Keijo Mattila

Comparison of implementations of the lattice-Boltzmann method

Simplicity of coding is usually an appealing feature of the lattice-Boltzmann method (LBM). Conventional implementations of LBM are often based on the two-lattice or the two-step algorithm, which however suffer from high memory consumption and poor computational performance, respectively. The aim of this work was to identify implementations of LBM that would achieve high computational performance with low memory consumption. Effects of memory addressing schemes were investigated in particular. Data layouts for velocity distribution values were also considered, and they were found to be related to computational performance. A novel bundle data layout was therefore introduced. Addressing schemes and data layouts were implemented for the Lagrangian, compressed-grid (shift), swap, two-lattice, and two-step algorithms. Implementations were compared for a wide range of fluid volume fractions. Simulation results indicated that indirect addressing implementations yield high computational performance. However, they achieved low memory consumption only for very low fluid volume fractions. Semi-direct addressing implementations could also provide high computational performance. The bundle data layout was found to be competitive, sometimes by a wide margin, in all the cases considered.

An efficient swap algorithm for the lattice Boltzmann method

During the last decade, the lattice-Boltzmann method (LBM) as a valuable tool in computational fluid dynamics has been increasingly acknowledged. The widespread application of LBM is partly due to the simplicity of its coding. The most well-known algorithms for the implementation of the standard lattice-Boltzmann equation (LBE) are the two-lattice and two-step algorithms. However, implementations of the two-lattice or the two-step algorithm suffer from high memory consumption or poor computational performance, respectively. Ultimately, the computing resources available decide which of the two disadvantages is more critical. Here we introduce a new algorithm, called the swap algorithm, for the implementation of LBE. Simulation results demonstrate that implementations based on the swap algorithm can achieve high computational performance and have very low memory consumption. Furthermore, we show how the performance of its implementations can be further improved by code optimization.

Transport properties of heterogeneous materials. Combining computerised X-ray micro-tomography and direct numerical simulations

Feasibility of a method for finding flow permeability of porous materials, based on combining computerised X-ray micro-tomography and numerical simulations, is assessed. The permeability is found by solving fluid flow through the complex 3D pore structures obtained by tomography for actual material samples. We estimate overall accuracy of the method and compare numerical and experimental results. Factors contributing to uncertainty of the method include numerical error arising from the finite resolution of tomographic images and the rather small sample size available with the present tomographic techniques. The total uncertainty of computed values of permeability is, however, not essentially larger than that of experimental results. We conclude that the method provides a feasible alternative for finding fluid flow properties of the kind of materials studied. It can be used to estimate all components of permeability tensor and is useful in cases where direct measurements are not achievable. Analogous methods can be applied to other modes of transport, such as diffusion and heat conduction.

A prospect for computing in porous materials research: Very large fluid flow simulations

Abstract Properties of porous materials, abundant both in nature and industry, have broad influences on societies via, e.g. oil recovery, erosion, and propagation of pollutants. The internal structure of many porous materials involves multiple scales which hinders research on the relation between structure and transport properties: typically laboratory experiments cannot distinguish contributions from individual scales while computer simulations cannot capture multiple scales due to limited capabilities. Thus the question arises how large domain sizes can in fact be simulated with modern computers. This question is here addressed using a realistic test case; it is demonstrated that current computing capabilities allow the direct pore-scale simulation of fluid flow in porous materials using system sizes far beyond what has been previously reported. The achieved system sizes allow the closing of some particular scale gaps in, e.g. soil and petroleum rock research. Specifically, a full steady-state fluid flow simulation in a porous material, represented with an unprecedented resolution for the given sample size, is reported: the simulation is executed on a CPU-based supercomputer and the 3D geometry involves 16,384 3 lattice cells (around 590 billion of them are pore sites). Using half of this sample in a benchmark simulation on a GPU-based system, a sustained computational performance of 1.77 PFLOPS is observed. These advances expose new opportunities in porous materials research. The implementation techniques here utilized are standard except for the tailored high-performance data layouts as well as the indirect addressing scheme with a low memory overhead and the truly asynchronous data communication scheme in the case of CPU and GPU code versions, respectively.

