## Section: New Results

### Mathematical modeling of multi-physics involving wave equations

#### Elasto-acoustic coupling

Participants : Hélène Barucq, Lionel Boillot, Henri Calandra, Julien Diaz, Simon Ettouati, Conrad Hillairet, Elvira Shishenina.

In the framework of her Master thesis, Elvira Shishenina developed a Discontinuous Galerkin Method for the elastoacoustic coupling in time domain. The proposed solution methodology in general and can be applied to any kind of fluxes. We have implemented and validated in Elasticus a centered flux version and an upwind flux version in two dimensions. The time discretization is achieved thanks to Runge Kutta schemes of second and fourth orders.

In frequency domain, Conrad Hillairet developed a 3D elasto-coupling IPDG scheme, in the framework of his Master thesis. It has been implemented and validated in Hou10ni. Moreover, the code is able to handle $p$-adaptivity and we have proposed a strategy in order to determine the order of the cell as a function of the size of the cell and of the physical parameters. The results of this work have been presented to the Siam Conference on Geosciences in Stanford [39] and to the XXIV Congress on Differential Equations and Applications in Cadiz [32] .

Finally, we have considered elastoacoustic coupling with curved interfaces and we have proposed a solution methodology based on Finite Element techniques, which allows for a flexible coupling between the fluid and the solid domain by using non-conforming meshes and curved elements. Since characteristic waves travel at different speeds through different media, specific levels of granularity for the mesh discretization are required on each domain, making impractical a possible conforming coupling in between. Advantageously, physical domains may be independently discretized in our framework due to the non-conforming feature. Consequently, an important increase in computational efficiency may be achieved compared to other implementations based on non-conforming techniques, namely by reducing the total number of degrees of freedom. Differently from other non-conforming approaches proposed so far, our technique is relatively simpler and requires only a geometrical adjustment at the coupling interface at a preprocessing stage, so that no extra computations are necessary during the time evolution of the simulation. On the other hand, as an advantage of using curvilinear elements, the geometry of the coupling interface between the two media of interest is faithfully represented up to the order of the scheme used. In other words, higher order schemes are in consonance with higher order approximations of the geometry. Concerning the time discretization, we analyzed both explicit and implicit schemes. These schemes are energy conserving and, for the explicit case, the stability is guaranteed by a CFL condition.

This work, which has been achieved in collaboration with Angel Rodriguez Rozas, former post-doc of the team, was published in Journal of Computational Physics [27] .

#### Atmospheric boundary conditions for helioseismology

Participants : Hélène Barucq, Juliette Chabassier, Marc Duruflé, Victor Péron.

The sun does not have a clear boundary like a solid ball, but it has an atmosphere which can be modeled as an exponential decay of the density. We have studied the replacement of this atmosphere by an equivalent boundary condition in order to avoid meshing the atmosphere. When we assume that the exponential decay is large enough, asymptotic modeling can be performed with respect to this large parameter. Equivalent boundary conditions have been obtained for order 1, 2 and 3, and they substantially improve Dirichlet condition (order 0) for low frequencies. However for high frequencies, these conditions are no longer relevant. We have developed a first-order absorbing boundary condition adapted to an exponential decay of the density, this last condition provides good results for the tested range of frequency. These conditions have been used by the team of Laurent Gizon (Max Planck institute) to retrieve experimental dispersion curves, so called “power spectrum”.

#### Absorbing Boundary Conditions for 3D elastic TTI modeling

Participants : Hélène Barucq, Lionel Boillot, Julien Diaz.

We propose stable low-order Absorbing Boundary Conditions (ABC) for elastic TTI modeling. Their derivation is justified in elliptic TTI media but it turns out that they are directly usable to non-elliptic TTI configurations. Numerical experiments are performed by using a new elastic tensor source formula which generates P-waves only in an elliptic TTI medium. Numerical results have been performed in 3D to illustrate the performance of the ABCs.

This work has been presented to the conferences PANACM [38] and SEG [33] .

#### The airfoil equation on near disjoint intervals : Approximate models and polynomial solutions

Participants : Leandro Farina, Marcos Ferreira, Victor Péron.

In [26] , the airfoil equation is considered over two disjoint intervals. Assuming the distance between the intervals is small an approximate solution is found and relationships between this approximation and the solution of the classical airfoil equation are obtained. Numerical results show the convergence of the approximation to the solution of the original problem. Polynomial solutions for an approximate model are obtained and a spectral method for the generalized airfoil equation on near disjoint intervals is proposed.

#### Finite element subproblem method

Participants : Patrick Dular, Christophe Geuzaine, Laurent Krähenbühl, Victor Péron.

In [25] , progressive refinements of inductors are done with a subproblem method, from their wire or filament representations with Biot-Savart models up to their volume finite-element models, from statics to dynamics. The reaction fields of additional magnetic and/or conducting regions are also considered. Accuracy improvements are efficiently obtained for local fields and global quantities, i.e., inductances, resistances, Joule losses, and forces.

#### Asymptotic study for Stokes-Brinkman model with Jump embedded transmission conditions

Participants : Philippe Angot, Gilles Carbou, Victor Péron.

In [18] , one considers the coupling of a Brinkman model and Stokes equations with jump embedded transmission conditions. Assuming that the viscosity in the porous region is very small, we derive a Wentzel-Kramers-Brillouin (WKB) expansion in power series of the square root of this small parameter for the velocity and the pressure which are solution of the transmission problem. This WKB expansion is justified rigorously by proving uniform errors estimates.

#### On the solution of the Laplace equation in 3-D domains with cracks and elliptical edges

Participants : Victor Péron, Samuel Shannon, Zohar Yosibash.

An explicit asymptotic solution to the elasticity system in a three-dimensional domain in the vicinity of an elliptical crack front, or for an elliptical sharp V-notch is still unavailable. Towards its derivation we first consider in [30] the explicit asymptotic solutions of the Laplace equation in the vicinity of an elliptical singular edge in a three-dimensional domain. Both homogeneous Dirichlet and Neumann boundary conditions on the surfaces intersecting at the elliptical edge are considered. The dual singular solution is also provided to be used in a future study to extract the edges flux intensity functions by the quasi-dual function method. We show that just as for the circular edge case, the solution in the vicinity of an elliptical edge is composed of three series, with eigenfunctions being functions of two coordinates.

In [29] the singular solution of the Laplace equation with a straight-crack is represented by a series of eigenpairs, shadows and their associated edge flux intensity functions (EFIFs). We address the computation of the EFIFs associated with the integer eigenvalues by the quasi dual function method (QDFM).The QDFM is based on the dual eigenpairs and shadows, and we show that the dual shadows associated with the integer eigenvalues contain logarithmic terms. These are then used with the QDFM to extract EFIFs from p-version finite element solutions. Numerical examples are provided.