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In this paper we consider a fluid-structure interaction problem given by the steady Navier Stokes equations coupled with linear elasticity taken from (Lasiecka et al. in Nonlinear Anal 44:54–85, 2018). An elastic body surrounded by a liquid in a rectangular domain is deformed by the flow which can be controlled by the Dirichlet boundary condition at the inlet. On the walls along the channel homogeneous Dirichlet boundary conditions and on the outflow boundary do-nothing conditions are prescribed. We recall existence results for the nonlinear system from that reference and analyze the control to state mapping generalizing the results of (Wollner and Wick in J Math Fluid Mech 21:34, 2019) to the setting of the nonlinear Navier-Stokes equation for the fluid and the situation of mixed boundary conditions in a domain with corners.

We construct gradient structures for free boundary problems with moving capillary interfaces with nonlinear (hyper)elasticity and study the impact of moving contact lines. In this context, we numerically analyze how phase-field models converge to certain sharp-interface models when the interface thickness tends to zero. In particular, we study the scaling of the Cahn--Hilliard mobility with certain powers of the interfacial thickness. In the presence of interfaces, it is known that the intended sharp-interface limit holds only for a particular range of powers However, in the presence of moving contact lines we show that some scalings that are valid for interfaces produce significant errors and the effective range of valid powers of the interfacial thickness in the mobility reduces.

We consider a non-linear system modelling the dynamics of a linearly elastic body immersed in an incompressible viscous fluid. We prove local existence of strong solutions and global existence and uniqueness for small data. The main result is the characterization of long-time behaviour of the elastic displacement. We show convergence either to a rest state or rigid motion, or to a time-periodic pressure wave that may occur only in specific geometric settings., Comment: 39 pages, 1 figure

We propose a numerical methodology for the simultaneous numerical simulation of four states of matter: gas, liquid, elastoplastic solids, and plasma. The distinct, interacting physical processes are described by a combination of compressible, inert, and reactive forms of the Euler equations, multi-phase equations, elastoplastic equations, and resistive MHD equations. Combinations of systems of equations are usually solved by coupling finite element for solid modelling and CFD models for fluid modelling or including material effects through boundary conditions rather than full material discretisation. Our simultaneous solution methodology lies on the recasting of all the equations in the same, hyperbolic form allowing their solution on the same grid with the same finite volume numerical schemes. We use a combination of sharp- and diffuse-interface methods to track or capture material interfaces, depending on the application. The communication between the distinct systems of equations (i.e., materials separated by sharp interfaces) is facilitated by means of mixed-material Riemann solvers at the boundaries of the systems, which represent physical material boundaries. To this end, we derive approximate mixed-material Riemann solvers for each pair of the above models based on characteristic equations. To demonstrate the applicability of the new methodology, we consider a case study, where we investigate the possibility of ignition of a combustible gas that lies over a liquid in a metal container that is struck by a plasma arc akin to a lightning strike. We study the effect of the metal container material and its conductivity on the ignition of the combustible gas, as well as the effects of an additional dielectric coating, the sensitivity of the gas, and differences between scenarios with sealed and pre-damaged metal surfaces.

For many biological systems that involve elastic structures immersed in fluid, small length scales mean that inertial effects are also small, and the fluid obeys the Stokes equations. One way to solve the model equations representing such systems is through the Stokeslet, the fundamental solution to the Stokes equations, and its regularized counterpart, which treats the singularity of the velocity at points where force is applied. In two dimensions, an additional complication arises from Stokes' paradox, whereby the velocity from the Stokeslet is unbounded at infinity when the net hydrodynamic force within the domain is nonzero, invalidating the solutions. A straightforward computationally inexpensive method is presented for obtaining valid solutions to the Stokes equations for net nonzero forcing. The approach is based on imposing a mean zero velocity condition on a large curve that surrounds the domain of interest. The condition is shown to be equivalent to a net-zero force condition, where the opposite forces are applied on the large curve. The numerical method is applied to models of cellular motility and blebbing., Comment: 33 pages, 9 figures, submitted to Communications in Applied Mathematics and Computational Science

