methods for solving the Cauchy problem in linear elasticity[END_REF][START_REF] Marin | Conjugate gradient-boundary element method for the Cauchy problem in elasticity[END_REF][START_REF] Marin | BEM first-order regularization method in linear elasticity for boundary identification[END_REF][START_REF] Marin | The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity[END_REF][START_REF] Marin | A meshless method for solving the Cauchy problem in threedimensional elastostatics[END_REF][START_REF] Marin | Boundary element-Landweber method for the Cauchy problem in linear elasticity[END_REF][START_REF] Marin | The minimal error method for the Cauchy problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity[END_REF][START_REF] Schnur | Finite element solution of two-dimensional elastic problems using spatial smoothing[END_REF][START_REF] Yeih | An inverse problem in elasticity with partially overspecified boundary conditions. I. Theoretical approach[END_REF][START_REF] Zabaras | Spatially regularized solution of inverse elasticity problems using the BEM[END_REF] deal with the Cauchy problem in linear elasticity. These methods can be classified as Tikhonov type methods [START_REF] Engl | Regularization of Inverse Problems[END_REF][START_REF] Koya | An inverse problem in elasticity with partially overspecified boundary conditions. II. Numerical details[END_REF][START_REF] Maniatty | Finite element analysis of some elasticity problems[END_REF][START_REF] Marin | Regularized boundary element solution for an inverse boundary value problem in linear elasticity[END_REF][START_REF] Marin | BEM first-order regularization method in linear elasticity for boundary identification[END_REF][START_REF] Marin | The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity[END_REF][START_REF] Marin | A meshless method for solving the Cauchy problem in threedimensional elastostatics[END_REF][START_REF] Schnur | Finite element solution of two-dimensional elastic problems using spatial smoothing[END_REF][START_REF] Tikhonov | Solution of ill-posed problems[END_REF][START_REF] Yeih | An inverse problem in elasticity with partially overspecified boundary conditions. I. Theoretical approach[END_REF][START_REF] Zabaras | Spatially regularized solution of inverse elasticity problems using the BEM[END_REF], quasi-reversibility type methods [START_REF] Bourgeois | A mixed formulation of quasi-reversibility to solve the Cauchy problem for the Laplace equation[END_REF][START_REF] Klibanov | A computational quasi-reversibility method for Cauchy problems for Laplace's equation[END_REF][START_REF] Lattes | Methode de quasi-reversibilite et applications[END_REF], iterative methods [START_REF] Andrieux | Solving Cauchy problems by minimizing an energy-like functional[END_REF][START_REF] Andrieux | An energy error-based method for the resolution of the Cauchy problem in 3D linear elasticity[END_REF][START_REF] Baranger | An optimization approach for the Cauchy problem in linear elasticity[END_REF][START_REF] Cimetiere | Une methode inverse avec regularisation evanescente[END_REF][START_REF] Cimetiere | Solution of the Cauchy problem using iterated Tikhonov regularisation[END_REF][START_REF] Cimetiere | Une methode inverse d'ordre un pour les problèmes de complétion de données[END_REF][START_REF] Delvare | An iterative boundary element method for Cauchy inverse problems[END_REF][START_REF] Delvare | Completion de donnees par methode inverse en elasticite lineaire lineaire[END_REF][START_REF] Delvare | A first order method for the Cauchy problem for the Laplace equation using BEM[END_REF][START_REF] Engl | Regularization of Inverse Problems[END_REF][START_REF] Jourhmane | Relaxation procedures for an iterative algorithm for solving the cauchy problem for the laplace equation[END_REF][START_REF] Kozlov | An iterative method for solving the Cauchy problem for elliptic equations[END_REF][START_REF] Lesnic | An iterative boundary element method for solving the Cauchy problem for the Laplace equations[END_REF][START_REF] Marin | Boundary element method for the Cauchy problem in linear elasticity[END_REF][START_REF] Marin | Conjugate gradient-boundary element method for the Cauchy problem in elasticity[END_REF][START_REF] Marin | Boundary element-Landweber method for the Cauchy problem in linear elasticity[END_REF][START_REF] Marin | The minimal error method for the Cauchy problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity[END_REF],... Quasi reversibility methods and Tikhonov regularization methods present the advantage of leading to well posed problems after modifying the equilibrium equations. Some iterative methods are based on the use of a sequence of well-posed problems and others on the minimization of an energy-like functional. Numerical algorithms are implemented using different numerical methods, such as the finite element method (FEM) [START_REF] Andrieux | Solving Cauchy problems by minimizing an energy-like functional[END_REF][START_REF] Andrieux | An energy error-based method for the resolution of the Cauchy problem in 3D linear elasticity[END_REF][START_REF] Baranger | An optimization approach for the Cauchy problem in linear elasticity[END_REF][START_REF] Bourgeois | A mixed formulation of quasi-reversibility to solve the Cauchy problem for the Laplace equation[END_REF][START_REF] Cimetiere | Une methode inverse avec regularisation evanescente[END_REF][START_REF] Cimetiere | Solution of the Cauchy problem using iterated Tikhonov regularisation[END_REF][START_REF] Cimetiere | Une methode inverse d'ordre un pour les problèmes de complétion de données[END_REF][START_REF] Delvare | Completion de donnees par methode inverse en elasticite lineaire lineaire[END_REF][START_REF] Maniatty | Finite element analysis of some elasticity problems[END_REF][START_REF] Schnur | Finite element solution of two-dimensional elastic problems using spatial smoothing[END_REF], the boundary element method (BEM) [START_REF] Delvare | An iterative boundary element method for Cauchy inverse problems[END_REF][START_REF] Delvare | A first order method for the Cauchy problem for the Laplace equation using BEM[END_REF][START_REF] Jourhmane | Relaxation procedures for an iterative algorithm for solving the cauchy problem for the laplace equation[END_REF][START_REF] Koya | An inverse problem in elasticity with partially overspecified boundary conditions. II. Numerical details[END_REF][START_REF] Lesnic | An iterative boundary element method for solving the Cauchy problem for the Laplace equations[END_REF][START_REF] Kozlov | An iterative method for solving the Cauchy problem for elliptic equations[END_REF][START_REF] Marin | Boundary element method for the Cauchy problem in linear elasticity[END_REF][START_REF] Marin | Regularized boundary element solution for an inverse boundary value problem in linear elasticity[END_REF][START_REF] Marin | Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition[END_REF][START_REF] Marin | Boundary element regularization methods for solving the Cauchy problem in linear elasticity[END_REF][START_REF] Marin | Conjugate gradient-boundary element method for the Cauchy problem in elasticity[END_REF][START_REF] Marin | BEM first-order regularization method in linear elasticity for boundary identification[END_REF][START_REF] Marin | Boundary element-Landweber method for the Cauchy problem in linear elasticity[END_REF][START_REF] Marin | The minimal error method for the Cauchy problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity[END_REF][START_REF] Yeih | An inverse problem in elasticity with partially overspecified boundary conditions. I. Theoretical approach[END_REF][START_REF] Zabaras | Spatially regularized solution of inverse elasticity problems using the BEM[END_REF], the finite difference method [START_REF] Klibanov | A computational quasi-reversibility method for Cauchy problems for Laplace's equation[END_REF] or meshless methods [START_REF] Marin | The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity[END_REF][START_REF] Marin | A meshless method for solving the Cauchy problem in threedimensional elastostatics[END_REF]. Some papers present comparisons between different numerical methods [START_REF] Cimetiere | An inversion method for harmonic functions reconstruction[END_REF][START_REF] Marin | Boundary element regularization methods for solving the Cauchy problem in linear elasticity[END_REF][START_REF] Marin | The minimal error method for the Cauchy problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity[END_REF]. In previous works [START_REF] Cimetiere | Une methode inverse avec regularisation evanescente[END_REF][START_REF] Cimetiere | Solution of the Cauchy problem using iterated Tikhonov regularisation[END_REF][START_REF] Delvare | An iterative boundary element method for Cauchy inverse problems[END_REF] a somewhat different resolution approach was first introduced in [START_REF] Cimetiere | Une methode inverse avec regularisation evanescente[END_REF] for the Cauchy problem associated with the Laplace equation. This approach reduced the resolution of the Cauchy problem to the resolution of a sequence of optimization problems under equality constraints. The functional is composed of two terms. At each step of the resolution, the first term gives the gap between the optimal element and the overspecified boundary data (relaxation term), the second one the gap between the optimal element and the previous optimal element (regularization term). The equality constraints are the equilibrium equations. So, at each step an optimal element is obtained which is an exact solution to the equilibrium equations and is nearer to the overspecified data than the previous optimal element calculated. In the case of compatible data, it was also proved that the sequence converges and its limit is the solution to the Cauchy problem. The additional regularization term tends to zero as iterations continue. It is the reason why this method is called the evanescent regularization method. In this paper, this inverse technique is extended to the Cauchy problem in linear elasticity. The next section is devoted to the formulation of the Cauchy problem in linear elasticity. The third section describes the iterative inverse method and the fourth section is devoted to its numerical implementation using the finite element method. In the fifth section, several numerical simulations are presented.

