Amel Ben Abda

Identification of planar cracks by complete overdetermined data: inversion formulae

The problem of determining a crack by overspecified boundary data is considered. When complete data are available on the external boundary, a reciprocity gap concept is introduced. This concept formalizes the comparison of the response of the safe body to the response of the cracked one of the same physical characteristics. If the crack is known (or assumed) to be planar, explicit inversion formulae are derived determining the host plane equation and the length of an emerging crack in two-dimensional (2D) situations. A reciprocity gap functional is designed and exploited to establish a complete identification result. Numerical trials of the identification methods proposed show very good accuracies and insignificant computational costs.

Reciprocity principle and crack identification

In this paper we are concerned with the planar crack identification problem defined by a unique complete elastostatic overdetermined boundary datum. Based on the reciprocity gap principle, we give a direct process for locating the host plane and we establish a new constuctive identifiability result for 3D planar cracks.

A semi-explicit algorithm for the reconstruction of 3D planar cracks

This paper deals with a semi-explicit algorithm to reconstruct two-dimensional (2D) segment cracks, or three-dimensional (3D) planar cracks, in the framework of overspecified boundary data. The algorithm is based on the reciprocity gap concept, introduced by Andrieux and Ben Abda, which provides explicitly the line (or the plane) support of the cracks. A numerical reconstruction of the cracks, which are actually the support of the solution jump across this plane, is then performed by computing the Fourier expansion of the solution jump itself. After the numerical analysis of the method, some numerical results are presented and commented on.

Identification of 2D cracks by elastic boundary measurements

The purpose of this work is to identify two-dimensional (2D) cracks by means of elastic boundary measurements. A uniqueness result is first proved in the general case, as well as the local Lipschitzian stability in the case of line segment emergent cracks. In this last case, the search for the unique zero of the reciprocity gap functional related to the singular solution of the elasticity problem provides a fast algorithm to determine the unknown crack tip.

Line segment crack recovery from incomplete boundary data

We are concerned with non-destructive control issues, namely detection and recovery of cracks in a planar (2D) isotropic conductor from partial boundary measurements of a solution to the Laplace–Neumann problem. We first build an extension of that solution to the whole boundary, using constructive approximation techniques in classes of analytic and meromorphic functions, and then use localization algorithms based on boundary computations of the reciprocity gap.

On the inverse emergent plane crack problem

This paper deals with the detection of emerging plane cracks, by using boundary measurements. An identifiability result (uniqueness of the solution) is first proved. Then, we look at the stability of this solution with respect to the measurement. A weak stability result is proved, as well as a local Lipshitz stability result for straight cracks, by using domain-derivative techniques.

Sources recovery from boundary data: A model related to electroencephalography

We are concerned with an inverse problem related to sources detection from boundary data in a 2D medium with piecewise constant conductivity. It stands as a 2D version of the inverse problem of electroencephalography, where pointwise sources model epilepsy foci, with the so-called multi-layer spherical model of the head (scalp, skull, brain). When overdetermined electrical measurements (potential and current flux) are available on the scalp, one wants to recover the current sources (conductivity defaults) located in the brain (inner boundary). This recovery issue reduces to a number of inverse problems, where the sources identification process makes use of best rational approximation in the disk, whereas the preliminary cortical mapping step (Cauchy type issue) relies on best constrained harmonic or analytic approximation in an annulus (bounded extremal problems).

On a Non Linear Geometrical Inverse Problem of Signorini type : Identifiability and Stability

This report deals with a non linear inverse problem of identification of unknown boundaries, on which the prescribed conditions are of Signorini type. We first prove an identifiability result, in both frameworks of steady state thermal and elastostatics testing. Local Lipschitz stability of the solutions with respect to the boundary measurements is also established, in case of unknown boundaries which are parts of

Boundary data completion: the method of boundary value problem factorization

{\cal C}^{1,\beta}

Data completion via an energy error functional

Jordan curves, with