In this paper we consider a fluid-structure interaction problem given by the steady Navier Stokes equations coupled with linear elasticity taken from [Lasiecka, Szulc, and Zochoswki, Nonl. Anal.: Real World Appl., 44, 2018]. An elastic body surrounded by a liquid in a rectangular domain is deformed by the flow which can be controlled by the Dirichlet boundary condition at the inlet. On the walls along the channel homogeneous Dirichlet boundary conditions and on the outflow boundary do-nothing conditions are prescribed. We recall existence results for the nonlinear system from that reference and analyze the control to state mapping generaziling the results of [Wollner and Wick, J. Math. Fluid Mech., 21, 2019] to the setting of the nonlinear Navier-Stokes equation for the fluid and the situation of mixed boundary conditions in a domain with corners.

We present a framework to systematically derive variational formulations for fluid-structure interaction problems based on thermodynamical driving functionals and geometric structures in different coordinate systems by suitable transformations within this formulation. Our approach provides a promising basis to construct structure-preserving discretization strategies.

We prove existence of a weak solution for a nonlinear, multi-physics, multi-scale problem of magnetorheological suspensions introduced in Nika & Vernescu (Z. Angew. Math. Phys., 71(1):1--19, '20). The hybrid model couples the Stokes' equation with the quasi-static Maxwell's equations through the Lorentz force and the Maxwell stress tensor. The proof of existence is based on: i) the augmented variational formulation of Maxwell's equations, ii) the definition of a new function space for the magnetic induction and the proof of a Poincaré type inequality, iii) the Altman--Shinbrot fixed point theorem when the magnetic Reynold's number, Rm, is small.

A derivation of the model for a poroelastic elliptic membrane shell is undertaken. The flow and deformation in a three-dimensional shell domain is described by the quasi-static Biot equations of linear poroelasticity. We consider the limit when the shell thickness goes to zero and look for the limit equations. Using the technique developed in the seminal articles by Ciarlet, Lods, Miara et al.and the recent results on the rigorous derivation of the equations for poroelastic plates and flexural poroelastic shells by Marciniak-Czochra, Mikelić, and Tambača, we present a rigorous derivation of the linear poroelastic elliptic membrane shell model. After rescaling, the corresponding velocity and the pressure field are close in the C([0, T] ; (H1x)2×(L2x)2) norm and the stresses in C([0, T] ; (L2x)9) norm. We note the major difference with respect to the flexural case: (i) it is not anymore the rescaled total stress divided by the scaling parameter, but the rescaled total stress itself which converges ; (ii) the same comment applies to the pore fluid pressure ; and (iii) there is a deterioration of the convergence for the vertical component of the rescaled displacement. Consequence of the above differences is that the effective model remains of the 2nd order in space. In the case of a spherical membrane shell, we confirm the results by Taber from the literature.

Flexible filaments and fibres are essential components of important complex fluids that appear in many biological and industrial settings. Direct simulations of these systems that capture the motion and deformation of many immersed filaments in suspension remain a formidable computational challenge due to the complex, coupled fluid--structure interactions of all filaments, the numerical stiffness associated with filament bending, and the various constraints that must be maintained as the filaments deform. In this paper, we address these challenges by first describing filament kinematics using quaternions to resolve both bending and twisting, applying implicit time-integration to alleviate numerical stiffness, and using quasi-Newton methods to obtain solutions to the resulting system of nonlinear equations. In particular, we employ geometric time integration to ensure that the quaternions remain unit as the filaments move. We also show that our framework can be used with a variety of models and methods, including matrix-free fast methods, that resolve low Reynolds number hydrodynamic interactions. We provide a series of tests and example simulations to demonstrate the performance and possible applications of our method. Finally, we provide a link to a MATLAB/Octave implementation of our framework that can be used to learn more about our approach and as a tool for filament simulation.

