In this project, we investigate theoretical and numerical mathematical issues concerning heterogeneous physical systems. The heterogeneities we consider result from the fact that the studied systems involve subsystems of different physical nature. In this wide class of problems, we study two types of systems: fluid-structure interaction systems (FSIS) and complex wave systems (CWS). In both situations, one has to develop specific methods to take the coupling between the subsystems into account.

(FSIS) Fluid-structure interaction systems appear in many applications: medicine (motion of the blood in veins and arteries), biology (animal locomotion in a fluid, such as swimming fishes or flapping birds but also locomotion of microorganisms, such as amoebas), civil engineering (design of bridges or any structure exposed to the wind or the flow of a river), naval architecture (design of boats and submarines, researching into new propulsion systems for underwater vehicles by imitating the locomotion of aquatic animals).
FSIS can be studied by modeling their motions through Partial Differential Equations (PDE) and/or Ordinary Differential Equations (ODE), as is classical in fluid mechanics or in solid mechanics.
This leads to the study of difficult nonlinear free boundary problems which have constituted a rich and active domain of research over the last decades.

(CWS) Complex wave systems are involved in a large number of applications in several areas of science and engineering: medicine (breast cancer detection, kidney stone destruction, osteoporosis diagnosis, etc.), telecommunications (in urban or submarine environments, optical fibers, etc.), aeronautics (target detection, aircraft noise reduction, etc.) and, in the longer term, quantum supercomputers.
Direct problems, that is finding a solution with respect to parameters of the problem, for instance the propagation of waves with respect to the knowledge of speed of propagation of the medium, most theoretical issues are now widely understood. However, substantial efforts remain to be undertaken concerning the simulation of wave propagation in complex media. Such situations include heterogeneous media with strong local variations of the physical properties (high frequency scattering, multiple scattering media) or quantum fluids (Bose-Einstein condensates). In the first case for instance, the numerical simulation of such direct problems is a hard task, as it generally requires solving ill-conditioned possibly indefinite large size problems, following from space or space-time discretizations of linear or nonlinear evolution PDE set on unbounded domains. Inverse problems are the converse problem of the direct problems, as they aim to find properties of the direct problem, for instance the speed of propagation in a medium, with respect to the solution or a partial observation of the solution. These problems are often ill-posed and many questions are open at both the theoretical (identifiability, stability and robustness, etc.) and practical (reconstruction methods, approximation and convergence analysis, numerical algorithms, etc.) levels.

Fluid-Structure Interaction Systems (FSIS) are present in many physical problems and applications. Their study involves solving several challenging mathematical problems:

In order to control such FSIS, one has first to analyze the corresponding system of PDE. The oldest works on FSIS go back to the pioneering contributions of Thomson, Tait and Kirchhoff in the 19th century and Lamb in the 20th century, who considered simplified models (potential fluid or Stokes system). The first mathematical studies in the case of a viscous incompressible fluid modeled by the Navier-Stokes system and a rigid body whose dynamics is modeled by Newton's laws appeared much later 118, 112, 91, and almost all mathematical results on such FSIS have been obtained in the last twenty years.

The most studied FSIS is the problem modeling a rigid body moving in a viscous incompressible fluid
( 73, 70, 110, 81, 85, 114, 117, 100, 83).
Many other FSIS have been studied as well. Let us mention 102, 88, 84, 75, 61, 80, 62, 79 for different fluids.
The case of deformable structures has also been considered, either for a fluid inside a moving structure (e.g. blood motion in arteries)
or for a moving deformable structure immersed in a fluid (e.g. fish locomotion).
The obtained coupled FSIS is a complex system and its study raises several difficulties. The main one comes from the fact that we gather two systems of different nature. Some studies have been performed for approximations of this system: 66, 61, 94, 74, 64).
Without approximations, the only known results 71, 72 were obtained with very strong assumptions on the regularity of the initial data.
Such assumptions are not satisfactory but seem inherent to this coupling between two systems of different natures. In order to study self-propelled motions of structures in a fluid, like fish locomotion, one can assume that the deformation of the structure is prescribed and known, whereas its displacement remains unknown (107). This permits to start the mathematical study of a challenging problem: understanding the locomotion mechanism of aquatic animals.
This is related to control or stabilization problems for FSIS. Some first results in this direction were obtained in 89, 63, 104.

