13. Lattice-Boltzmann

For an implicit treatment of a solvent, ESPResSo can couple the molecular dynamics simulation to a lattice-Boltzmann fluid. The lattice-Boltzmann method (LBM) is a fast, lattice-based method that, in its “pure” form, allows to calculate fluid flow in different boundary conditions of arbitrarily complex geometries. Coupled to molecular dynamics, it allows for the computationally efficient inclusion of hydrodynamic interactions into the simulation. The focus of the ESPResSo implementation of the LBM is, of course, the coupling to MD and therefore available geometries and boundary conditions are somewhat limited in comparison to “pure” LB codes.

Here we restrict the documentation to the interface. For a more detailed description of the method, please refer to the literature.

Note

Please cite [Arnold et al., 2013] (BibTeX key arnold13a in doc/bibliography.bib) if you use the LB fluid and [Roehm and Arnold, 2012] (BibTeX key rohm12a in doc/bibliography.bib) if you use the GPU implementation.

13.1. Setting up a LB fluid

The following minimal example illustrates how to use the LBM in ESPResSo:

import espressomd
import espressomd.lb
system = espressomd.System(box_l=[10, 20, 30])
system.time_step = 0.01
system.cell_system.skin = 0.4
lb = espressomd.lb.LBFluid(agrid=1.0, dens=1.0, visc=1.0, tau=0.01)
system.actors.add(lb)
system.integrator.run(100)

To use the GPU-accelerated variant, replace line 6 in the example above by:

lb = espressomd.lb.LBFluidGPU(agrid=1.0, dens=1.0, visc=1.0, tau=0.01)

Note

Feature CUDA required for the GPU-accelerated variant

To use the (much faster) GPU implementation of the LBM, use LBFluidGPU in place of LBFluid. Please note that the GPU implementation uses single precision floating point operations. This decreases the accuracy of calculations compared to the CPU implementation. In particular, due to rounding errors, the fluid density decreases over time, when external forces, coupling to particles, or thermalization is used. The loss of density is on the order of \(10^{-12}\) per time step.

The command initializes the fluid with a given set of parameters. It is also possible to change parameters on the fly, but this will only rarely be done in practice. Before being able to use the LBM, it is necessary to set up a box of a desired size. The parameter is used to set the lattice constant of the fluid, so the size of the box in every direction must be a multiple of agrid.

In the following, we discuss the parameters that can be supplied to the LBM in ESPResSo. The detailed interface definition is available at LBFluid.

The LB scheme and the MD scheme are not synchronized: In one LB time step typically several MD steps are performed. This allows to speed up the simulations and is adjusted with the parameter tau, the LB time step. The parameters dens and visc set up the density and (kinematic) viscosity of the LB fluid in (usual) MD units. Internally the LB implementation works with a different set of units: all lengths are expressed in agrid, all times in tau and so on. LB nodes are located at 0.5, 1.5, 2.5, etc. (in terms of agrid). This has important implications for the location of hydrodynamic boundaries which are generally considered to be halfway between two nodes for flat, axis-aligned walls. For more complex boundary geometries, the hydrodynamic boundary location deviates from this midpoint and the deviation decays to first order in agrid. The LBM should not be used as a black box, but only after a careful check of all parameters that were applied.

In the following, we describe a number of optional parameters. Thermalization of the fluid (and particle coupling later on) can be activated by providing a non-zero value for the parameter kT. Then, a seed has to be provided for the fluid thermalization:

lbfluid = espressomd.lb.LBFluid(kT=1.0, seed=134, ...)

The parameter ext_force_density takes a three dimensional vector as an array_like of float, representing a homogeneous external body force density in MD units to be applied to the fluid. The parameter bulk_visc allows one to tune the bulk viscosity of the fluid and is given in MD units. In the limit of low Mach number, the flow does not compress the fluid and the resulting flow field is therefore independent of the bulk viscosity. It is however known that the value of the viscosity does affect the quality of the implemented link-bounce-back method. gamma_even and gamma_odd are the relaxation parameters for the kinetic modes. These fluid parameters do not correspond to any macroscopic fluid properties, but do influence numerical properties of the algorithm, such as the magnitude of the error at boundaries. Unless you are an expert, leave their defaults unchanged. If you do change them, note that they are to be given in LB units.

