Methods and Method Creation (lbmpy.methods)¶
This module defines the classes defining all types of lattice Boltzmann methods available in lbmpy, together with a set of factory functions used to create their instances. The factory functions are organized in three levels of abstraction. Objects of the method classes should be created only through these factory functions, never manually.
- Methods in lbmpy can be distinguished into three categories:
Raw Moment-based methods, including the classical single relaxation-time (SRT, BGK), two relaxation-time (TRT) and multiple relaxation-time (MRT) methods, as well as entropic variants of the TRT method.
Central Moment-based methods, which are multiple relaxation-time methods using relaxation in central moment space.
Cumulant-based methods, multiple relaxation-time methods using relaxation in cumulant space.
Abstract LB Method Base Class¶
- class LbmCollisionRule(lb_method, *args, **kwargs)¶
A pystencils AssignmentCollection that additionally holds an
AbstractLbMethod
- class AbstractLbMethod(stencil)¶
Abstract base class for all LBM methods.
- property stencil¶
Discrete set of velocities, represented by
lbmpy.stencils.LBStencil
- property dim¶
The method’s spatial dimensionality
- property pre_collision_pdf_symbols¶
Tuple of symbols representing the pdf values before collision
- property post_collision_pdf_symbols¶
Tuple of symbols representing the pdf values after collision
- abstract property relaxation_rates¶
Tuple containing the relaxation rates of each moment
- property relaxation_matrix¶
Returns a qxq diagonal matrix which contains the relaxation rate for each moment on the diagonal
- property symbolic_relaxation_matrix¶
Returns a qxq diagonal matrix which contains the relaxation rate for each moment on the diagonal. In contrast to the normal relaxation matrix all numeric values are replaced by sympy symbols
- property subs_dict_relxation_rate¶
returns a dictonary which maps the replaced numerical relaxation rates to its original numerical value
- abstract property conserved_quantity_computation¶
Returns an instance of class
lbmpy.methods.AbstractConservedQuantityComputation
- abstract property weights¶
Returns a sequence of weights, one for each lattice direction
- abstract get_equilibrium()¶
Returns equation collection, to compute equilibrium values. The equations have the post collision symbols as left hand sides and are functions of the conserved quantities
- abstract get_collision_rule()¶
Returns an LbmCollisionRule i.e. an equation collection with a reference to the method. This collision rule defines the collision operator.
Conserved Quantity Computation¶
The classes of the conserved quantity computation (CQC) submodule define an LB Method’s conserved quantities and
the equations to compute them. For hydrodynamic methods, lbmpy.methods.DensityVelocityComputation
is
the typical choice. For custom methods, however, a custom CQC class might have to be created.
- class AbstractConservedQuantityComputation¶
This class defines how conserved quantities are computed as functions of the pdfs. Conserved quantities are used for output and as input to the equilibrium in the collision step
Depending on the method they might also be computed slightly different, e.g. due to a force model.
An additional method describes how to get the conserved quantities for the equilibrium for initialization. In most cases the inputs can be used directly, but for some methods they have to be altered slightly. For example in zero centered hydrodynamic schemes with force model, the density has to be decreased by one, and the given velocity has to be shifted dependent on the force.
- abstract property conserved_quantities¶
Dict, mapping names (symbol) to dimensionality (int) For example: {‘density’ : 1, ‘velocity’ : 3} The naming strings can be used in
output_equations_from_pdfs()
andequilibrium_input_equations_from_init_values()
- defined_symbols(order='all')¶
Returns a dict, mapping names of conserved quantities to their symbols
- abstract property default_values¶
Returns a dict of symbol to default value, where “default” means that the equilibrium simplifies to the weights if these values are inserted. Hydrodynamic example: rho=1, u_i = 0
- abstract equilibrium_input_equations_from_pdfs(pdfs, **kwargs)¶
Returns an equation collection that defines all necessary quantities to compute the equilibrium as functions of the pdfs. For hydrodynamic LBM schemes this is usually the density and velocity.
- Parameters
pdfs – values or symbols for the pdf values
- abstract output_equations_from_pdfs(pdfs, output_quantity_names_to_symbols, **kwargs)¶
Returns an equation collection that defines conserved quantities for output. These conserved quantities might be slightly different that the ones used as input for the equilibrium e.g. due to a force model.
- Parameters
pdfs – values for the pdf entries
output_quantity_names_to_symbols – dict mapping of conserved quantity names (See
conserved_quantities()
) to symbols or field accesses where they should be written to
- abstract equilibrium_input_equations_from_init_values(**kwargs)¶
Returns an equation collection that defines all necessary quantities to compute the equilibrium as function of given conserved quantities. Parameters can be names that are given by symbol names of
conserved_quantities()
. For all parameters not specified each implementation should use sensible defaults. For example hydrodynamic schemes use density=1 and velocity=0.