The Cauchy problem in linear elasticity

Let us consider a linear elastic material which occupies an open set Ω in IR 2 or IR 3 , with a smooth boundary Γ, assuming that the boundary is divided in two parts Γ d and Γ u , where

Γ d ∪ Γ u = Γ and Γ d ∩ Γ u = Ø.
The subscript d denotes the data and the subscript u denotes the unknowns. With no body force, the equilibrium equations are given by:

∂σ i j (x) ∂x j = 0 x ∈ Ω (1)
where σ i j is the Cauchy stress tensor. This tensor is energy conjugated to the infinitesimal strain tensor ε i j by the following constitutive equations:

σ i j = 2 µ ε i j + λ δ i j ε kk (2)
where λ and µ are the Lamé constants. These are related to Young's modulus E and Poisson's ratio ν as:

λ = νE (1 + ν) (1 -2 ν) µ = E 2 (1 + ν)
The linear strain tensor conponents ε i j , as function of the displacement gradients, are given by:

ε i j = 1 2 ∂u j ∂x i + ∂u i ∂x j (3)
By substituting Hooke's law (2) into the governing equations (1), the Lamé equations are obtained:

L(u) i = E 2(1 + ν) ∂ 2 u i ∂x 2 j + E 2(1 + ν)(1 -2ν) ∂ 2 u j ∂x i ∂x j = 0 (4)
At a point x ∈ Γ, n(x) the outward unit normal vector is defined as well as the traction vector p(x) whose components are defined by:

p i (x) = p i u(x) = σ i j u(x) n j (x) x ∈ Γ
It is assumed that both the displacement vector and the traction vector are given or known on the part of boundary Γ d but no condition is prescribed on the remaining part Γ u :

u(x) = φ d x ∈ Γ d p(x) = ψ d x ∈ Γ d (5)
where φ d and ψ d are prescribed vector functions. The Lamé (or Navier) system (4) and the boundary conditions (5) lead to the formulation of the Cauchy problem in linear elasticity:

           L(u) = 0 x ∈ Ω u(x) = φ d x ∈ Γ d p(x) = ψ d x ∈ Γ d ( 6
)
This problem is difficult to solve, since it is ill-posed. When it admits a solution, its solution is unique [START_REF] Yeih | An inverse problem in elasticity with partially overspecified boundary conditions. I. Theoretical approach[END_REF], but it is known to be very sensitive [START_REF] Hadamard | Lectures on Cauchy's problem in linear partial differential equations[END_REF] to small perturbations on boundary conditions (5).

The evanescent regularization method

Let us introduce the space H(Ω) of solutions of the equilibrium equation ( 4):

H(Ω) = v satisfying L(v) = 0 in Ω
Next, let us denote H(Γ) the space composed of couples of restrictions on Γ of elements v in H(Ω) and of their associated tractions p(v). An equivalent formulation of problem (6) reads:

Find U = (u, p) ∈ H(Γ) such as : U = Φ d on Γ d ( 7
)
where

Φ d =(φ d , ψ d ).
The problem [START_REF] Castem | Code de calcul pour l'analyse de structures par la méthode des éléments finis[END_REF] is also ill-posed even if it admits a unique solution. So then an iterative regularizing method is introduced to solve it. This method is a generalization of the inverse technique introduced by Cimetière et al [START_REF] Cimetiere | Une methode inverse avec regularisation evanescente[END_REF][START_REF] Cimetiere | Solution of the Cauchy problem using iterated Tikhonov regularisation[END_REF] to solve the Cauchy problem for the Laplace equation. It can be considered as an iterative Tikhonov-type method. Given c > 0 and U 0 ∈ H(Γ) the iterative algorithm reads:

Find U k+1 ∈ H(Γ) such as :

J k c (U k+1 ) ≤ J k c (V) ∀V ∈ H(Γ) with J k c (V) = V -Φ d 2 Γ d + c V -U k 2 Γ ( 8
)
where the norms are defined by:

V 2 Γ d = Γ d v 2 ds + 1 E 2 Γ d p 2 ds V 2 Γ = Γ v 2 ds + 1 E 2 Γ p 2 ds
In this iterative process, the equilibrium equation ( 4) is taken into account exactly since at each step the search for the optimal element is performed in space H(Γ). The functional is composed of two terms which play different roles. The first one acts only on Γ d and represents the gap between the optimal element and the overspecified boundary data. It relaxes the overspecified data which can be possibly blurred by measurement noises (relaxation term). The second one acts on the whole boundary Γ and not only on the boundary Γ u where the boundary conditions are to be completed. This term is a regularization term and controls the distance between the new optimal element and the previous optimal element. This term tends to zero as iterations go on. So, at each step the optimal element obtained is an exact solution of the equilibrium equation ( 4) and is near to the overspecified data Φ d =(φ d , ψ d ). This unique optimal element U k+1 is characterized by:

U k+1 -Φ d , V Γ d + c U k+1 -U k , V Γ = 0 ∀ V ∈ H(Γ) (9)
T: Convergence of the sequence Let Φ d be compatible data associated with the compatible pair U e ∈ H(Γ). Then the sequence produced by the iterative scheme ( 8) strongly converges to Φ d on Γ d and weakly to U e on Γ.

The proof of this theorem is based on the following lemma.

L:

For all n ∈ IN, the sequence (U k ) corresponding of optimal elements verifies:

U n+1 -U e 2 Γ + n k=0 U k+1 -U k 2 Γ + 2 c n k=0 U k+1 -Φ d 2 Γ d = U 0 -U e 2 Γ ( 10
)
where U e is the solution of Cauchy problem [START_REF] Bui | Fracture mechanics. Inverse problems and Solutions[END_REF] and

Φ d = U e | Γ d .
The proof of the lemma and consequently those of the theorem is similar to that established for the algorithm introduced in [START_REF] Cimetiere | Une methode inverse avec regularisation evanescente[END_REF] to solve the Cauchy problem associated with the Laplace equation. This proof is valid for all c > 0. The c value only influences the convergence rate of the algorithm.

