Three-dimensional temperature simulation of a generator/motor armature under periodic boundary conditions

1987 ◽  
Vol 107 (3) ◽  
pp. 46-55
Author(s):  
Mitsuru Fukushima ◽  
Masato Teraoka
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Three Dimensional ◽  
Periodic Boundary Conditions

Optimal feedback control problem for Bingham media motion with periodic boundary conditions

2019 ◽  
Vol 485 (2) ◽  
pp. 139-141
Author(s):  
V. G. Zvyagin ◽  
M. V. Turbin
Keyword(s):  
Feedback Control ◽  
Boundary Conditions ◽  
Control Problem ◽  
Periodic Boundary ◽  
Three Dimensional ◽  
Degree Theory ◽  
Periodic Boundary Conditions ◽  
Optimal Feedback Control ◽  
Operator Inclusion ◽  
Optimal Feedback

We study the optimal feedback control problem for the motion of Bingham media with periodic boundary conditions in two- and three-dimensional cases. First, the considered problem is interpreted as an operator inclusion with a multivalued right-hand side. Then, the approximation-topological approach to hydrodynamic problems and the degree theory for a class of multivalued maps are used to prove the existence of solutions of this inclusion. Finally, we prove that, among the solutions of the considered problem, there exists one minimizing the given cost functional.

scholarly journals Thermodynamic limit for a system with dipole-dipole interactions

1975 ◽  
Vol 19 (1) ◽  
pp. 116-128 ◽  
Author(s):  
E. R. Smith ◽  
J. W. Perram
Keyword(s):  
Free Energy ◽  
Variational Principle ◽  
Boundary Conditions ◽  
Thermodynamic Limit ◽  
Periodic Boundary ◽  
Three Dimensional ◽  
Periodic Boundary Conditions ◽  
Free Energies ◽  
Dipole Interactions ◽  
Simple Boundary

AbstractIt is shown that for the three dimensional Ising model with dipole-dipole interactions, the thermodynamic limit of the free energy with simple boundary conditions is not the same as the thermodynamic limit of the free energy with periodic boundary conditions. A variational principle is developed to connect the two free energies.

A Simplified CFD Model With Multi-Periodic Boundary Conditions for Cross Wavy Channels

2011 ◽  
Author(s):  
L. X. Du ◽  
M. Zeng ◽  
Q. W. Wang
Keyword(s):  
Heat Transfer ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Flow Direction ◽  
Wave Length ◽  
Three Dimensional ◽  
Periodic Boundary Conditions ◽  
Navier Stokes ◽  
The Cross ◽  
Wavy Channels

The compact and efficient primary surface heat exchangers are often used as recuperators in microturbine regenerative cycle systems. In the present study, the flow and the heat transfer performance of the cross wavy (CW) ducts have been simulated by three-dimensional models. The hydrodynamic diameters of the models are 1.689mm. Navier-Stokes and energy equations are solved by COMSOL3.5. Because one single wavy cell will overlap more than one adjacent channel, multi-periodic boundary conditions are especially adopted to simplify the calculations. Multi-periodic boundary conditions have been proved to have more reasonable flow field and heat transfer coefficient compared with the literature results. A dimensionless parameter L/A (wave length L, internal height of the corrugation in flow direction A) is defined as the optimization target. The numerical results indicated that when L/A = 6, the CW channel has the best comprehensive performance in all the cases. The comprehensive performances of the CW ducts are evaluated by the j/f (heat transfer factor j and friction factor f). The flow and heat transfer characteristics are much more complex in the cross wavy channels, especially when L/A is small.

The Harmonic Solid

2019 ◽  
pp. 331-349
Author(s):  
Robert H. Swendsen
Keyword(s):  
Boundary Conditions ◽  
Specific Heat ◽  
Periodic Boundary ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Normal Modes ◽  
Periodic Boundary Conditions ◽  
One Dimension ◽  
Debye Approximation

In Chapter 26 we return to calculating the contributions to the specific heat of a crystal from the vibrations of the atoms. The vibrations of a model of a solid, for which the interactions are quadratic in form, is investigated. Calculations are restricted to one dimension for simplicity in the derivations of the Fourier modes and the equations of motion. Both pinned and periodic boundary conditions are discussed. The representation of the Hamiltonian in terms of normal modes and the solution in terms of the equations of motion are derived. The Debye approximation is then introduced for three-dimensional systems.

Phase Transitions in Three Dimensional Ising Systems under Free and Periodic Boundary Conditions

2004 ◽  
Vol 301 (1) ◽  
pp. 91-95
Author(s):  
M FELISA MARTÍNEZ RUIZ ◽  
FRANCISCA PAJUELO ◽  
JORGE GARCÍA ◽  
CARMEN ARAGÓ ◽  
JULIO. A. GONZALO
Keyword(s):  
Phase Transitions ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Three Dimensional ◽  
Periodic Boundary Conditions ◽  
Ising Systems

More About Periodic Boundary Conditions

2006 ◽  
Author(s):  
Vasily Bulatov ◽  
Wei Cai
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Three Dimensional ◽  
Periodic Boundary Conditions ◽  
Good News ◽  
Infinite Domain ◽  
Dynamic Simulations ◽  
Linear Elasticity Theory ◽  
Additional Strain ◽  
Simulation Cell

We have already used periodic boundary conditions (PBC) for the static and dynamic simulations described in Chapters 2 and 3. There, PBC were applied along one or two directions of the simulation cell. Application of PBC in all three directions holds an important advantage when one’s goal is to examine the behavior in the bulk: under fully three-dimensional (3D) PBC, the simulated solid can be free of any surfaces. By comparison, the simulations discussed in the previous chapters all contained free surfaces or artificial interfaces in the directions where PBC were not applied. Full 3D PBC are easy to implement in an atomistic simulation through the use of scaled coordinates. However, there are important technical issues specific to simulations of lattice dislocations. First, a fully periodic simulation cell can accommodate only such dislocation arrangements whose net Burgers vector is zero. Thus, the minimal number of dislocations that can be introduced in a periodic supercell is two, i.e. a dislocation dipole. Two dislocations forming a dipole are bound to interact with each other, as well as with their periodic images. Associated with these interactions are additional strain, energy, and forces whose effects can “pollute” the calculated results. The good news is that, in most cases, the artifacts of PBC can be quantified through the use of linear elasticity theory so that physical properties of dislocations can be accurately extracted. Given the simplicity and robustness of PBC, the extrawork required to extract physical results is well worth it. This chapter describes how to evaluate and eliminate the artifacts that inevitably appear when 3D PBC are used for atomistic simulations of dislocations. In the following three sections, we show how to take full advantage of PBC when one wants to calculate the displacement field induced by a dislocation (Section 5.1), the dislocation’s core energy (Section 5.2) and Peierls stress (Section 5.3). The common theme for all three case studies is an attempt to construct a solution of the elasticity equations in a periodic domain by superimposing a periodic array of solutions of an infinite domain.

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>

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