- class DensityVelocityComputation(stencil, compressible, zero_centered, force_model=None, background_density=1, density_symbol=rho, density_deviation_symbol=delta_rho, velocity_symbols=(u_0, u_1, u_2), c_s_sq=c_s ** 2, second_order_moment_symbols=(p_0, p_1, p_2, p_3, p_4, p_5, p_6, p_7, p_8), zeroth_order_moment_symbol=None, first_order_moment_symbols=None)¶
This class emits equations for in- and output of the conserved quantities of regular hydrodynamic lattice Boltzmann methods, which are density and velocity. The following symbolic quantities are manages by this class:
Density \(\rho\), background density \(\rho_0\) (typically set to \(1\)) and the density deviation \(\delta \rho\). They are connected through \(\rho = \rho_0 + \delta\rho\).
Velocity \(\mathbf{u} = (u_0, u_1, u_2)\).
Furthermore, this class provides output functionality for the second-order moment tensor \(p\).
- Parameters
stencil – see
lbmpy.stencils.LBStencil
compressible –
True
indicates the usage of a compressible equilibrium (seelbmpy.equilibrium.ContinuousHydrodynamicMaxwellian
), and sets the reference density to \(\rho\).False
indicates an incompressible equilibrium, using \(\rho\) as reference density.zero_centered – Indicates whether or not PDFs are stored in regular or zero-centered format.
force_model – Employed force model. See
lbmpy.forcemodels
.
- property conserved_quantities¶
Dict, mapping names (symbol) to dimensionality (int) For example: {‘density’ : 1, ‘velocity’ : 3} The naming strings can be used in
output_equations_from_pdfs()
andequilibrium_input_equations_from_init_values()
- property compressible¶
Indicates whether a compressible or incompressible equilibrium is used.
- defined_symbols(order='all')¶
Returns a dict, mapping names of conserved quantities to their symbols
- property zero_centered_pdfs¶
Whether regular or zero-centered storage is employed.
- property zeroth_order_moment_symbol¶
Symbol corresponding to the zeroth-order moment of the stored PDFs, i.e.
density_deviation_symbol
if zero-centered, elsedensity_symbol
.
- property first_order_moment_symbols¶
Symbol corresponding to the first-order moment vector of the stored PDFs, divided by the reference density.
- property density_symbol¶
Symbol for the density.
- property density_deviation_symbol¶
Symbol for the density deviation.
- property background_density¶
Symbol or value of the background density.
- property velocity_symbols¶
Symbols for the velocity.
- property default_values¶
Returns a dict of symbol to default value, where “default” means that the equilibrium simplifies to the weights if these values are inserted. Hydrodynamic example: rho=1, u_i = 0
- equilibrium_input_equations_from_pdfs(pdfs, force_substitution=True)¶
Returns an equation collection that defines all necessary quantities to compute the equilibrium as functions of the pdfs. For hydrodynamic LBM schemes this is usually the density and velocity.
- Parameters
pdfs – values or symbols for the pdf values
- equilibrium_input_equations_from_init_values(density=1, velocity=(0, 0, 0), force_substitution=True)¶
Returns an equation collection that defines all necessary quantities to compute the equilibrium as function of given conserved quantities. Parameters can be names that are given by symbol names of
conserved_quantities()
. For all parameters not specified each implementation should use sensible defaults. For example hydrodynamic schemes use density=1 and velocity=0.
- output_equations_from_pdfs(pdfs, output_quantity_names_to_symbols, force_substitution=True)¶
Returns an equation collection that defines conserved quantities for output. These conserved quantities might be slightly different that the ones used as input for the equilibrium e.g. due to a force model.
- Parameters
pdfs – values for the pdf entries
output_quantity_names_to_symbols – dict mapping of conserved quantity names (See
conserved_quantities()
) to symbols or field accesses where they should be written to
Raw Moment-based methods¶
These methods are represented by instances of lbmpy.methods.momentbased.MomentBasedLbMethod
and will derive
collision equations either in population space (SRT, TRT, entropic TRT), or in raw moment space (MRT variants).
Creation Functions¶
The following factory functions create raw moment-based methods using variants of the regular hydrodynamic maxwellian equilibrium.
- create_srt(stencil, relaxation_rate, continuous_equilibrium=True, **kwargs)¶
Creates a single relaxation time (SRT) lattice Boltzmann model also known as BGK model.