Before running a simulation at least the following parameters must be set up: agrid, tau, visc, dens. For the other parameters, the following are taken: bulk_visc=0, gamma_odd=0, gamma_even=0, ext_force_density=[0, 0, 0].

13.2. Checkpointing

lb.save_checkpoint(path, binary)
lb.load_checkpoint(path, binary)

The first command saves all of the LB fluid nodes’ populations to an ASCII (binary=False) or binary (binary=True) format respectively. The second command loads the LB fluid nodes’ populations. In both cases path specifies the location of the checkpoint file. This is useful for restarting a simulation either on the same machine or a different machine. Some care should be taken when using the binary format as the format of doubles can depend on both the computer being used as well as the compiler. One thing that one needs to be aware of is that loading the checkpoint also requires the user to reuse the old forces. This is necessary since the coupling force between the particles and the fluid has already been applied to the fluid. Failing to reuse the old forces breaks momentum conservation, which is in general a problem. It is particularly problematic for bulk simulations as the system as a whole acquires a drift of the center of mass, causing errors in the calculation of velocities and diffusion coefficients. The correct way to restart an LB simulation is to first load in the particles with the correct forces, and use:

system.integrator.run(steps=number_of_steps, reuse_forces=True)

upon the first call integrator.run. This causes the old forces to be reused and thus conserves momentum.

13.3. Interpolating velocities

To get interpolated velocity values between lattice nodes, the function:

lb.get_interpolated_velocity(pos=[1.1, 1.2, 1.3])

with a single position pos as an argument can be used. For the GPU fluid espressomd.lb.LBFluidGPU, a method get_interpolated_fluid_velocity_at_positions() is also available, which expects a numpy array of positions as an argument.

By default, the interpolation is done linearly between the nearest 8 LB nodes, but for the GPU implementation also a quadratic scheme involving 27 nodes is implemented (see eqs. 297 and 301 in [Dünweg and Ladd, 2009]). You can choose by calling one of:

lb.set_interpolation_order('linear')
lb.set_interpolation_order('quadratic')

A note on boundaries: both interpolation schemes don’t take into account the physical location of the boundaries (e.g. in the middle between two nodes for a planar wall) but will use the boundary node slip velocity at the node position. This means that every interpolation involving at least one boundary node will introduce an error.

13.4. Coupling LB to a MD simulation

MD particles can be coupled to a LB fluid through frictional coupling. The friction force

\[F_{i,\text{frict}} = - \gamma (v_i(t)-u(x_i(t),t))\]

depends on the particle velocity \(v\) and the fluid velocity \(u\). It acts both on the particle and the fluid (in opposite direction). Because the fluid is also affected, multiple particles can interact via hydrodynamic interactions. As friction in molecular systems is accompanied by fluctuations, the particle-fluid coupling has to be activated through the Lattice-Boltzmann thermostat (see more detailed description there). A short example is:

system.thermostat.set_lb(LB_fluid=lbf, seed=123, gamma=1.5)

where lbf is an instance of either LBFluid or LBFluidGPU, gamma the friction coefficient and seed the seed for the random number generator involved in the thermalization.

13.5. Reading and setting properties of single lattice nodes

Appending three indices to the lb object returns an object that represents the selected LB grid node and allows one to access all of its properties:

lb[x, y, z].density              # fluid density (one scalar for LB and CUDA)
lb[x, y, z].velocity             # fluid velocity (a numpy array of three floats)
lb[x, y, z].pressure_tensor      # fluid pressure tensor (a symmetric 3x3 numpy array of floats)
lb[x, y, z].pressure_tensor_neq  # nonequilibrium part of the pressure tensor (as above)
lb[x, y, z].boundary             # flag indicating whether the node is fluid or boundary (fluid: boundary=0, boundary: boundary != 0)
lb[x, y, z].population           # 19 LB populations (a numpy array of 19 floats, check order from the source code)

All of these properties can be read and used in further calculations. Only the property population can be modified. The indices x, y, z are integers and enumerate the LB nodes in the three Cartesian directions, starting at 0. To modify boundary, refer to Setting up boundary conditions.