We study a recently introduced formulation for fluid-structure interaction problems which makes use of a distributed Lagrange multiplier in the spirit of the fictitious domain method. The time discretization of the problem leads to a mixed problem for which a rigorous stability analysis is provided. The finite element space discretization is discussed and optimal convergence estimates are proved.

Flexible filaments and fibres are essential components of important complex fluids that appear in many biological and industrial settings. Direct simulations of these systems that capture the motion and deformation of many immersed filaments in suspension remain a formidable computational challenge due to the complex, coupled fluid–structure interactions of all filaments, the numerical stiffness associated with filament bending, and the various constraints that must be maintained as the filaments deform. In this paper, we address these challenges by describing filament kinematics using quaternions to resolve both bending and twisting, applying implicit time-integration to alleviate numerical stiffness, and using quasi-Newton methods to obtain solutions to the resulting system of nonlinear equations. In particular, we employ geometric time integration to ensure that the quaternions remain unit as the filaments move. We also show that our framework can be used with a variety of models and methods, including matrix-free fast methods, that resolve low Reynolds number hydrodynamic interactions. We provide a series of tests and example simulations to demonstrate the performance and possible applications of our method. Finally, we provide a link to a MATLAB/Octave implementation of our framework that can be used to learn more about our approach and as a tool for filament simulation.

We consider periodically heterogeneous fluid-saturated poroelastic media described by the Biot model with inertia effects. The weak and semistrong formulations for displacement, seepage and pressure fields involve three equations expressing the momentum and mass balance and the Darcy law. Using the two-scale homogenization method, we obtain the limit two-scale problem and prove the existence and uniqueness of its weak solutions. The Laplace transformation in time is used to decouple the macroscopic and microscopic scales. It is shown that the seepage velocity is eliminated from the macroscopic equations involving strain and pressure fields only. The plane harmonic wave propagation is studied using an example of layered medium. Illustrations show some influence of the orthotropy on the dispersion phenomena.

We study a coupled system of equations describing the movement of a rigid body which is immersed in a viscoelastic fluid. It is shown that under natural assumptions on the data and for general goemetries of the rigid body, excluding contact scenarios, a unique local-in-time strong solution exists.

International audience; Thanks to the Cagniard-de Hoop we derive the solution to the problem of wave propagation in an infinite bilayered acoustic/poroelastic media, where the poroelastic layer is modelled by the biphasic Biot's model. This first part is dedicated to solution to the two dimensional problem. We illustrate the interest of the solution by using it to validate a numerical code.

The problem of feedback controlled Rayleigh-Bénard convection is considered. For this problem with the simple flow structure in the vertical direction, a Galerkin method that uses only a few basis functions in this direction is presented. This approximation yields considerable simplification of the problem, explicitly incorporates the non-classical boundary conditions at the horizontal boundaries of the fluid layer resulting from feedback control and reduces the dimension of the original problem by one. This method is in spirit very similar to lubrication theory, where the simple laminar flow in the vertical direction is integrated out across the height of the fluid layer. Using a minimal set of appropriate basis functions to capture the nonlinear behaviour of the flow, we investigate the effects of feedback control on amplitude, wavelength and selection of patterns via weakly nonlinear analysis and numerical simulations of the resulting dimension-reduced problems in two and three dimensions. In the second part of this study we discuss the derivation of the appropriate basis functions and prove convergence of the Galerkin scheme. © 2008 IOP Publishing Ltd and London Mathematical Society.

We consider the problem of phase separation in eutectic alloy such as e.g. SnPb. For this we derive a phase field model from an atomistic point of view. We find the surface energy to be anisotropic, having in general a nonlinear dependence on concentration. We use matched asymptotic analysis to obtain a corresponding sharp-interface model. The resulting expression for the surface tension agrees with that found on the basis of classical thermodynamics for jump conditions at singular interfaces. A boundary integral formulation of the sharp-interface model enables us to numerically describe the motion and deformation of the binary alloy.