Internally calls either
create_with_discrete_maxwellian_equilibrium()
orcreate_with_continuous_maxwellian_equilibrium()
, depending on the value ofcontinuous_equilibrium
.If not specified otherwise, collision equations will be derived in population space.
- Parameters
stencil – instance of
lbmpy.stencils.LBStencil
relaxation_rate – relaxation rate (inverse of the relaxation time) usually called \(\omega\) in LBM literature
continuous_equilibrium – determines if the discrete or continuous maxwellian equilibrium is used to compute the equilibrium moments
- Returns
- create_trt(stencil, relaxation_rate_even_moments, relaxation_rate_odd_moments, continuous_equilibrium=True, **kwargs)¶
Creates a two relaxation time (TRT) lattice Boltzmann model, where even and odd moments are relaxed differently. In the SRT model the exact wall position of no-slip boundaries depends on the viscosity, the TRT method does not have this problem.
Parameters are similar to
lbmpy.methods.create_srt()
, but instead of one relaxation rate there are two relaxation rates: one for even moments (determines viscosity) and one for odd moments. If unsure how to choose the odd relaxation rate, use the functionlbmpy.methods.create_trt_with_magic_number()
Internally calls either
create_with_discrete_maxwellian_equilibrium()
orcreate_with_continuous_maxwellian_equilibrium()
, depending on the value ofcontinuous_equilibrium
.If not specified otherwise, collision equations will be derived in population space.
- Returns
- create_trt_with_magic_number(stencil, relaxation_rate, magic_number=3 / 16, **kwargs)¶
Creates a two relaxation time (TRT) lattice Boltzmann method, where the relaxation time for odd moments is determines from the even moment relaxation time and a “magic number”. For possible parameters see
lbmpy.methods.create_trt()
Internally calls either
create_with_discrete_maxwellian_equilibrium()
orcreate_with_continuous_maxwellian_equilibrium()
, depending on the value ofcontinuous_equilibrium
.If not specified otherwise, collision equations will be derived in population space.
- Parameters
stencil – instance of
lbmpy.stencils.LBStencil
relaxation_rate – relaxation rate (inverse of the relaxation time) usually called \(\omega\) in LBM literature
magic_number – magic number which is used to calculate the relxation rate for the odd moments.
- Returns
- create_mrt_orthogonal(stencil, relaxation_rates, continuous_equilibrium=True, weighted=None, nested_moments=None, **kwargs)¶
Returns an orthogonal multi-relaxation time model for the stencils D2Q9, D3Q15, D3Q19 and D3Q27. These MRT methods are just one specific version - there are many MRT methods possible for all these stencils which differ by the linear combination of moments and the grouping into equal relaxation times. To create a generic MRT method use
create_with_discrete_maxwellian_equilibrium
Internally calls either
create_with_discrete_maxwellian_equilibrium()
orcreate_with_continuous_maxwellian_equilibrium()
, depending on the value ofcontinuous_equilibrium
.If not specified otherwise, collision equations will be derived in raw moment space.
- Parameters
stencil – instance of
lbmpy.stencils.LBStencil
relaxation_rates – relaxation rates for the moments
continuous_equilibrium – determines if the discrete or continuous maxwellian equilibrium is used to compute the equilibrium moments
weighted – whether to use weighted or unweighted orthogonality
nested_moments – a list of lists of modes, grouped by common relaxation times. If this argument is not provided, Gram-Schmidt orthogonalization of the default modes is performed. The default modes equal the raw moments except for the separation of the shear and bulk viscosity.
- create_trt_kbc(dim, shear_relaxation_rate, higher_order_relaxation_rate, method_name='KBC-N4', continuous_equilibrium=True, **kwargs)¶
Creates a method with two relaxation rates, one for lower order moments which determines the viscosity and one for higher order moments. In entropic models this second relaxation rate is chosen subject to an entropy condition. Which moments are relaxed by which rate is determined by the method_name
Internally calls either
create_with_discrete_maxwellian_equilibrium()
orcreate_with_continuous_maxwellian_equilibrium()
, depending on the value ofcontinuous_equilibrium
.If not specified otherwise, collision equations will be derived in population space.
- Parameters
dim – 2 or 3, leads to stencil D2Q9 or D3Q27
shear_relaxation_rate – relaxation rate that determines viscosity
higher_order_relaxation_rate – relaxation rate for higher order moments
method_name – string ‘KBC-Nx’ where x can be an number from 1 to 4, for details see “Karlin 2015: Entropic multi relaxation lattice Boltzmann models for turbulent flows”
continuous_equilibrium – determines if the discrete or continuous maxwellian equilibrium is used to compute the equilibrium moments.
