Behaviour of a thermodynamic model system under time-dependent periodic boundary conditions

1982 ◽  
Vol 5 (3) ◽  
pp. 332-358 ◽  
Author(s):  
R. S. Berry ◽  
F. D'Isep ◽  
L. Sertorio
Keyword(s):  
Boundary Conditions ◽  
Thermodynamic Model ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Model System ◽  
Time Dependent

Wavelet Based Solution to Time-Dependent Higher Order Non-Linear Two-Point Initial Boundary Value Problems With Non-Periodic Boundary Conditions: KdV, Boussinesq Equations

2003 ◽  
Author(s):  
H. N. Narang ◽  
Rajiv K. Nekkanti
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Periodic Boundary Conditions ◽  
Initial Boundary ◽  
Time Dependent ◽  
Initial Boundary Value Problems ◽  
Non Linear ◽  
Initial Boundary Value

The Wavelet solution for boundary-value problems is relatively new and has been mainly restricted to the solutions in data compression, image processing and recently to the solution of differential equations with periodic boundary conditions. This paper is concerned with the wavelet-based Galerkin’s solution to time dependent higher order non-linear two-point initial-boundary-value problems with non-periodic boundary conditions. The wavelet method can offer several advantages in solving the initial-boundary-value problems than the traditional methods such as Fourier series, Finite Differences and Finite Elements by reducing the computational time near singularities because of its multi-resolution character. In order to demonstrate the wavelet, we extend our prior research of solution to parabolic equations and problems with non-linear boundary conditions to non-linear problems involving KdV Equation and Boussinesq Equation. The results of the wavelet solutions are examined and they are found to compare favorably to the known solution. This paper on the whole indicates that the wavelet technique is a strong contender for solving partial differential equations with non-periodic conditions.

Analytically derived space time-based boundary condition (STBC) to account for stress wave propagation in a heterogeneous micromechanical model at hypervelocity impact

2019 ◽  
Author(s):  
Zhiye Li ◽  
Somnath Ghosh
Keyword(s):  
Boundary Condition ◽  
Wave Propagation ◽  
Boundary Conditions ◽  
High Strain Rate ◽  
Material Properties ◽  
Periodic Boundary ◽  
High Strain ◽  
Periodic Boundary Conditions ◽  
Time Dependent ◽  
Stress Wave Propagation

Abstract Recent years have seen a surge in research on material and structural response of composites using the homogenization based hierarchical modeling method. The microstructural representative volume element (RVE) is a small micro-region for which the volume average of variables is the same as those for the entire body. Representations of the microstructure are used for micromechanical simulations in determination of effective material properties by homogenization. Conventionally, periodic boundary conditions (PBC) are applied on the RVE boundary. However, when the heterogeneous microstructure is under very high strain rate loading conditions (105s−1−107s−1), periodic boundary conditions (PBC) do not accurately represent the effect of stress wave propagation. Improper boundary conditions can lead to significant error in homogenized material properties. In order to increase the accuracy of the homogenization model, this study introduces a new space-time dependent boundary condition (STBC) for a 3D microscopic RVE subjected to high strain rate deformation in explicit FEM simulation by using the characteristics method of traveling waves. The advantages of the STBC are discussed in comparison with time-dependent averaged results of examples using PBC. The proposed STBC offers significant advantages over conventional PBC in the RVE-based analysis of heterogeneous materials.

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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