Consider a time-harmonic acoustic plane wave incident onto a doubly periodic (biperiodic) surface from above. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid fluid of constant mass density, whereas the region below is occupied by an isotropic and linearly elastic solid body characterized by its Lamé constants. This paper is concerned with a variational approach to the fluid-solid interaction problems with unbounded biperiodic Lipschitz interfaces between the domains of the acoustic and elastic waves. The existence of quasi-periodic solutions in Sobolev spaces is established at arbitrary frequency of incidence, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. A finite element scheme coupled with Dirichlet-to-Neumann mappings is proposed. The Dirichlet-to-Neumann mappings are approximated by truncated Rayleigh series expansions, and, finally, numerical tests in 2D are performed.

We consider the system of equations modeling the free motion of a rigid body with a cavity filled by a viscous (Navier-Stokes) liquid. We give a rigorous proof of Zhukovskiy's Theorem, which states that in the limit of time going to infinity, the relative fluid velocity tends to zero and the rigid velocity of the full structure tends to a steady rotation around one of the principle axes of inertia. The existence of global weak solutions for this system was established previously. In particular, we prove that every weak solution of this type is subject to Zhukovskiy's Theorem. Independently of the geometry and of parameters, this shows that the presence of fluid prevents precession of the body in the limit. In general, we cannot predict which axis will be attained, but we show stability of the largest axis and provide criteria on the initial data which are decisive in special cases., Comment: 18 pages, 0 figures

We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in-plane motions on a flexible flat part of the boundary. The main peculiarity of the model is the assumption that the transversal displacements of the plate are negligible relative to in-plane displacements. This kind of models arises in the study of blood flows in large arteries. Our main result states the existence of a compact global attractor of finite dimension. We also show that the corresponding linearized system generates exponentially stable $C_0$-semigroup. We do not assume any kind of mechanical damping in the plate component. Thus our results means that dissipation of the energy in the fluid due to viscosity is sufficient to stabilize the system., Comment: 18 pages

We introduce a framework that is capable to model the appearance of mechanical stresses due to inelastic deformations. Among these we consider in particular thermal expansions, diffusion and phase transitions. Among the quantities of central importance are the eigenstrain and the misfit strain. They describe the phenomenon that different material volumes of a compact body may not be compatible to each other in a stress-free reference configuration, so that here a compact body may not exist. We shall show that it is possible to find a further reference configuration, where the body is compact but not free of stress. A typical example where misfit appears concerns a body whose local parts differently transform their phase. This might be a change of the crystal lattice from the ferrite to the austenite symmetry in steel, or the formation of liquid droplets in crystalline gallium arsenide. In both cases the new interior phase has with respect to the parent phase different volume or shape in its state that is free of stress. In this study we consider the eigenstrain problem for pure substances as well as for mixtures. In the latter case subtle arguments are needed for an appropriate description. Special focus is given to the equivalence of interface boundaries with discontinues and continues displacement vectors.

This work deals with a class of evolution problems consisting of a pair of coupled equations for modelling propagation of elastic waves in fluid-saturated porous media. The type of the first equation depends on two physical parameters (density and secondary consolidation) which can vanish while the second one is always parabolic. In case the density never vanishes, the first equation is second-order hyperbolic type and a weak solution to the problem is constructed using a variational method in a Sobolev framework. Next, the proof of uniqueness involves Ladyzenskaja's test-functions used to compensate a lack of regularity that would be required in a standard energy method. This approach gives rise to a priori estimates which are useful to prove that the linearized thermoelasticity and the quasi-static systems are defined as asymptotic models of the Biot problem when the secondary consolidation coefficient or the density is small.

The paper is devoted to the thermodynamic construction of a two-component model of poroelastic media undergoing, in contrast to earlier works on this subject, nonisothermal processes. Under the constitutive dependence on partial mass densities, deformation gradient of skeleton, relative velocity, temperature, temperature gradient and porosity (simple poroelastic material) as well as the assumption of small deviations from the thermodynamic equilibrium we construct explicit relations for fluxes, prove the splitting of the free energy into partial contributions without mechanical couplings and construct a chemical potential for the fluid component important for the formulation of boundary conditions on permeable boundaries. We discuss as well a modification of the porosity balance equation in which we account for time changes of equilibrium porosity. This modification yields the behavior of the model characteristic for granular materials.

