def test_forward_transform(cumulants, stencil):
stencil = LBStencil(stencil)
if cumulants == 'monomial':
moment_polynomials = get_default_moment_set_for_stencil(stencil)
elif cumulants == 'polynomial':
moment_polynomials = [
item for sublist in cascaded_moment_sets_literature(stencil)
for item in sublist
]
pdfs = sp.symbols(f"f_:{stencil.Q}")
rho = sp.Symbol('rho')
u = sp.symbols(f"u_:{stencil.D}")
matrix_transform = PdfsToCentralMomentsByMatrix(stencil,
moment_polynomials, rho, u)
fast_transform = FastCentralMomentTransform(stencil, moment_polynomials,
rho, u)
shift_transform = PdfsToCentralMomentsByShiftMatrix(
stencil, moment_polynomials, rho, u)
assert shift_transform.moment_exponents == fast_transform.moment_exponents
if cumulants == 'monomial' and not have_same_entries(
stencil, LBStencil(Stencil.D3Q15)):
assert fast_transform.mono_to_poly_matrix == sp.eye(stencil.Q)
assert shift_transform.mono_to_poly_matrix == sp.eye(stencil.Q)
else:
assert not fast_transform.mono_to_poly_matrix == sp.eye(stencil.Q)
assert not shift_transform.mono_to_poly_matrix == sp.eye(stencil.Q)
f_to_k_matrix = matrix_transform.forward_transform(pdfs)
f_to_k_matrix = f_to_k_matrix.new_without_subexpressions(
).main_assignments_dict
f_to_k_fast = fast_transform.forward_transform(pdfs)
f_to_k_fast = f_to_k_fast.new_without_subexpressions(
).main_assignments_dict
f_to_k_shift = shift_transform.forward_transform(pdfs,
simplification=False)
f_to_k_shift = f_to_k_shift.new_without_subexpressions(
).main_assignments_dict
cm_symbols = matrix_transform.pre_collision_symbols
for moment_symbol in cm_symbols:
rhs_matrix = f_to_k_matrix[moment_symbol].expand()
rhs_fast = f_to_k_fast[moment_symbol].expand()
rhs_shift = f_to_k_shift[moment_symbol].expand()
assert (
rhs_matrix - rhs_fast
) == 0, f"Mismatch between matrix and fast transform at {moment_symbol}."
assert (
rhs_matrix - rhs_shift
) == 0, f"Mismatch between matrix and shift-matrix transform at {moment_symbol}."
def __init__(
self,
stencil,
cumulant_to_relaxation_info_dict,
conserved_quantity_computation,
force_model=None,
galilean_correction=False,
central_moment_transform_class=PdfsToCentralMomentsByShiftMatrix,
cumulant_transform_class=CentralMomentsToCumulantsByGeneratingFunc
):
assert isinstance(conserved_quantity_computation,
AbstractConservedQuantityComputation)
super(CenteredCumulantBasedLbMethod, self).__init__(stencil)
if force_model is not None:
assert (isinstance(force_model, CenteredCumulantForceModel)
or isinstance(force_model, Simple) or isinstance(
force_model,
Luo)), "Given force model currently not supported."
for m in moments_up_to_order(1, dim=self.dim):
if exponent_to_polynomial_representation(
m) not in cumulant_to_relaxation_info_dict.keys():
raise ValueError(
f'No relaxation info given for conserved cumulant {m}!')
self._cumulant_to_relaxation_info_dict = cumulant_to_relaxation_info_dict
self._conserved_quantity_computation = conserved_quantity_computation
self._force_model = force_model
self._weights = None
self._galilean_correction = galilean_correction
if galilean_correction:
if not have_same_entries(stencil, LBStencil(Stencil.D3Q27)):
raise ValueError(
"Galilean Correction only available for D3Q27 stencil")
if not contains_corrected_polynomials(
cumulant_to_relaxation_info_dict):
raise ValueError(
"For the galilean correction, all three polynomial cumulants"
"(x^2 - y^2), (x^2 - z^2) and (x^2 + y^2 + z^2) must be present!"
)
self._cumulant_transform_class = cumulant_transform_class
self._central_moment_transform_class = central_moment_transform_class
self.force_model_rr_override = False
if isinstance(self._force_model, CenteredCumulantForceModel) and \
self._force_model.override_momentum_relaxation_rate is not None:
self.set_first_moment_relaxation_rate(
self._force_model.override_momentum_relaxation_rate)
self.force_model_rr_override = True
def get_stencil_name(stencil):
for name in ('D2Q9', 'D3Q15', 'D3Q19', 'D3Q27'):
if have_same_entries(stencil, get_stencil(name, 'walberla')):
return name
def __eq__(self, other):
return self.ordering == other.ordering and have_same_entries(
self._stencil_entries, other.stencil_entries)
def cascaded_moment_sets_literature(stencil):
"""
Returns default groups of 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
Args:
stencil: instance of :class:`lbmpy.stencils.LBStencil`. Can be D2Q9, D3Q7, D3Q15, D3Q19 or D3Q27
"""
x, y, z = MOMENT_SYMBOLS
if have_same_entries(stencil, LBStencil(Stencil.D2Q9)):