Example:

print(lb[0, 0, 0].velocity)
lb[0, 0, 0].density = 1.2

The first line prints the fluid velocity at node (0 0 0) to the screen. The second line sets this fluid node’s density to the value 1.2.

The nodes can be read and modified using slices. Example:

print(lb[0:4:2, 0:2, 0].velocity)
lb[0:4:2, 0:2, 0].density = [[[1.1], [1.2]], [[1.3], [1.4]]]

The first line prints an array of shape (2, 2, 1, 3) with the velocities of nodes (0 0 0), (0 1 0), (2 0 0), (2 1 0). The second line updates these nodes with densities ranging from 1.1 to 1.4. You can set either a value that matches the length of the slice (which sets each node individually), or a single value that will be copied to every node (e.g. a scalar for density, or an array of length 3 for the velocity).

13.6. Output for visualization

ESPResSo implements a number of commands to output fluid field data of the whole fluid into a file at once.

lb.write_vtk_velocity(path)
lb.write_vtk_boundary(path)
lb.write_velocity(path)
lb.write_boundary(path)

Currently supported fluid properties are the velocity, and boundary flag in ASCII VTK as well as Gnuplot compatible ASCII output.

The VTK format is readable by visualization software such as ParaView 1 or Mayavi2 2. If you plan to use ParaView for visualization, note that also the particle positions can be exported using the VTK format (see writevtk()).

The variant

lb.write_vtk_velocity(path, bb1, bb2)

allows you to only output part of the flow field by specifying an axis aligned bounding box through the coordinates bb1 and bb1 (lists of three ints) of two of its corners. This bounding box can be used to output a slice of the flow field. As an example, executing

lb.write_vtk_velocity(path, [0, 0, 5], [10, 10, 5])

will output the cross-section of the velocity field in a plane perpendicular to the \(z\)-axis at \(z = 5\) (assuming the box size is 10 in the \(x\)- and \(y\)-direction).

13.7. Choosing between the GPU and CPU implementations

ESPResSo contains an implementation of the LBM for NVIDIA GPUs using the CUDA framework. On CUDA-supporting machines this can be activated by compiling with the feature CUDA. Within the Python script, the LBFluid object can be substituted with the LBFluidGPU object to switch from CPU based to GPU based execution. For further information on CUDA support see section CUDA acceleration.

The following minimal example demonstrates how to use the GPU implementation of the LBM in analogy to the example for the CPU given in section Setting up a LB fluid:

import espressomd
system = espressomd.System(box_l=[10, 20, 30])
system.time_step = 0.01
system.cell_system.skin = 0.4
lb = espressomd.lb.LBFluidGPU(agrid=1.0, dens=1.0, visc=1.0, tau=0.01)
system.actors.add(lb)
system.integrator.run(100)

For boundary conditions analogous to the CPU implementation, the feature LB_BOUNDARIES_GPU has to be activated. Lees–Edwards boundary conditions are not supported by either LB implementation.

13.8. Electrohydrodynamics

Note

This needs the feature LB_ELECTROHYDRODYNAMICS.

If the feature is activated, the lattice-Boltzmann code can be used to implicitly model surrounding salt ions in an external electric field by having the charged particles create flow.

For that to work, you need to set the electrophoretic mobility (multiplied by the external \(E\)-field) \(\mu E\) on the particles that should be subject to the field. This effectively acts as a velocity offset between the particle and the LB fluid.

For more information on this method and how it works, read the publication [Hickey et al., 2010].

13.9. Using shapes as lattice-Boltzmann boundary

Note

Feature LB_BOUNDARIES required

Lattice-Boltzmann boundaries are implemented in the module espressomd.lbboundaries. You might want to take a look at the classes LBBoundary and LBBoundaries for more information.

