In this paper, we compare a biomechanics empirical model of the heart fibrous structure to two models obtained by a non-periodic homogenization process. To this end, the two homogenized models are simplified using the small amplitude homogenization procedure of Tartar, both in conduction and in elasticity. A new small amplitude homogenization expansion formula for a mixture of anisotropic elastic materials is also derived and allows us to obtain a third simplified model.

This work is devoted to the study of a two-dimensional vectorPoisson equation with the normal component of the unknown andthe value of the divergence of the unknown prescribed simultaneouslyon the entire boundary.These two scalar boundary conditions appear prima faciealternative in a standard variational framework. An originalvariational formulation of this boundary value problemis proposed here. Furthermore, an uncoupled solution algorithm isintroduced together with its finite element approximation.The numerical scheme has been implemented and appliedto solve a simple test problem.

This is the second part of the paper for a Non-Newtonian flow. Dual combined Finite Element Methods are used to investigate the littleparameter-dependent problem arising in a nonliner three field version of the Stokes system for incompressible fluids, where the viscosity obeys a general law including the Carreau's law and the Power law. Certain parameter-independent error bounds are obtained which solved the problem proposed by Baranger in [4] in a unifying way. We also give somestable finite element spaces by exemplifying the abstract B-B inequality. The continuous approximation for the extra stress is achieved as a by-product of the new method.

In this article, we prove convergence of the weakly penalized adaptive discontinuousGalerkin methods. Unlike other works, we derive the contraction property for variousdiscontinuous Galerkin methods only assuming the stabilizing parameters are large enoughto stabilize the method. A central idea in the analysis is to construct an auxiliarysolution from the discontinuous Galerkin solution by a simple post processing. Based onthe auxiliary solution, we define the adaptive algorithm which guides to the convergenceof adaptive discontinuous Galerkin methods.

We consider a non-conforming stabilized domaindecomposition technique forthe discretization of the three-dimensional Laplace equation.The aim is to extend the numerical analysis of residual error indicators tothis model problem. Two formulations of the problem are consideredand the error estimators are studied for both. In thefirst one, the error estimator provides upper and lower bounds forthe energy norm of the mortar finite element solution whereas inthe second case, it also estimates the error for the Lagrangemultiplier.

The paper is devoted to the convergence analysis of a well-knowncell-centered Finite Volume Method (FVM) for aconvection-diffusion problem in $\mathbb{R}^2$ . This FVM is based on Voronoi boxes andexponential fitting. To prove the convergence of the FVM, we usea new nonconforming Petrov-Galerkin Finite Element Method (FEM)for which the system of linear equations coincides completely withthat of the FVM. Thus, by proving convergence properties of theFEM we obtain similar ones for the FVM. For the error estimationof the FEM well-known statements have to be modified.

A modal synthesis method to solve the elastoacoustic vibration problemis analyzed. A two-dimensional coupled fluid-solid system is considered;the solid is described by displacement variables, whereas displacement potential is used for the fluid. A particular modal synthesis leading to a symmetric eigenvalue problem is introduced. Finite element discretizations with Lagrangian elements are considered for solving the uncoupled problems.Convergence for eigenvalues and eigenfunctions is proved, error estimates are given, and numerical experiments exhibiting the good performance of the method are reported.

Finite element semidiscrete approximations on nonlinear dynamicshallow shell models in considered. It is shown that the algorithmleads to global, optimal rates of convergence. The resultpresented in the paper improves upon the existing literature where therates of convergence were derived for small initial data only[19].

In the framework of an explicitly correlated formulation of the electronic Schrödingerequation known as the transcorrelated method, this work addresses some fundamental issuesconcerning the feasibility of eigenfunction approximation by hyperbolic wavelet bases.Focusing on the two-electron case, the integrability of mixed weak derivatives ofeigenfunctions of the modified problem and the improvement compared to the standardformulation are discussed. Elements of a discretization of the eigenvalue problem based onorthogonal wavelets are described, and possible choices of tensor product bases arecompared especially from an algorithmic point of view. The use of separable approximationsof potential terms for applying operators efficiently is studied in detail, and estimatesfor the error due to this further approximation are given.

We consider a quasistatic system involving a Volterra kernel modellingan hereditarily-elastic aging body. We are concerned with the behavior of displacement and stress fields in the neighborhood of cracks. In this paper, we investigate the case of a straight crack in a two-dimensional domain with a possiblyanisotropic material law.We study the asymptotics of the time dependent solution near the crack tips. We prove that, depending on the regularity of the materiallaw and the Volterra kernel, these asymptotics contain singular functions which are simple homogeneousfunctions of degree $\frac12$ or have a more complicated dependence onthe distance variable r to the crack tips. In the latter situation,we observe a novel behavior of the singular functions, incompatible withthe usual fracture criteria, involving super polynomial functions of ln r growing in time.

We consider plasma tearing mode instabilities when the resistivity depends on a flux function (ψ), for the plane slab model.This problem, represented by the MHD equations, is studied as a bifurcation problem. Forso doing, it is written in the form (I(.)-T(S,.)) = 0, whereT(S,.) is a compact operator in a suitable space and S is the bifurcationparameter.In this work, the resistivity is not assumed to be a given quantity (as usuallydone in previous papers, see [1,2,5,7,8,9,10], but it depends nonlinearly of the unknowns of the problem; this is the main difficulty, with newmathematical results.We also develop in this paper a 1D code to compute bifurcation points from the trivialbranch (equilibrium state).

This paper deals with the flow problem of aviscous plastic fluid in a cylindrical pipe. In order toapproximate this problem governed by a variational inequality, we apply the nonconforming mortar finite element method. By using appropriate techniques, we are able to prove the convergence of the method and to obtain the same convergence rate as in the conforming case.

