Sébastien Imperiale

In time-harmonic elastography, the shear modulus is typically inferred from full-field displacement data by solving an inverse problem based on the time-harmonic elastodynamic equation. In this paper, we focus on nearly incompressible media, which pose robustness challenges, especially in the presence of noisy data. Restricting ourselves to 2D and considering an isotropic, linearly deforming medium, we reformulate the problem as a non-autonomous hyperbolic system and, through theoretical analysis, establish existence, uniqueness, and stability of the inverse problem. To ensure robustness with noisy data, we propose a least-squares approach with regularization. The convergence properties of the method are verified numerically using in silico data.

In this work, we develop an efficient numerical method for solving 3D Maxwell's equations in non-cylindrical coaxial cables. The main challenge arises from the elongated geometry of the computational domain, which induces strong anisotropy between the longitudinal direction (along the cable) and the transverse directions (within the cross-sections). This leads to the use of highly anisotropic meshes, where the longitudinal mesh size is much larger than the transverse one. Our objective is to design a numerical scheme that is explicit in the longitudinal direction, with a CFL stability condition depending only on the longitudinal mesh size. In a previous work, we achieved this for cylindrical cables by employing prismatic edge elements, 1D quadrature for longitudinal mass lumping, and a hybrid explicit/implicit time discretization. The present paper extends this approach to non-cylindrical cables, addressing several new difficulties with the following key ingredients: (1) representing the cable as a deformation of a reference cylindrical cable and employing mapping techniques between the physical and reference domains; (2) using an anisotropic space discretization that combines an interior penalty discontinuous Galerkin (IPDG) method in the transverse directions with a conforming finite element method in the longitudinal direction; (3) utilizing prismatic edge elements on a prismatic mesh of the reference cable; and (4) adapting the construction of the hybrid explicit-implicit time discretization to the new structure of the semi-discrete problem. From a theoretical perspective, the main difficulty lies in the stability analysis, which requires extending and adapting standard techniques for DG methods in space and energy methods in time.

This work addresses the identification of nonhomogeneous constitutive parameters from full-field measurements in both linear and nonlinear elasticity, considering incompressible as well as compressible materials. The inverse identification procedure relies on the Virtual Fields Method (VFM), which is based on the principle of virtual work with specifically chosen virtual fields. We propose an optimal class of virtual fields, designed to optimize the reconstruction stability with respect to measurement noise. A series of numerical experiments illustrate the effectiveness of the proposed approach. The method exhibits moderate sensitivity to measurement noise and remains robust even when the boundary conditions are only partially known.

In this work, we are interested by the identification in a wave equation of a space dependent source term multiplied by a known time and space dependent function, from internal velocity or field measurements. The first part of the work consists in proving stability inequalities associated with this inverse problem from adapted Carleman estimates. Then, we present a sequential reconstruction strategy which is proved to be equivalent to the minimization of a cost functional with Tikhonov regularization. Based on the obtained stability estimates, the reconstruction error is evaluated with respect to the noise intensity. Finally, the proposed method is illustrated with numerical simulations, both in the case of regular source terms and of piecewise constant source terms.

In this work, we develop an efficient numerical method for solving 3D Maxwell's equations in non-cylindrical coaxial cables. The main challenge arises from the elongated geometry of the computational domain, which induces strong anisotropy between the longitudinal direction (along the cable) and the transverse directions (within the cross-sections). This leads to the use of highly anisotropic meshes, where the longitudinal mesh size is much larger than the transverse one.Our objective is to design a numerical scheme that is explicit in the longitudinal direction, with a CFL stability condition depending only on the longitudinal mesh size. In a previous work, we achieved this for cylindrical cables by employing prismatic edge elements, 1D quadrature for longitudinal mass lumping, and a hybrid explicit/implicit time discretization. The present paper extends this approach to non-cylindrical cables, addressing several new difficulties with the following key ingredients: (1) representing the cable as a deformation of a reference cylindrical cable and employing mapping techniques between the physical and reference domains; (2) using an anisotropic space discretization that combines an interior penalty discontinuous Galerkin (IPDG) method in the transverse directions with a conforming finite element method in the longitudinal direction; (3) utilizing prismatic edge elements on a prismatic mesh of the reference cable; and (4) adapting the construction of the hybrid explicit-implicit time discretization to the new structure of the semidiscrete problem. From a theoretical perspective, the main difficulty lies in the stability analysis, which requires extending and adapting standard techniques for DG methods in space and energy methods in time.

This work addresses the mathematical analysis, by means of asymptotic analysis, of a class of linear wave propagation problems with singular stiff terms represented by a single small parameter. This abstract setting is defined using linear operators in Hilbert spaces. In this setting, we show, under some assumptions on the structure of the wave propagation problems, weak and strong convergence of solutions with respect to the small parameter towards the solution of a well-defined limit problem. Applications in nearly incompressible elastodynamics, elastic waves in thin plates, piezoelectricity, and homogenization are presented.

The numerical approximation of wave propagation problems in nearly or pure incompressible solids faces several challenges such as locking and stability constraints. In this work we propose a stabilized Leapfrog scheme based on the use of Chebyshev polynomials to relax the stability condition, which is strongly limited by the enforcement of incompressibility. The scheme is fully explicit, second order accurate and energy-preserving. For the space discretization we use a mixed formulation with high-order spectral elements and mass-lumping. A strategy is proposed for an efficient and accurate computation of the pressure contribution with a new definition of the discrete Grad-div operator. Finally, we consider linear wave propagation problems in nearly-incompressible hyperelastic solids subject to static preload.

The objective of this work is to propose a method using observers to estimate a source term of a wave equation from internal measurements in a subdomain. The first part of the work consists in proving an identifiability result from classical observability conditions for wave equations. We show that the source reconstruction is an ill-posed inverse problem of degree 1 or 2 depending on the measurements type. This inverse problem is solved using observers -- a sequential strategy -- that is proven to be equivalent to a minimization of a cost functional with Tikhonov regularization.