Adding a shape-based boundary is straightforward:

lbb = espressomd.lbboundaries.LBBoundary(shape=my_shape, velocity=[0, 0, 0])
system.lbboundaries.add(lbb)

or:

lbb = espressomd.lbboundaries.LBBoundary()
lbb.shape = my_shape
lbb.velocity = [0, 0, 0]
system.lbboundaries.add(lbb)

13.9.1. Minimal usage example

Note

Feature LB_BOUNDARIES or LB_BOUNDARIES_GPU required

In order to add a wall as boundary for a lattice-Boltzmann fluid you could do the following:

wall = espressomd.shapes.Wall(dist=5, normal=[1, 0, 0])
lbb = espressomd.lbboundaries.LBBoundary(shape=wall, velocity=[0, 0, 0])
system.lbboundaries.add(lbb)

13.9.2. Setting up boundary conditions

The following example sets up a system consisting of a spherical boundary in the center of the simulation box acting as a no-slip boundary for the LB fluid that is driven by 4 walls with a slip velocity:

import espressomd
import espressomd.lb
import espressomd.lbboundaries
import espressomd.shapes

system = espressomd.System(box_l=[64, 64, 64])
system.time_step = 0.01
system.cell_system.skin = 0.4

lb = espressomd.lb.LBFluid(agrid=1.0, dens=1.0, visc=1.0, tau=0.01)
system.actors.add(lb)

v = [0, 0, 0.01]  # the boundary slip
walls = [None] * 4

wall_shape = espressomd.shapes.Wall(normal=[1, 0, 0], dist=1)
walls[0] = espressomd.lbboundaries.LBBoundary(shape=wall_shape, velocity=v)

wall_shape = espressomd.shapes.Wall(normal=[-1, 0, 0], dist=-63)
walls[1] = espressomd.lbboundaries.LBBoundary(shape=wall_shape, velocity=v)

wall_shape = espressomd.shapes.Wall(normal=[0, 1, 0], dist=1)
walls[2] = espressomd.lbboundaries.LBBoundary(shape=wall_shape, velocity=v)

wall_shape = espressomd.shapes.Wall(normal=[0, -1, 0], dist=-63)
walls[3] = espressomd.lbboundaries.LBBoundary(shape=wall_shape, velocity=v)

for wall in walls:
    system.lbboundaries.add(wall)

sphere_shape = espressomd.shapes.Sphere(radius=5.5, center=[33, 33, 33], direction=1)
sphere = espressomd.lbboundaries.LBBoundary(shape=sphere_shape)
system.lbboundaries.add(sphere)

system.integrator.run(4000)

print(sphere.get_force())

After integrating the system for a sufficient time to reach the steady state, the hydrodynamic drag force exerted on the sphere is evaluated.

The LB boundaries use the same shapes objects to specify their geometry as constraints do for particles. This allows the user to quickly set up a system with boundary conditions that simultaneously act on the fluid and particles. For a complete description of all available shapes, refer to espressomd.shapes.

Intersecting boundaries are in principle possible but must be treated with care. In the current implementation, all nodes that are within at least one boundary are treated as boundary nodes.

Currently, only the so-called “link-bounce-back” algorithm for wall nodes is available. This creates a boundary that is located approximately midway between the lattice nodes, so in the above example wall[0] corresponds to a boundary at \(x=1.5\). Note that the location of the boundary is unfortunately not entirely independent of the viscosity. This can be seen when using the sample script with a high viscosity.

The bounce back boundary conditions permit it to set the velocity at the boundary to a non-zero value via the v property of an LBBoundary object. This allows to create shear flow and boundaries moving relative to each other. The velocity boundary conditions are implemented according to [Succi, 2001] eq. 12.58. Using this implementation as a blueprint for the boundary treatment, an implementation of the Ladd-Coupling should be relatively straightforward. The LBBoundary object furthermore possesses a property force, which keeps track of the hydrodynamic drag force exerted onto the boundary by the moving fluid.

1

https://www.paraview.org/

2

http://code.enthought.com/projects/mayavi/