Class¶
- class MomentBasedLbMethod(stencil, equilibrium, relaxation_dict, conserved_quantity_computation=None, force_model=None, zero_centered=False, moment_transform_class=<class 'lbmpy.moment_transforms.rawmomenttransforms.PdfsToMomentsByChimeraTransform'>)¶
Moment based LBM is a class to represent the single (SRT), two (TRT) and multi relaxation time (MRT) methods. These methods work by transforming the pdfs into moment space using a linear transformation. In the moment space each component (moment) is relaxed to an equilibrium moment by a certain relaxation rate. These equilibrium moments can e.g. be determined by taking the equilibrium moments of the continuous Maxwellian.
- Parameters
stencil – see
lbmpy.stencils.LBStencil
equilibrium – Instance of
lbmpy.equilibrium.AbstractEquilibrium
, defining the equilibrium distribution used by this method.relaxation_dict – a dictionary mapping moments in either tuple or polynomial formulation to their relaxation rate.
conserved_quantity_computation – instance of
lbmpy.methods.AbstractConservedQuantityComputation
. This determines how conserved quantities are computed, and defines the symbols used in the equilibrium moments like e.g. density and velocity.force_model – Instance of
lbmpy.forcemodels.AbstractForceModel
, or None if no forcing terms are requiredzero_centered – Determines the PDF storage format, regular or centered around the equilibrium’s background distribution.
moment_transform_class – transformation class to transform PDFs to the moment space (subclass of
lbmpy.moment_transforms.AbstractRawMomentTransform
), orNone
if equations are to be derived in population space.
- property force_model¶
Force model employed by this method.
- property relaxation_info_dict¶
Dictionary mapping this method’s moments to their relaxation rates and equilibrium values. Beware: Changes to this dictionary are not reflected in the method. For changing relaxation rates, use
relaxation_rate_dict
instead.
- property conserved_quantity_computation¶
Returns an instance of class
lbmpy.methods.AbstractConservedQuantityComputation
- property equilibrium_distribution¶
Returns this method’s equilibrium distribution (see
lbmpy.equilibrium.AbstractEquilibrium
- property moment_transform_class¶
The transform class (subclass of
lbmpy.moment_transforms.AbstractRawMomentTransform
defining the transformation of populations to moment space.
- property moment_space_collision¶
Returns whether collision is derived in terms of moments or in terms of populations only.
- property moments¶
Moments relaxed by this method.
- property relaxation_rate_dict¶
Dictionary mapping moments to relaxation rates. Changes are reflected by the method.
- property zeroth_order_equilibrium_moment_symbol¶
Returns a symbol referring to the zeroth-order moment of this method’s equilibrium distribution, which is the area under it’s curve (see
lbmpy.equilibrium.AbstractEquilibrium.zeroth_order_moment_symbol
).
- property first_order_equilibrium_moment_symbols¶
Returns a vector of symbols referring to the first-order moment of this method’s equilibrium distribution, which is its mean value. (see
lbmpy.equilibrium.AbstractEquilibrium.first_order_moment_symbols
).
- property weights¶
Returns a sequence of weights, one for each lattice direction
- override_weights(weights)¶
Manually set this method’s lattice weights.
- get_equilibrium(conserved_quantity_equations=None, include_force_terms=False, pre_simplification=False, subexpressions=False, keep_cqc_subexpressions=True)¶
Returns equation collection, to compute equilibrium values. The equations have the post collision symbols as left hand sides and are functions of the conserved quantities
- get_equilibrium_terms()¶
Returns this method’s equilibrium populations as a vector.
- get_collision_rule(conserved_quantity_equations=None, pre_simplification=True)¶
Returns an LbmCollisionRule i.e. an equation collection with a reference to the method. This collision rule defines the collision operator.
- set_zeroth_moment_relaxation_rate(relaxation_rate)¶
Alters the relaxation rate of the zeroth-order moment.
- set_first_moment_relaxation_rate(relaxation_rate)¶
Alters the relaxation rates of the first-order moments.
- set_conserved_moments_relaxation_rate(relaxation_rate)¶
Alters the relaxation rates of the zeroth- and first-order moments.
- set_force_model(force_model)¶
Updates this method’s force model.
Central Moment-based methods¶
These methods are represented by instances of lbmpy.methods.momentbased.CentralMomentBasedLbMethod
and will derive
collision equations in central moment space.
