Asymptotic behavior of Structures made of Plates

The aim of this work is to study the asymptotic behavior of a structure made of plates of thickness $2\delta$ when $\delta\to 0$. This study is carried on within the frame of linear elasticity by using the unfolding method. It is based on several decompositions of the structure displacements and on the passing to the limit in fixed domains. We begin with studying the displacements of a plate. We show that any displacement is the sum of an elementary displacement concerning the normal lines on the middle surface of the plate and a residual displacement linked to these normal lines deformations. An elementary displacement is linear with respect to the variable $x$3. It is written $U(^x)+R(^x)\land x3e3$ where U is a displacement of the mid-surface of the plate. We show a priori estimates and convergence results when $\delta \to 0$. We characterize the limits of the unfolded displacements of a plate as well as the limits of the unfolded of the strained tensor. Then we extend these results to the structures made of plates. We show that any displacement of a structure is the sum of an elementary displacement of each plate and of a residual displacement. The elementary displacements of the structure (e.d.p.s.) coincide with elementary rods displacements in the junctions. Any e.d.p.s. is given by two functions belonging to $H1(S;R3)$ where S is the skeleton of the structure (the plates mid-surfaces set). One of these functions : U is the skeleton displacement. We show that U is the sum of an extensional displacement and of an inextensional one. The first one characterizes the membrane displacements and the second one is a rigid displacement in the direction of the plates and it characterizes the plates flexion. Eventually we pass to the limit as $\delta \to 0$ in the linearized elasticity system, on the one hand we obtain a variational problem that is satisfied by the limit extensional displacement, and on the other hand, a variational problem satisfied by the limit of inextensional displacements

Theory and implementation of H\mathcal{H}-matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels

In this work, we study the accuracy and efficiency of hierarchical matrix ($\mathcal{H}$-matrix) based fast methods for solving dense linear systems arising from the discretization of the 3D elastodynamic Green's tensors. It is well known in the literature that standard $\mathcal{H}$-matrix based methods, although very efficient tools for asymptotically smooth kernels, are not optimal for oscillatory kernels. $\mathcal{H}^2$-matrix and directional approaches have been proposed to overcome this problem. However the implementation of such methods is much more involved than the standard $\mathcal{H}$-matrix representation. The central questions we address are twofold. (i) What is the frequency-range in which the $\mathcal{H}$-matrix format is an efficient representation for 3D elastodynamic problems? (ii) What can be expected of such an approach to model problems in mechanical engineering? We show that even though the method is not optimal (in the sense that more involved representations can lead to faster algorithms) an efficient solver can be easily developed. The capabilities of the method are illustrated on numerical examples using the Boundary Element Method

Stress intensity factors computation for bending plates with extended finite element method

The modelization of bending plates with through-the-thickness cracks is investigated. We consider the Kirchhoff–Love plate model, which is valid for very thin plates. Reduced Hsieh–Clough–Tocher triangles and reduced Fraejis de Veubeke–Sanders quadrilaterals are used for the numerical discretization. We apply the eXtended Finite Element Method strategy: enrichment of the finite element space with the asymptotic bending singularities and with the discontinuity across the crack. The main point, addressed in this paper, is the numerical computation of stress intensity factors. For this, two strategies, direct estimate and J-integral, are described and tested. Some practical rules, dealing with the choice of some numerical parameters, are underlined

Quantum Integration in Sobolev Classes

We study high dimensional integration in the quantum model of computation. We develop quantum algorithms for integration of functions from Sobolev classes $W^r_p([0,1]^d)$ and analyze their convergence rates. We also prove lower bounds which show that the proposed algorithms are, in many cases, optimal within the setting of quantum computing. This extends recent results of Novak on integration of functions from H\"older classes.Comment: Paper submitted to the Journal of Complexity. 28 page

The Fourier Singular Complement Method for the Poisson problem. Part II: axisymmetric domains

This paper is the second part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In the first part of this series, the Fourier Singular Complement Method was introduced and analysed, in prismatic domains. In this second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges or vertices of the domain, nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used

The Fourier Singular Complement Method for the Poisson problem. Part I: prismatic domains

This is the first part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In this first part, the Fourier Singular Complement Method is introduced and analysed, in prismatic domains. In the second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges of the domain nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used

Continuum Electromechanical Modeling of Protein-Membrane Interaction

A continuum electromechanical model is proposed to describe the membrane curvature induced by electrostatic interactions in a solvated protein-membrane system. The model couples the macroscopic strain energy of membrane and the electrostatic solvation energy of the system, and equilibrium membrane deformation is obtained by minimizing the electro-elastic energy functional with respect to the dielectric interface. The model is illustrated with the systems with increasing geometry complexity and captures the sensitivity of membrane curvature to the permanent and mobile charge distributions.Comment: 5 pages, 12 figure

The boundary value problem for discrete analytic functions

This paper is on further development of discrete complex analysis introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discrete analytic, if for each face the difference quotients along the two diagonals are equal. We prove that the Dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution. In the case when each face has orthogonal diagonals we prove that this solution uniformly converges to a harmonic function in the scaling limit. This solves a problem of S. Smirnov from 2010. This was proved earlier by R. Courant-K. Friedrichs-H. Lewy and L. Lusternik for square lattices, by D. Chelkak-S. Smirnov and implicitly by P.G. Ciarlet-P.-A. Raviart for rhombic lattices. In particular, our result implies uniform convergence of the finite element method on Delaunay triangulations. This solves a problem of A. Bobenko from 2011. The methodology is based on energy estimates inspired by alternating-current network theory.Comment: 22 pages, 6 figures. Several changes: Theorem 1.2 generalized, several assertions added, minor correction in the proofs of Lemma 2.5, 3.3, Example 3.6, Corollary 5.