Section: New Results

Asymptotic methods and approximate models

Homogenization and interfaces

Participants : Sonia Fliss, Valentin Vinoles.

This topic is developed in collaboration with Xavier Claeys (LJLL, Paris VI).

The mathematical modelling of electromagnetic metamaterials and the homogenization theory are intimately related because metamaterials are precisely constructed by a periodic assembly of small resonating micro-structures involving dielectric materials presenting a high contrast with respect to a reference medium. In the framework of the ANR Metamath (see 6.2.6 ), we wish to look carefully at the treatment of boundaries and interfaces that are generally poorly taken into account by the first order homogenization.

This question is already relevant for standard homogenization (ie without high contrast). Indeed, the presence of a boundary induces a loss of accuracy due to the inadequateness of the standard homogenization approach to take into account boundary layer effects. Our objective is to construct approximate effective boundary conditions that would restore the desired accuracy.

We first considered a plane interface between a homogeneous and the periodic media in the standard case without high-contrast. We obtained high order transmission conditions between the homogeneous media and the periodic media. The technique we used involves matched asymptotic expansions combined with standard homogenization ansatz. Those conditions are non standard : they involve Laplace-Beltrami operators at the interface and requires to solve cell problems in infinite periodic waveguides. The derivation of the corresponding error estimates is in progress. The analysis is based on a original combination of Floquet-Bloch and a periodic version of Kondratiev technique.

The next step will be to consider the same problem but with a high-contrast periodic media in collaboration with Guy Bouchitté, a french expert in high contrast homogenization.

Effective boundary conditions for thin periodic coatings

Participants : Mathieu Chamaillard, Patrick Joly.

This topic is the object of a collaboration with Houssem Haddar (CMAP École Polytechnique). We are interested in the construction of "equivalent” boundary condition for the diffraction of waves by an obstacle with smooth boundary $\Gamma$ covered with a thin coating of width $\delta$ whose physical characteristics vary "periodically” along $\Gamma$ with a period proportional to the small parameter $\delta$. For a general boundary $\Gamma$, the notion of periodicity is ambiguous: we have chosen to define the coating as the image, or the deformation, by a smooth mapping ${\psi }_{\Gamma }$ of a flat layer of width $\delta$ (the reference configuration) that preserves the normals, which appears consistent with a manufacturing process. The electromagnetic parameters in the coating are then defined as the images through ${\psi }_{\Gamma }$ of periodic functions in the reference configuration.

We have first considered the case of the scalar wave equation when the homogeneous Neumann condition is applied on the boundary of the obstacle. Using an asymptotic analysis in $\delta$, which combines homogenization and matched asymptotic expansions, we have been able to establish a second order boundary condition of the form

where $B_{\Gamma }^1$ and $B_{\Gamma }^2$ are second order tangential differential operators along $\Gamma$. The coefficients of these operators depend on both the geometrical characteristics of $\Gamma$ (through the curvature tensor), the deformation mapping ${\psi }_{\Gamma }$ and the material properties of the coating, through the resolution of particular unbounded cell problems in the flat reference configuration. When the coating is homogeneous, we have checked that one recovers the well known second order thin layer condition. We have moreover proven that this approximate condition provides in $𝒪\left({\delta }^3\right)$.

Thin Layers in Isotropic Elastodynamics

Participants : Marc Bonnet, Aliénor Burel, Patrick Joly.

This research is concerned with the numerical modelling of non-destructive testing experiments using ultrasonic waves. Some materials, e.g. composite materials, involve thin layers of resin. The numerical modelling of such thin layers can be problematic as they result in very small spatial mesh sizes. To alleviate this difficulty, we develop an approach based on an asymptotic analysis with respect to the layer thickness $\epsilon$, aiming to model the thin layer by approximate effective transmission conditions (ETCs), which remove the need to mesh the layer. So far, ETCs that are second-order accurate in $\epsilon$ have been formulated, justified, implemented and numerically validated, for 2-D and 3-D configurations involving planar interfaces of constant thickness. In particular, the continuous evolution problem is shown to be stable, and a time-stepping scheme that essentially preserves the stability requirement on the time step is proposed. Extension of this work to 2-D and 3-D configurations involving a curved layer is ongoing.