# Cumulants of the D2Q9 stencil up to third order are equal to
# the central moments; only the fourth-order cumulant x**2 * y**2
# has a more complicated form. They can be arranged into groups
# for the preservation of rotational invariance as described by
# Martin Geier in his dissertation.
#
# Reference: Martin Geier. Ab inito derivation of the cascaded Lattice Boltzmann
# Automaton. Dissertation. University of Freiburg. 2006.
return [
[sp.sympify(1)], # density is conserved
[x, y], # momentum is relaxed for cumulant forcing
[x * y, x ** 2 - y ** 2], # shear
[x ** 2 + y ** 2], # bulk
[x ** 2 * y, x * y ** 2],
[x ** 2 * y ** 2]
]
elif have_same_entries(stencil, LBStencil(Stencil.D3Q7)):
# D3Q7 moments: https://arxiv.org/ftp/arxiv/papers/1611/1611.03329.pdf
return [
[sp.sympify(1)], # density is conserved
[x, y, z], # momentum might be affected by forcing
[x ** 2 - y ** 2,
x ** 2 - z ** 2], # shear
[x ** 2 + y ** 2 + z ** 2], # bulk
]
elif have_same_entries(stencil, LBStencil(Stencil.D3Q15)):
# D3Q15 central moments by Premnath et al. https://arxiv.org/pdf/1202.6081.pdf.
return [
[sp.sympify(1)], # density is conserved
[x, y, z], # momentum might be affected by forcing
[x * y,
x * z,
y * z,
x ** 2 - y ** 2,
x ** 2 - z ** 2], # shear
[x ** 2 + y ** 2 + z ** 2], # bulk
[x * (x ** 2 + y ** 2 + z ** 2),
y * (x ** 2 + y ** 2 + z ** 2),
z * (x ** 2 + y ** 2 + z ** 2)],
[x * y * z],
[x ** 2 * y ** 2 + x ** 2 * z ** 2 + y ** 2 * z ** 2]
]
elif have_same_entries(stencil, LBStencil(Stencil.D3Q19)):
# D3Q19 cumulants are obtained by pruning the D3Q27 cumulant set as
# described by Coreixas, 2019.
return [
[sp.sympify(1)], # density is conserved
[x, y, z], # momentum might be affected by forcing
[x * y,
x * z,
y * z,
x ** 2 - y ** 2,
x ** 2 - z ** 2], # shear
[x ** 2 + y ** 2 + z ** 2], # bulk
[x * y ** 2 + x * z ** 2,
x ** 2 * y + y * z ** 2,
x ** 2 * z + y ** 2 * z],
[x * y ** 2 - x * z ** 2,
x ** 2 * y - y * z ** 2,
x ** 2 * z - y ** 2 * z],
[x ** 2 * y ** 2 - 2 * x ** 2 * z ** 2 + y ** 2 * z ** 2,
x ** 2 * y ** 2 + x ** 2 * z ** 2 - 2 * y ** 2 * z ** 2],
[x ** 2 * y ** 2 + x ** 2 * z ** 2 + y ** 2 * z ** 2]
]
elif have_same_entries(stencil, LBStencil(Stencil.D3Q27)):
# Cumulants grouped to preserve rotational invariance as described by Geier et al, 2015
return [
[sp.sympify(1)], # density is conserved
[x, y, z], # momentum might be affected by forcing
[x * y,
x * z,
y * z,
x ** 2 - y ** 2,
x ** 2 - z ** 2], # shear
[x ** 2 + y ** 2 + z ** 2], # bulk
[x * y ** 2 + x * z ** 2,
x ** 2 * y + y * z ** 2,
x ** 2 * z + y ** 2 * z],
[x * y ** 2 - x * z ** 2,
x ** 2 * y - y * z ** 2,
x ** 2 * z - y ** 2 * z],
[x * y * z],
[x ** 2 * y ** 2 - 2 * x ** 2 * z ** 2 + y ** 2 * z ** 2,
x ** 2 * y ** 2 + x ** 2 * z ** 2 - 2 * y ** 2 * z ** 2],
[x ** 2 * y ** 2 + x ** 2 * z ** 2 + y ** 2 * z ** 2],
[x ** 2 * y * z,
x * y ** 2 * z,
x * y * z ** 2],
[x ** 2 * y ** 2 * z,
x ** 2 * y * z ** 2,
x * y ** 2 * z ** 2],
[x ** 2 * y ** 2 * z ** 2]
]
else:
raise ValueError("No default set of cascaded moments is available for this stencil. "
"Please specify your own set of cascaded moments.")
def 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.
