An Effective Finite Element Iterative Solver for a Poisson--Nernst--Planck Ion Channel Model with Periodic Boundary Conditions

2020 ◽  
Vol 42 (6) ◽  
pp. B1490-B1516
Author(s):  
Dexuan Xie ◽  
Benzhuo Lu
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Ion Channel ◽  
Periodic Boundary ◽  
Channel Model ◽  
Periodic Boundary Conditions ◽  
Iterative Solver ◽  
Ion Channel Model

scholarly journals Image Charge Method for Reaction Fields in a Hybrid Ion-Channel Model

2011 ◽  
Vol 9 (4) ◽  
pp. 1056-1070 ◽  
Author(s):  
Zhenli Xu ◽  
Wei Cai ◽  
Xiaolin Cheng
Keyword(s):  
Boundary Conditions ◽  
Ion Channel ◽  
Electrostatic Interactions ◽  
Channel Model ◽  
Field Potential ◽  
Image Method ◽  
Optimization Approach ◽  
Inhomogeneous System ◽  
Reaction Field ◽  
Ion Channel Model

AbstractA multiple-image method is proposed to approximate the reaction-field potential of a source charge inside a finite length cylinder due to the electric polarization of the surrounding membrane and bulk water. When applied to a hybrid ion-channel model, this method allows a fast and accurate treatment of the electrostatic interactions of protein with membrane and solvent. To treat the channel/membrane interface boundary conditions of the electric potential, an optimization approach is used to derive image charges by fitting the reaction-field potential expressed in terms of cylindric harmonics. Meanwhile, additional image charges are introduced to satisfy the boundary conditions at the planar membrane interfaces. In the end, we convert the electrostatic interaction problem in a complex inhomogeneous system of ion channel/membrane/water into one in a homogeneous free space embedded with discrete charges (the source charge and image charges). The accuracy of this method is then validated numerically in calculating the solvation self-energy of a point charge.

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.

The Energetics of Dislocation-Obstacle Interactions by 3-D Quasicontinuum Simulations

1999 ◽  
Vol 578 ◽  
Author(s):  
Kedar Hardikar ◽  
R. Phillips
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Point Defects ◽  
Periodic Boundary ◽  
Boundary Effect ◽  
Periodic Boundary Conditions ◽  
Boundary Effects ◽  
Quasicontinuum Method ◽  
Finite Element Calculations ◽  
Infinite Extent

AbstractThe goal of this work is to study the interaction of dislocations with local obstacles to glide such as point defects, precipitates and other dislocations. The quasicontinuum method is used as the basis of this study. It is demonstrated that two types of boundary effects are of concern in the calculation of hardening parameters using finite sized simulation cells. A recently developed technique to incorporate periodic boundary conditions in the quasicontinuum method is used to eliminate surface effects which were present in earlier implementations and to simulate a dislocation of infinite extent interacting with an array of obstacles. The second type of boundary effect is due to the boundary conditions on the lateral boundaries. A method based on finite element calculations is proposed for quantifying the effect of lateral boundaries in these simulations. Preliminary results for the validation of the method are presented as well as a simulation of the interaction between a conventional edge dislocation in Al with an array of clusters of Ni atoms.

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