The aim of this work is twofold. We show the construction of an objective relative acceleration for a two-component mixture and prove that its incorporation in the momentum source requires additional terms in partial stresses and in the energy. This may be interpreted as an influence of tortuosity in the theory of saturated poroelastic materials and a connection of tortuosity with fluctuations of the kinetic energy on a mesoscopic level of observation. The linearization of such a model yields Biot's equations used in poroacoustics. We demonstrate as well that results for the propagation of acoustic waves in saturated poroelastic media are qualitatively similar for Biot's model and for the simple mixture model in which both the tortuosity and the Biot's coupling between partial stresses are neglected. It is also indicated that the coupling constant of Biot's model obtained by means of the Gassmann relation may be too large as it leads to very small differences in the speed of propagation of the P1-wave for small and large frequencies which contradicts the data for soils.

The main aim of this article is to present a review of most important acoustic surface waves which are described by linear one- and two-component models. It has been written for the CISM-course: Surface waves in Geomechanics (Udine, September 6-10, 2004). Among the waves in one-component linear elastic media we present the classical Rayleigh waves on a plane boundary, Rayleigh waves on a cylindrical surface, Love waves, Stoneley waves (solid/solid and fluid/solid interface). In the second part of the article we discuss two two-component models of porous materials (Biot's model and a simple mixture model). We indicate basic differences of the models and demonstrate qualitative similarities. We introduce as well some fundamental notions yielding the description of surface waves in two-component systems (saturated porous materials) and review certain (porous materials with impermeable boundaries) asymptotic results for such waves. However, the full discussion of this subject including numerous results of computer calculations can be found in the article of B. Albers also included in this volume.

The article contains a brief review of elements of thermodynamic modeling in theoretical geophysics. We motivate the existence of the second law of thermodynamics in macroscopic theoretical physics and demonstrate its evaluation. In particular we show its consequences in the construction of constitutive laws for a two-component poroelastic medium. This construction is also related to microstructural properties verified by means of the second law.

We study the behavior of weak solutions to a time-harmonic fluid-structure interaction problem in the exterior of a thin, elastic domain, in the limit of vanishing thickness. Boundary integral operators are used to reduce the exterior problem. Formal asymptotic expansions are developed and justified. The asymptotic procedure is simple, and involves the solution of a reduced exterior problem. Exact representation formulae for the elastic displacements in the thin domain are also provided.

This work contains the material presented in the key lecture during the Congress Cancam 2003 (Calgary, Canada). It contains a review of the recent development of thermodynamic modeling of porous and granular materials. We present briefly main features of the thermodynamic construction of a nonlinear poroelastic model but the emphasis is put on the analysis of a linear two-component model. In particular we indicate similarities and differences of the thermodynamic model with the classical Biot's model of porous materials. We analyze jacketed and ujacketed Gedankenexperiments which provide a micro-macrotransition procedure for compressibilities. This gives rise to Gassmann-like relations which are incorporated in wave analysis. An acoustic waves analysis is presented in some details. In particular we show the construction of bulk monochromatic waves as well as some surface waves and indicate their practical applications in testing of soils.

In this paper we present a comparison of the linear wave analysis for four models of poroelastic materials. As shown in a paper by Wilmanski (Arch. Mech. 2002) a nonlinear thermodynamical construction of a two-component model of such materials requires a dependence on the porosity gradient. In the linear version this dependence may or may not be present (WIAS-Preprint No. 868). Consequently, we may work with the model without a dependence on this gradient which is identical with Biot's model or we can use the so-called full model. In both cases we can construct simplified models without a coupling between partial stresses introduced by Biot. These simplified models have the advantage that their application to, for instance, surface wave analysis yields much simpler mathematical problems. In the present work we show that such a simplification for granular materials leads to a good qualitative agreement of all four models in ranges of porosity and Poisson's ratio commonly appearing in geotechnical applications. Quantitative differences depend on the mode of propagation and vary between 10% and 20%. We illustrate the analysis with a numerical example corresponding to data for sands. Simultaneously we demonstrate severe limitations of the applicability of Gassmann relations which yield an instability of models in a wide range of practically important values of parameters.

