Entropy of point defects calculated within periodic boundary conditions

2004 ◽  
Vol 69 (15) ◽  
Author(s):  
E. Rauls ◽  
Th. Frauenheim
Keyword(s):  
Boundary Conditions ◽  
Point Defects ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions

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.

Molecular Dynamics Using Non-Variational Polarizable Force Fields: Theory, Periodic Boundary Conditions Implementation and Application to the Bond Capacity Model

2019 ◽  
Author(s):  
Pier Paolo Poier ◽  
Louis Lagardere ◽  
Jean-Philip Piquemal ◽  
Frank Jensen
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Force Fields ◽  
Periodic Boundary Conditions ◽  
Computational Effort ◽  
Polarizable Force Fields ◽  
Energy Models ◽  
Capacity Model ◽  
Energy Gradient ◽  
Polarization Models

<div> <div> <div> <p>We extend the framework for polarizable force fields to include the case where the electrostatic multipoles are not determined by a variational minimization of the electrostatic energy. Such models formally require that the polarization response is calculated for all possible geometrical perturbations in order to obtain the energy gradient required for performing molecular dynamics simulations. </p><div> <div> <div> <p>By making use of a Lagrange formalism, however, this computational demanding task can be re- placed by solving a single equation similar to that for determining the electrostatic variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p><div><div><div> </div> </div> </div> <p> </p><div> <div> <div> <p>variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p> </div> </div> </div> </div> </div> </div> </div> </div> </div>

scholarly journals Sharp Lyapunov-type inequalities for second-order half-linear difference equations with different kinds of boundary conditions

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Difference Equation ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

AbstractIn this work, we establish optimal Lyapunov-type inequalities for the second-order difference equation with p-Laplacian $$\begin{aligned} \Delta (\left| \Delta u(k-1)\right| ^{p-2}\Delta u(k-1))+a(k)\left| u(k)\right| ^{p-2}u(k)=0 \end{aligned}$$ Δ ( Δ u ( k - 1 ) p - 2 Δ u ( k - 1 ) ) + a ( k ) u ( k ) p - 2 u ( k ) = 0 with Dirichlet, Neumann, mixed, periodic and anti-periodic boundary conditions.

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