## Section: New Results

### Integral equations and boundary element methods (BEMs)

#### Accelerated and adapted BEMs for wave propagation

Participants : Faisal Amlani, Stéphanie Chaillat.

*This work is done in collaboration with Adrien Loseille (EPI Gamma3).*

We extend to high-order curved elements a recently introduced metric-based anisotropic mesh adaptation strategy for accelerated boundary element methods (e.g. Fast Multipole(FM-) BEM) applied to exterior boundary value problems. This method derives from an adaptation framework for volumetric finite element methods and is based on an iterative procedure that completely remeshes at each refinement step and that leads to a strategy that is independent of discretization technique (e.g., collocation or Galerkin) and integral representation (e.g., single- or double-layer). In effect, it results in a truly anisotropic adaptation that alters the size, shape and orientation of each element according to an optimal metric based on a numerically recovered Hessian of the boundary solution. The algorithm is principally characterized by its ability to recover optimal convergence rates for both flat and curved discretizations (e.g. ${P}_{0}$-, ${P}_{1}$- or ${P}_{2}$-elements) of a geometry containing singularities such as corners and edges. This is especially powerful for realistic geometries that include engineering detail (whose solutions often entail severe singular behavior).

Additionally, we address — by way of introducing hierarchical ($\mathscr{H}$-) matrix preconditioning applied to fast multipole methods via a Flexible GMRES (FGMRES) routine — the computational difficulties that arise when resolving highly anisotropic (and hence highly ill-conditioned) linear systems. The new technique, which uses a very coarse $\mathscr{H}$-matrix system (constructed rapidly via high-performance parallelization) to precondition the full Fast Multipole Method system, drastically reduces the overall computation time as well as the iterative solve time, further improving the tractability of addressing even larger and more complex geometries by FM-BEM.

#### Preconditioned $\mathscr{H}$-matrix based BEMs for wave propagation

Participants : Stéphanie Chaillat, Patrick Ciarlet, Félix Kpadonou.

We are interested with fast boundary element methods (BEMs) for the solution of acoustic and elastodynamic problems.

The discretisation of the boundary integral equations, using BEM, yields to a linear system, with a fully-populated matrix. Standard methods to solve this system are prohibitive in terms of memory requirements and solution time. Thus one is rapidly limited in terms of complexity of problems that can be solved. The $\mathscr{H}$-matrix based BEMs is commonly used to address these limitations. It is a purely algebraic approach.

The starting point is that the BEM matrix can be partitioned into some blocks which can either be of low or full rank. Memory can be saved by using low-rank revealing technique such as the Adaptive Cross Approximation. We have already study the efficiency of this approach for wave propagation problems. The purpose being the applications to large scale problems, we are now interested in an efficient implementation of the solver in a high performance computing setting. Thus, a bottleneck, with an hierarchical matrix data-sparse representation, is the management of the memory and its (prior) estimation for array allocations.

The first part of our work has been devoted to the proposition of an a priori estimation of the ranks of the blocks in the hierarchical matrix. Afterwards, we have implemented a parallel construction of the $\mathscr{H}$-matrix representation and H-matrix vector product (basic operation in any iterative solver), using a multi-threading OpenMP parallelization. The solution is then computed through the GMRES iterative solver. A crucial point is then the solution time of that solver and the number of iterations as the problem complexity increases. We have developed a two-level, nested outer-inner, iterative solver strategy. The inner solver preconditioned the outer. The preconditioner is a coarse data-sparse representation of the BEM system matrix.

#### Coupling integral equations and high-frequency methods

Participants : Marc Bonnet, Marc Lenoir, Eric Lunéville, Laure Pesudo.

This theme concerns wave propagation phenomena which involve two different space scales, namely, on the one hand, a medium scale associated with lengths of the same order of magnitude as the wavelength (medium-frequency regime) and on the other hand, a long scale related to lengths which are large compared to the wavelength (high-frequency regime). Integral equation methods are known to be well suited for the former, whereas high-frequency methods such as geometric optics are generally used for the latter. Because of the presence of both scales, both kinds of simulation methods are simultaneously needed but these techniques do not lend themselves easily to coupling.

The scattering of an acoustic wave by two sound-hard obstacles: a large obstacle subject to high-frequency regime relatively to the wavelength and a small one subject to medium-frequency regime has been investigated by Marc Lenoir, Eric Lunéville and Laure Pesudo. The technique proposed in this case consists in an iterative method which allows to decouple the two obstacles and to use Geometric Optics or Physical Optics for the large obstacle and Boundary Element Method for the small obstacle. This approach has been validated on various situations using the XLife++ library developed in the lab. When the obstacles are not sticked, even if they are very close, the iterative method coupling BEM and some high-frequency methods (ray approximation or Kirchoff approximation) works very well. When the obstacle are sticked, the "natural" iterative method is no longer convergent. We are currently looking for some improved methods to deal with these cases that have a practical interest.

#### The eddy current model as a low-frequency, high-conductivity asymptotic form of the Maxwell transmission problem

Participant : Marc Bonnet.

In this work, done in collaboration with Edouard Demaldent (CEA LIST), we study the relationship between the Maxwell and eddy current (EC) models for three-dimensional configurations involving highly-conducting bounded bodies in air and sources placed remotely from those bodies. Such configurations typically occur in the numerical simulation of eddy current non destructive testing (ECT). The underlying Maxwell transmission problem is formulated using boundary integral formulations of PMCHWT type. In this context, we derive and rigorously justify an asymptotic expansion of the Maxwell integral problem with respect to the non-dimensional parameter $\gamma :=\sqrt{\omega {\epsilon}_{0}/\sigma}$. The EC integral problem is shown to constitute the limiting form of the Maxwell integral problem as $\gamma \to 0$, i.e. as its low-frequency and high-conductivity limit. Estimates in $\gamma $ are obtained for the solution remainders (in terms of the surface currents, which are the primary unknowns of the PMCHWT problem, and the electromagnetic fields) and the impedance variation measured at the extremities of the excitating coil. In particular, the leading and remainder orders in $\gamma $ of the surface currents are found to depend on the current component (electric or magnetic, charge-free or not). Three-dimensional illustrative numerical simulations corroborate these theoretical findings.

#### Modelling the fluid-structure coupling caused by a far-field underwater explosion

Participants : Marc Bonnet, Stéphanie Chaillat, Damien Mavaleix-Marchessoux.

This work, funded by Naval Group and a CIFRE PhD grant, addresses the computational modelling of the mechanical effect on ships of remote underwater explosions. We aim at a comprehensive modelling approach that accounts for the effect of the initial (fast) wave impinging the ship as well as that of later, slower, water motions. Both fluid motion regimes are treated by boundary element methods (respectively for the wave and potential flow models), while the structure is modelled using finite elements. To cater for large and geometrically complex structures, the BEM-FEM interface requires large numbers of DOFs, which entails the use of a fast BEM solver. Accordingly, the wave-like fluid motions are to be computed by means of the convolution quadrature method (CQM) implemented in the in-house fast BEM code COFFEE. This work is in progress (the thesis having started in Dec. 2017). Work accomplished so far has mainly consisted in (a) thoroughly examinating the physical modelling issues, (b) formulating the mathematical and computational model that takes relevant physical features into account, and (c) implementing and assessing the CQM under conditions similar to those of the aimed application.