## Section: New Results

### Integral equations

#### Fast BEM solvers based on $\mathscr{H}$-matrices for 3-D frequency-domain elastodynamics

Participants : Stéphanie Chaillat, Patrick Ciarlet, Luca Desiderio.

The main advantage of the Boundary Element Method (BEM) is that only the domain boundaries are discretized leading to a drastic reduction of the total number of degrees of freedom. In traditional BE implementation the dimensional advantage with respect to domain discretization methods is offset by the fully-populated nature of the BEM coefficient matrix. Using the $\mathscr{H}$-matrix arithmetic and low-rank approximations (performed with Adaptive Cross Approximation) it is possible to derive fast iterative and direct solvers for the BEM system. We extend the method to 3-D frequency-domain elastodynamics. To this end, the Adaptive Cross Approximation is adapted to deal with vectorial problems. To validate the accuracy of the solution of the LU based direct solver, we derive an error estimate. Finally, we check numerically the theoretical estimate of the storage costs. In particular, we study the efficiency of low-rank approximations when the frequency is increased. This is done in partnership with SHELL company in the framework of the PhD of Luca Desiderio.

#### OSRC preconditioner for 3D elastodynamics

Participant : Stéphanie Chaillat.

This work is done in collaboration with Marion Darbas from University of Picardie and Frédérique Le Louer from Technological University of Compiègne. The fast multipole accelerated boundary element method (FM-BEM) is a possible approach to deal with scattering problems of time-harmonic elastic waves by a three-dimensional rigid obstacle. In 3D elastodynamics, the FM-BEM has been shown to be efficient with solution times of order $O(NlogN)$ per iteration (where $N$ is the number of BE degrees of freedom). However, the number of iterations in GMRES can significantly hinder the overall efficiency of the FM-BEM. To reduce the number of iterations, we propose a clever integral representation of the scattered field which naturally incorporates a regularizing operator. When considering Dirichlet boundary value problems, the regularizing operator is a high-frequency approximation to the Dirichlet-to-Neumann operator. For a spherical obstacle, the approximation of the DtN is a linear combination of the tangential and normal parts. The numerical efficiency of the preconditioned integral equation (i.e. the independence of the number of iterations from the mesh size and frequency) is verified for spherical obstacles, validating the concept of analytical preconditioners for 3D elastodynamics FM-BEM. For more general shapes, this approximation of the DtN is more complex to derive. As a first step, we construct and validate the approximation in the framework of the On-Surface Radiation Condition (OSRC) method.

#### A wideband Fast Multipole Method for oscillatory kernels

Participant : Stéphanie Chaillat.

This work is done in collaboration with Francis Collino. We derive a new Fast Multipole Method (FMM) based on plane wave expansions (PWFMM), combining the advantages of the low and high frequency formulations. We revisit the method of Greengard et al. (1998) devoted to the low frequency regime and based on the splitting of the Green’s function into a propagative and an evanescent part. More precisely, we give an explicit formula of the filtered translation function for the propagative part, we derive a new formula for the evanescent part and we provide a new interpolation algorithm. At all steps, we check the accuracy of the method by providing error estimates. These theoretical developments are used to propose a wideband FMM based entirely on plane wave expansions. The numerical efficiency and accuracy of this broadband PWFMM are illustrated with a numerical example.

#### Coupling integral equations and high-frequency methods for ultrasonic NDT modelling

Participants : Marc Bonnet, Laure Pesudo.

This work, in partnership with CEA LIST and in collaboration with Francis Collino, is undertaken in the context of the PhD thesis of Laure Pesudo. Modelling ultrasonic non destructive testing (NDT) experiments simultaneously involves the scattering of waves by defects of moderate size (for which discretization-based methods such as the BEM are appropriate) and propagation over large distances (requiring high-frequency approximations). Those two types of simulation methods are therefore simultaneously needed in NDT modelling but do not lend themselves easily to coupling. The coupling approach proposed here takes advantage of the fact that the far-field asymptotic approximation of integral representation formulas (which accurately account for the scattering by defects) yields a superposition of rays (satisfying the leading-order equations arising from high-frequency asymptotics). This allows to convert incoming rays into plane waves, compute their scattering by obstacles, and convert the scattered field into rays. A defect of given shape and characteristics becomes (approximately) represented as a point-like scatterer with anisotropic reflection properties that are computed (offline) from BEM solutions of near-field problems. Using a partition of unity on the obstacle boundary allows to approximate the obstacle by a set of point-like reflectors, thereby enlarging the size of obstacles amenable to this approach. Preliminary tests on 2D scalar wave propagation problems show that sufficient far-field accuracy is achieved for wavelength-sized defects.

#### Dynamic soil-structure interaction

Participants : Marc Bonnet, Stéphanie Chaillat, Zouhair Adnani.

This work, undertaken in the context of the PhD thesis of Zouhair Adnani (CIFRE partnership with EDF), concerns the simulation of dynamic soil-structure interaction (SSI) in connection with seismic assessment of civil engineering structures. The main goal is to formulate, implement, and evaluate on realistic test examples, a computational strategy that combines the fast multipole integral equation method for elastic wave propagation in unbounded regions (COFFEE FMM-accelerated BEM solver), and finite elements for modelling civil engineering structures and neighboring soil regions (the EDF in-house code Code_Aster). In a preliminary phase, the evaluation of transient elastodynamic responses via the Fourier synthesis of frequency-domain solutions computed using COFFEE (see Section 6.1 ) has been studied on several test problems, achieving substantial improvements of computational efficiency for this component of SSI analysis.

#### Volume Integral Formulations

Participant : Marc Bonnet.

Volume integral equations (VIEs), also known as Lippmann-Schwinger integral equations, arise naturally when considering the scattering of waves by penetrable, and possibly heterogeneous, inhomogeneities embedded in a homogeneous background medium. In contrast with the vast existing literature on boundary integral equations, comparatively few studies are available regarding the mathematical properties of VIEs. In this work, we investigate the solvability of VIE formulations arising in elastodynamic scattering by penetrable obstacles. The elasticity tensor and mass density are allowed to be smoothly heterogeneous inside the obstacle and may be discontinuous across the background-obstacle interface, the background elastic material being homogeneous. Both materials may be anisotropic, within certain limitations for the background medium. The VIE associated with this problem is derived, relying on known properties of the background fundamental tensor. To avoid difficulties associated with existing radiation conditions for anisotropic elastic media, we propose an alternative definition of the radiating character of transmission solutions. The unique solvability of the volume integral equation (and of the scattering problem) is established. For the important special case of isotropic background properties, our definition of a radiating solution becomes equivalent to the classical Sommerfeld-Kupradze radiation conditions.