## Section: New Results

### High-order numerical methods for modeling wave propagation in complex media: development and implementation

#### High order discretization of seismic waves-problems based upon DG-SE methods

Participants : Hélène Barucq, Julien Diaz, Aurélien Citrain.

Accurate wave propagation simulations require selecting numerical schemes capable of taking features of the medium into account. In case of complex topography, unstructured meshes are the most adapted and in that case, Discontinuous Galerkin Methods (DGm) have demonstrated great performance. Off-shore exploration involves propagation media which can be well represented by hybrid meshes combining unstructured meshes with structured grids that are best for representing homogeneous media like water layers. Then it has been shown that Spectral Element Methods (SEm) deliver very accurate simulations on structured grids with much lower computational costs than DGms.

We have developed a DG-SEm numerical method for solving time-dependent elasto- acoustic wave problems. We consider the first-order coupled formulation for which we propose a DG-SEm formulation which turns out to be stable.

While the 2D case is almost direct, the 3D case requires a particular attention on the coupling boundary on which it is necessary to manage the possible positions of the faces of the tetrahedrons with respect to that of the neighboring hexaedra.

In the framework of this DG-SEm coupling, we are also interested in the Perfectly Matched Layer (PML) in particular the use of the SEm inside it to stabilize it in cases where the use of DGm leads to instabilities.

These results have been obtained in collaboration with Henri Calandra (TOTAL) and Christian Gout (INSA Rouen) and have been presented at Journées Ondes Sud-Ouest (JOSO) in Le Barp, the 14th International Conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES) in Vienna (Austria) and MATHIAS conference in Paris [21], [26]

#### Isogeometric analysis of sharp boundaries in full waveform inversion

Participants : Hélène Barucq, Julien Diaz, Stefano Frambati.

Efficient seismic full-waveform inversion simultaneously demands a high efficiency per degree of freedom in the solution of the PDEs, and the accurate reproduction of the geometry of sharp contrasts and boundaries. Moreover, it has been shown that the stability constant of the FWI minimization grows exponentially with the number of unknowns. Isogeometric analysis has been shown to possess a higher efficiency per degree of freedom, a better convergence in high energy modes (Helmholtz) and an improved CFL condition in explicit-time wave propagation, and it seems therefore a good candidate for FWI.

In the first part of the year, we have focused on a small-scale one-dimensional problem, namely the inversion over a multi-step velocity model using the Helmholtz equation. By exploiting a relatively little-known connection between B-splines ad Dirichlet averages, we have added the knot positions as degrees of freedom in the inversion. We have shown that arbitrarily-placed discontinuities in the velocity model can be recovered using a limited amount of degrees of freedom, as the knots can coalesce at arbitrary positions, obviating the need for a very fine mesh and thus improving the stability of the inversion.

In order to reproduce the same results in two and three dimensions, the usual tensor-product structure of B-splines cannot be used. We have therefore focused on the construction of (unstructured) multivariate B-spline bases. We have generalized a known B-spline basis construction through the language of oriented matroids, showing that multivariate spline bases can be easily constructed with repeated knots and that the construction algorithm can be extended to three dimensions. This gives the freedom to locally reduce the regularity of the basis functions and to place internal boundaries in the domain. The resulting mass matrix is block-dagonal, with adjustable block size, providing an avenue for a simple unstructured multi-patch DG-IGA scheme that is being investigated. With this goal in mind, more efficient quadrature schemes for multivariate B-splines exploiting the connection to oriented matroids are also being investigated.

A research report is in preparation.

#### Seismic wave propagation in carbonate rocks at the core scale

Participants : Julien Diaz, Florian Faucher, Chengyi Shen.

Reproduction of large-scale seismic exploration at lab-scale with controllable sources is a promising approach that could not only be applied to study small-scale physical properties of the medium, but also contribute to significant progress in wave-propagation understanding and complex media imaging at exploration scale via upscaling methods. We propose to apply a laser-generated seismic point source for core-scale new geophysical experiments. This consists in generating seismic waves in various media by well-calibrated pulsed-laser impacts and measuring precisely the wavefield (displacement) by Laser Doppler Vibrometer (LDV). The point-source-LDV configuration is convenient to model numerically. It can also favor the incertitude estimate of the source and receiver locations. Parallel 2D/3D simulations featuring the Discontinuous Galerkin discretization method with Interior Penalties (IPDG) are done to match the experimental data. The IPDG method is of particular interest when it comes to solve wave propagation problems in highly heterogeneous media, such as the limestone cores that we are studying.

