## Section: New Results

### Numerical Methods

#### Numerical analysis for an energy-stable total discretization of a poromechanics model with inf-sup stability

Participants : Dominique Chapelle [correspondant] , Philippe Moireau.

In this joint work with Bruno Burtschell [16], we consider a previously proposed general nonlinear poromechanical formulation, and we derive a linearized version of this model. For this linearized model, we obtain an existence result and we propose a complete discretization strategy–in time and space–with a special concern for issues associated with incompressible or nearly-incompressible behavior. We provide a detailed mathematical analysis of this strategy, the main result being an error estimate uniform with respect to the compressibility parameter. We then illustrate our approach with detailed simulation results and we numerically investigate the importance of the assumptions made in the analysis, including the fulfillment of specific inf-sup conditions.

#### Conservative and entropy controlled remap for multi-material ALE simulations

Participant : Patrick Le Tallec.

For many multi-material problems such as fluid-structure interaction, impact or implosion problems, materials are in very large strains due to their nature or to the applied forces. In our situations of interest, we also have a strong coupling between energy and momentum conservation laws, due to intense transfers between internal and kinetic energies and to strong advection effects. Such situations are classically governed by the Euler's equations, written in Lagrangian form, and using a multi-material, single velocity framework, but their numerical solution demands a strict control of energy conservation and entropy production, which is hard to achieve in situations where dynamic remeshing is mandatory. In this framework, our approach deals with the analysis of the impact of a second-order staggered remap using an intersection-based approach on conservation properties and on the entropy control. We show that an accurate remap with exact mesh intersections and exact integrations affects both the momentum and the kinetic energy because of node mass re-localizations and node velocity remap. We propose therefore a staggered remapping strategy in order to take into account these discrepancies at a low computational cost. While preserving the strict conservation of total energy, our strategy allows to recover a proper entropy control at the expense of strict momentum conservation and monotonicity losses. This work [32] is done in collaboration with Alexandra Claisse (CEA DAM) and Alexis Marboeuf (École Polytechnique and CEA DAM).

#### Multipatch isogeometric analysis for complex structures

Participant : Patrick Le Tallec.

This work – done in collaboration with Nicolas Adam (École Polytechnique and PSA) and Malek Zarroug (PSA) – introduces, analyzes and validates isogeometric mortar methods for the solution of thick shells problems which are set on a multipatch geometry. It concerns industrial parts of complex geometries for which the effects of transverse shear cannot be neglected. For this purpose, Reissner-Mindlin model was retained and rotational degrees of freedom (DOF) of the normal are taken into account. A particular attention is devoted to the introduction of a proper formulation of the coupling conditions at patches interfaces, with a particular interest on augmented lagrangian formulations, to the choice and validation of mortar spaces, and to the derivation of adequate integration rules. The relevance of the proposed approach is assessed numerically on various significative examples of industrial relevance. This work has been submitted for publication in an international journal.

#### Mathematical and numerical study of transient wave scattering by obstacles with the Arlequin Method

Participant : Sébastien Imperiale.

In this work [14] we extend the Arlequin method to overlapping domain decomposition technique for transient wave equation scattering by obstacles. The main contribution of this work is to construct and analyze from the continuous level up to the fully discrete level some variants of the Arlequin method. The constructed discretizations allow to solve wave propagation problems while using non-conforming and overlapping meshes for the background propagating medium and the surrounding of the obstacle respectively. Hence we obtain a flexible and stable method in terms of the space discretization – an inf-sup condition is proven – while the stability of the time discretization is ensured by energy identities.

#### Construction and analysis of fourth-order, energy consistent, family of explicit time discretizations for dissipative linear wave equations

Participants : Juliette Chabassier [MAGIQUE-3D] , Julien Diaz [MAGIQUE-3D] , Sébastien Imperiale [correspondant] .

This work and the corresponding article [19], deal with the construction of a family of fourth order, energy consistent, explicit time discretizations for dissipative linear wave equations. The schemes are obtained by replacing the inversion of a matrix, that comes naturally after using the technique of the Modified Equation on the second order Leap Frog scheme applied to dissipative linear wave equations, by explicit approximations of its inverse. The stability of the schemes are studied using an energy analysis and a convergence analysis is carried out. Numerical results in 1D illustrate the space/time convergence properties of the schemes and their efficiency is compared to more classical time discretizations.

#### Energy decay and stability of a perfectly matched layer For the wave equation

Participants : Sébastien Imperiale [correspondant] , Maryna Kachanovska [POEMS] .