The implementation using the Finite Element Method

The implementation of the inverse method using the finite element method is presented in 2D situations.

Discrete solutions space

The first issue in this section is to discretize space H(Γ). Our main concern has been to make use of any ordinary finite element code, not to work out a specific one. As a matter of fact, computations were run using Cast3M 1 [START_REF] Castem | Code de calcul pour l'analyse de structures par la méthode des éléments finis[END_REF] and piecewise linear finite elements, which means a piecewise constant approximation for the traction vector. Let us now discretize the domain Ω, 1 The FE code Cast3M is developed by the Department of Mechanics and Technology (DMT) of the French Atomic Energy Agency (CEA -DEN/DM2S/SEMT), http://www-cast3m.cea.fr h being the discretization parameter standing for the element size, leading to n nodes and n elements on the boundary, and m nodes inside the domain. Let V h be the space of continuous piecewise linear functions with respect to the mesh, and let us define V h (Γ) and W h (Γ) as the space of continuous piecewise linear functions and the space of piecewise constant functions on the boundary. Traces of functions belonging to V h (Ω) span the space V h (Γ), whereas the associated traction vectors belong to the space W h (Γ) of piecewise constant functions. Defining U and P as the 2n vectors standing respectively for the 2n nodal values of u and the 2n discrete values of p on the boundary, and U * the 2m-vector of internal nodal values of u, the discrete equilibrium equations read as:

A ii A T ei A ei A ee U * U = 0 -BP (11)
A ii is the stiffness matrix corresponding to the Dirichlet problem and thus is invertible. Expressing the internal unknowns U * in terms of the boundary ones U, i.e performing a condensation, equation ( 11) reduces to:

( A ee -A ei A -1 ii A T ei ) U + B P = 0 (12)
The matrix form of ( 12) reads:

A B U P = 0 (13)
The finite element method leads to the definition of the following discrete compatible pairs space which reads:

H h (Γ) =        (U, P) ∈ IR 2n × IR 2n such that E h (U, P) = AU + BP = 0        (14)
where E h denotes a linear operator mapping IR 2n × IR 2n onto IR 2n .

Discretization of the (k + 1) th iteration

Given now c > 0 and (U k , P k ) ∈ H h (Γ), iteration (k + 1) of the discretized iterative algorithm reads as follows: 15) is a minimization problem in R 2n × R 2n under the 2n equality constraints expressed by [START_REF] Delvare | Completion de donnees par methode inverse en elasticite lineaire lineaire[END_REF]. Its solution is given by: Initializing with (U 0 = 0 , P 0 = 0) Find U k+1 , P k+1 , λ k+1 ∈ IR 2n × IR 2n × IR 2n such that : ∇J k c U k+1 , P k+1 + λ k+1 T ∇E h U k+1 , P k+1 = 0 E h U k+1 , P k+1 = 0 (16) where λ k+1 is a 2n-vector of Lagrange multipliers introduced to take the equality constraints (13) into account. Each iteration in the iterative algorithm needs to solve a system of 6n linear equations with 6n unknowns. The matrix of this linear system is independant of the iterations and needs to be computed only once. For this reason, a direct algorithm (the Crout factorization) has been preferred to iterative methods. The factorization, which is obtained at the first step, is also used at each following step.

Find (U k+1 , P k+1 ) ∈ IR 2n × IR 2n such that : J k c U k+1 , P k+1 ≤ J k c (V, Q) ∀(V, Q) ∈ IR 2n × IR 2n under the 2n scalar equality constraints E h (V, Q) = 0 (15) Problem (
In the case of compatible data, the proof of the convergence of the discrete algorithm is similar to that established for the corresponding algorithm used to solve the Cauchy problem for the Laplace equation [START_REF] Cimetiere | Une methode inverse avec regularisation evanescente[END_REF]. This proof is valid for all c > 0 and the c value only influences the convergence rate of the algorithm.

Numerical results

The purpose of this section is to present numerical results obtained with the method introduced. Firstly, different cases are considered for which an analytical solution u an is known. These cases have been previously studied, among others, by [START_REF] Marin | Boundary element method for the Cauchy problem in linear elasticity[END_REF][START_REF] Marin | Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition[END_REF][START_REF] Marin | The minimal error method for the Cauchy problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity[END_REF]. The following control quantities are used to estimate the accuracy of the method and to determine a stopping criteria for the iterative process:

• the norm of the error made on u:

u error = Γ U -u an 2 ds Γ u an 2 ds
• the norm of the error made on p:

p error = Γ P -p an 2 ds Γ p an 2 ds
• the first term (relaxation term) of the functional

J Γ d (U k+1 ) = U k+1 -Φ d 2 Γ d
• the second term (regularization term) of the functional

J Γ (U k+1 ) = c U k+1 -U k 2 Γ • and the value J(U k+1 ) = J Γ d (U k+1 ) + J Γ (U k+1
) of the functional for the optimal element U k+1 .

