def elbm_d3q15_equilibrium(grid):
"""
Form of equilibrium defined in PRL 97, 010201 (2006).
See also Chikatamarla, PhD Thesis, Eq. (5.7), (5.8).
"""
rho = S.rho
prefactor = Symbol('prefactor')
coeff1 = Symbol('coeff1')
coeff2 = Symbol('coeff2')
coeff3 = Symbol('coeff3')
vsq = Symbol('vsq')
vx, vy, vz = grid.v
lvars = []
lvars.append(Eq(vsq, grid.v.dot(grid.v)))
lvars.append(Eq(prefactor, poly_factorize(
rho * (1 - 3 * vsq / 2 + 9 * vsq**2 / 8 +
Rational(27, 16) * (-vsq**3 + 2 * (vy**2 + vz**2) *
(vsq * vx**2 + vy**2 * vz**2) +
20 * vx**2 * vy**2 * vz**2) +
Rational(81, 128) * vsq**4 +
Rational(81, 32) * (vx**8 + vy**8 + vz**8
- 36 * vx**2 * vy**2 * vz**2 * vsq
- vx**4 * vy**4
- vy**4 * vz**4
- vx**4 * vz**4)))))
cfs = [coeff1, coeff2, coeff3]
for i, coeff in enumerate(cfs):
tmp = (1 + 3 * grid.v[i] + 9 * grid.v[i]**2 / 2 +
9 * grid.v[i]**3 / 2 + 27 * grid.v[i]**4 / 8)
# Cyclic permutation.
x = i
y = (i + 1) % 3
z = (i + 2) % 3
tmp += Rational(27, 8) * (grid.v[x]**5
- 4 * grid.v[x] * grid.v[y]**2 * grid.v[z]**2)
tmp += Rational(81, 16) * (grid.v[x]**6
- 8 * grid.v[x]**2 * grid.v[y]**2 * grid.v[z]**2)
tmp += Rational(81, 16) * (grid.v[x]**7
- 10 * grid.v[x]**3 * grid.v[y]**2 * grid.v[z]**2
+ 2 * grid.v[x] * grid.v[y]**2 * grid.v[z]**2 * vsq)
tmp += Rational(243, 128) * (grid.v[x]**8
+ 16 * grid.v[x]**2 * grid.v[y]**2 * grid.v[z]**2
* (grid.v[y]**2 + grid.v[z]**2))
lvars.append(Eq(coeff, poly_factorize(tmp)))
out = []
for ei, weight in zip(grid.basis, grid.entropic_weights):
t = prefactor * weight
for j, comp in enumerate(ei):
t *= cfs[j]**comp
out.append(t)
return EqDef(out, lvars)
def shallow_water_equilibrium(grid, config):
"""Get expressions for the BGK equilibrium distribution for the shallow
water equation."""
if grid.dim != 2 or grid.Q != 9:
raise TypeError('Shallow water equation requires the D2Q9 grid.')
out = [S.rho - grid.weights[0] * S.rho * (
Rational(15, 8) * S.gravity * S.rho - 3 * grid.v.dot(grid.v))]
for ei, weight in zip(grid.basis[1:], grid.weights[1:]):
out.append(weight * (
S.rho * poly_factorize(Rational(3,2) * S.rho * S.gravity + 3*ei.dot(grid.v) +
Rational(9,2) * (ei.dot(grid.v))**2 - Rational(3, 2) * grid.v.dot(grid.v))))
return EqDef(out, local_vars=[])
def guo_external_force(grid, grid_num=0):
"""Gets expressions for the external body force correction in the BGK model.
This implements the external force as in Eq. 20 from PhysRevE 65, 046308.
:param grid: the grid class to be used
:rtype: list of sympy expressions (in the same order as the current grid's basis)
"""
pref = Symbol('pref')
ea = accel_vector(grid, grid_num)
ret = []
for i, ei in enumerate(grid.basis):
ret.append(pref * grid.weights[i] *
poly_factorize((ei - grid.v + ei.dot(grid.v)*ei*3).dot(ea)))
return ret
def bgk_equilibrium(grid, config, rho=None, rho0=None):
"""Get expressions for the BGK equilibrium distribution.
:param grid: the grid class to be used
"""
out = []
if rho is None:
rho = S.rho
if rho0 is None:
if config.incompressible:
rho0 = S.rho0
elif config.minimize_roundoff:
rho0 = rho + 1.0
else:
rho0 = rho
for ei, weight in zip(grid.basis, grid.weights):
out.append(weight * (rho + rho0 * poly_factorize(
3 * ei.dot(grid.v) + Rational(9, 2) * (ei.dot(grid.v))**2 -
Rational(3, 2) * grid.v.dot(grid.v))))
return EqDef(out, local_vars=[])