Rectangular Lattice-Boltzmann Schemes with BGK-Collision Operator

The usual lattice-Boltzmann schemes for fluid flow simulations operate with square and cubic lattices. Instead of relying on square lattices it is possible to use rectangular and orthorombic lattices as well. Schemes using rectangular lattices can be constructed in several ways. Here we construct a rectangular scheme, with the BGK collision operator, by introducing 2 additional discrete velocities into the standard D2Q9 stencil and show how the same procedure can be applied in three dimensions by extending the D3Q19 stencil. The weights and scaling factors for the new stencils are found as the solutions of the well-known Hermite quadrature problem, assuring isotropy of the lattice tensors up to rank four (Philippi et al., Phys. Rev. E 73(5):056702, 2006) This isotropy is a necessary and sufficient condition for assuring the same second order accuracy of lattice-Boltzmann equation with respect to the Navier–Stokes hydrodynamic equations that is found with the standard D2Q9 and D3Q19 stencils. The numerical validation is done, in the two-dimensional case, by using the new rectangular scheme with D2R11 stencil for simulating the Taylor–Green vortex decay. The D3R23 stencil is numerically validated with three-dimensional simulations of cylindrical sound waves propagating from a point source.

High-order regularization in lattice-Boltzmann equations

A lattice-Boltzmann equation (LBE) is the discrete counterpart of a continuous kinetic model. It can be derived using a Hermite polynomial expansion for the velocity distribution function. Since LBEs are characterized by discrete, finite representations of the microscopic velocity space, the expansion must be truncated and the appropriate order of truncation depends on the hydrodynamic problem under investigation. Here we consider a particular truncation where the non-equilibrium distribution is expanded on a par with the equilibrium distribution, except that the diffusive parts of high-order non-equilibrium moments are filtered, i.e., only the corresponding advective parts are retained after a given rank. The decomposition of moments into diffusive and advective parts is based directly on analytical relations between Hermite polynomial tensors. The resulting, refined regularization procedure leads to recurrence relations where high-order non-equilibrium moments are expressed in terms of low-order ones. T...

Using microtomography, image analysis and flow simulations to characterize soil surface seals

Raindrops that impact on soil surface affect the pore structure and form compact soil surface seals. Damaged pore structure reduces water infiltration which can lead to increased soil erosion. We introduce here methods to characterize the properties of surface seals in a detailed manner. These methods include rainfall simulations, x-ray microtomography, image analysis and pore-scale flow simulations. Methods were tested using clay soil samples, and the results indicate that the sealing process changes several properties of the pore structure.

HIGH-ORDER LATTICE-BOLTZMANN EQUATIONS AND STENCILS FOR MULTIPHASE MODELS

The lattice Boltzmann (LB) method, based on mesoscopic modeling of transport phenomena, appears to be an attractive alternative for the simulation of complex fluid flows. Examples of such complex dynamics are multiphase and multicomponent flows for which several LB models have already been proposed. However, due to theoretical or numerical reasons, some of these models may require application of high-order lattice-Boltzmann equations (LBEs) and stencils. Here, we will present a derivation of LBEs from the discrete-velocity Boltzmann equation (DVBE). By using the method of characteristics, high-order accurate equations are conveniently formulated with standard numerical methods for ordinary differential equations (ODEs). In particular, we will derive implicit LB schemes due to their stability properties. A simple algorithm is presented which enables implementation of the implicit schemes without resorting to, e.g. change of variables. Finally, some numerical experiments with high-order equations and stencils together with two specific multiphase models are presented.

Mass-flux-based outlet boundary conditions for the lattice Boltzmann method

We present outlet boundary conditions for the lattice Boltzmann method. These boundary conditions are constructed with a mass-flux-based approach. Conceptually, the mass-flux-based approach provides a mathematical framework from which specific boundary conditions can be derived by enforcing given physical conditions. The object here is, in particular, to explain the mass-flux-based approach. Furthermore, we illustrate, transparently, how boundary conditions can be derived from the emerging mathematical framework. For this purpose, we derive and present explicitly three outlet boundary conditions. By construction, these boundary conditions have an apparent physical interpretation which is further demonstrated with numerical experiments.

High-accuracy approximation of high-rank derivatives: isotropic finite differences based on lattice-Boltzmann stencils.

We propose isotropic finite differences for high-accuracy approximation of high-rank derivatives. These finite differences are based on direct application of lattice-Boltzmann stencils. The presented finite-difference expressions are valid in any dimension, particularly in two and three dimensions, and any lattice-Boltzmann stencil isotropic enough can be utilized. A theoretical basis for the proposed utilization of lattice-Boltzmann stencils in the approximation of high-rank derivatives is established. In particular, the isotropy and accuracy properties of the proposed approximations are derived directly from this basis. Furthermore, in this formal development, we extend the theory of Hermite polynomial tensors in the case of discrete spaces and present expressions for the discrete inner products between monomials and Hermite polynomial tensors. In addition, we prove an equivalency between two approaches for constructing lattice-Boltzmann stencils. For the numerical verification of the presented finite differences, we introduce 5th-, 6th-, and 8th-order two-dimensional lattice-Boltzmann stencils.