While alternans in a single cardiac cell appears through a simpleperiod-doubling bifurcation, in extended tissue the exact natureof the bifurcation is unclear. In particular, the phase ofalternans can exhibit wave-like spatial dependence, eitherstationary or travelling, which is known as discordantalternans. We study these phenomena in simple cardiac modelsthrough a modulation equation proposed by Echebarria-Karma. Asshown in our previous paper, the zero solution of their equationmay lose stability, as the pacing rate is increased, througheither a Hopf or steady-state bifurcation. Which bifurcationoccurs first depends on parameters in the equation, and for onecritical case both modes bifurcate together at a degenerate(codimension 2) bifurcation. For parameters close to thedegenerate case, we investigate the competition between modes,both numerically and analytically. We find that at sufficientlyrapid pacing (but assuming a 1:1 response is maintained), steadypatterns always emerge as the only stable solution. However, inthe parameter range where Hopf bifurcation occurs first, theevolution from periodic solution (just after the bifurcation) tothe eventual standing wave solution occurs through an interestingseries of secondary bifurcations.

Development of user-friendly and flexible scientific programs is a key to their usage, extension and maintenance. This paper presents an OOP (Object-Oriented Programming) approach for design of finite element analysis programs. General organization of the developed software system, called FER/SubDomain, is given which includes the solver and the pre/post processors with a friendly GUI (Graphical User Interfaces). A case study with graphical representations illustrates some functionalities of the program.

The change in the electric potential due to lightning is evaluated.The potential along the lightning channel is a constant which isthe projection of the pre-flash potential along a piecewise harmoniceigenfunction which is constant along the lightning channel.The change in the potential outside the lightning channel is a harmonicfunction whose boundary conditionsare expressed in terms of the pre-flash potential andthe post-flash potential along the lightning channel.The expression for the lightning induced electric potential change isderived both for the continuous equations, and for a spatially discretizedformulation of the continuous equations.The results for the continuous equations are based on the properties ofthe eigenvalues and eigenfunctions of the following generalized eigenproblem:Find $u \in H_0^1 (\Omega)$ , $u \ne 0$ ,and $\lambda \in \mathbb{R}$ such that $\langle \nabla u, \nabla v \rangle_{\mathcal{L}} =\lambda \langle \nabla u, \nabla v \rangle_{\Omega}$ for all $v \in H_0^1 (\Omega)$ , where $\Omega \subset \mathbb{R}^n$ is a bounded domain (a box containing the thunderstorm), $\mathcal{L}$ is a subdomain (the lightning channel),and $\langle \cdot, \cdot \rangle_{\Omega}$ isthe inner product $\langle \nabla u,\nabla v\rangle_\Omega =\int_{\Omega}\nabla u\cdot\nabla v \; {{\rm d}x}.$

Although the intellectual merits of computational modelling across various length and time scales are generally well accepted, good illustrative examples are often lacking. One way to begin appreciating the benefits of the multiscale approach is to first gain experience in probing complex physical phenomena at one scale at a time. Here we discuss materials modelling at two characteristic scales separately, the atomistic level where interactions are specified through classical potentials and the electronic level where interactions are treated quantum mechanically. The former is generally sufficient for dealing with mechanical deformation at large strain, whereas the latter is necessary for treating chemical reactions or electronic transport. We will discuss simulations of defect nucleation using molecular dynamics, the study of water-silica reactions using a tight-binding approach, the design of a semiconductor-oxide interface using density functional theory, and the analysis of conjugated polymer in molecular actuation using Hartree-Fock calculations. The diversity of the problems discussed notwithstanding, our intent is to lay the groundwork for future problems in materials research, a few will be mentioned, where modelling at the electronic and atomistic scales are needed in an integrated fashion. It is in these problems that the full potential of multiscale modelling can be realized.

Let $\mathcal{Q}$ be a partition of a polygonal domain of the planinto convexe quadrilaterals. Given a regular function f , we construct afunction πƒ ∈ C2 (Ω), interpolating position values andderivatives of f up of order 2 at vertices of $\mathcal{Q}.$ On eachquadrilateral $Q\in\mathcal{Q},$ πƒ|Q is a finite element obtainedfrom a polynomial scheme of FVS type by adding some rational functions.

In this paper we show how abstract physical requirements are enoughto characterize the classical collision kernels appearing in kinetic equations. In particular Boltzmann and Landau kernels are derived.

The dynamics of dendritic growth of a crystal in an undercooled melt isdetermined by macroscopic diffusion-convection of heat and by capillary forcesacting on the nanometer scale of the solid-liquid interface width.Its modelling is useful for instance in processing techniques based on casting.The phase-field method is widely used to study evolution of such microstructural phase transformations ona continuum level; it couples the energy equation to a phenomenological Allen-Cahn/Ginzburg-Landauequation modelling the dynamics of an order parameter determining the solid and liquid phases,including also stochastic fluctuations to obtain the qualitatively correctresult of dendritic side branching.This work presents a method to determine stochastic phase-field models from atomisticformulations by coarse-graining molecular dynamics. It has three steps: (1) a precisequantitative atomistic definition of the phase-field variable, based on the localpotential energy;(2) derivation of its coarse-graineddynamics model, from microscopic Smoluchowski molecular dynamics (that is Brownian or over damped Langevin dynamics);and (3) numerical computation of the coarse-grained model functions. The coarse-grained model approximates Gibbs ensemble averages of the atomistic phase-field, bychoosing coarse-grained drift and diffusion functions that minimize the approximation error of observables in thisensemble average.