Mathematical modelling of electromagnetic wave propagation in electric networks.

Participants : Geoffrey Beck, Patrick Joly.

This topic is developed in collaboration with S. Imperiale (Inria Saclay) in the framework of the ANR project SODDA, in collaboration with CEA-LETI, about the non destructive testing of electric networks. This is the subject of the PhD thesis of G. Beck.

We investigate the question of the electromagnetic propagation in thin electric cables from a mathematical point of view via an asymptotic analysis with respect to the (small) transverse dimension of the cable: as it has been done in the past in mechanics for the beam theory from 3D elasticity, we use such an approach for deriving simplified effective 1D models from 3D Maxwell's equations.

During last year, we have achieved some progress in various directions:

• Single wire coaxial cables. This is the direct continuation of what has been done last year. Concerning the lowest order, the telegraphist's model, we have extended the error analysis, previously restricted to non lossy cylindrical cables to very general cases. Technically, this relies on time Laplace transform and new, parameter dependent, Poincaré-Friedrichs inequalities. From the numerical point of view, in collaboration with M. Duruflé, we have initiated a quantitative comparison between the full 3D model and our 1D model. Furthermore we have derived and studied a higher order generalized telegraphist's equation that include dispersive effects through nonlocal capacity and inductance operators. The corresponding mathematical analysis is in progress.

• Multiple wires cables. The objective here was to extend our approach to cables containing $N$ conducting wires. Our results into a vectorial generalized telegraphist's model with $2N$ (2 for each wire) 1D unknowns, N electrical potentials and N currents. This model involves in particular a capacity matrix C, an inductance matrix L, a resistance matrix R and a conductance matrix G, whose properties have been deeply investigated, which allowed us to justify rigorously and extend some results from the electrical engineering literature. In the most general case, the effective models also involve time memory terms with matrix valued convolution kernels.

• Junction of cables. This is a new and essential step towards the modelling of networks. We have started the case of junctions of single wire cables via the method of matched asymptotic expansions in the spirit of the PhD thesis of A. Semin.

Elastic wave propagation in strongly heterogeneous media

Participants : Simon Marmorat, Patrick Joly.

This subject enters our long term collaboration with CEA-LIST on the development of numerical methods for time-domain non destructive testing experiments using ultra-sounds, and is realized in collaboration with Xavier Claeys (LJLL, Paris VI). We aim at developing an efficient numerical approach to simulate the propagation of waves in a medium made of many small heterogeneities, embedded in a smooth (or piecewise smooth) background medium, without any particular assumption (such as periodicity) on the spatial distribution of these heterogeneities. The figure 5 is a snapshot of a simulation inside such a medium, computed thanks to classical simulation tools: to reach satisfying accuracy, one has to use mesh refinement in the vicinity of the heterogeneities, which greatly increases the computational cost of the method.

By considering the medium with defects as a perturbation of the smooth one, we have derived an auxiliary model in the acoustic case, involving the defect-free wave operator and some volume Lagrange multipliers which account for the presence of the defects. These Lagrange multipliers are unknown functions defined on the defects and live in some infinite dimensional functional space. Exploiting the smallness of the defects, we have shown thanks to matched asymptotic analysis that the aforementioned functional space may be well described by a finite number $N$ of profile functions: we propose an asymptotic model by looking for the Lagrange multipliers into the space spanned by these $N$ profile functions, and we have shown that the error hence made is controlled by ${\epsilon }^N$, $\epsilon$ being the characteristic size of the defects, assumed to be small.

On a computational point of view, the asymptotic model is much easier to solve than the original one since it can be discretized using a computation mesh that ignores the presence of the heterogeneities, the Lagrangian multipliers being computed by solving a linear system of size $N$. A resolution of this model has been implemented in the 1D and in the 2D case, and a rigorous error estimate has been established.