Periodic boundary conditions

Periodic boundary conditions are selected in analytical or numerical model calculations in order to avoid a separate treatment of edges or to the area to which it refers the bill shrink. Periodic boundary conditions in dimensions can be interpreted as a compactification of the space to a torus in dimensions.

Application areas are in solid state physics of crystalline materials, molecular dynamics, Monte Carlo simulations as well as simulations on grids such as lattice gauge theories.

In continuous Particle Simulations with periodic boundary conditions, particles emerge at one edge of the simulation box and enter the opposite edge again.

Mathematical definition

A partial differential equation is an equation of an unknown function in the derivatives

Occur for an open subset. One speaks of periodic boundary conditions, if the shape

And is one demands that there is a continuous extension with

To give.

Need not be a cuboid, but allow a full periodic coverage of the space (see space filling and tiling ).

Compactification

Periodic boundary conditions in all dimensions correspond to a flat torus in dimensions, ie, not curved surface. The surface is therefore not described as a subset of the -dimensional space, but as a Cartesian product of circles.

Conserved quantities

Periodic boundary conditions allow energy and momentum conservation, but violate the conservation of angular momentum. Formal: conservation of angular momentum is a result of not given here, invariance of physical laws with respect to rotations of the reference system. Clearly: A locally -started eddy grows by diffusion pulse down to the length scale of the simulation box and then systematically destroyed.

Examples

Periodic solution for a periodic problem

In the first example, periodic boundary conditions with various non- periodic boundary conditions are combined. It involves the simulation of flow around turbine blades in an annular gap. Suppose that the assumption ( or approximation) that the flow around all N blades is the same. The problem therefore has a n-fold rotational symmetry, which is to be used by a sector is selected as the simulation domain, which contains only one of the blades. In the end, the solution adopted for the sector N times is reproduced be assembled into the ring. The computational effort is thus reduced to a fraction of the total area and total solution automatically has the expected symmetry, is periodic. Thus, the solution may be physically meaningful but it must match at the seams, as smooth as it is required by the solution in the interior of the simulation area. This requirement makes the periodic boundary conditions. In addition, in this particular example has the solution to the solid inner and outer surfaces corresponding boundary conditions satisfy (eg, flow rate = wall velocity, if the boundary layer is modeled in detail) and upstream and downstream fit with the solutions for other turbine stages ( it is to find a common solution ).

How to reach solutions that satisfy the periodic boundary conditions, depends on how ever the physical quantities of the problem (in this case pressure, temperature and velocity components ) are to be represented. One possibility would be the sum of basis functions, which are individually and periodically each smooth, that is sinusoidally dependent on the angle around the axis of rotation, for example, see Fourier series. An edge then does not occur. One other, more appropriate here class of methods uses numerous grid points ( grid points ), see, eg, finite-volume method. In this case, the grid points are defined at a side area of the simulation of such on the other side to be adjacent to the boundary condition to satisfy.

Ignoring surface waves

In a block-shaped single crystal phonons are suggestions mechanical standing waves. Ignoring that the atoms in the periphery have different neighborhood relations, the wave equation simplifies to the infinitely extended lattice. The solutions are superpositions of plane waves of the type

Therein, the components of which the deflections of all the atoms of the crystallographic unit cell out of their positions of rest, and location coordinates (for simplicity, only two dimensions ), the time and the frequency of which is dependent on the wave vector and the polarization contained in the amplitude factor.

The only considered surface effect is the restriction to discrete wave vectors:

Natural numbers and the size of the crystal in terms of the size of the unit cell of the crystal. These solutions satisfy the periodicity condition

In the Born-von Kármán model - periodic boundary conditions are also called Born-von Kármán boundary conditions.

Compression without vice

The picture of the vise stands for large external forces. If you were to leave on simulated vice jaws on the atoms of a small simulation box, the results would be useless. Very large forces are needed to investigate materials at not realizable in the laboratory conditions such as those in the core of the Earth (eg, to determine its elasticity tensor ). By using periodic boundary conditions, this problem is circumvented by ( mechanical stresses) not specified forces and deformations are observed, but vice versa. The simulation box contains to a single unit cell of the crystal and is arbitrarily deformed ( edge lengths and angles ). For each geometry of the unit cell, the positions of the atoms is varied and in each case, the electronic energy calculated (see Born- Oppenheimer approximation ), the density functional theory is used with the periodic basis functions. From the function of the electronic energy of the geometry of the unit cell of relaxed in arrangement of atoms, the external mechanical stresses arise as a result of calculation.

Application areas with non-periodic problem

The advantage of Wandlosigkeit by periodic boundary conditions can also be used in the simulation of systems that are not periodic actually. The size of the simulation box is then set arbitrarily, but not arbitrary: it must be greater than the distances that occur on the correlations. This can be a large molecule in solution, a material distortion or a density fluctuation near a phase transition.

For short-range interactions cutoff radii can be introduced, from which no explicit interaction between the particles is more charged, however analytically obtained additional terms for the cut interactions can still be taken into account. In the case of using this cutoff radii can specify a criterion for a not to be border size of the simulation box, which often use other criteria must be considered, the much larger box size force: The smallest diameter of the simulation should be at least twice as large, like most such used cutoff radius, otherwise a particle in the simulation box (Central box ) provides a copy of its neighboring themselves in a box. Furthermore, this is the criterion, so that one may use the minimum image convention for these interactions. This convention states that you have to consider only interactions with particles in the next adjacent boxes of the Central Box.

Thermodynamic limit and the continuum limit

In many molecular dynamics simulations of the thermodynamic limit is, i.e., is determined. For this purpose, the measured sizes for different sizes of the simulation can be approximated to infinite volume. In general, the effects of the periodic boundary conditions disappear in this limit.

Similarly, simulations of the mesh in the continuum limit, i.e. the limit case of a lattice spacing is determined.

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