Three properties of the functional terms in the minimizing sequence can be easily established without the assumption that the data Φ d is compatible:

• The first term of the functional J Γ d (U k ) is monotonically decreasing :

U k+1 -Φ d 2 Γ d ≤ U k -Φ d 2 Γ d ( 17
)
• The second one J Γ (U k ) is monotonically decreasing as soon as c > 0:

U k+1 -U k 2 Γ ≤ U k -U k-1 2 Γ ( 18
)
• The sequence defined by the values of the functional J for each optimal element U k is also monotonically decreasing as soon as c > 0:

J(U k+1 ) ≤ J(U k ). (19)
In the last example, the compression of an infinite cylinder between two rigid planes is investigated and shows that the iterative inverse method allows us to reconstruct the contact zones and the contact pressure distribution from the knowledge of the displacement on a part of the free force boundary.

The procedure used during the numerical simulations is as follows:

• The meshing of the boundary is made using SEG2 elements. The SEG2 element is a finite element with two nodes which leads to a linear interpolation of the displacements. This induces a piecewise constant interpolation of the traction vector components.

• The user specifies the meshing of the boundary specifying the number and the distribution of the finite elements on each part of the boundary.

• Then, the mesh of the entire domain is generated automatically by a routine included in the software Cast3m [START_REF] Castem | Code de calcul pour l'analyse de structures par la méthode des éléments finis[END_REF]. This mesh is constituted of 4-node quadrilaterals and 3node triangles.

• The computation and the assembly of the stiffness matrix corresponding to the domain is performed thanks to the standard routines of Cast3m software [START_REF] Castem | Code de calcul pour l'analyse de structures par la méthode des éléments finis[END_REF].

• This software creates a superelement based on the boundary and computes the corresponding stiffness matrix. This leads to the condensed stiffness matrix A.

• The stiffness matrix A is then used by the specific code that implements the inverse method introduced.

Note that the inverse method only involves values on the boundary. Therefore, for the different numerical tests only the discretization of the boundary Γ will be specified. Moreover, note that all numerical computations have been performed on a machine with a 2.20GHz Intel Core T M 2 Duo processor T7500.

Hollow cylinder under internal and external uniform pressures: Reconstruction on the inner boundary

A two-dimensional isotropic linear elastic medium in a plane stress state characterized by the material constants E = 97.9 GPa and ν = 0.34 corresponding to a copper alloy is studied. The annular domain Ω is defined by:

Ω = x = (x 1 , x 2 ) / r 2 i < x 2 1 + x 2 2 < r 2 0 with r i = 1 m and r 0 = 4 m.
The boundary part Γ d is defined by:

Γ d = x = (x 1 , x 2 ) ∈ Γ/x 2 1 + x 2 2 = r 2 0
Γ d is discretized using N finite elements SEG2. All finite elements have the same length and the nodes are uniformly distributed. The boundary part Γ u is defined by:

Γ u = x = (x 1 , x 2 ) ∈ Γ/x 2 1 + x 2 2 = r 2 i
and discretized using a regular mesh with N finite elements SEG2. It is used to find the displacement and the pressure on Γ u , from the knowledge of the displacement and the pressure on Γ d . The data are built using the analytical solution for the displacements:

u an j (x 1 , x 2 ) = V 1 -ν E x j -W 1 + ν E x j x 2 1 + x 2 2
x j [START_REF] Jourhmane | Relaxation procedures for an iterative algorithm for solving the cauchy problem for the laplace equation[END_REF] where

j = 1, 2; V = - σ 0 r 2 0 -σ i r 2 i r 2 0 -r 2 i ; W = (σ 0 -σ i ) r 2 0 r 2 i r 2 0 -r 2 i ( 21
)
with σ i = 10 GPa and σ 0 = 20 GPa, which corresponds to constant internal and external pressures for which the stress tensor is given by: In a first step, a reliable stopping criterion is needed to stop the iterative process. In the present case, the data on Γ d are generated from the analytical solution (equations ( 20) and ( 22)) and are not blurred by measurement noise. The boundaries Γ d and Γ u are both approximated with 160 SEG2 finite elements.

σ 11 (x 1 , x 2 ) = V + W x 2 1 -x 2 2 (x 2 1 + x 2 2 ) 2 σ 22 (x 1 , x 2 ) = V -W x 2 1 -x 2 2 (x 2 1 + x 2 2 ) 2 σ 12 (x 1 , x 2 ) = 2W x 1 x 2 (x 2 1 + x 2 2 ) 2 . (22)