Creation Functions¶
The following factory functions create central moment-based methods using variants of the regular hydrodynamic maxwellian equilibrium.
- create_central_moment(stencil, relaxation_rates, nested_moments=None, continuous_equilibrium=True, **kwargs)¶
Creates moment based LB method where the collision takes place in the central moment space.
Internally calls either
create_with_discrete_maxwellian_equilibrium()
orcreate_with_continuous_maxwellian_equilibrium()
, depending on the value ofcontinuous_equilibrium
.- Parameters
stencil – instance of
lbmpy.stencils.LBStencil
relaxation_rates – relaxation rates (inverse of the relaxation times) for each moment
nested_moments – a list of lists of modes, grouped by common relaxation times.
continuous_equilibrium – determines if the discrete or continuous maxwellian equilibrium is used to compute the equilibrium moments.
- Returns
lbmpy.methods.momentbased.CentralMomentBasedLbMethod
instance
Class¶
- class CentralMomentBasedLbMethod(stencil, equilibrium, relaxation_dict, conserved_quantity_computation=None, force_model=None, zero_centered=False, central_moment_transform_class=<class 'lbmpy.moment_transforms.centralmomenttransforms.BinomialChimeraTransform'>)¶
Central Moment based LBM is a class to represent the single (SRT), two (TRT) and multi relaxation time (MRT) methods, where the collision is performed in the central moment space. These methods work by transforming the pdfs into moment space using a linear transformation and then shiftig them into the central moment space. In the central moment space each component (moment) is relaxed to an equilibrium moment by a certain relaxation rate. These equilibrium moments can e.g. be determined by taking the equilibrium moments of the continuous Maxwellian.
- Parameters
stencil – see
lbmpy.stencils.LBStencil
equilibrium – Instance of
lbmpy.equilibrium.AbstractEquilibrium
, defining the equilibrium distribution used by this method.relaxation_dict – a dictionary mapping moments in either tuple or polynomial formulation to their relaxation rate.
conserved_quantity_computation – instance of
lbmpy.methods.AbstractConservedQuantityComputation
. This determines how conserved quantities are computed, and defines the symbols used in the equilibrium moments like e.g. density and velocity.force_model – Instance of
lbmpy.forcemodels.AbstractForceModel
, or None if no forcing terms are requiredzero_centered – Determines the PDF storage format, regular or centered around the equilibrium’s background distribution.
central_moment_transform_class – transformation class to transform PDFs to central moment space (subclass of
lbmpy.moment_transforms.AbstractCentralMomentTransform
)
- property force_model¶
Force model employed by this method.
- property relaxation_info_dict¶
Dictionary mapping this method’s moments to their relaxation rates and equilibrium values. Beware: Changes to this dictionary are not reflected in the method. For changing relaxation rates, use
relaxation_rate_dict
instead.
- property conserved_quantity_computation¶
Returns an instance of class
lbmpy.methods.AbstractConservedQuantityComputation
- property equilibrium_distribution¶
Returns this method’s equilibrium distribution (see
lbmpy.equilibrium.AbstractEquilibrium
- property central_moment_transform_class¶
The transform class (subclass of
lbmpy.moment_transforms.AbstractCentralMomentTransform
defining the transformation of populations to central moment space.
- property moments¶
Central moments relaxed by this method.
- property relaxation_rate_dict¶
Dictionary mapping moments to relaxation rates. Changes are reflected by the method.
- property zeroth_order_equilibrium_moment_symbol¶
Returns a symbol referring to the zeroth-order moment of this method’s equilibrium distribution, which is the area under it’s curve (see
lbmpy.equilibrium.AbstractEquilibrium.zeroth_order_moment_symbol
).
- property first_order_equilibrium_moment_symbols¶
Returns a vector of symbols referring to the first-order moment of this method’s equilibrium distribution, which is its mean value. (see
lbmpy.equilibrium.AbstractEquilibrium.first_order_moment_symbols
).
- property weights¶
Returns a sequence of weights, one for each lattice direction
- property relaxation_matrix¶
Returns a qxq diagonal matrix which contains the relaxation rate for each moment on the diagonal
- get_equilibrium(conserved_quantity_equations=None, subexpressions=False, pre_simplification=False, keep_cqc_subexpressions=True, include_force_terms=False)¶
Returns equation collection, to compute equilibrium values. The equations have the post collision symbols as left hand sides and are functions of the conserved quantities
- Parameters
conserved_quantity_equations – equations to compute conserved quantities.
subexpressions – if set to false all subexpressions of the equilibrium assignments are plugged into the main assignments
pre_simplification – with or without pre_simplifications for the calculation of the collision
keep_cqc_subexpressions – if equilibrium is returned without subexpressions keep_cqc_subexpressions determines if also subexpressions to calculate conserved quantities should be plugged into the main assignments
- get_collision_rule(conserved_quantity_equations=None, pre_simplification=False)¶
Returns an LbmCollisionRule i.e. an equation collection with a reference to the method. This collision rule defines the collision operator.
