Mathematical Models and Methods in Applied Sciences

We consider mixed finite elements for linear elasticity with weakly symmetric stress. We propose a low-order three-dimensional rectangular element with optimal O(h) rate of convergence for all the unknowns. The element is a rectangular analogue of the simplified Arnold–Falk–Winther element. Instead of the elasticity complex approach, our stability analysis is based on new mesh-dependent norms.

We propose a new numerical scheme designed for a wide class of structured population models based on the idea of operator splitting and particle approximations. This scheme is related to the Escalator Boxcar Train (EBT) method commonly used in biology, which is in essence an analogue of particle methods used in physics. Our method exploits the split-up technique, thanks to which the transport step and the nonlocal integral terms in the equation can be separately considered. The order of convergence of the proposed method is obtained in the natural space of finite non-negative Radon measures equipped with the flat metric. This convergence is studied even adding reconstruction and approximation steps in the particle simulation to keep the number of approximation particles under control. We validate our scheme in several test cases showing the theoretical convergence error. Finally, we use the scheme in situations in which the EBT method does not apply showing the flexibility of this new method to cope with the different terms in general structured population models.

The purpose of this paper is to propose a deterministic method for optimizing a structure with respect to its worst possible behavior when a "small" uncertainty exists over some of its features. The main idea of the method is to linearize the considered cost function with respect to the uncertain parameters, then to consider the supremum function of the obtained linear approximation, which can be rewritten as a more "classical" function of the design, owing to standard adjoint techniques from optimal control theory. The resulting "linearized worst-case" objective function turns out to be the sum of the initial cost function and of a norm of an adjoint state function, which is dual with respect to the considered norm over perturbations. This formal approach is very general, and can be justified in some special cases. In particular, it allows to address several problems of considerable importance in both parametric and shape optimization of elastic structures, in a unified framework.

We present a quantitative analysis of the effect of rough hydrophobic surfaces on viscous Newtonian flows. We use a model introduced by Ybert and coauthors in [Achieving large with superhydrophobic surfaces: Scaling laws for generic geometries, Phys. Fluids 19 (2007) 123601], in which the rough surface is replaced by a flat plane with alternating small areas of slip and no-slip. We investigate the averaged slip generated at the boundary, depending on the ratio between these areas. This problem reduces to the homogenization of a nonlocal system, involving the Dirichlet to Neumann map of the Stokes operator, in a domain with small holes. Pondering on the works of Allaire [Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes, Arch. Rational Mech. Anal. 113 (1990) 209–259; Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes. II. Noncritical sizes of the holes for a volume distribution and a surface distribution of holes, Arch. Rational Mech. Anal. 113 (1990) 261–298]. We compute accurate scaling laws of the averaged slip for various types of roughness (riblets, patches). Numerical computations complete and confirm the analysis.

We present and analyze two mathematical models for the self-consistent quantum transport of electrons in a graphene layer. We treat two situations. First, when the particles can move in all the plane ℝ², the model takes the form of a system of massless Dirac equations coupled together by a self-consistent potential, which is the trace in the plane of the graphene of the 3D Poisson potential associated to surface densities. In this case, we prove local in time existence and uniqueness of a solution in H^s(ℝ²), for which includes in particular the energy space H^1/2(ℝ²). The main tools that enable to reach are the dispersive Strichartz estimates that we generalized here for mixed quantum states. Second, we consider a situation where the particles are constrained in a regular bounded domain Ω. In order to take into account Dirichlet boundary conditions which are not compatible with the Dirac Hamiltonian H₀, we propose a different model built on a modified Hamiltonian displaying the same energy band diagram as H₀ near the Dirac points. The well-posedness of the system in this case is proved in , the domain of the fractional order Dirichlet Laplacian operator, for .

We present a new hydrodynamic model for the interactions between collision-free Cucker–Smale flocking particles and a viscous incompressible fluid. Our proposed model consists of two hydrodynamic models. For the Cucker–Smale flocking particles, we employ the pressureless Euler system with a non-local flocking dissipation, whereas for the fluid, we use the incompressible Navier–Stokes equations. These two hydrodynamic models are coupled through a drag force, which is the main flocking mechanism between the particles and the fluid. The flocking mechanism between particles is regulated by the Cucker–Smale model, which accelerates global flocking between the particles and the fluid. We show that this model admits the global-in-time classical solutions, and exhibits time-asymptotic flocking, provided that the initial data is appropriately small. In the course of our analysis for the proposed system, we first consider the hydrodynamic Cucker–Smale equations (the pressureless Euler system with a non-local flocking dissipation), for which the global existence and the time-asymptotic behavior of the classical solutions are also investigated.