Stopping criterion

The value of the parameter c is arbitrarily fixed at 0.1. Figure 1 represents the evolution of different control quantities J(U k+1 ), J Γ d (U k+1 ), J Γ (U k+1 ), u error and p error versus the number of iterations k. We notice from the 130 th iteration that the quantities u error , p error , J(U k+1 ) and J Γ d (U k+1 ) remain constant. This confirms that the method converges. The determination of the iteration to stop the iterative process is made using the quantity J(U k+1 ) where U k+1 is the optimal element obtained at the (k + 1) th step. The evolution of J follows an L-curve [START_REF] Hansen | Analysis of discrete ill-posed problems by means of the Lcurve[END_REF]. As expected from relation [START_REF] Johnson | Contact mechanics[END_REF], the control quantity J decreases when k < 130 and then becomes almost constant when k > 130 and never reincreases. So the iterative process is stopped when k = 130. This stopping criterion is blind because when calculating the quantity J it does not need to know the analytical solution. Indeed, it only needs to know two successive optimal elements and the data on Γ d . As expected from relation [START_REF] Hansen | Analysis of discrete ill-posed problems by means of the Lcurve[END_REF], we may also notice that J Γ (regularization term) decreases as iterations go on. This term becomes negligible compared to J Γ d and tends to zero. This proves that the algorithm converges. As expected from relation [START_REF] Hadamard | Lectures on Cauchy's problem in linear partial differential equations[END_REF], the residual error J Γ d (relaxation term) decreases during the iterative process. After convergence this term remains constant and corresponds to the approximation error of the finite element method. It can also be observed that the errors in the numerical tractions obtained (p error ) using the iterative method are larger than those corresponding to the reconstructed displacements (u error ). This last remark is also valid for all the following numerical examples which will be analysed.

Influence of parameter c

It is necessary to look at the influence of parameter c which defines the relative weight of the regularization term J Γ compared to the relaxation term J Γ d . For the same discretization as in the previous example, different values of the parameter c are tested. Table 1 lists the results obtained for each value of parameter c by specifying the number of iterations necessary to achieve convergence, the u error , the p error and the CPU time. The errors on u and p are quite identical for each value of the parameter c. This confirms that the algorithm converges to the same solution whatever the value of c. However the choice of the parameter c affects the number k of iterations needed to obtain convergence. The evolution of the number of iterations necessary to achieve convergence seems to evolve linearly with c where 0.1 < c < 10. From the evolution of the CPU time with c (or with the number of iterations k) it can be deduced that the CPU time taken by each iteration for k > 1 is roughly 4.4 10 -3 s. This CPU time is lower than the CPU time taken to achieve both the preliminary computations and the first iteration (roughly 2.93 s). For a small value of c, the convergence only takes a few iterations but the p error increases a little. This may be explained by the fact that the regularization term becomes too weak and induces some instabilities on the reconstructions. Moreover, it is preferable to have a greater value of the parameter c which induces few additional CPU time in order to have more accurate reconstructions. Subsequently, the value of c and the number of iterations required to achieve convergence will be no longer specified. It is necessary to see how the reconstructions are influenced when the mesh refinement increases (depending on the number of finite elements N on each boundary). Figure 2 shows the reconstructions on Γ u of the u 2 component of the displacement obtained with N = 20, N = 80 and N = 160. Figure 3 shows the corresponding reconstructions of the p 2 component of the traction vector. Table 2 lists the errors u error and p error for different mesh refinements. It can be observed that the error level decreases as the mesh refinement increases. This confirms that the inverse method is stable with respect to the mesh refinement. It can also be observed that reconstruction using a small number of data (N = 20) is also accurate.

c

Influence of mesh refinement

φ d = φ d an + δ η φ d max ( 23
)
where -1 ≤ η ≤ 1 is a random value, δ is the noise level in % and φ d max is the maximal value of the data of Γ d . The first step is to verify that the stopping criterion is still reliable when the data are noisy. The boundary parts Γ d and Γ u are both approximated with 160 finite elements. The value of parameter c is arbitrarily fixed at 1. noise level in % u error in % p error in % 0

3.20E-7 4.96E-6 1 1.96E-3 8.25E-3 2 6.85E-3 2.84E-2 3 1.49E-2 6.11E-2 5 4.03E-2 1.65E-1 10 1.50E-1 5.82E-1 20 5.05E-1 1.79
Table 3: influence of the noise level on the u error and on the p error Figure 4 represents the evolution of the different control quantities J(U k+1 ), J Γ d (U k+1 ), J Γ (U k+1 ), u error and p error versus the number of iterations k when the noise level is at 5%. It can be noticed that from the 200 th iteration the quantities J and J Γ d remain constant. The iterative process is stopped at this step. However if the iterations continue, it can be seen that the quantities u error and p error continue to decrease, reaching a minimum at around the 400 th iteration and then rising again. The stopping criterion, defined in section 5.1.1 is still valid because it leads us to choose a stable solution but it does not allow us to achieve the optimal solution (the closest to the analytical solution). The non monotonicity of the quantities u error and p error do not come out from the non compatibility part of the noise (this part of the noise is deleted because each optimal element must be compatible by definition in [START_REF] Cimetiere | Une methode inverse avec regularisation evanescente[END_REF] for the continuous case and in [START_REF] Engl | Regularization of Inverse Problems[END_REF] for the discrete case) but is due to the fact that the compatible data are dense in the space of all possible data. This compatible part of the noise probably leads to a different solution of the Γ u part. It is the reason why we must stop the iterative process with the L-curve criterion.

