Mathematical Models and Methods in Applied Sciences

The reflection and transmission of a time-harmonic plane wave in an isotropic elastic medium by a three-dimensional diffraction grating is investigated. If the diffractive structure involves an impenetrable surface, we study the first, second, third and fourth kind boundary value problems for the Navier equation in an unbounded domain by the variational approach. A radiation condition based on the Rayleigh expansion of the quasi-periodic solutions is presented. Existence of solutions in Sobolev spaces is established if the grating profile is a two-dimensional Lipschitz surface, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. Similar solvability results are obtained for multilayered transmission gratings in the case of an incident pressure wave. Moreover, by a periodic Rellich identity, uniqueness of the solution to the first kind (Dirichlet) boundary value problem is established for all frequencies under the assumption that the impenetrable surface is given by the graph of a Lipschitz function.

This paper studies a hyperbolic modification of Maxwell's equations to ensure Gauss' law. This correction was obtained by adding a parameter-dependent new unknown and is of great interest for the numerical simulation in plasma physics since the discretization of the Maxwell–Vlasov system does not grant straightforwardly the physical conservation of the charge. Such problems are encountered while using Particle-In-Cell schemes. In this paper the new proposed system has the interest of still being a Friedrichs' one. Its asymptotic behavior with respect to the parameter and the link between modified and original Maxwell's systems are thus investigated. At last, we look for some boundary conditions, granting the well-posedness of the system. Generalizations of the Silver–Müller condition, perfect electric and magnetic conductors, as well as impedance and admittance representation of materials are detailed.

We propose kinetic models to describe dust particles in a rarefied atmosphere in order to model the beginning of a Loss of Vacuum Accident (LOVA) in the framework of safety studies in the International Thermonuclear Experimental Reactor (ITER). After having studied characteristic time and length scales at the beginning of a LOVA in ITER and underlined that these characteristic scales justify a kinetic approach, we firstly propose a kinetic model by supposing that the collisions between dust particles and gas molecules are inelastic and are given by a diffuse reflexion mechanism on the surface of dust particles. This collision mechanism allows us to take into account the macroscopic character of dust particles compared to gas molecules. This leads to establish new Boltzmann type kinetic operators that are non-classical. Then, by noting that the mass of a dust particle is huge compared to the mass of a gas molecule, we perform an asymptotic expansion to one of the dust–molecule kinetic operators with respect to the ratio of mass between a gas molecule and a dust particle. This allows us to obtain a dust–molecule kinetic operator of Vlasov type whose any numerical discretization is less expensive than any numerical discretization of the original Boltzmann type operator. At last, we perform numerical simulations with Monte–Carlo and Particle-In-Cell (PIC) methods which validate and justify the derivation of the Vlasov operator. Moreover, examples of 3D numerical simulations of a LOVA in ITER using these kinetic models are presented.

Propagation of elastic waves is studied in a 1D medium containing two cracks. The latter are modeled by smooth nonlinear jump conditions accounting for the finite, non-null compressibility of real cracks. The evolution equations are written in the form of a system of two nonlinear neutral delay differential equations, leading to a well-posed Cauchy problem. Perturbation analysis indicates that, under periodic excitation, the periodic solutions oscillate around positive mean values, which increase with the forcing level. This typically nonlinear phenomenon offers non-destructive means to evaluate the cracks. Existence, uniqueness and attractivity of periodic solutions is then examined. At some particular values of the ratio between the wave travel time and the period of the source, results are obtained whatever the forcing level. With a much larger set of ratios but at small forcing levels, results are obtained under a Diophantine condition. Lastly, numerical experiments are proposed to illustrate the behavior of the periodic diffracted waves.

We present a new class of macroscopic models for pedestrian flows. Each individual is assumed to move towards a fixed target, deviating from the best path according to the instantaneous crowd distribution. The resulting equation is a conservation law with a nonlocal flux. Each equation in this class generates a Lipschitz semigroup of solutions and is stable with respect to the functions and parameters defining it. Moreover, key qualitative properties such as the boundedness of the crowd density are proved. Specific models are presented and their qualitative properties are shown through numerical integrations. In particular, the present model accounts for the possibility of reducing the exit time from a room by carefully positioning obstacles that direct the crowd flow.