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 Asymptotic solution of Sturm-Liouville problem with periodic boundary conditions for relativistic finite-difference Schrödinger equation

2020 ◽  
Vol 28 (3) ◽  
pp. 230-251
Author(s):  
Ilkizar V. Amirkhanov ◽  
Irina S. Kolosova ◽  
Sergey A. Vasilyev
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Asymptotic Series ◽  
Periodic Boundary Conditions ◽  
Singularly Perturbed ◽  
Half Line ◽  
Asymptotic Convergence ◽  
Relativistic Particles ◽  
Positive Half

The quasi-potential approach is very famous in modern relativistic particles physics. This approach is based on the so-called covariant single-time formulation of quantum field theory in which the dynamics of fields and particles is described on a space-like three-dimensional hypersurface in the Minkowski space. Special attention in this approach is paid to methods for constructing various quasi-potentials. The quasipotentials allow to describe the characteristics of relativistic particles interactions in quark models such as amplitudes of hadron elastic scatterings, mass spectra, widths of meson decays and cross sections of deep inelastic scatterings of leptons on hadrons. In this paper SturmLiouville problems with periodic boundary conditions on a segment and a positive half-line for the 2m-order truncated relativistic finite-difference Schrdinger equation (LogunovTavkhelidzeKadyshevsky equation, LTKT-equation) with a small parameter are considered. A method for constructing of asymptotic eigenfunctions and eigenvalues in the form of asymptotic series for singularly perturbed SturmLiouville problems with periodic boundary conditions is proposed. It is assumed that eigenfunctions have regular and boundary-layer components. This method is a generalization of asymptotic methods that were proposed in the works of A. N. Tikhonov, A. B. Vasilyeva, and V. F Butuzov. We present proof of theorems that can be used to evaluate the asymptotic convergence for singularly perturbed problems solutions to solutions of degenerate problems when 0 and the asymptotic convergence of truncation equation solutions in the case m. In addition, the SturmLiouville problem on the positive half-line with a periodic boundary conditions for the quantum harmonic oscillator is considered. Eigenfunctions and eigenvalues are constructed for this problem as asymptotic solutions for 4-order LTKT-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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