Keywords: lattice Boltzmann, finite element, thermal, efficiency, coupling, accuracy.

The lattice Boltzmann (LB) equation has emerged as a method of choice for simulating complex fluid behaviour, particularly in porous media. A general introduction, including many applications, is given by [1]. However, accurate simulation of many systems must include complex behaviour of the solid phase, for which LB is ill-suited. To retain LB's advantages for the liquid phase in such simulations, a coupling between LB and another computational method to model the solid is therefore required.

This paper applies Lattice Boltzmann and Finite Element methods for the heat equation to a benchmark bar-heating problem, and their efficiency and accuracy are compared. The FE models follow Smith-Griffiths [2], and the LB models are the popular lattice Bhatnagar-Gross-Krook (LBGK) D1Q3, D2Q9 and D3Q19 variants [1]. An interface condition, coupling FE to LB using finite differences, is introduced and applied to the same benchmark problem. The accuracy of the coupled method is found to be approximately second order in space, and comparable to the individual methods across the entire computational domain. For the considered problem, the LB solver is more efficient than the alternatives, due to its local nodal updates. These results, achieved without substantial increases in computational cost, demonstrate for the first time simulation of a physical parameter across a mixed FE-LB domain.

Coupling finite elements (FE) and LB is a potentially powerful combination which has been suggested in the literature, but to the authors' knowledge, although upscaling and surface-stress simulations have been reported, no study of a combined model simulating a physical parameter across the entire domain has been reported, and no consideration of accuracy has been mentioned. This paper presents a first step towards a capable multi-physics FE-LB solver.

Finally, responding to the desire to present working code segments at ACME conferences, a complete and concise exemplary matlab script implementation of diffusion across a mixed FE-LB domain is provided, illustrating the coupling described in the paper.

Lattice Boltzmann methods are widely used for detailed porous domain simulations. Coupling LB with FE, demonstrated in this paper for a simple problem, opens the path for simulations of double-population [3] hydrothermal LB models interacting with a FE skeleton. The presented coupling is valid for thermal and moisture transport between the domains. By incorporating stresses between the hydrodynamic LB and FE models, it will be possible to create a composite model capable of investigating hygro-thermo-mechanical processes in porous media.

1

S. Succi, "The Lattice Boltzmann Equation for Fluid Dynamics and Beyond", Oxford University Press, 2001.

2

I. M. Smith and D. V. Griffiths, "Programming the Finite Element Method (4th Ed.)", Wiley, 2004.

3

X. He, S. Chen and G. Doolen, A novel thermal model for the lattice Boltzmann method in incompressible limit, J. Comp. Phys., 144, 282-300, 1998. doi:10.1006/jcph.1998.6057

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £75 +P&P)