Overdetermined Boundary Data Research Articles

Determination of rigid inclusions immersed in an isotropic elastic body from boundary measurement

We study the determination of some rigid inclusions immersed in an isotropic elastic medium from overdetermined boundary data. We propose an accurate approach based on the topological sensitivity technique and the reciprocity gap concept. We derive a higher-order asymptotic formula, connecting the known boundary data and the unknown inclusion parameters. The obtained formula is interesting and useful tool for developing accurate and robust numerical algorithms in geometric inverse problems.

The Carleman Contraction Mapping Method for Quasilinear Elliptic Equations with Over-determined Boundary Data

The Carleman Contraction Mapping Method for Quasilinear Elliptic Equations with Over-determined Boundary Data

On the crack inverse problem for pressure waves in half-space

After formulating the pressure wave equation in half-space minus a crack with a zero Neumann condition on the top plane, we introduce a related inverse problem. That inverse problem consists of identifying the crack and the unknown forcing term on that crack from overdetermined boundary data on a relatively open set of the top plane. This inverse problem is not uniquely solvable unless some additional assumption is made. However, we show that we can differentiate two cracks Γ1 and Γ2 under the assumption that R3∖Γ1∪Γ2‾ is connected. If that is not the case we provide counterexamples that demonstrate non-uniqueness, even if Γ1 and Γ2 are smooth and “almost” flat. Finally, we show in the case where R3∖Γ1∪Γ2‾ is not necessarily connected that after excluding a discrete set of frequencies, Γ1 and Γ2 can again be differentiated from overdetermined boundary data.

A Carleman-based numerical method for quasilinear elliptic equations with over-determined boundary data and applications

We propose a new iterative scheme to compute the numerical solution to an over-determined boundary value problem for a general quasilinear elliptic PDE. The main idea is to repeatedly solve its linearization by using the quasi-reversibility method with a suitable Carleman weight function. The presence of the Carleman weight function allows us to employ a Carleman estimate to prove the convergence of the sequence generated by the iterative scheme above to the desired solution. The convergence of the iteration is fast at an exponential rate without the need of an initial good guess. We apply this method to compute solutions to some general quasilinear elliptic equations and a large class of first-order Hamilton-Jacobi equations. Numerical results are presented.

Cavities identification in linear elasticity: energy-gap versus L2-gap cost functionals

We study the geometric inverse problem of cavities identification in linear elasticity equation with partially over-determined boundary data. This work proposes a way to compute the topological derivative based on the topological sensitivity analysis concepts, adopting the energy-gap and the -gap as cost functionals. Then, a comparison of both functionals is presented to determine the best choice for numerical studies. Further, several numerical experiences are shown to explore the efficiency of this method.

Solution for the Geometric Elastostatic Inverse Problem by Means of Not Completely Overdetermined Boundary Data

The problem of detecting and localizing a cavity, a crack, or an inclusion in an elastic body by means of partially overdetermined data given on its outer boundary is considered. The input data can be obtained in a single static experiment. A new approach for solving this problem is proposed. The capabilities of the proposed approach are illustrated by the example of the problem for detecting a cavity in an elastic plate.

Topological sensitivity analysis for identification of voids under Navier’s boundary conditions in linear elasticity

This article is concerned with a geometric inverse problem related to the two-dimensional linear elasticity system. Thereby, voids under Navier’s boundary conditions are reconstructed from the knowledge of partially over-determined boundary data. The proposed approach is based on the so-called energy-like error functional combined with the topological sensitivity method. The topological derivative of the energy-like misfit functional is computed through the topological-shape sensitivity method. Firstly, the shape derivative of the corresponding misfit function is presented briefly from previous work Méjri (2018 J. Inverse Ill-Posed Problems). Then, an explicit solution of the fundamental boundary-value problem in the infinite plane with a circular hole is calculated by the Muskhelishvili formulae. Finally, the asymptotic expansion of the topological gradient is derived explicitly with respect to the nucleation of a void. Numerical tests are performed in order to point out the efficiency of the developed approach.

Identification of multiple cracks in an anisotropic elastic plate by boundary data

A linear elastic, anisotropic plate, which can contain multiple, rectilinear cracks is considered. It is assumed that the loads applied to the boundary of the plate and the displacements of the boundary are measured in a single static experiment. A method for the cracks identification using the known overdetermined boundary data is developed. The developed method is based on the reciprocity gap concept, generalized functions theory and methods of cluster analysis. Numerical examples are considered.

Shape sensitivity analysis for identification of voids under Navier’s boundary conditions in linear elasticity

Abstract This work is devoted to the study of the void identification problem from partially overdetermined boundary data in the 2D-elastostatic case. In a first part, a shape identifiability result from a Cauchy data is presented, i.e. with traction field and boundary displacement as measurements. Then this geometric inverse problem is tackled by the minimization of two cost functionals, an energy gap functional and an L 2 {L^{2}} -gap functional, which enable the reconstruction of voids under Navier’s boundary conditions. The shape derivatives of these cost functionals are computed for the purpose of sensitivity analysis.

Symmetry problem 1

A symmetry problem is solved. A new method is used. The idea of this methodis to reduce to a contradiction the PDE and the over-determined boundary data on the boundary.The new method allows one to solve other symmetry problems.

