257 research outputs found
A new approach for approximating linear elasticity problems
No full textInternational audienceIn this Note, we present and analyze a new method for approximating linear elasticity problems in dimension two or three. This approach directly provides approximate strains, i.e., without simultaneously approximating the displacements, in finite element spaces where the Saint Venant compatibility conditions are exactly satisfied in a weak form. To cite this article: P.G. Ciarlet, P. Ciarlet, Jr., C. R. Acad. Sci. Paris, Ser. I 346 (2008). © 2008 Académie des sciences
Tools for solving the div-curl problem with mixed boundary conditions in a polygonal domain
No full textFollowing [Assous, Ciarlet, Jr., Sonnendrücker, Resolution of the Maxwell equations in a domain with reentrant corners (1998)], we continue to study the resolution of two-dimensional problems in non convex domains. In this previous paper, we considered several methods for solving Maxwell’s equations (stationary or instationary) with a perfectly conducting boundary condition
Explicit T-coercivity for the Stokes problem: a coercive finite element discretization: Explicit T -coercivity for Stokes
No full textUsing the T -coercivity theory as advocated in Chesnel-Ciarlet [Numer. Math., 2013], we propose a new variational formulation of the Stokes problem which does not involve nonlocal operators. With this new formulation, unstable finite element pairs are stabilized. In addition, the numerical scheme is easy to implement, and a better approximation of the velocity and the pressure is observed numerically when the viscosity is smal
Characterization of the kernel of the operator CURL CURL
No full textIn a simply-connected domain Ω in R3, the kernel of the operator CURLCURL acting on symmetric matrix fields from L2s (Ω) to H−2 s (Ω) coincides with the space of linearized strain tensor fields. For not simply-connected domains, Volterra has characterized this kernel for smooth fields. Here we extend this result for domains with a Lipschitz-continuous boundary for fields in L2s (Ω). To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007). © 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved
A justification of Peek's empirical law in electrostatics [Justification de la loi de Peek en électrostatique]
No full textInternational audienceWe consider the computation of the electrostatic charge density at the tip of a rounded corner. The relation between the curvature radius and the electrostatic field is given by Peek's empirical law which is valid only for thin, cylindrical or spherical, geometries. In this Note, we justify mathematically this law and extend it to other geometries. With the help of multiscaled asymptotic expansions, we derive an expression for the charge density for geometries which coincide at infinity with a cone. A numerical illustration is provided. To cite this article: P. Ciarlet Jr., S. Kaddouri, C. R. Acad. Sci. Paris, Ser. I 343 (2006). © 2006 Académie des sciences
Finite Element Heterogeneous Multiscale Method for the Helmholtz Equation
No full textAbstract We show that standard Finite Element Heterogeneous Multiscale Method (FE-HMM) can be used to approximate the effective behavior of solutions of the classical Helmholtz equation in highly oscillatory media. Using a novel combination of well known results about FE-HMM and the notion of T -coercivity we derive an a priori error bound. Numerical experiments corroborate the analytical findings. To cite this article: P. Ciarlet Jr., C. Stohrer, C. R. Acad. Sci. Paris, Ser
On Saint Venant's compatibility conditions and Poincaré's lemma
No full textSaint Venant's theorem constitutes a classical characterization of smooth matrix fields as linearized strain tensor fields. This theorem has been extended to matrix fields with components in L2 by the second author and P. Ciarlet, Jr. in 2005. One objective of this Note is to further extend this characterization to matrix fields whose components are only in H-1. Another objective is to demonstrate that Saint Venant's theorem is in fact nothing but the matrix analog of Poincaré's lemma
Système de Stokes avec flux de vitesse et pression imposés
No full textInternational audienceDans cette Note, nous étudions le système de Stokes avec flux de vitesse et pression imposés, dans un domaine borné, à bord régulier par morceaux
Electrowetting of a 3D drop : numerical modelling with electrostatic vector fields
No full textInternational audienceThe electrowetting process is commonly used to handle very small amounts of liquid on a solid surface. This process can be modelled mathematically with the help of the shape optimization theory. However, solving numerically the resulting shape optimization problem is a very complex issue, even for reduced models that occur in simplified geometries. Recently, the second author obtained convincing results in the 2D axisymmetric case. In this paper, we propose and analyze a method that is suitable for the full 3D case. © EDP Sciences, SMAI, 2010
Study of a degenerate non-elliptic equation to model plasma heating
No full textInternational audienceIn this manuscript, we study solutions to resonant Maxwell's equations in heterogeneous plasmas. We concentrate on the phenomenon of upper-hybrid heating, which occurs in a localized region where electromagnetic waves transfer energy to the particles. In the 2D case, it can be modelled mathematically by the partial differential equation − div (α∇u) − ω 2 u = 0, where the coefficient α is a smooth, sign-changing, real-valued function. Since the locus of the sign change is located within the plasma, the equation is non-elliptic, and degenerate. On the other hand, using the limiting absorption principle, one can build a family of elliptic equations that approximate the degenerate equation. Then, a natural question is to relate the solution of the degenerate equation, if it exists, to the family of solutions of the elliptic equations. For that, we assume that the family of solutions converges to a limit, which can be split into a regular part and a singular part, and that this limiting absorption solution is governed by the non-elliptic equation introduced above. One of the difficulties lies in the definition of appropriate norms and function spaces in order to be able to study the non-elliptic equation and its solutions. As a starting point, we revisit a prior work [13] on this topic by A. Nicolopoulos, M. Campos Pinto, B. Després and P. Ciarlet Jr., who proposed a variational formulation for the plasma heating problem. We improve the results they obtained, in particular by establishing existence and uniqueness of the solution, by making a different choice of function spaces. Also, we propose a series a numerical tests, comparing the numerical results of Nicolopoulos et al to those obtained with our numerical method, for which we observe better convergence
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