Args:
stencil: instance of :class:`lbmpy.stencils.LBStencil`. Can be D2Q9, D3Q15, D3Q19 or D3Q27
is_weighted: whether to use weighted or unweighted orthogonality
MRT schemes as described in the following references are used
"""
x, y, z = MOMENT_SYMBOLS
one = sp.Rational(1, 1)
if have_same_entries(stencil, LBStencil(Stencil.D2Q9)) and is_weighted:
# Reference:
# Duenweg, B., Schiller, U. D., & Ladd, A. J. (2007). Statistical mechanics of the fluctuating
# lattice Boltzmann equation. Physical Review E, 76(3)
sq = x ** 2 + y ** 2
all_moments = [one, x, y, 3 * sq - 2, 2 * x ** 2 - sq, x * y,
(3 * sq - 4) * x, (3 * sq - 4) * y, 9 * sq ** 2 - 15 * sq + 2]
nested_moments = list(sort_moments_into_groups_of_same_order(all_moments).values())
return nested_moments
elif have_same_entries(stencil, LBStencil(Stencil.D3Q15)) and is_weighted:
sq = x ** 2 + y ** 2 + z ** 2
nested_moments = [
[one, x, y, z], # [0, 3, 5, 7]
[sq - 1], # [1]
[3 * sq ** 2 - 9 * sq + 4], # [2]
[(3 * sq - 5) * x, (3 * sq - 5) * y, (3 * sq - 5) * z], # [4, 6, 8]
[3 * x ** 2 - sq, y ** 2 - z ** 2, x * y, y * z, x * z], # [9, 10, 11, 12, 13]
[x * y * z]
]
elif have_same_entries(stencil, LBStencil(Stencil.D3Q19)) and is_weighted:
# This MRT variant mentioned in the dissertation of Ulf Schiller
# "Thermal fluctuations and boundary conditions in the lattice Boltzmann method" (2008), p. 24ff
# There are some typos in the moment matrix on p.27
# The here implemented ordering of the moments is however different from that reference (Eq. 2.61-2.63)
# The moments are weighted-orthogonal (Eq. 2.58)
# Further references:
# Duenweg, B., Schiller, U. D., & Ladd, A. J. (2007). Statistical mechanics of the fluctuating
# lattice Boltzmann equation. Physical Review E, 76(3)
# Chun, B., & Ladd, A. J. (2007). Interpolated boundary condition for lattice Boltzmann simulations of
# flows in narrow gaps. Physical review E, 75(6)
sq = x ** 2 + y ** 2 + z ** 2
nested_moments = [
[one, x, y, z], # [0, 3, 5, 7]
[sq - 1], # [1]
[3 * sq ** 2 - 6 * sq + 1], # [2]
[(3 * sq - 5) * x, (3 * sq - 5) * y, (3 * sq - 5) * z], # [4, 6, 8]
[3 * x ** 2 - sq, y ** 2 - z ** 2, x * y, y * z, x * z], # [9, 11, 13, 14, 15]
[(2 * sq - 3) * (3 * x ** 2 - sq), (2 * sq - 3) * (y ** 2 - z ** 2)], # [10, 12]
[(y ** 2 - z ** 2) * x, (z ** 2 - x ** 2) * y, (x ** 2 - y ** 2) * z] # [16, 17, 18]
]
elif have_same_entries(stencil, LBStencil(Stencil.D3Q27)) and not is_weighted:
xsq, ysq, zsq = x ** 2, y ** 2, z ** 2
all_moments = [
sp.Rational(1, 1), # 0
x, y, z, # 1, 2, 3
x * y, x * z, y * z, # 4, 5, 6
xsq - ysq, # 7
(xsq + ysq + zsq) - 3 * zsq, # 8
(xsq + ysq + zsq) - 2, # 9
3 * (x * ysq + x * zsq) - 4 * x, # 10
3 * (xsq * y + y * zsq) - 4 * y, # 11
3 * (xsq * z + ysq * z) - 4 * z, # 12
x * ysq - x * zsq, # 13
xsq * y - y * zsq, # 14
xsq * z - ysq * z, # 15
x * y * z, # 16
3 * (xsq * ysq + xsq * zsq + ysq * zsq) - 4 * (xsq + ysq + zsq) + 4, # 17
3 * (xsq * ysq + xsq * zsq - 2 * ysq * zsq) - 2 * (2 * xsq - ysq - zsq), # 18
3 * (xsq * ysq - xsq * zsq) - 2 * (ysq - zsq), # 19
3 * (xsq * y * z) - 2 * (y * z), # 20
3 * (x * ysq * z) - 2 * (x * z), # 21
3 * (x * y * zsq) - 2 * (x * y), # 22
9 * (x * ysq * zsq) - 6 * (x * ysq + x * zsq) + 4 * x, # 23
9 * (xsq * y * zsq) - 6 * (xsq * y + y * zsq) + 4 * y, # 24
9 * (xsq * ysq * z) - 6 * (xsq * z + ysq * z) + 4 * z, # 25
27 * (xsq * ysq * zsq) - 18 * (xsq * ysq + xsq * zsq + ysq * zsq) + 12 * (xsq + ysq + zsq) - 8, # 26
]
nested_moments = list(sort_moments_into_groups_of_same_order(all_moments).values())
else:
raise NotImplementedError("No MRT model is available (yet) for this stencil. "
"Create a custom MRT using 'create_with_discrete_maxwellian_eq_moments'")
return nested_moments