Lattice gas automaton

Lattice gas automata (LGA) or lattice gas cellular automata (LGCA) methods are a series of cellular automata methods used to simulate fluid flows. It was the precursor to the lattice Boltzmann methods. From the LGCA, it is possible to derive the macroscopic Navier-Stokes equations. [Succi, section 2.3 describes the process] Interest in the LGCA methods levelled off in the early 1990s, as the interest in the lattice Boltzmann started to rise. [Succi, section 2.6]

Basic principles

As a cellular automata, these models comprise of a lattice, where the sites on the lattice can take a certain number of different states. In lattice gas, the various states are particles with certain velocities. Evolution of the simulation is done in discrete time steps. After each time step, the state at a given site can be determined by the state of the site itself and neighboring sites, "before" the time step.

The state at each site is purely boolean. At a given site, there either "is" or there "is not" a particle which is moving up.

At each time step, two processes are carried out, propagation and collision. [Buick, section 3.4]

In the propagation step, each particle will move to a neighboring site determined by the velocity that particle had. Barring any collisions, a particle with an upwards velocity will after the time step maintain that velocity, but be moved to the neighboring site above the original site. The so-called exclusion principle prevents two or more particles from travelling on the same link in the same direction.

In the collision step, collision rules are used to determine what happens if multiple particles reach the same site. These collision rules are required to maintain mass conservation, and conserve the total momentum. Note that the exclusion principle does not prevent two particles from travelling on the same link in "opposite" directions, when this happens, the two particles pass each other without colliding.

Early attempts with a square lattice

In papers published in 1973 and 1976, Hardy, Pomeau and de Pazzis introduced the first Lattice Boltzmann model, which is called the HPP model after the authors. In this model, the lattice is square, and particles can move to any of the four sites whose cells share a common edge. Particles cannot move diagonally.

If two particles collide head-on, for example a particle moving to the left meets a particle moving to the right, the outcome will be two particles leaving the site at right angles to the direction they came in. [Buick, section 3.2.1]

The HPP model lacked rotational invariance, which made the model highly anisotropic. This means for example, that the vortices produced by the HPP model are square-shaped. [Succi, footnote p. 22]

Hexagonal grids

The hexagonal grid model was first introduced in 1986, in a paper by Uriel Frisch, Brosl Hasslacher and Yves Pomeau, and this has become known as the FHP model after its inventors. The model has six or seven velocities, depending on which variation which is used. In any case, six of the velocities represent movement to each of the neighboring sites. In some models (called FHP-II), a seventh velocity representing particles "at rest" is introduced. The "at rest" particles do not propagate to neighboring sites, but they are capable of colliding with other particles. [Buick, section 3.2.2]

The collision rules in the FHP model are not deterministic, some input situations produce two possible outcomes, and when this happens, one of them is picked at random.

The hexagonal grid does not suffer as large anisotropy troubles as those which plague the HPP square grid model, a fortunate fact which is not entirely obvious, and which prompted Frisch to remark that "the symmetry gods are benevolent" [Succi, footnote p. 23] .

Three dimensions

For a three-dimensional grid, the only regular polytope which fills the whole space is the cube, while the only regular polytopes with a sufficiently large symmetry group are the dodecahedron and icosahedron (without the second constraint the model will suffer the same drawbacks as the HPP model). In order to make a model which tackles three dimensions, it is therefore necessary to increase the number of dimensions, such as the 1986 model by D'Humières, Lallemand and Frisch which employed a face-centered hypercube model.Wolf-Gladrow, sections 3.4 - 3.5]

Obtaining macroscopic quantities

The density at a site can be found by counting the number of particles at each site. If the particles are multiplied with the unit velocity before being summed, one can obtain the momentum at the site. [Buick, section 3.5.1]

However, calculating the density, momentum, and velocity for individual sites is subject to a large amount of noise, and in practice, one would average over a larger region in order to obtain more reasonable results. Ensemble averaging is often used to reduce the statistical noise further. [Buick, section 3.8]

Advantages and disadvantages

The main assets held by the lattice gas model are that the boolean states mean there will be exact computing without any round-off error due to floating-point precision, and that the cellular automata system makes it possible to run LGCA simulations with parallel computing. [Succi, section 2.4]

Disadvantages of the lattice gas include the lack of Galilean invariance, and the statistical noise which can arise. [Succi, section 2.5] Another problem is the difficulty in expanding the model to handle three dimensional problems, requiring the use of more dimensions to maintain a sufficiently symmetric grid to tackle such issues.

Notes

References

*cite book|author=Sauro Succi|title=The Lattice Boltzmann Equation, for fluid dynamics and beyond|publisher=Oxford Science Publications|year=2001|isbn=0 19 850398 9 (Chapter 2 is about lattice gas Cellular Automata)
*James Maxwell Buick (1997). Lattice Boltzmann Methods in Interfacial Wave Modelling. PhD Thesis, University of Edinburgh. (Chapter 3 is about the lattice gas model.) [http://www-personal.une.edu.au/~jbuick/Publications/PDF/Thesis/tot.html Online]
*cite book|author=Dieter A. Wolf-Gladrow|title=Lattice-Gas Cellular Automata and Lattice Boltzmann Models|publisher=Springer|year=2000|isbn=3-540-66973-6