Finite Element Methods for Eigenvalue Problems on a Rectangle with (Semi-) Periodic Boundary Conditions on a Pair of Adjacent Sides

Computing ◽  
2000 ◽  
Vol 64 (3) ◽  
pp. 191-206 ◽  
Author(s):  
Hennie De Schepper
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Finite Element Methods ◽  
Periodic Boundary ◽  
Eigenvalue Problems ◽  
Periodic Boundary Conditions

The Electrostatic Potential inside the Electron-Optical Systen with Periodic Boundary-Value Conditions

2014 ◽  
Vol 933 ◽  
pp. 534-537
Author(s):  
Ting Hang Pei ◽  
Kuen Yu Tsai ◽  
Jia Han Li
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Optical System ◽  
Finite Element Methods ◽  
Electrostatic Potential ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Electron Optical System ◽  
Large Domain

A program based on the SOR method have been setup for calculating the electrostatic potential inside the electron optical system (EOS). This method can deal with large domain calculation more efficiently than by using the finite difference or finite element methods. Since the MEBDW system is composed of an array of EOSs, periodic boundary conditions in thexandydirections are applied. A case of EOS is demonstrated in this paper.

scholarly journals Spectrum of discrete 2n-th order difference operator with periodic boundary conditions and its applications

2021 ◽  
Vol 41 (4) ◽  
pp. 489-507
Author(s):  
Abdelrachid El Amrouss ◽  
Omar Hammouti
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Critical Point Theory ◽  
Periodic Boundary ◽  
Eigenvalue Problems ◽  
Difference Operator ◽  
Periodic Boundary Conditions ◽  
Point Theory ◽  
Existence Of A Solution ◽  
Order Difference

Let \(n\in\mathbb{N}^{*}\), and \(N\geq n\) be an integer. We study the spectrum of discrete linear \(2n\)-th order eigenvalue problems \[\begin{cases}\sum_{k=0}^{n}(-1)^{k}\Delta^{2k}u(t-k) = \lambda u(t) ,\quad & t\in[1, N]_{\mathbb{Z}}, \\ \Delta^{i}u(-(n-1))=\Delta^{i}u(N-(n-1)),\quad & i\in[0, 2n-1]_{\mathbb{Z}},\end{cases}\] where \(\lambda\) is a parameter. As an application of this spectrum result, we show the existence of a solution of discrete nonlinear \(2n\)-th order problems by applying the variational methods and critical point theory.

The interlacing of eigenvalues for periodic multi-parameter problems

1978 ◽  
Vol 80 (3-4) ◽  
pp. 357-362 ◽  
Author(s):  
Patrick J. Browne
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Periodic Boundary ◽  
Eigenvalue Problems ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Image Position ◽  
Periodic Equation

SynopsisThis paper studies a linked system of second order ordinary differential equationswhere xx ∈ [ar, br] and the coefficients qrars are continuous, real valued and periodic of period (br − ar), 1 ≤ r,s ≤ k. We assume the definiteness condition det{ars(xr)} > 0 and 2k possible multiparameter eigenvalue problems are then formulated according as periodic or semi-periodic boundary conditions are imposed on each of the equations of (*). The main result describes the interlacing of the 2k possible sets of eigentuples thus extending to the multiparameter case the well known theorem concerning 1-parameter periodic equation.

A numerical homogenization technique for unidirectional composites using polygonal generalized finite elements

2021 ◽  
pp. 146442072110463
Author(s):  
Murilo Sartorato
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Boundary Conditions ◽  
Computational Efficiency ◽  
Periodic Boundary ◽  
Deformation Gradient ◽  
Reaction Force ◽  
Periodic Boundary Conditions ◽  
Unit Cell ◽  
Unit Cells

The present study proposes a computational methodology to obtain the homogenized effective elastic properties of unidirectional fibrous composite materials by using the generalized finite-element method and penalization techniques to impose periodic boundary conditions on non-uniform polygonal unit cells. Each unit cell is described by a single polygonal finite element using Wachspress functions as base shape functions and different families of enrichment functions to account for the internal fiber influence on stresses and strains fields. The periodic boundary conditions are imposed using reflection laws between two parallel opposing faces using a Lagrange multiplier approach; this reflection law creates a distributed reaction force over the edges of the [Formula: see text]-gon from the direct application of a given deformation gradient, which simulates different macroscopic load cases on the macroscopic body the unit cell is part of. The methodology is validated through a comparison with results for similar unit cells found in the literature and its computational efficiency is compared to simple cases solved using a classic finite-element approach. This methodology showed computational advantages over the classic finite elements in both computational efficiency and total number of degrees of freedom for convergence and flexibility on the shape of the unit cell used. Finally, the methodology provides an efficient way to introduce non-circular fiber shapes and voids.

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