Iterative momentum relaxation for fast lattice-Boltzmann simulations
Introduction
The lattice-Boltzmann method (LBM) is a mesoscopic approach based on the kinetic Boltzmann equation for simulating fluid flow [1], [2], [3], [4], [14], [15]. In this method fluid is modeled by particles moving on a regular lattice. At each time step particles propagate to neighboring lattice points and re-distribute their velocities in a local collision phase. This inherent spatial and temporal locality of the update rules makes this method ideal for parallel computing [5]. During the recent years, LBM has been successfully used for simulating many complex fluid-dynamical problems, such as suspension flows, multi-phase flows, and fluid flow in porous media. All these problems are quite difficult to simulate by conventional methods [3], [6], [7], [8].
However, as most numerical algorithms, the standard lattice-Boltzmann scheme also has its shortcomings. For instance, in a recently performed comparative study between the finite element and the lattice-Boltzmann method for simulating steady-state fluid flow in an SMRX static mixer reactor, it became evident that the computational time (on a sequential machine) required by the lattice-Boltzmann method was higher than that of the finite element method for obtaining the same level of accuracy. The memory requirements on the other hand were lower for the lattice-Boltzmann simulations (details can be found in [9]). It can be argued that the longer computational time of LBM is a direct consequence of the transient nature of this scheme. In this paper, we will present a new technique, namely the iterative momentum relaxation technique (IMR), which can significantly reduce the saturation time. In this technique the body-force which is often used to drive a flow in lattice-Boltzmann simulations, is adjusted dynamically by calculating the average loss of momentum due to viscous forces.
We first review the basics of the lattice-Boltzmann method and the IMR technique. Next, we discuss the results obtained with the IMR technique and finally the conclusions are presented.
Section snippets
The lattice-Boltzmann BGK method
Basically, the time evolution of the lattice-Boltzmann model consists of a propagation phase, where particles move along lattice bonds from a lattice node to one of its neighbors, and a collision phase with a local redistribution of the particle densities subject to conservation of mass and momentum [10], [11], [12]. The simplest and currently the most widely used lattice-Boltzmann model is the so-called lattice-BGK (Bhatnagar–Gross–Krook) model. Here the collision operator is based on a
Simulation results
To validate the IMR technique, we simulated three benchmark problems, namely fluid flow around a spherical obstacle, flow in a random fiber mat (see Fig. 1a) and flow in an SMRX static mixer reactor (see Fig. 1b). We included the last benchmark, as it is one of the very few cases of fluid flow in complex geometries with well documented results from traditional numerical methods and experimental data.
In our first benchmark the sphere radius was a0=5.5 lattice points and the lattice dimensions
Conclusions
In many lattice-Boltzmann simulations, the complete time evolution of the system is computed with a constant body-force starting from some initial velocity and pressure fields. The number of time steps which is required to reach the steady state can then be very large in some cases. We presented a new technique for reducing the number of time steps that is needed to reach the steady state for body-force driven flows. This strategy does not influence the explicit character of the
Acknowledgements
This work was partly carried out within the massive parallel computing (MPR) project “Many Particle Systems” funded by the Dutch foundation for basic research. We would like to thank Robert Belleman, David Vidal, Huub Hoefsloot, Markku Kataja and Jussi Timonen for many useful discussions concerning the IMR technique and the benchmark application.
Drona Kandhai holds his PhD in computational sciences from the University of Amsterdam on large scale lattice-Boltzmann simulations of fluid flow using massively parallel systems. Currently he is a post-doctoral researcher at the Section of Computational Science of the University of Amsterdam. His research interests pertain to fluid flow in porous media, lattice-Boltzmann methods, grid refinement and parallel computing for particle based models and blood flow in arteries.
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A parallel workload balanced and memory efficient lattice-Boltzmann algorithm with single unit BGK relaxation time for laminar Newtonian flows
2010, Computers and FluidsCitation Excerpt :It also justifies the use of an LBM implementation based on a single unit relaxation time for simulating laminar Newtonian fluid flows. Note that one way to alleviate the constraint on CPU time resulting from the choice τ* = 1 and speed up convergence to steady state, would be to use the iterative momentum relaxation technique proposed by Kandhai et al. [36]. This technique was reported to cut down the number of iterations required to reach steady state by 45–97%.
Steady state convergence acceleration of the generalized lattice Boltzmann equation with forcing term through preconditioning
2009, Journal of Computational PhysicsAcceleration of lattice Boltzmann models through state extrapolation: a reaction-diffusion example
2008, Applied Numerical Mathematics
Drona Kandhai holds his PhD in computational sciences from the University of Amsterdam on large scale lattice-Boltzmann simulations of fluid flow using massively parallel systems. Currently he is a post-doctoral researcher at the Section of Computational Science of the University of Amsterdam. His research interests pertain to fluid flow in porous media, lattice-Boltzmann methods, grid refinement and parallel computing for particle based models and blood flow in arteries.
Antti Koponen holds his PhD in physics from the University of Jyväskylä in Finland. Currently he is a part-time post-doctoral researcher at the Department of Physics, University of Jyväskylä, Finland and a Lecturer of physics at the Jyväskylä Polytechnic School of Engineering and Technology. His research interests pertain to fluid flow in porous media, multiphase flows, multicomponent flows and suspension flows.
Alfons Hoekstra holds his PhD in computational sciences from the University of Amsterdam on large scale simulations of elastic light scattering using massively parallel systems. Currently he is an Assistant Professor at the Section of Computational Science of the University of Amsterdam. His research interests include particle based modeling and massively parallel particle based simulations.
Peter Sloot received his PhD in computer science from the University of Amsterdam in 1988. He is a full-time professor in computational physics since 1997. He is the co-founder of the Amsterdam Center for Computational Science and group leader of the Section of Computational Science of the Department for Informatics of the University of Amsterdam. He published over 180 papers on different topics of computer simulation. His interest is in the modeling and simulation of dynamical complex systems. More information can be found at http://carol.wins.uva.nl/∼sloot/.