Current seismic data allowed us to retrieve ${V}_{p}$ tomography slices. Further more, qualitative/quantitative comparisons between simulations and experimental data validated the experiment protocol and vice-versa the high-order FEM schemes, opening the possibility of performing FWI on dense, high frequency and large band-width data.

This work is in collaboration with Clarisse Bordes, Daniel Brito, Federico Sanjuan and Deyuan Zhang (LFCR, UPPA) and with Stéphane Garambois (ISTerre). Ii is one of the topic of the PhD. thesis of Chengyi Shen.

#### Simulation of electro-seismic waves using advanced numerical methods

Participants : Hélène Barucq, Julien Diaz, Ha Howard Faucher, Rose-Cloé Meyer.

We study time-harmonic waves propagation in conducting poroelastic media, in order to obtain accurate images for complex media with high-order methods. In these kind of media, we observe the coupling between electromagnetic and seismic wave fields, which is called seismokinetic effect. The converted waves are very interesting because they are heavily sensitive to the medium properties, and the modeling of seismo-electric conversion can allow to detect interfaces in the material where the seismic field would be blind. To the best of our knowledge, the numerical simulation of this phenomenon has never been achieved with high-order finite element methods. Simulations are difficult to perform in time domain, because the time step and the mesh size have to be adapted to the huge variations of wave velocities. To ease the numerical implementation, we work in the frequency domain. We can then include physical parameters that depend non-linearly on the frequencies. Then, we have developed a new Hybridizable Discontinuous Galerkin method for discretizing the equations. This allows us to reduce the computational costs by considering only degrees of freedom on the skeleton of the mesh. We have validated the numerical method thanks to comparison with analytical solutions. We have obtained numerical results for 2D realistic poroelastic media and conducting poroelastic media is under investigation.

Results on analytical solutions for poroelasticity are presented in the research report [44].

#### Quasinormal mode expansion of electromagnetic Lorentz dispersive materials

Participants : Marc Duruflé, Alexandre Gras.

We have studied the electromagnetic scattering of optical waves by dispersive materials governed by a Drude-Lorentz model. The electromagnetic fields can be decomposed onto the eigenmodes of the system, known as quasinormal modes. In [51], a common formalism is proposed to obtain different formulas for the coefficients of the modal expansion. In this paper, it is also explained how to handle dispersive Perfectly Matched Layers and degenerate eigenvectors. Lately, we have investigated the use of an interpolation method in order to compute quickly the diffracted field for a large number of frequencies.

#### A Hybridizable Galerkin Discontinuous formulation for elasto-acoustic coupling in frequency-domain

Participants : Hélène Barucq, Julien Diaz, Vinduja Vasanthan.

We are surrounded by many solid-fluid interactions, such as the seabed or red blood cells. Indeed, the seabed represents the ocean floors immersed in water, and red blood cells are coreless hemoglobin-filled cells. Hence, when wanting to study the propagation of waves in such domains, we need to take into account the interactions at the solid-fluid interface. Therefore, we need to implement an elasto-acoustic coupling. Many methods have already tackled with the elasto-acoustic coupling, particularly the Discontinuous Galerkin method. However, this method needs a large amount of degrees of freedom, which increases the computational cost. It is to overcome this drawback that the Hybridizable Discontinuous Galerkin (HDG) has been introduced. The implementations of HDG for the elastic wave equations, as well as partially for the acoustic ones, have been done previously. Using these, we have performed in this work the elasto-acoustic coupling for the HDG methods in 1D, 2D and 3D. The results are presented in Vinduja Vasanthan’s master's thesis [54].

#### Absorbing Radiation Conidition in elongated domains

Participants : Hélène Barucq, Sébastien Tordeux.