The area of inverse problems covers a large class of theoretical and practical issues which are important in many applications (see for instance the books of Isakov 90 or Kaltenbacher, Neubauer, and Scherzer 92). Roughly speaking, an inverse problem is a problem where one attempts to recover an unknown property of a given system from its response to an external probing signal. For systems described by evolution PDE, one can be interested in the reconstruction from partial measurements of the state (initial, final or current), the inputs (a source term, for instance) or the parameters of the model (a physical coefficient for example). For stationary or periodic problems (i.e. problems where the time dependency is given), one can be interested in determining from boundary data a local heterogeneity (shape of an obstacle, value of a physical coefficient describing the medium, etc.). Such inverse problems are known to be generally ill-posed and their study raises the following questions:

We can split our research in inverse problems into two classes which both appear in FSIS and CWS:

Identification for evolution PDE.

Driven by applications, the identification problem for systems of infinite dimension described by evolution PDE has seen in the last three decades a fast and significant growth. The unknown to be recovered can be the (initial/final) state (e.g. state estimation problems 56, 82, 87, 113 for the design of feedback controllers), an input (for instance source inverse problems 53, 65, 76) or a parameter of the system. These problems are generally ill-posed and many regularization approaches have been developed. Among the different methods used for identification, let us mention optimization techniques ( 69), specific one-dimensional techniques (like in 57) or observer-based methods as in 97.

In the last few years, we have developed observers to solve initial data inverse problems for a class of linear systems of infinite dimension. Let us recall that observers, or Luenberger observers 96, have been introduced in automatic control theory to estimate the state of a dynamical system of finite dimension from the knowledge of an output (for more references, see for instance 101 or 115). Using observers, we have proposed in 103, 86 an iterative algorithm to reconstruct initial data from partial measurements for some evolution equations. We are deepening our activities in this direction by considering more general operators or more general sources and the reconstruction of coefficients for the wave equation. In connection with this problem, we study the stability in the determination of these coefficients. To achieve this, we use geometrical optics, which is a classical albeit powerful tool to obtain quantitative stability estimates on some inverse problems with a geometrical background, see for instance 59, 58.

Geometric inverse problems.

We investigate some geometric inverse problems that appear naturally in many applications, like medical imaging and non destructive testing. A typical problem we have in mind is the following: given a domain

where

Within the team, we have developed in the last few years numerical codes for the simulation of FSIS and CWS. We plan to continue our efforts in this direction.

Below, we explain in detail the corresponding scientific program.

Some companies aim at building biomimetic robots that can swim in an aquarium, as toys but also for medical purposes. An objective of Sphinx is to model and to analyze several models of these robotic swimmers. For the moment, we focus on the motion of a nanorobot. In that case, the size of the swimmers leads us to neglect the inertia forces and to only consider the viscosity effects. Such nanorobots could be used for medical purposes to deliver some medicine or perform small surgical operations. In order to get a better understanding of such robotic swimmers, we have obtained control results via shape changes and we have developed simulation tools (see 67, 68, 98, 95). Among all the important issues, we aim to consider the following ones:

The main tools for this investigation are the 3D codes that we have developed for simulation of fish in a viscous incompressible fluid (SUSHI3D) or in an inviscid incompressible fluid (SOLEIL).

We will develop robust and efficient solvers for problems arising in aeronautics (or aerospace) like electromagnetic compatibility and acoustic problems related to noise reduction in an aircraft. Our interest for these issues is motivated by our close contacts with companies like Airbus or “Thales Systèmes Aéroportés”. We will propose new applications needed by these partners and assist them in integrating these new scientific developments in their home-made solvers. In particular, in collaboration with C. Geuzaine (Université de Liège), we are building a freely available parallel solver based on Domain Decomposition Methods that can handle complex engineering simulations, in terms of geometry, discretization methods as well as physics problems, see here.

