Primal and Shadow functions, Dual and Dual-Shadow functions for a circular crack and a circular 90 degree V-notch with Neumann boundary conditions

This report presents explicit analytical expressions for the primal, primal shadows, dual and dual shadows functions for the Laplace equation in the vicinity of a circular singular edge with Neumann boundary conditions on the faces that intersect at the singular edge. Two configurations are investigated: a penny-shaped crack and a 90^o V-notch

Analytic Regularity for Linear Elliptic Systems in Polygons and Polyhedra

We prove weighted anisotropic analytic estimates for solutions of second order elliptic boundary value problems in polyhedra. The weighted analytic classes which we use are the same as those introduced by Guo in 1993 in view of establishing exponential convergence for hp finite element methods in polyhedra. We first give a simple proof of the known weighted analytic regularity in a polygon, relying on a new formulation of elliptic a priori estimates in smooth domains with analytic control of derivatives. The technique is based on dyadic partitions near the corners. This technique can successfully be extended to polyhedra, providing isotropic analytic regularity. This is not optimal, because it does not take advantage of the full regularity along the edges. We combine it with a nested open set technique to obtain the desired three-dimensional anisotropic analytic regularity result. Our proofs are global and do not require the analysis of singular functions.Comment: 54 page

Continuity properties of the inf-sup constant for the divergence

The inf-sup constant for the divergence, or LBB constant, is explicitly known for only few domains. For other domains, upper and lower estimates are known. If more precise values are required, one can try to compute a numerical approximation. This involves, in general, approximation of the domain and then the computation of a discrete LBB constant that can be obtained from the numerical solution of an eigenvalue problem for the Stokes system. This eigenvalue problem does not fall into a class for which standard results about numerical approximations can be applied. Indeed, many reasonable finite element methods do not yield a convergent approximation. In this article, we show that under fairly weak conditions on the approximation of the domain, the LBB constant is an upper semi-continuous shape functional, and we give more restrictive sufficient conditions for its continuity with respect to the domain. For numerical approximations based on variational formulations of the Stokes eigenvalue problem, we also show upper semi-continuity under weak approximation properties, and we give stronger conditions that are sufficient for convergence of the discrete LBB constant towards the continuous LBB constant. Numerical examples show that our conditions are, while not quite optimal, not very far from necessary

Weighted analytic regularity in polyhedra

International audienceWe explain a simple strategy to establish analytic regularity for solutions of second order linear elliptic boundary value problems. The abstract framework presented here helps to understand the proof of analytic regularity in polyhedral domains given in "http://hal.archives-ouvertes.fr/hal-00454133" the authors' paper published in Math. Models Methods Appl. Sci. 22 (8) (2012). We illustrate this strategy by considering problems set in smooth domains, in corner domains and in polyhedra

A fast semi-analytic method for the computation of elastic edge singularities

International audienceThe singularities that we consider are the characteristic non-smooth solutions of the equations of linear elasticity in piecewise homogeneous media near two dimensional corners or three dimensional edges. We describe here a method to compute their singularity exponents and the associated angular singular functions. We present the implementation of this method in a program whose input data are geometrical data, the elasticity coefficients of each material involved and the type of boundary conditions (Dirichlet, Neumann or mixed conditions). Our method is particularly useful with anisotropic materials and allows to ''follow" the dependency of singularity exponents along a curved edge

Ultrafast light-induced response of photoactive yellow protein chromophore analogues

The fluorescence decays of several analogues of the photoactive yellow protein (PYP) chromophore in aqueous solution have been measured by femtosecond fluorescence up-conversion and the corresponding time-resolved fluorescence spectra have been reconstructed. The native chromophore of PYP is a thioester derivative of p-coumaric acid in its trans deprotonated form. Fluorescence kinetics are reported for a thioester phenyl analogue and for two analogues where the thioester group has been changed to amide and carboxylate groups. The kinetics are compared to those we previously reported for the analogues bearing ketone and ester groups. The fluorescence decays of the full series are found to lie in the 1–10 ps range depending on the electron-acceptor character of the substituent, in good agreement with the excited-state relaxation kinetics extracted from transient absorption measurements. Steady-state photolysis is also examined and found to depend strongly on the nature of the substituent. While it has been shown that the ultrafast light-induced response of the chromophore in PYP is controlled by the properties of the protein nanospace, the present results demonstrate that, in solution, the relaxation dynamics and pathway of the chromophore is controlled by its electron donor–acceptor structure: structures of stronger electron donor–acceptor character lead to faster decays and less photoisomerisation

Les premiers recensements au Sénégal et l’évolution démographique : partie I: présentation des documents

La documentation démographique concernant les populations rurales de l’Ouest-Africain ne s’est vraiment développée qu’à une époque très récente. On remarque une disparité entre les données relatives aux villes et escales, et celles qui ont trait à la situation rurale. La population urbaine a pu faire l’objet de véritables études démographiques, dès avant les indépendances, alors qu’en milieu rural, on s’est contenté des dénombrements effectués à des fins administratives et fiscales

Circular edge singularities for the Laplace equation and the elasticity system in 3-D domains

International audienceAsymptotics of solutions to the Laplace equation with Neumann or Dirichlet conditions in the vicinity of a circular singular edge in a three-dimensional domain are derived and provided in an explicit form. These asymptotic solutions are represented by a family of eigen-functions with their shadows, and the associated edge flux intensity functions (EFIFs), which are functions along the circular edge. We provide explicit formulas for a penny-shaped crack for an axisymmetric case as well as a case in which the loading is non-axisymmetric. Explicit formulas for other singular circular edges such as a circumferential crack and an external crack are also derived. The mathematical machinery developed in the framework of the Laplace operator is extended to derive the asymptotic solution (three-component displacement vector) for the elasticity system in the vicinity of a circular edge in a three-dimensional domain. As a particular case we present explicitly the series expansion for a traction free or clamped penny-shaped crack in an axisymmetric or a non-axisymmetric situation. The precise representation of the asymptotic series is required for constructing benchmark problems with analytical solutions against which numerical methods can be assessed, and to develop new extraction techniques for the edge flux/intensity functions which are of practical engineering importance in predicting crack propagation. <br

On Friedrichs constant and Horgan-Payne angle for LBB condition

In dimension 2, the Horgan-Payne angle serves to construct a lower bound for the inf-sup constant of the divergence arising in the so-called LBB condition. This lower bound is equivalent to an upper bound for the Friedrichs constant. Explicit upper bounds for the latter constant can be found using a polar parametrization of the boundary. Revisiting carefully the original paper which establishes this strategy, we found out that some proofs need clarification, and some statements, replacement