Cavities Identification from Partially Overdetermined Boundary Data in Linear Elasticity

Our main interest in this work is an analysis of geometrical inverse problem related to the detection of cavities, in elasticity framework from partially overdetermined boundary data in two spatial dimensions. For the reconstruction, we have only access to the displacement field and to the normal component of the normal stress. We propose an identification method based on the Kohn-Vogelius formulation combined with the topological gradient method. An asymptotic expansion for an energy function is derived with respect to the creation of a small hole. A one-shot reconstruction algorithm based on the topological sensitivity analysis is implemented. Some numerical experiments concerning the cavities identification are finally reported, highlighting the ability of the method to identify multiple cavities.

Time domain linear sampling method for qualitative identification of buried cavities from elastodynamic over-determined boundary data

A time domain version of linear sampling method (LSM) is developed for elastic wave imaging of media including scatterers with arbitrary geometries. The LSM is an effective approach to image the geometrical features of unknown targets from multi-view data collected from measurement of casual waves. This study emphasizes the exploitation of the LSM using spectral finite element method (SFEM). A comprehensive set of numerical simulations on two-dimensional elastodynamic problems is presented to highlight many efficient features of the proposed fast qualitative LSM identification method such as its ability to locate an inclusion (e.g., a crack) and estimate its dimensions.

Stability Estimates for an Inverse Hyperbolic Initial Boundary Value Problem with Unknown Boundaries

In this paper we consider an inverse initial boundary value problem for the anisotropic wave equation. We prove that the unknown portion of the boundary of a domain in $\mathbb{R}^n$ depends logarithmically on the overdetermined boundary data.

Topological sensitivity of energy cost functional for wave-based defect identification

This article is concerned with establishing the topological sensitivity (TS) against the nucleation of small trial inclusions of an energy-like cost function. The latter measures the discrepancy between two time-harmonic elastodynamic states (respectively defined, for cases where overdetermined boundary data is available for identification purposes, in terms of Dirichlet or Neumann boundary data for the same reference solid) as the strain energy of their difference. Such cost function constitutes a particular form of error in constitutive relation and may be used for e.g. defect identification. The TS is expressed in terms of four elastodynamic fields, namely the free and adjoint solutions for Dirichlet or Neumann data. A similar result is also given for the linear acoustic scalar case. A synthetic numerical example where the TS result is used for the qualitative identification of an inclusion is presented for a simple 2D acoustic configuration.

A FEM-based topological sensitivity approach for fast qualitative identification of buried cavities from elastodynamic overdetermined boundary data

A time-domain topological sensitivity (TS) approach is developed for elastic-wave imaging of media of arbitrary geometry. The TS, which quantifies the sensitivity of the misfit cost functional to the creation at a specified location of an infinitesimal hole, is expressed in terms of the time convolution of the free field and a supplementary adjoint field as a function of that specified location. Following previous studies performed under (mostly) static or time-harmonic conditions, the TS field is here considered as a natural and computationally efficient approach for defining a defect indicator function. This study emphasizes the implementation and exploitation of TS fields using standard displacement-based FEM approaches, a straightforward task once the correct sensitivity formulation is available. A comprehensive set of numerical experiments on 3D and 2D elastodynamic and acoustic configurations is reported, allowing to assess and highlight many features of the proposed TS-based fast qualitative identification such as its ability to identify multiple defects and its robustness against data noise.

Unique continuation of boundary over-determined Stokes and Oseen eigenproblems

We provide several radically different proofs of the following unique continuation result: The Oseen eigenvalue problem with over-determined homogeneous Cauchy data for the velocity field on the boundary implies the zero solution, at least if the equilibrium solution is sufficiently 'small.' In particular, this unique continuation result from over-determined boundary data holds true for the Stokes problem.

Imagerie électromagnétique de petites inhomogénéités

We give a short review of some recent results concerning asymptotic representation formulas for the perturbations in the electromagnetic fields caused by the presence of small inhomogeneities. We also discuss how these representation formulas may be used to design very effective numerical algorithms to recover the inhomogeneities for overdetermined boundary data.

A uniqueness result for a nonlinear hyperbolic equation

In this paper we consider the problem of identifying a coefficient function in the principal part of a nonlinear hyperbolic PDE from overdetermined boundary data of the PDE solution. Assuming that the coefficient function can be written as a linear combination of finitely many ansatz functions, we derive a stability and uniqueness result that is based on an identifiability condition in terms of the initial data as well as on the smallness of the time interval. In doing so, we distinguish between the two- and higher dimensional case where functionals on the boundary can be measured and the one-dimensional situation with only point measurements at one boundary point. Our results also give perspectives to the case of an infinite dimensional coefficient function.

Cavity identification in linear elasticity and thermoelasticity

AbstractThe aim of this paper is an analysis of geometric inverse problems in linear elasticity and thermoelasticity related to the identification of cavities in two and three spatial dimensions. The overdetermined boundary data used for the reconstruction are the displacement and temperature on a part of the boundary.We derive identifiability results and directional stability estimates, the latter using the concept of shape derivatives, whose form is known in elasticity and newly derived for thermoelasticity. For numerical reconstructions we use a least‐squares formulation and a geometric gradient descent based on the associated shape derivatives. The directional stability estimates guarantee the stability of the gradient descent approach, so that an iterative regularization is obtained. This iterative scheme is then regularized by a level set approach allowing the reconstruction of multiply connected shapes. Copyright © 2007 John Wiley & Sons, Ltd.

Lipschitz stability for a stationary 2D inverse problem with unknown polygonal boundary

We consider the stability issue for the inverse problem of determining an unknown portion Σ of a two-dimensional simply connected domain from overdetermined boundary data for the Laplace equation. In this paper, we study the case in which Σ is a polygonal line. We prove a Lipschitz stability estimate under further a priori geometric assumptions on Σ.