Figure 5 shows reconstructions on Γ u of the u 2 component of the displacement obtained with different noise levels (δ = 2%, δ = 5% and δ = 10%). Figure 6 gives the corresponding reconstructions of the component p 2 . The reconstructions obtained can be seen to be very accurate. Table 3 lists the errors on u and on p for the different noise levels considered. The functional is composed of two terms with different roles.

As in most inverse methods, there is a regularization term which tends to zero as the iterations go on. But, in the present method, there is also a relaxation term that allows data blurred by noise to be taken into account. We therefore seek a solution which is close to the data but not a solution that exactly fits the data. The algorithm then recomputes, at each step, a solution on the whole boundary. Figure 7 represents the reconstruction of u 1 on Γ d and the noisy data φ d used (δ = 10%). It can be noticed that the reconstruction corresponds to the analytical solution u an 1 and that the noise in the data has been deleted by the algorithm. The regularization term J Γ is negligible. The residual term J is equal to the relaxation term J Γ d . This term corresponds to the distance on Γ d between the deblurred reconstruction and the noisy data. It can also be noted that the relations [START_REF] Hadamard | Lectures on Cauchy's problem in linear partial differential equations[END_REF][START_REF] Hansen | Analysis of discrete ill-posed problems by means of the Lcurve[END_REF][START_REF] Johnson | Contact mechanics[END_REF] are also verified when the data φ d is noisy. It is not surprising because the inequalities were established without assumption on the data (for instance that Φ d is compatible).

ψ d = ψ d an + δ η ψ d max ( 24
)
Figure 8 represents the evolution of the different control quantities J(U k+1 ), J Γ d (U k+1 ), J Γ (U k+1 ), u error and p error versus the number of iterations k when the noise level is 5%.

It can be noticed that from the 400 th iteration the quantities J and J Γ d remain constant. The iterative process will be stopped at this step. The relations [START_REF] Hadamard | Lectures on Cauchy's problem in linear partial differential equations[END_REF][START_REF] Hansen | Analysis of discrete ill-posed problems by means of the Lcurve[END_REF][START_REF] Johnson | Contact mechanics[END_REF] are once more verified although the data ψ d is noisy. It is also noticed that the quantities u error and p error reach their minimum at almost the 400 th iteration then rise again. The stopping criterion chosen in section 5.1.1 is still valid. Figure 9 gives the reconstructions of the component u 1 of the displacement respectively obtained with a noise level of 2%, 5% and 10%. Figure 10 gives the corresponding reconstructions of p 2 . The reconstructions obtained can be seen to be very accurate.

Square plate under traction

A two-dimensional isotropic linear elastic medium in a plane stress state characterized by the material constants E = 97.9 GPa and ν = 0.34 corresponding to a copper alloy has been studied. In this situation, the domain Ω is defined by:

Ω = {x = (x 1 , x 2 )/ -1 < x 1 , x 2 < 1}
The boundary part Γ d = Γ 1 ∪ Γ 3 is defined by:

Γ 1 = {x = (x 1 , x 2 )/ -1 < x 1 < 1, x 2 = -1} and Γ 3 = {x = (x 1 , x 2 )/ -1 < x 1 < 1, x 2 = 1}
The boundary part Γ u = Γ 2 ∪ Γ 4 defined by:

Γ 2 = {x = (x 1 , x 2 )/x 1 = 1, -1 < x 2 < 1} and
Γ 4 = {x = (x 1 , x 2 )/x 1 = -1, -1 < x 2 < 1}
Γ 1 , Γ 2 , Γ 3 and Γ 4 are discretized using 100 finite elements SEG2.

All finite elements have the same length and the nodes are uniformly distributed. The analytical solution under consideration for the displacements reads:

u an 1 (x 1 , x 2 ) = 1 E σ 0 x 1 , u an 2 (x 1 , x 2 ) = - ν E σ 0 x 2 (25)
It corresponds to the uniform traction stress given by:

σ 11 (x 1 , x 2 ) = σ 0 , σ 22 (x 1 , x 2 ) = σ 12 (x 1 , x 2 ) = 0 ( 26
)
with σ 0 = 15 GPa. Figure 11 gives the reconstructions of u 1 on the whole boundary Γ when the displacement data φ d is exact or blurred by a noise level of 1% and 5%. Figure 12 gives the corresponding reconstructions of p 1 . The reconstructions obtained are very accurate and prove the stability of the inverse method with respect to noisy data. On the boundary part Γ d supporting the data, it can be noted that the computed reconstruction is no longer noisy. Among others, the reconstructions of p 1 allow the tensile stress applied on Γ u to be identified.