Cumulant-based methods¶
These methods are represented by instances of lbmpy.methods.cumulantbased.CumulantBasedLbMethod
and will derive
collision equations in cumulant space.
Creation Functions¶
The following factory functions create cumulant-based methods using the regular continuous hydrodynamic maxwellian equilibrium.
- create_cumulant(stencil, relaxation_rates, cumulant_groups, **kwargs)¶
Creates a cumulant-based lattice Boltzmann method.
- Parameters
stencil – instance of
lbmpy.stencils.LBStencil
relaxation_rates – relaxation rates for each cumulant group. If None are provided a list of symbolic relaxation rates is created and used. If only a list with one entry is provided this relaxation rate is used for determine the viscosity of the simulation. All other cumulants are relaxed with one. If a cumulant force model is provided the first order cumulants are relaxed with two to ensure that the force is applied correctly to the momentum groups
cumulant_groups – Nested sequence of polynomial expressions defining the cumulants to be relaxed. All cumulants within one group are relaxed with the same relaxation rate.
kwargs – See
create_with_continuous_maxwellian_equilibrium()
- Returns
- create_with_monomial_cumulants(stencil, relaxation_rates, **kwargs)¶
Creates a cumulant lattice Boltzmann model using the given stencil’s canonical monomial cumulants.
- Parameters
stencil – instance of
lbmpy.stencils.LBStencil
relaxation_rates – relaxation rates for each cumulant group. If None are provided a list of symbolic relaxation rates is created and used. If only a list with one entry is provided this relaxation rate is used for determine the viscosity of the simulation. All other cumulants are relaxed with one. If a cumulant force model is provided the first order cumulants are relaxed with two to ensure that the force is applied correctly to the momentum groups
kwargs – See
create_cumulant()
- Returns
- create_with_default_polynomial_cumulants(stencil, relaxation_rates, **kwargs)¶
Creates a cumulant lattice Boltzmann model based on the default polynomial set of [GSchonherrPK15].
Args: See
create_cumulant()
.- Returns
Class¶
- class CumulantBasedLbMethod(stencil, equilibrium, relaxation_dict, conserved_quantity_computation=None, force_model=None, zero_centered=False, central_moment_transform_class=<class 'lbmpy.moment_transforms.centralmomenttransforms.BinomialChimeraTransform'>, cumulant_transform_class=<class 'lbmpy.moment_transforms.cumulanttransforms.CentralMomentsToCumulantsByGeneratingFunc'>)¶
This class implements cumulant-based lattice boltzmann methods which relax all the non-conserved quantities as either monomial or polynomial cumulants. It is mostly inspired by the work presented in [GSchonherrPK15].
This method is implemented modularily as the transformation from populations to central moments to cumulants is governed by subclasses of
lbmpy.moment_transforms.AbstractMomentTransform
which can be specified by constructor argument. This allows the selection of the most efficient transformation for a given setup.- Parameters
stencil – see
lbmpy.stencils.LBStencil
equilibrium – Instance of
lbmpy.equilibrium.AbstractEquilibrium
, defining the equilibrium distribution used by this method.relaxation_dict – a dictionary mapping cumulants in either tuple or polynomial formulation to their relaxation rate.
conserved_quantity_computation – instance of
lbmpy.methods.AbstractConservedQuantityComputation
. This determines how conserved quantities are computed, and defines the symbols used in the equilibrium moments like e.g. density and velocity.force_model – Instance of
lbmpy.forcemodels.AbstractForceModel
, or None if no forcing terms are requiredzero_centered – Determines the PDF storage format, regular or centered around the equilibrium’s background distribution.
galilean_correction – if set to True the galilean_correction is applied to a D3Q27 cumulant method
central_moment_transform_class – transformation class to transform PDFs to central moment space (subclass of
lbmpy.moment_transforms.AbstractCentralMomentTransform
)cumulant_transform_class – transform class to get from the central moment space to the cumulant space
- property force_model¶
Force model employed by this method.
- property relaxation_info_dict¶
Dictionary mapping this method’s cumulants to their relaxation rates and equilibrium values. Beware: Changes to this dictionary are not reflected in the method. For changing relaxation rates, use
relaxation_rate_dict
instead.