Control

Controlling coupled systems is a complex issue depending on the coupling conditions and the equations themselves. Our team has a strong expertise to tackle these kind of problems in the context of fluid-structure interaction systems. More precisely, we obtained the follwing results.

In 40, Badra and Takahashi consider the controllability of an abstract parabolic system by using switching controls. More precisely, we show that under general hypotheses, if a parabolic system is null-controllable for any positive time with N controls, then it is also null-controllable with the property that at each time, only one of these controls is active. The main difference with previous results in the literature is that we can handle the case where the main operator of the system is not self-adjoint. We give several examples to illustrate our result: coupled heat equations with terms of orders 0 and 1, the Oseen system or the Boussinesq system.

In 43, the authors prove an Hölder type inequality reflecting the unique continuation property at one time for the heat equation with a potential and Neumann boundary condition. The main feature of the proof is to overcome the propagation of smallness by a global approach using a refined parabolic frequency function method. It relies on a Carleman commutator estimate to obtain the logarithmic convexity property of the frequency function.

In 21, Imene Djebour
shows the local null controllability of a fluid-solid interaction system by
using a distributed control located in the fluid.
The fluid is modeled by the incompressible Navier-Stokes system with Navier
slip boundary conditions and the rigid body is governed by Newton's laws.
Her main result yields that one can drive the velocities of the fluid and of
the structure to 0 and one can control exactly the position of the rigid
body.
One important ingredient of the proof consists in a new Carleman estimate
for a linear fluid-rigid body system with Navier boundary conditions.

In 41, we prove a Lebeau-Robbiano spectral inequality for the Oseen operator in a two dimensional channel, that is, the linearized Navier-Stokes operator around a laminar flow, with no-slip boundary conditions. This is done by deriving a proper Carleman estimate by handling the vorticity near the boundary using two different characteristic sets in the different microlocal regions of the cotangent space. As a consequence of the spectral inequality, we derive a new estimate of the cost of the control for the small-time null-controllability.

In 42, we are interested in the controllability of a fluid-structure interaction system where the fluid is viscous and incompressible and where the structure is elastic and located on a part of the boundary of the fluid's domain. In this article, we simplify this system by considering a linearization and by replacing the wave/plate equation for the structure by a heat equation. We show that the corresponding system coupling the Stokes equations with a heat equation at its boundary is null-controllable. The proof is based on Carleman estimates and interpolation inequalities. One of the Carleman estimates corresponds to the case of Ventcel boundary conditions. This work can be seen as a first step to handle the real system where the structure is modeled by the wave or the plate equation.

The convergence of numerical controls for the wave equation is investigated for a Galerkin semi-discretization. The convergence of the numerical approximation for this equation is notoriously difficult as usual discretization schemes introduce spurious high frequencies. Filtering techniques are known in the literature for finite element methods. We introduced for the first time in 26 low cost filtering techniques for Galerkin approximations.

In 45, the controllability properties of the ground state solitary wave is studied for the mass critical and subcritical focusing Schrödinger equation. Using a fine description of the blow-up profile, Gagnon proves the local controllability between the ground state with two different scaling in a minimal time. This result provides insight on the technique needed to disrupt the stability of the ground state to gain controllability.

In 24, the dynamics of a particle trapped on a network in presence of an external electromagnetic field is adressed. The controllability of the motion is studied when the intensity of the field changes over time and plays the role of control. From a mathematical point of view, the dynamics of the particle is modeled by the so-called bilinear Schrödinger equation defined on a graph representing the network. The main purpose of this work is to extend the existing theory for bilinear quantum systems on bounded intervals to the framework of graphs. To this end, we introduce a suitable mathematical setting where to address the controllability of the equation from a theoretical point of view. More precisely, we determine assumptions on the network and on the potential field ensuring its global exact controllability in suitable spaces. Finally, we discuss two applications of our results and their practical implications to two specific problems involving a star-shaped network and a tadpole graph.