We follow a previous work where PML formulations was proposed for the wave equation in its standard second-order form. In the present work [15], energy decay and ${L}^{2}$ stability bounds in two and three space dimensions are rigorously proved both for continuous and discrete formulations with constant damping coefficients. Numerical results validate the theory.

#### A high-order spectral element fast Fourier transform for the poisson equation

Participants : Federica Caforio, Sébastien Imperiale [correspondant] .

The aim of this work [17] is to propose a novel, fast solver for the Poisson problem discretised with High-Order Spectral Element Methods (HO-SEM) in canonical geometries (rectangle in 2D, rectangular parallelepiped in 3D). This method is based on the use of the Discrete Fourier Transform to reduce the problem to the inversion of the symbol of the operator in the frequency space. The proposed solver is endowed with several properties. First, it preserves the efficiency of the standard FFT algorithm; then, the matrix storage is drastically reduced (in particular, it is independent of the space dimension); a pseudo-explicit Singular Value Decomposition (SVD) is used for the inversion of the symbols; finally, it can be extended to non-periodic boundary conditions. Furthermore, due to the underlying HO-SEM discretisation, the multi-dimensional symbol of the operator can be efficiently computed from the one-dimensional symbol by tensorisation.

#### Thermodynamic properties of muscle contraction models and associated discrete-time principles

Participants : François Kimmig, Dominique Chapelle [correspondant] , Philippe Moireau.

Considering a large class of muscle contraction models accounting for actin-myosin interaction, we present a mathematical setting in which solution properties can be established, including fundamental thermodynamic balances. Moreover, we propose a complete discretization strategy for which we are also able to obtain discrete versions of the thermodynamic balances and other properties. Our major objective is to show how the thermodynamics of such models can be tracked after discretization, including when they are coupled to a macroscopic muscle formulation in the realm of continuum mechanics. Our approach allows to carefully identify the sources of energy and entropy in the system, and to follow them up to the numerical applications. See [30] for more detail.

#### Mechanical and imaging models-based image registration

Participants : Radomir Chabiniok, Martin Genet [correspondant] .

Image registration plays an increasingly important role in many fields such as biomedical or mechanical engineering. Generally speaking, it consists in deforming a (moving) source image to match a (fixed) template image. Many approaches have been proposed over the years; if new model-free machine learning-based approaches are now beginning to provide robust and accurate results, extracting motion from images is still most commonly based on combining some statistical analysis of the images intensity and some model of the underlying deformation as initial guess or regularizer. These approaches may be efficient even for complex type of motion; however, any artifact in the source image (e.g., partial voluming, local decrease of signal-to-noise ratio or even local signal void), drastically deteriorates the registration. This work introduces a novel approach of extracting motion from biomedical image series, based on a model of the imaging modality. It is, to a large extent, independent of the type of model and image data – the pre-requisite is to incorporate biomechanical constraints into the motion of the object (organ) of interest and being able to generate data corresponding to the real image, i.e., having an imaging model at hand. We will illustrate the method with examples of synthetically generated 2D tagged magnetic resonance images. This work was presented at the VipIMAGE 2019 conference. It also represents a part of the objectives supported by the Inria-UTSW Associated Team TOFMOD. See [44] for more detail. This work was done in collaboration with Katerina Skardova (Czech Technical University in Prague) and Matthias Rambausek (École Polytechnique).

#### Validation of finite element image registration-based cardiac strain estimation from magnetic resonance images

Participants : Martin Genet [correspondant] , Philippe Moireau.

Accurate assessment of regional and global function of the heart is an important readout for the diagnosis and routine evaluation of cardiac patients. Indeed, recent clinical and experimental studies suggest that compared to global metrics, regional measures of function could allow for more accurate diagnosis and early intervention for many cardiac diseases. Although global strain measures derived from tagged magnetic resonance (MR) imaging have been shown to be reproducible for the majority of image registration techniques, the measurement of regional heterogeneity of strain is less robust. Moreover, radial strain is underestimated with the current techniques even globally. Finite element (FE)-based techniques offer a mechanistic approach for the regularization of the ill-posed registration problem. This work presents the validation of a recently proposed FE-based image registration method with mechanical regularization named equilibrated warping. For this purpose, synthetic 3D-tagged MR images are generated from a reference biomechanical model of the left ventricle (LV). The performance of the registration algorithm is consequently tested on the images with different signal-to-noise ratios (SNRs), revealing the robustness of the method. See [35] for more detail.