- property conserved_quantity_computation¶
Returns an instance of class
lbmpy.methods.AbstractConservedQuantityComputation
- property equilibrium_distribution¶
Returns this method’s equilibrium distribution (see
lbmpy.equilibrium.AbstractEquilibrium
- property central_moment_transform_class¶
The transform class (subclass of
lbmpy.moment_transforms.AbstractCentralMomentTransform
defining the transformation of populations to central moment space.
- property cumulant_transform_class¶
The transform class defining the transform from central moment to cumulant space.
- property cumulants¶
Cumulants relaxed by this method.
- property relaxation_rate_dict¶
Dictionary mapping cumulants to relaxation rates. Changes are reflected by the method.
- property zeroth_order_equilibrium_moment_symbol¶
Returns a symbol referring to the zeroth-order moment of this method’s equilibrium distribution, which is the area under it’s curve (see
lbmpy.equilibrium.AbstractEquilibrium.zeroth_order_moment_symbol
).
- property first_order_equilibrium_moment_symbols¶
Returns a vector of symbols referring to the first-order moment of this method’s equilibrium distribution, which is its mean value. (see
lbmpy.equilibrium.AbstractEquilibrium.first_order_moment_symbols
).
- property weights¶
Returns a sequence of weights, one for each lattice direction
- get_equilibrium(conserved_quantity_equations=None, subexpressions=False, pre_simplification=False, keep_cqc_subexpressions=True, include_force_terms=False)¶
Returns equation collection, to compute equilibrium values. The equations have the post collision symbols as left hand sides and are functions of the conserved quantities
- Parameters
conserved_quantity_equations – equations to compute conserved quantities.
subexpressions – if set to false all subexpressions of the equilibrium assignments are plugged into the main assignments
pre_simplification – with or without pre_simplifications for the calculation of the collision
keep_cqc_subexpressions – if equilibrium is returned without subexpressions keep_cqc_subexpressions determines if also subexpressions to calculate conserved quantities should be plugged into the main assignments
- get_collision_rule(conserved_quantity_equations=None, pre_simplification=False)¶
Returns an LbmCollisionRule i.e. an equation collection with a reference to the method. This collision rule defines the collision operator.
Default Moment sets¶
The following functions provide default sets of polynomial raw moments, central moments and cumulants to be used in MRT-type methods.
- cascaded_moment_sets_literature(stencil)¶
Returns default groups of central moments or cumulants to be relaxed with common relaxation rates as stated in literature. Groups are ordered like this:
First group is density
Second group are the momentum modes
Third group are the shear modes
Fourth group is the bulk mode
Remaining groups do not govern hydrodynamic properties
- Parameters
stencil – instance of
lbmpy.stencils.LBStencil
. Can be D2Q9, D3Q7, D3Q15, D3Q19 or D3Q27
- mrt_orthogonal_modes_literature(stencil, is_weighted)¶
Returns a list of lists of modes, grouped by common relaxation times. This is for commonly used MRT models found in literature.
- Parameters
stencil – instance of
lbmpy.stencils.LBStencil
. Can be D2Q9, D3Q15, D3Q19 or D3Q27is_weighted – whether to use weighted or unweighted orthogonality
MRT schemes as described in the following references are used
Low-Level Method Creation Interface¶
The following classes and factory functions constitute the lower levels of abstraction in method creation. They are called from the higher-level functions described above, or, in special cases, by the user directly.
Custom method variants in population space, raw and central moment space based on the hydrodynamic Maxwellian
equilibrium may be created using either
lbmpy.methods.creationfunctions.create_with_discrete_maxwellian_equilibrium()
or
create_with_continuous_maxwellian_equilibrium()
.
Methods may also be created using custom equilibrium distributions using
lbmpy.methods.creationfunctions.create_from_equilibrium()
.
The desired collision space, and the transform classes defining the manner of transforming populations to that
space and back, are defined using lbmpy.enums.CollisionSpace
and lbmpy.methods.CollisionSpaceInfo
.
Collision Space Info¶
Low-Level Creation Functions¶
- create_with_discrete_maxwellian_equilibrium(stencil, moment_to_relaxation_rate_dict, compressible=False, zero_centered=False, delta_equilibrium=False, force_model=None, equilibrium_order=2, c_s_sq=1 / 3, **kwargs)¶
Creates a moment-based LBM by taking a dictionary of moments with corresponding relaxation rates.