In 52, the controllability properties of a system of $m$ coupled Stokes systems or $m$ coupled Navier-Stokes systems are studied. The null-controllability of such systems is proved in the case where the coupling is in a cascade form and when the control acts only on one of the systems. Moreover, we impose that this control has a vanishing component so that we control a

Stabilization

Stabilization of infinite dimensional systems governed by PDE is a challenging problem. In our team, we have investigated this issue for different kinds of systems (fluid systems and wave systems) using different techniques.

In 48, Guerrero and Takahashi consider the controllability of a viscous incompressible fluid modeled by the Navier-Stokes system with a nonlinear viscosity. To prove the controllability to trajectories, we linearize around a trajectory and the corresponding linear system includes a nonlocal spatial term. Our main result is a Carleman estimate for the adjoint of this linear system. This estimate yields in a standard way the null controllability of the linear system and the local controllability to trajectories. Our method to obtain the Carleman estimate is completely general and can be adapted to other parabolic systems when a Carleman estimate is available.

In 22, Imene Djebour, Takéo Takahashi and Julie Valein consider the stabilization of parabolic systems with a finite-dimensional control subjected to a constant delay. Their main result shows that the Fattorini-Hautus criterion yields the existence of such a feedback control, as in the case of stabilization without delay. The proof consists in splitting the system into a finite dimensional unstable part and a stable infinite-dimensional part and in applying the Artstein transformation on the finite-dimensional system to remove the delay in the control. Using this abstract result, they can prove new results for the stabilization of parabolic systems with constant delay: the
N
-dimensional linear reaction-convection-diffusion equation with

The aim of 36 is to study the asymptotic stability of the nonlinear Korteweg-de Vries equation in the presence of a delayed term in the internal feedback. First, the case where the weight of the term with delay is smaller than the weight of the term without delay is considered and a semiglobal stability result for any length is proved. Secondly, the case where the support of the term without delay is not included in the support of the term with delay is considered. In this case, a local exponential stability result is proved provided the weight of the delayed term is small enough. These results are illustrated by some numerical simulations. The above results on the stabilization of delay systems, added to other contributions on the control and stabilization of PDE constitute the material of the habilitation thesis 116 of Julie Valein, defended on November 4th 2020.

In 25, Ludovick Gagnon, Pierre Lissy and Swann Marx prove the exponential decay of a degenerate parabolic equation. The equation has a degeneracy at

In 46, the backstepping method with the Fredholm alternative is thoroughly studied for the Laplacian operator on the torus. A sharp functional setting is presented in this setting and the stabilization of the heat equation on the torus with two feedback laws is presented as an application.

In 44, Imene Djebour considers a fluid-structure interaction system composed by a three-dimensional viscous incompressible fluid and an elastic plate located on the upper part of the fluid boundary. The fluid motion is governed by the Navier-Stokes system whereas the structure displacement satisfies the damped plate equation. We consider here the Navier slip boundary conditions. The main result of this work is the feedback stabilization of the strong solutions of the corresponding system around a stationary state for any exponential decay rate by means of a time delayed control localized on the fixed fluid boundary. The strategy here is based on the Fattorini-Hautus criterion. Then, the main tool in this work is to show the unique continuation property of the associate solution to the adjoint system.

Optimization

We have also considered optimization issues for fluid-structure interaction systems.

J.F. Scheid, V. Calesti and I. Lucardesi study an optimal shape problem for an elastic structure immersed in a viscous incompressible fluid. They aim to establish the existence of an optimal elastic domain associated with an energy-type functional for a Stokes-Elasticity system. They want to find an optimal reference domain (the domain before deformation) for the elasticity problem that minimizes an energy-type functional. This problem is concerned with 2D geometry and is an extension of 111 for a 1D problem. The optimal domain is searched for in a class of admissible open sets defined with a diffeomorphism of a given domain. The main difficulty lies in the coupling between the Stokes problem written in a eulerian frame and the linear elasticity problem written in a lagrangian form. The shape derivative of an energy-type functional has been formally obtained. This will allow us to numerically determine an optimal elastic domain which minimizes the energy-type functional under consideration. The rigorous proof of the derivability of the energy-type functional with respect to the domain is still in progress.