We develop and analyse a high-order outgoing radiation boundary condition for solving three-dimensional scattering problems by elongated obstacles. This Dirichlet-to-Neumann condition is constructed using the classical method of separation of variables that allows one to define the scattered field in a truncated domain. It reads as an infinite series that is truncated for numerical purposes. The radiation condition is implemented in a finite element framework represented by a large dense matrix. Fortunately, the dense matrix can be decomposed into a full block matrix that involves the degrees of freedom on the exterior boundary and a sparse finite element matrix. The inversion of the full block is avoided by using a Sherman–Morrison algorithm that reduces the memory usage drastically. Despite being of high order, this method has only a low memory cost. This work has been published in [13].

#### Discontinuous Galerkin Trefftz type method for solving the Maxwell equations

Participants : Margot Sirdey, Sébastien Tordeux.

Trefftz type methods have been developed in Magique 3D to solve Helmholtz equation and it has been presented in [25].These methods reduce the numerical dispersion and the condition number of the linear system. This work aims in pursuing this development for electromagnetic scattering. We have adapted and tested the method for an academical 2D configuration. This is the topic of the PhD thesis of Margot Sirdey.

#### Reduced models for multiple scattering of electromagnetic waves

Participants : Justine Labat, Victor Péron, Sébastien Tordeux.

In this project, we develop fast, accurate and efficient numerical methods for solving the time-harmonic scattering problem of electromagnetic waves by a multitude of obstacles for low and medium frequencies in 3D. First, we consider a multi-scale diffraction problem in low-frequency regimes in which the characteristic length of the obstacles is small compared to the incident wavelength. We use the matched asymptotic expansion method which allows for the model reduction. Then, small obstacles are no longer considered as geometric constraints and can be modelled by equivalent point-sources which are interpreted in terms of electromagnetic multipoles. Second, we justify the Generalized Multiparticle Mie-solution method (Xu, 1995) in the framework of spherical obstacles at medium-frequencies as a spectral boundary element method based on the Galerkin discretization of a boundary integral equation into local basis composed of the vector spherical harmonics translated at the center of each obstacle. Numerically, a clever algorithm is implemented in the context of periodic structures allowing to avoid the global assembling of the matrix and so, reduce memory usage. The reduced asymptotic models of the first problem can be adapted for this regime by incorporating non-trivial corrections appearing in the Mie theory. Consequently, a change in variable between the two formulations can be made explicit, and an inherent advantage of the asymptotic formulation is that the basis and the shape can be separated with a semi-analytical expression of the polarizability tensors. A comparison of these different methods in terms of their accuracy has been carried out. Finally, for both methods and in the context of large numbers of obstacles, we implement an iterative resolution with preconditionning in a GMRES framework.

These results have been presented at Journées Ondes Sud-Ouest (JOSO) in Le Barp (France) and the 14th International Conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES) in Vienna (Austria), see [22], [39]. Part of this work has been published in Wave Motion [17].

#### Boundary Element Method for 3D Conductive Thin Layer in Eddy Current Problems

Participant : Victor Péron.

Thin conducting sheets are used in many electric and electronic devices. Solving numerically the eddy current problems in presence of these thin conductive sheets requires a very fine mesh which leads to a large system of equations, and becoming more problematic in case of high frequencies. In this work we show the numerical pertinence of asymptotic models for 3D eddy current problems with a conductive thin layer of small thickness based on the replacement of the thin layer by its mid-surface with impedance transmission conditions that satisfy the shielding purpose, and by using an efficient discretization with the Boundary Element Method in order to reduce the computational cost. These results have been obtained in collaboration with M. Issa, R. Perrussel and J-R. Poirier (LAPLACE, CNRS/INPT/UPS, Univ. de Toulouse) and O. Chadebec (G2Elab, CNRS/INPG/UJF, Institut Polytechnique de Grenoble). This work has been published in COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, [16].

#### Asymptotic Models and Impedance Conditions for Highly Conductive Sheets in the Time-Harmonic Eddy Current Model

Participant : Victor Péron.

This work is concerned with the time-harmonic eddy current problem for a medium with a highly conductive thin sheet. We present asymptotic models and impedance conditions up to the second order of approximation for the electromagnetic field. The conditions are derived asymptotically for vanishing sheet thickness where the skin depth is scaled like thickness parameter. The first order condition is the perfect electric conductor boundary condition. The second order condition turns out to be a Poincaré-Steklov map between tangential components of the magnetic field and the electric field. This work has been published in SIAM Journal on Applied Mathematics, [18].