Application of the inverse method to a contact problem

As a final example, a contact problem without friction has been considered. An elastic cylinder with an infinite length is compressed between two parallel rigid plane supports (Figure 13). The fact the cylinder has an infinite length justifies a twodimensional study with a plane strain assumption. The response of the elastic medium corresponding to a resin epoxy (Young's modulus E = 4 GPa and ν = 0.4) is studied. The compression load is generated by an imposed vertical displacement u 0 = 0.002 m of the above support, the lower support standing still. The cylinder has a radius R = 0.1 m. A numerical reference solution is determined using a standard finite element method. The FEM computing code used is Cast3m [START_REF] Castem | Code de calcul pour l'analyse de structures par la méthode des éléments finis[END_REF]. The boundary mesh used has 360 non-uniformly distributed nodes, the density being higher on the Γ u part (120 uniformly distributed nodes) corresponding to 1 8 of Γ. The boundary part Γ d corresponds to 8 9 of Γ and is distributed on both sides of both contact areas (Figure 13). Noticing that Γ d is a free edge of the cylinder, given ψ d is zero. Data φ d is the restriction to Γ d of the numerical solution obtained by the standard finite element method described above. Figures 14 and 15 compare reconstructions of the vertical displacement u 2 and the vertical nodal forces p 2 on half of the boundary Γ with the numerical solution obtained by the finite element method. The reconstruction of the displacement (Figure 14) obtained is quite accurate and allows to identify a posteriori the extensions of the contact zones. The reconstruction of the nodal force (Figure 15) is less accurate and gives with a relative precision the maximal value of the contact pressure distribution. At the end of the boundary part Γ u some instabilities appear in the reconstruction of the vertical nodal forces.

Conclusion

This paper introduces an iterative method for solving the Cauchy problem in linear elasticity. This problem consists in recovering missing informations (displacements and forces) on some part of a boundary domain from the knowledge of overspecified data (displacements and forces) on the remaining part. This approach reduced the resolution of the Cauchy problem to the resolution of a sequence of optimization problems under equality constraints. The algorithm reads as a least square fitting. The functional is composed of two terms. The first is a relaxation term which represents the gap between the optimal element and the overspecified boundary data. The second is a regularization term which represents the gap between the optimal element and the previous optimal element. At each step, the optimal element obtained is an exact solution of the equilibrium equations and is near to the overspecified data. The regularization term vanishes when iterations go on. Numerical simulations using the finite element method have highlighted the efficiency, accuracy and robustness of the inverse method to noisy data as well as its ability to deblur noisy data. For all analysed situations, it can be observed that the errors in the numerical tractions obtained using the iterative method are larger than those corresponding to the reconstructed displacements. For the Cauchy problem associated with the Laplace equation, a first order method was introduced [START_REF] Cimetiere | Une methode inverse d'ordre un pour les problèmes de complétion de données[END_REF][START_REF] Delvare | A first order method for the Cauchy problem for the Laplace equation using BEM[END_REF] in order to improve the reconstruction of the normal derivatives. It also gave accurate reconstructions when the boundary part Γ u had corners. An extension to the Cauchy problem in linear elasticity could be useful to improve the efficiency of the inverse method presented in this paper when boundary singularities occur.

Figure 1 :

 1 Figure 1: evolution of control quantities versus the number of iterations k

Figure 2 :Figure 3 :

 23 Figure 2: Reconstruction of u 2 on Γ u

Figure 4 :Figure 5 :Figure 6 :

 456 Figure 4: Evolution of control quantities versus the number of iterations k

Figure 7 :

 7 Figure 7: Noisy data φ d and reconstruction of u 1 on Γ d

Figure 8 :

 8 Figure 8: Evolution of control quantities versus the number of iterations k

1 Figure 9 :Figure 10 :

 1910 Figure 9: Reconstruction of u 1 on Γ u

1 Figure 11 :

 111 Figure 11: Reconstruction of u 1 on the whole boundary Γ

1 Figure 12 :Figure 13 :Figure 14 : 2 Figure 15 :

 1121314215 Figure 12: Reconstruction of p 1 on the whole boundary Γ

Table 1 :

 1 influence of c on the number of iterations k to achieve convergenceinfluence of c on the u error and on the p error

		k	u error in % p error in % CPU time in s
	10.	12810 3.20E-7	4.95E-6	59.93
	1.	1278	3.20E-7	4.95E-6	8.61
	0.1	130	3.20E-7	4.95E-6	3.50
	0.01	20	3.21E-7	4.96E-6	3.01
	0.001	5	3.28E-7	4.95E-6	2.95
	0.0001	3	3.29E-7	5.05E-6	2.94
	0.00001 2	3.36E-7	5.22E-6	2.94

Table 2 :

 2 The reconstructions of the u 1 component of the displacement and of the p 1 component are not presented because the results are similar to those obtained for the u 2 and p 2 components. influence of mesh refinement on the u error and on the p error5.1.4. Reconstruction with noisy displacement data φ dIt is necessary to see how the reconstructions are influenced when data are noisy. The noisy displacement data φ d are generated by:

	N	u error in % p error in %
	20	1.05E-3	1.72E-2
	40	6.38E-6	4.00E-5
	80	2.22E-6	1.95E-5
	160 3.20E-7	4.96E-6

acknowledgements

The authors would like to thank the reviewers for their valuable comments.