Diffusion in Potential Fields: Time-Dependent Capture on Radial and Rectangular Substrates

2004 ◽  
Vol 859 ◽  
Author(s):  
J. A. Venables ◽  
P. Yang
Keyword(s):  
Periodic Boundary ◽  
Energy Surface ◽  
Periodic Boundary Conditions ◽  
Nucleation And Growth ◽  
Time Dependent ◽  
Potential Fields ◽  
Diffusion Field ◽  
Rectangular Array ◽  
Diffusion Controlled ◽  
And Diffusion

ABSTRACTIn models of nucleation and growth on surfaces, it is usually assumed that the energy surface of the substrate is flat, that diffusion is isotropic, and that capture numbers can be calculated in the diffusion-controlled limit. We lift these restrictions analytically, and introduce a hybrid discrete FFT method of solving for the 2D time-dependent diffusion field of adparticles on general non-uniform substrates. The method, with periodic boundary conditions, is appropriate, for example, following nucleation on a regular (rectangular) array of defects. The programs, which have been realized in Matlab®6.5, are instructive for visualizing potential and diffusion fields, and for producing illustrative movies of crystal growth under various conditions. Here we demonstrate the time-dependence of capture numbers in the initial stages of annealing at high adparticle concentration in the presence of repulsive adparticle-cluster interactions; however it is clear that the method works in general for deposition, growth and annealing at all times.

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.

scholarly journals Description of Shapiro steps on the potential energy surface of a Frenkel–Kontorova model Part I: The chain in a variable box

2021 ◽  
Vol 94 (3) ◽  
Author(s):  
W. Quapp ◽  
J. M. Bofill
Keyword(s):  
Potential Energy ◽  
Boundary Conditions ◽  
Potential Energy Surface ◽  
Periodic Boundary ◽  
Energy Surface ◽  
Periodic Boundary Conditions ◽  
Shapiro Steps ◽  
Kontorova Model

AbstractWe explain the vibrations of a Frenkel–Kontorova (FK) model under Shapiro steps by the action of an external alternating force. We demonstrate Shapiro steps for a case with soft ‘springs’ between an 8-particles FK chain. Shapiro steps start with a single jump over the highest $$\hbox {SP}_4$$ SP 4 in the global valley through the PES. They finish with doubled, and again doubled oscillations. We study in this part I a traditional FK model with periodic boundary conditions.

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 Global classical solutions to the elastodynamic equations with damping

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mengmeng Liu ◽  
Xueyun Lin
Keyword(s):  
Initial Data ◽  
Periodic Boundary ◽  
Energy Estimate ◽  
Periodic Boundary Conditions ◽  
Deformation Tensor ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Weighted Energy ◽  
Global Classical Solutions ◽  
Weighted Energy Estimate

AbstractIn this paper, we show the global existence of classical solutions to the incompressible elastodynamics equations with a damping mechanism on the stress tensor in dimension three for sufficiently small initial data on periodic boxes, that is, with periodic boundary conditions. The approach is based on a time-weighted energy estimate, under the assumptions that the initial deformation tensor is a small perturbation around an equilibrium state and the initial data have some symmetry.

scholarly journals Normal periodic solutions for the fractional abstract Cauchy problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jennifer Bravo ◽  
Carlos Lizama
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Periodic Solution ◽  
Periodic Boundary ◽  
Abstract Cauchy Problem ◽  
Periodic Boundary Conditions ◽  
Sectorial Operator ◽  
Closed Linear Operator ◽  
Resolvent Set ◽  
Spectral Angle

AbstractWe show that if A is a closed linear operator defined in a Banach space X and there exist $t_{0} \geq 0$ t 0 ≥ 0 and $M>0$ M > 0 such that $\{(im)^{\alpha }\}_{|m|> t_{0}} \subset \rho (A)$ { ( i m ) α } | m | > t 0 ⊂ ρ ( A ) , the resolvent set of A, and $$ \bigl\Vert (im)^{\alpha }\bigl(A+(im)^{\alpha }I \bigr)^{-1} \bigr\Vert \leq M \quad \text{ for all } \vert m \vert > t_{0}, m \in \mathbb{Z}, $$ ∥ ( i m ) α ( A + ( i m ) α I ) − 1 ∥ ≤ M  for all  | m | > t 0 , m ∈ Z , then, for each $\frac{1}{p}<\alpha \leq \frac{2}{p}$ 1 p < α ≤ 2 p and $1< p < 2$ 1 < p < 2 , the abstract Cauchy problem with periodic boundary conditions $$ \textstyle\begin{cases} _{GL}D^{\alpha }_{t} u(t) + Au(t) = f(t), & t \in (0,2\pi ); \\ u(0)=u(2\pi ), \end{cases} $$ { D t α G L u ( t ) + A u ( t ) = f ( t ) , t ∈ ( 0 , 2 π ) ; u ( 0 ) = u ( 2 π ) , where $_{GL}D^{\alpha }$ D α G L denotes the Grünwald–Letnikov derivative, admits a normal 2π-periodic solution for each $f\in L^{p}_{2\pi }(\mathbb{R}, X)$ f ∈ L 2 π p ( R , X ) that satisfies appropriate conditions. In particular, this happens if A is a sectorial operator with spectral angle $\phi _{A} \in (0, \alpha \pi /2)$ ϕ A ∈ ( 0 , α π / 2 ) and $\int _{0}^{2\pi } f(t)\,dt \in \operatorname{Ran}(A)$ ∫ 0 2 π f ( t ) d t ∈ Ran ( A ) .

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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