The paper is devoted to the thermodynamic derivation of a two-component poroelastic model with balance equation of porosity as a prototype of Biot's model. It is shown that a constitutive dependence on the porosity gradient yields the possibility of the construction of the linear Biot's model of poroelastic materials provided one negelcts the relaxation of porosity. A procedure of micro-macrotransition for a homogeneous microstructure yields Gassmann relations as an approximation of the thermodynamic model. Simultaneously it is shown that the model with the porosity balance equation can be applied to porous materials with a rather rigid skeleton. In the case of soft granular materials there exists a correction to porosity changes which follow from an appropriate modification of the porosity source.

In the paper we consider a micro-macro transition for a linear thermodynamical model of poroelastic media which yields the Biot's model. We investigate a two-component poroelastic linear model in which a constitutive dependence on the porosity gradient is incorporated and this is compared with the classical Biot's model without added mass effects. We analyze three Gedankenexperiments: jacketed undrained, jacketed drained and unjacketed and derive a generalization of classical Gassmann relations between macroscopic material parameters and microscopic compressibility moduli of the solid, and of the fluid. Dependence on the porosity is particularly exposed due to its importance in acoustic applications of the model. In particular we show that Gassmann relations follow as one of two physically justified solutions of the full set of micro-macro compatibility relations. In this solution the coupling to the porosity gradient is absent. Simultaneously, we demonstrate the second solution which lies near the Gassmann results but admits the coupling. In both models couplings are weak enough to admit, within the class of problems of acoustic wave analysis, an approximation by a "simple mixture" model in which coupling of stresses is fully neglected.

We consider two problems of propagation of weak discontinuity waves in porous materials. In the first part we present basic properties of bulk waves in fully saturated materials. These materials are modelled by a two-component immiscible mixture. We present general propagation conditions for such a model which yield three modes of propagation: P1-, S-, and P2-waves. Then we discuss the dispersion relation and we show that results are strongly dependent on the way in which waves are excited. In the second part we present some properties of surface waves. We begin with the classical Rayleigh and Love problems and then we extend them on heterogeneous materials important in practical applications. Subsequently we proceed to surface waves in two-component porous materials on the contact surface with vacuum (impermeable boundary) and with a liquid (permeable boundary). We show the existence of different modes of surface waves in the high frequency limit as well as the degeneration of the problem in the low frequency limit.

Asymptotic behavior of the Biot slow wave is investigated. Formulae for short- and long-wave approximations of phase velocity of the P2 wave are presented. These asymptotic expansions are compared with exact solution, constructed numerically. It is shown that both expansions fit very well the real velocity of the P2 mode. Procedure for matching of short- and long-wave asymptotic expansions is suggested.

Existence and asymptotic behavior of the Stoneley surface wave at vacuum/porous medium interface are investigated in the low frequency range. It is shown that the Stoneley wave possesses a bifurcation in the vicinity of critical wave number 푘cr. It is proven also that within the 푘-domain of existence, the Stoneley wave cannot appear for certain values of elastic moduli of the solid phase. Asymptotic formulae for the phase velocity of the Stoneley wave are presented.

The paper contains the analysis of the propagation of acoustic waves in two-component poroelastic media. It is shown that the existence of P2-mode as a wave in the range of low frequencies depends on the way in which the wave is excited. This property as well as properties of other bulk modes are discussed on practical examples of soil mechanics.