These moments are relaxed against the moments of the discrete Maxwellian distribution (see
lbmpy.equilibrium.DiscreteHydrodynamicMaxwellian
).Internally, this method calls
lbmpy.methods.create_from_equilibrium()
.- Parameters
stencil – instance of
lbmpy.stencils.LBStenil
moment_to_relaxation_rate_dict – dict that has as many entries as the stencil. Each moment, which can be represented by an exponent tuple or in polynomial form (see
lbmpy.moments
), is mapped to a relaxation rate.compressible – If
False
, the incompressible equilibrium formulation is used to better approximate the incompressible Navier-Stokes equations. Otherwise, the default textbook equilibrium is used.zero_centered – If
True
, the zero-centered storage format for PDFs is used, storing only their deviation from the background distribution (given by the lattice weights).delta_equilibrium – Takes effect only if zero-centered storage is used. If
True
, the equilibrium distribution is also given only as its deviation from the background distribution.force_model – instance of
lbmpy.forcemodels.AbstractForceModel
, or None if no external forces are present.equilibrium_order – approximation order of macroscopic velocity \(\mathbf{u}\) in the equilibrium
c_s_sq – Speed of sound squared
kwargs – See
lbmpy.methods.create_from_equilibrium()
- Returns
Instance of a subclass of
lbmpy.methods.AbstractLbMethod
.
- create_with_continuous_maxwellian_equilibrium(stencil, moment_to_relaxation_rate_dict, compressible=False, zero_centered=False, delta_equilibrium=False, force_model=None, equilibrium_order=2, c_s_sq=1 / 3, **kwargs)¶
Creates a moment-based LBM by taking a dictionary of moments with corresponding relaxation rates. These moments are relaxed against the moments of the continuous Maxwellian distribution (see
lbmpy.equilibrium.ContinuousHydrodynamicMaxwellian
).Internally, this method calls
lbmpy.methods.create_from_equilibrium()
.- Parameters
stencil – instance of
lbmpy.stencils.LBStenil
moment_to_relaxation_rate_dict – dict that has as many entries as the stencil. Each moment, which can be represented by an exponent tuple or in polynomial form (see
lbmpy.moments
), is mapped to a relaxation rate.compressible – If
False
, the incompressible equilibrium formulation is used to better approximate the incompressible Navier-Stokes equations. Otherwise, the default textbook equilibrium is used.zero_centered – If
True
, the zero-centered storage format for PDFs is used, storing only their deviation from the background distribution (given by the lattice weights).delta_equilibrium – Takes effect only if zero-centered storage is used. If
True
, the equilibrium distribution is also given only as its deviation from the background distribution.force_model – Instance of
lbmpy.forcemodels.AbstractForceModel
, or None if no external forces are present.equilibrium_order – approximation order of macroscopic velocity \(\mathbf{u}\) in the equilibrium
c_s_sq – Speed of sound squared
kwargs – See
lbmpy.methods.create_from_equilibrium()
- Returns
Instance of a subclass of
lbmpy.methods.AbstractLbMethod
.
- create_from_equilibrium(stencil, equilibrium, conserved_quantity_computation, moment_to_relaxation_rate_dict, collision_space_info=CollisionSpaceInfo(collision_space=<CollisionSpace.POPULATIONS: 1>, raw_moment_transform_class=None, central_moment_transform_class=None, cumulant_transform_class=None), zero_centered=False, force_model=None)¶
Creates a lattice Boltzmann method in either population, moment, or central moment space, from a given discrete velocity set and equilibrium distribution.
This function takes a stencil, an equilibrium distribution, an appropriate conserved quantity computation instance, a dictionary mapping moment polynomials to their relaxation rates, and a collision space info determining the desired collision space. It returns a method instance relaxing the given moments against their equilibrium values computed from the given distribution, in the given collision space.
- Parameters
stencil – Instance of
lbmpy.stencils.LBStencil
equilibrium – Instance of a subclass of
lbmpy.equilibrium.AbstractEquilibrium
.conserved_quantity_computation – Instance of a subclass of
lbmpy.methods.AbstractConservedQuantityComputation
, which must provide equations for the conserved quantities used in the equilibrium.moment_to_relaxation_rate_dict – Dictionary mapping moment polynomials to relaxation rates.
collision_space_info – Instance of
CollisionSpaceInfo
, defining the method’s desired collision space and the manner of transformation to this space. Cumulant-based methods are not supported yet.zero_centered – Whether or not the zero-centered storage format should be used. If
True
, the given equilibrium must either be a deviation-only formulation, or must provide a background distribution for PDFs to be centered around.force_model – Instance of
lbmpy.forcemodels.AbstractForceModel
, or None if no external forces are present.