The article 31 is devoted to the mathematical analysis of a fluid-structure interaction system where the fluid is compressible and heat conducting and where the structure is deformable and located on a part of the boundary of the fluid domain. The fluid motion is modeled by the compressible Navier-Stokes-Fourier system and the structure displacement is described by a structurally damped plate equation. Our main results are the existence of strong solutions in an

In 39, Badra and Takahashi consider a viscous incompressible fluid interacting with an elastic structure located on a part of its boundary. The fluid motion is modeled by the bi-dimensional Navier-Stokes system and the structure follows the linear wave equation in dimension 1 in space. Our aim is to study the linearized system coupling the Stokes system with a wave equation and to show that the corresponding semigroup is analytic. In particular the linear system satisfies a maximal regularity property that allows us to deduce the existence and uniqueness of strong solutions for the nonlinear system. This result can be compared to the case where the elastic structure is a beam equation for which the corresponding semigroup is only of Gevrey class.

Direct problems

Negative materials are artificially structured composite materials (also known as metamaterials), whose dielectric permittivity and magnetic permeability are simultaneously negative in some frequency ranges. K. Ramdani continued his collaboration with R. Bunoiu on the homogenization of composite materials involving both positive and negative materials. Due to the sign-changing coefficients in the equations, classical homogenization theory fails, since it is based on uniform energy estimates which are known only for positive (more precisely constant sign) coefficients. In a joint work with L. Chesnel and M. Rihani, the authors studied 19 the vector case of Maxwell'es equations. In 20 and 35, both in collaboration with C. Timofte, two “degenerate” situations are respectively considered : the case of thin periodic domains and the one of extreme contrasts.

Inverse problems

Alexandre Munnier and Karim Ramdani have obtained a PhD funding from Université de Lorraine to supervise the PhD of Anthony Gerber-Roth. The thesis is devoted to the investigation of some geometric inverse problems, and can be seen as a continuation of the work initiated by the two supervisors in 99 and 8. In these papers, the authors addressed a particular case of Calderón’s inverse problem in dimension two, namely the case of a homogeneous background containing a finite number of cavities (i.e. heterogeneities of infinitely high conductivities). They proposed a non iterative method to reconstruct the cavities from the knowledge of the Dirichlet-to-Neumann map of the problem. The first contribution of Anthony Gerber-Roth is to extend the results obtained in 8 in dimension three. This work is in progress.

Besides these static inverse problems, we also investigate estimation issues for time-dependent problems.

Computational acoustics.

Artificial boundary conditions/PML.

New stable PML (Perfectly Matched Layers) have been proposed in 33 for solving the convected Helmholtz
equation for future industrial applications with Siemens (ongoing
CIFRE Ph.D. thesis of Philippe Marchner).

Numerical approximation by volume methods.

In 23, the authors propose a new high precision Iso-Geometric Analysis (IGA) B-Spline approximation of the high frequency scattering Helmholtz problem, which minimizes the numerical pollution effects that affect standard Galerkin finite element approaches when combined with HABC.

In the papers 32, 34, 50, we build and evaluate some new absorbing boundary conditions for the heterogeneous Helmholtz equation, two-dimensional Schrödinger equation in the presence of corners and the 2D peridynamics equations based on kernel analysis, respectively.

In 51, we develop the numerical analysis of discretization schemes with absorbing boundary conditions for the one-dimensional Schrödinger equation where the Laplacian is replaced by a nonlocal spatial operator.

Integral equation approximation.

In 12, an extensive review of recent methods for preconditioning fast integral equation solvers is mainly developed for time-harmonic acoustics, but also for electromagnetic and elastic waves.

In 11, we introduce a coupling algorithm between the integral equation and OSRC methods to solve scattering problems by non convex obstacles.

In 47, the mathematical analysis of the steepest descent methods is investigated for the acooustic single-layer integral operator.

Scattering by moving boundaries.