In most structural components in mechanical engineering, there are surface parts, which are particularly stressed. The aim of surface hardening is to increase the hardness of the corresponding boundary layers by rapid heating and subsequent quenching. This heat treatment leads to a change in the microstructure, which produces the desired hardening effect. The mathematical model accounts for electromagnetic effects that lead to the heating of the workpiece as well as thermomechanical effects that cause the hardening of the workpiece. The new contribution of this paper is that we put a special emphasis on the thermomechanical effects caused by the phase transitions. We formulate a consistent model which takes care of effects like transformation strain and transformation plasticity induced by the phase transitions and allows for physical parameters depending on the respective phase volume fractions. The coupling between the electromagnetic and the thermomechanical part of the model is given through the temperature-dependent electric conductivity on the one hand and through the Joule heating term on the other hand, which appears in the energy balance and leads to the rise in temperature. Owing to the quadratic Joule heat term and a quadratic mechanical dissipation term in the energy balance, we obtain a parabolic equation with L1 data. We prove existence of a weak solution to the complete system using a truncation argument.

We present in this paper an existence result as well a numerical method for a fluid-structure interaction problem. The structure motion is governed by the Vón Kármán equations and the fluid behavior by the viscous wave equation.

We consider a mathematical model of the noise reduction problem, which couples two hyperbolic equations: the wave equation in the interior ("chamber")---which describes the unwanted acoustic waves---and a (hyperbolic) Kirchoff equation ---which models the vibrations of the elastic wall. In past models, the elastic wall was modeled by an Euler-Bernoulli equation with Kelvin-Voight damping (parabolic model). Our main result is a sharp regularity result, in two dual versions, of the resulting system of two coupled hyperbolic P.D.E.'s. With this regularity result established, one can then invoke a wealth of abstract results from [14], [15], [16], [19], etc. on optimal control problems, min-max game theory (and $H^\infty$-problems), etc. The proof of the main result is based on combining technical results from [18] and [11].

This paper is concerned with a nonlinear integro-partial differential equation perturbed by a state dependent white noise arising from the aeroelastic panel vibration. With the aid of a stochastic energy equation, the existence, uniqueness and regularity of solutions are proved.

27 pages; International audience; The ascending branch of the aorta is one of the most stressed organ of the arterial system. We aim to design a biomechanical model for analysing the aorta dynamics under a shock. The model includes the aorta layers and the influence of the blood pressure. First, we undertake a modal analysis of the coupled aorta-blood system. For the analytical solving we adopt the Helmholtz decomposition. The finite element model is then validated by these original analytical solutions. The second part focuses on aorta-blood dynamics, corresponding to an acceleration/deceleration during car crash. Numerical and analytical solutions are compared.

We study the feedback stabilization of a system composed by an incompressible viscous fluid and a rigid body. We stabilize the position and the velocity of the rigid body and the velocity of the fluid around a stationary state by means of a Dirichlet control, localized on the exterior boundary of the fluid domain and with values in a finite dimensional space. Our first result concerns weak solutions in the two-dimensional case, for initial data close to the stationary state. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domain of the stationary state and of the stabilized solution are different. This additional difficulty leads to the assumption that the initial position of the rigid body is the position associated to the stationary state. Without this hypothesis, we work with strong solutions, and to deal with compatibility conditions at the initial time, we use finite dimensional dynamical controls. We again prove that for initial data close to the stationary state, we can stabilize the position and the velocity of the rigid body and the velocity of the fluid. In the three dimensional case, we also obtain the local stabilization of strong solutions with finite dimensional dynamical controls.

In this paper we study the acoustic properties of a microstructured material such as glass, wool, or foam. In our model, the solid matrix is governed by linear elasticity and the surrounding fluid obeys the Stokes equations. The microstructure is assumed to be periodic at some small scale ${\varepsilon}$ and the viscosity coefficient of the fluid is assumed to be of order ${\varepsilon}^2$. We consider the time-harmonic regime forced by vibrations applied on a part of the boundary. We use the two-scale convergence theory to prove the convergence of the displacements to the solution of a homogeneous problem as the size of the microstructure shrinks to 0.