A new frequency domain method has been introduced in 28 during the Ph.D. thesis of D. Gasperini to solve scattering problems by moving boundaries. This research was done during a contract with the company IEE (Luxembourg) for modeling the radar detection inside cars at very high frequency.

In 29, we propose an original coupled frequency domain approach for solving by the finite element method the scattering problem with a moving boundary for two- and three-dimensional problems.

The paper 37 introduces a new OSRC formulaion with phase reduction and approximated by IGA-NURBS to solve time-harmonic acoustic scattering problems.

In 18, a weak coupling finite element/boundary element method is introduced for solving 3D electromagnetic scattering problems.

Underwater acoustics.

In 49, we develop an efficient second-order scheme with HABC for the one-dimensional Green-Naghdi equation that arises in water waves. We propose an adaptive method so that the accuracy of the scheme is maintained while strongly accelerating the speed-up, in particular because of the presence of a nonlocal time convolution-type operator involved in the HABC.

Quantum theory.

In 27, we give an overview of the
BEC2HPC parallel solver developed in the BEC2HPC associated team for computing the stationary states of fast rotating BECs in 2D/3D. In 16, in collaboration with Q. Tang and J. Shen (Purdue University),
we propose some new efficient spectral schemes for the dynamics of the nonlinear Schrödinger
and Gross-Pitaevskii equations.

In 17, X. Antoine and X. Zhao (Wuhan University) introduce some
new locally smooth singular absorption profiles for the spectral numerical solution of the nonlinear Klein-Gordon equation.
In particular, this leads to an accuracy of the scheme that does not depend on the small parameter
arising in the non-relativistic regime. Applications are also given for the rotating Klein Gordon-equation
used in the modeling of the cosmic superfluid in a rotating frame.

Fractional PDE.

In 30, with S. Ji, G. Pang, and J. Zhang, Xavier Antoine is interested in the development
and analysis of artificial boundary conditions for nonlocal Schrödinger equations that are a generalization
of some fractional Schrödinger equations.

In 13, 14, the numerical computation of fractional linear systems involving several matrix power functions. We propose several gradient methods for solving these very computationally complex problems, which themselves require the solution to standard Fractional Linear Systems. The convergence study is developed and numerical experiments are proposed to illustrate and compare the methods.

The authors propose in 15 the construction and implementation of PML operators for the one- and two-dimensional fractional Laplacian, and some extensions.

In 38, a Schwarz waveform relaxation domain decomposition method has been introduced for solving space fractional PDE related to Schrödinger and heat equations.

Fluid mechanics.

Chaotic advection in a viscous fluid under an electromagnetic field. J.-F. Scheid, J.-P. Brancher (IECL) and J. Fontchastagner (GREEN) study the chaotic behavior of trajectories of a dynamical system arising from a coupling
system beetwen Stokes flow and an electromagnetic field. They consider an electrically conductive
viscous fluid crossed by a uniform electric current. The fluid is subjected to a magnetic field induced
by the presence of a set of magnets. The resulting electromagnetic force acts on the conductive fluid
and generates a flow in the fluid. According to a specific arrangement of the magnets surrounding the
fluid, vortices can be generated and the trajectories of the dynamical system associated to the stationary
velocity field in the fluid may have chaotic behavior. The aim of this study is to numerically show the
chaotic behavior of the flow for the proposed disposition of the magnets along the container of the fluid.
The flow in the fluid is governed by the Stokes equations with the Laplace force induced by the electric
current and the magnetic field. An article is in preparation.

Karim Ramdani and Julie Valein were invited to give a talk in the conference “Control and analysis of PDE systems” in honor of Marius Tucsnak 60th birthday (Bordeaux, Nov. 29–Dec. 1 2021).

Except L. Gagnon, K. Ramdani, T. Takahashi and J.-C. Vivalda, SPHINX members have teaching obligations at “Université de Lorraine” and are teaching at least 192 hours each year. They teach mathematics at different level (Licence, Master, Engineering school). Many of them have pedagogical responsibilities.

The following PhD thesis was defended this year:

The following PhD thesis are in progress: