"""The basic equations for the IISPH formulation of
M. Ihmsen, J. Cornelis, B. Solenthaler, C. Horvath, M. Teschner, "Implicit
Incompressible SPH," IEEE Transactions on Visualization and Computer
Graphics, vol. 20, no. 3, pp. 426-435, March 2014.
http://dx.doi.org/10.1109/TVCG.2013.105
"""
from numpy import sqrt, fabs
from compyle.api import declare
from pysph.base.particle_array import get_ghost_tag
from pysph.sph.equation import Equation
from pysph.sph.integrator_step import IntegratorStep
from pysph.base.reduce_array import serial_reduce_array, parallel_reduce_array
from pysph.sph.scheme import Scheme, add_bool_argument
GHOST_TAG = get_ghost_tag()
[docs]class IISPHStep(IntegratorStep):
"""A straightforward and simple integrator to be used for IISPH.
"""
[docs] def stage1(self, d_idx, d_x, d_y, d_z, d_u, d_v, d_w,
d_uadv, d_vadv, d_wadv, d_au, d_av, d_aw,
d_ax, d_ay, d_az, dt):
d_u[d_idx] = d_uadv[d_idx] + dt * d_au[d_idx]
d_v[d_idx] = d_vadv[d_idx] + dt * d_av[d_idx]
d_w[d_idx] = d_wadv[d_idx] + dt * d_aw[d_idx]
d_x[d_idx] += dt * d_u[d_idx]
d_y[d_idx] += dt * d_v[d_idx]
d_z[d_idx] += dt * d_w[d_idx]
[docs]class NumberDensity(Equation):
[docs] def initialize(self, d_idx, d_V):
d_V[d_idx] = 0.0
[docs] def loop(self, d_idx, d_V, WIJ):
d_V[d_idx] += WIJ
[docs]class SummationDensity(Equation):
[docs] def initialize(self, d_idx, d_rho):
d_rho[d_idx] = 0.0
[docs] def loop(self, d_idx, d_rho, s_idx, s_m, WIJ):
d_rho[d_idx] += s_m[s_idx]*WIJ
[docs]class SummationDensityBoundary(Equation):
def __init__(self, dest, sources, rho0):
self.rho0 = rho0
super(SummationDensityBoundary, self).__init__(dest, sources)
[docs] def loop(self, d_idx, d_rho, s_idx, s_V, WIJ):
d_rho[d_idx] += self.rho0/s_V[s_idx]*WIJ
[docs]class NormalizedSummationDensity(Equation):
[docs] def initialize(self, d_idx, d_rho, d_rho_adv, d_rho0, d_V):
d_rho0[d_idx] = d_rho[d_idx]
d_rho[d_idx] = 0.0
d_rho_adv[d_idx] = 0.0
d_V[d_idx] = 0.0
[docs] def loop(self, d_idx, d_rho, d_rho_adv, d_V, s_idx, s_m, s_rho0, WIJ):
tmp = s_m[s_idx]*WIJ
d_rho[d_idx] += tmp
d_rho_adv[d_idx] += tmp/s_rho0[s_idx]
d_V[d_idx] += WIJ
[docs] def post_loop(self, d_idx, d_rho, d_rho_adv):
d_rho[d_idx] /= d_rho_adv[d_idx]
[docs]class AdvectionAcceleration(Equation):
def __init__(self, dest, sources, gx=0.0, gy=0.0, gz=0.0):
self.gx = gx
self.gy = gy
self.gz = gz
super(AdvectionAcceleration, self).__init__(dest, sources)
[docs] def initialize(self, d_idx, d_au, d_av, d_aw, d_uadv, d_vadv, d_wadv):
d_au[d_idx] = self.gx
d_av[d_idx] = self.gy
d_aw[d_idx] = self.gz
d_uadv[d_idx] = 0.0
d_vadv[d_idx] = 0.0
d_wadv[d_idx] = 0.0
[docs] def post_loop(self, d_idx, d_au, d_av, d_aw, d_uadv, d_vadv, d_wadv,
d_u, d_v, d_w, dt=0.0):
d_uadv[d_idx] = d_u[d_idx] + dt*d_au[d_idx]
d_vadv[d_idx] = d_v[d_idx] + dt*d_av[d_idx]
d_wadv[d_idx] = d_w[d_idx] + dt*d_aw[d_idx]
[docs]class ViscosityAcceleration(Equation):
def __init__(self, dest, sources, nu):
self.nu = nu
super(ViscosityAcceleration, self).__init__(dest, sources)
[docs] def loop(self, d_idx, d_au, d_av, d_aw, s_idx, s_m, EPS,
VIJ, XIJ, RHOIJ1, R2IJ, DWIJ):
dwijdotxij = DWIJ[0]*XIJ[0] + DWIJ[1]*XIJ[1] + DWIJ[2]*XIJ[2]
fac = 2.0*self.nu*s_m[s_idx]*RHOIJ1*dwijdotxij/(R2IJ + EPS)
d_au[d_idx] += fac*VIJ[0]
d_av[d_idx] += fac*VIJ[1]
d_aw[d_idx] += fac*VIJ[2]
[docs]class ViscosityAccelerationBoundary(Equation):
"""The acceleration on the fluid due to a boundary.
"""
def __init__(self, dest, sources, rho0, nu):
self.nu = nu
self.rho0 = rho0
super(ViscosityAccelerationBoundary, self).__init__(dest, sources)
[docs] def loop(self, d_idx, d_au, d_av, d_aw, d_rho, s_idx, s_V, EPS,
VIJ, XIJ, R2IJ, DWIJ):
phi_b = self.rho0/(s_V[s_idx]*d_rho[d_idx])
dwijdotxij = DWIJ[0]*XIJ[0] + DWIJ[1]*XIJ[1] + DWIJ[2]*XIJ[2]
fac = 2.0*self.nu*phi_b*dwijdotxij/(R2IJ + EPS)
d_au[d_idx] += fac*VIJ[0]
d_av[d_idx] += fac*VIJ[1]
d_aw[d_idx] += fac*VIJ[2]
[docs]class ComputeDII(Equation):
[docs] def initialize(self, d_idx, d_dii0, d_dii1, d_dii2):
d_dii0[d_idx] = 0.0
d_dii1[d_idx] = 0.0
d_dii2[d_idx] = 0.0
[docs] def loop(self, d_idx, d_rho, d_dii0, d_dii1, d_dii2,
s_idx, s_m, DWIJ):
rho_1 = 1.0/d_rho[d_idx]
fac = -s_m[s_idx]*rho_1*rho_1
d_dii0[d_idx] += fac*DWIJ[0]
d_dii1[d_idx] += fac*DWIJ[1]
d_dii2[d_idx] += fac*DWIJ[2]
[docs]class ComputeDIIBoundary(Equation):
def __init__(self, dest, sources, rho0):
self.rho0 = rho0
super(ComputeDIIBoundary, self).__init__(dest, sources)
[docs] def loop(self, d_idx, d_dii0, d_dii1, d_dii2, d_rho,
s_idx, s_m, s_V, DWIJ):
rhoi1 = 1.0/d_rho[d_idx]
fac = -rhoi1*rhoi1*self.rho0/s_V[s_idx]
d_dii0[d_idx] += fac*DWIJ[0]
d_dii1[d_idx] += fac*DWIJ[1]
d_dii2[d_idx] += fac*DWIJ[2]
[docs]class ComputeRhoAdvection(Equation):
[docs] def initialize(self, d_idx, d_rho_adv, d_rho, d_p0, d_p, d_piter, d_aii):
d_rho_adv[d_idx] = d_rho[d_idx]
d_p0[d_idx] = d_p[d_idx]
d_piter[d_idx] = 0.5*d_p[d_idx]
[docs] def loop(self, d_idx, d_rho, d_rho_adv, d_uadv, d_vadv, d_wadv, d_u,
d_v, d_w, s_idx, s_m, s_uadv, s_vadv, s_wadv, DWIJ, dt=0.0):
vijdotdwij = (d_uadv[d_idx] - s_uadv[s_idx])*DWIJ[0] + \
(d_vadv[d_idx] - s_vadv[s_idx])*DWIJ[1] + \
(d_wadv[d_idx] - s_wadv[s_idx])*DWIJ[2]
d_rho_adv[d_idx] += dt*s_m[s_idx]*vijdotdwij
[docs]class ComputeRhoBoundary(Equation):
def __init__(self, dest, sources, rho0):
self.rho0 = rho0
super(ComputeRhoBoundary, self).__init__(dest, sources)
[docs] def loop(self, d_idx, d_rho, d_rho_adv, d_uadv, d_vadv, d_wadv,
s_idx, s_u, s_v, s_w, s_V, WIJ, DWIJ, dt=0.0):
phi_b = self.rho0/s_V[s_idx]
vijdotdwij = (d_uadv[d_idx] - s_u[s_idx])*DWIJ[0] + \
(d_vadv[d_idx] - s_v[s_idx])*DWIJ[1] + \
(d_wadv[d_idx] - s_w[s_idx])*DWIJ[2]
d_rho_adv[d_idx] += dt*phi_b*vijdotdwij
[docs]class ComputeAII(Equation):
[docs] def initialize(self, d_idx, d_aii):
d_aii[d_idx] = 0.0
[docs] def loop(self, d_idx, d_aii, d_dii0, d_dii1, d_dii2, d_m, d_rho,
s_idx, s_m, s_rho, DWIJ):
rho1 = 1.0/d_rho[d_idx]
fac = d_m[d_idx]*rho1*rho1
# The following is m_j (d_ii - d_ji) . DWIJ
# DWIJ = -DWJI
dijdotdwij = (d_dii0[d_idx] - fac*DWIJ[0])*DWIJ[0] + \
(d_dii1[d_idx] - fac*DWIJ[1])*DWIJ[1] + \
(d_dii2[d_idx] - fac*DWIJ[2])*DWIJ[2]
d_aii[d_idx] += s_m[s_idx]*dijdotdwij
[docs]class ComputeAIIBoundary(Equation):
""" This is important and not really discussed in the original IISPH paper.
"""
def __init__(self, dest, sources, rho0):
self.rho0 = rho0
super(ComputeAIIBoundary, self).__init__(dest, sources)
[docs] def loop(self, d_idx, d_m, d_aii, d_dii0, d_dii1, d_dii2, d_rho,
s_idx, s_m, s_V, DWIJ):
phi_b = self.rho0/s_V[s_idx]
rho1 = 1.0/d_rho[d_idx]
fac = d_m[d_idx]*rho1*rho1
dijdotdwij = ((d_dii0[d_idx] - fac*DWIJ[0])*DWIJ[0] +
(d_dii1[d_idx] - fac*DWIJ[1])*DWIJ[1] +
(d_dii2[d_idx] - fac*DWIJ[2])*DWIJ[2])
d_aii[d_idx] += phi_b*dijdotdwij
[docs]class ComputeDIJPJ(Equation):
[docs] def initialize(self, d_idx, d_dijpj0, d_dijpj1, d_dijpj2):
d_dijpj0[d_idx] = 0.0
d_dijpj1[d_idx] = 0.0
d_dijpj2[d_idx] = 0.0
[docs] def loop(self, d_idx, d_dijpj0, d_dijpj1, d_dijpj2,
s_idx, s_m, s_rho, s_piter, DWIJ):
rho1 = 1.0/s_rho[s_idx]
fac = -s_m[s_idx]*rho1*rho1*s_piter[s_idx]
d_dijpj0[d_idx] += fac*DWIJ[0]
d_dijpj1[d_idx] += fac*DWIJ[1]
d_dijpj2[d_idx] += fac*DWIJ[2]
[docs]class UpdateGhostProps(Equation):
def __init__(self, dest, sources=None):
super(UpdateGhostProps, self).__init__(dest, sources)
# We do this to ensure that the ghost tag is indeed 2.
# If not the initialize method will never work.
assert GHOST_TAG == 2
[docs] def initialize(self, d_idx, d_tag, d_orig_idx, d_dijpj0, d_dijpj1,
d_dijpj2, d_dii0, d_dii1, d_dii2, d_piter):
idx = declare('int')
if d_tag[d_idx] == 2:
idx = d_orig_idx[d_idx]
d_dijpj0[d_idx] = d_dijpj0[idx]
d_dijpj1[d_idx] = d_dijpj1[idx]
d_dijpj2[d_idx] = d_dijpj2[idx]
d_dii0[d_idx] = d_dii0[idx]
d_dii1[d_idx] = d_dii1[idx]
d_dii2[d_idx] = d_dii2[idx]
d_piter[d_idx] = d_piter[idx]
[docs]class PressureSolve(Equation):
def __init__(self, dest, sources, rho0, omega=0.5,
tolerance=1e-2, debug=False):
self.rho0 = rho0
self.omega = omega
self.compression = 0.0
self.debug = debug
self.tolerance = tolerance
super(PressureSolve, self).__init__(dest, sources)
[docs] def initialize(self, d_idx, d_p, d_compression):
d_p[d_idx] = 0.0
d_compression[d_idx] = 0.0
[docs] def loop(self, d_idx, d_p, d_piter, d_rho, d_m, d_dijpj0, d_dijpj1,
d_dijpj2, s_idx, s_m, s_dii0, s_dii1, s_dii2,
s_piter, s_dijpj0, s_dijpj1, s_dijpj2, DWIJ):
# Note that a good way to check this is to see that when
# d_idx == s_idx the contribution is zero, as is expected.
rho1 = 1.0/d_rho[d_idx]
fac = d_m[d_idx]*rho1*rho1*d_piter[d_idx]
djkpk0 = s_dijpj0[s_idx] - fac*DWIJ[0]
djkpk1 = s_dijpj1[s_idx] - fac*DWIJ[1]
djkpk2 = s_dijpj2[s_idx] - fac*DWIJ[2]
tmp0 = d_dijpj0[d_idx] - s_dii0[s_idx]*s_piter[s_idx] - djkpk0
tmp1 = d_dijpj1[d_idx] - s_dii1[s_idx]*s_piter[s_idx] - djkpk1
tmp2 = d_dijpj2[d_idx] - s_dii2[s_idx]*s_piter[s_idx] - djkpk2
tmpdotdwij = tmp0*DWIJ[0] + tmp1*DWIJ[1] + tmp2*DWIJ[2]
# This is corrected in the post_loop.
d_p[d_idx] += s_m[s_idx]*tmpdotdwij
[docs] def post_loop(self, d_idx, d_piter, d_p0, d_p, d_aii, d_rho_adv, d_rho,
d_compression, dt=0.0):
dt2 = dt*dt
# Recall that d_p now has \sum_{j\neq i} a_ij p_j
tmp = self.rho0 - d_rho_adv[d_idx] - d_p[d_idx]*dt2
dnr = d_aii[d_idx]*dt2
if fabs(dnr) > 1e-9:
# Clamp pressure to positive values.
p = max((1.0 - self.omega)*d_piter[d_idx] +
self.omega/dnr*tmp, 0.0)
else:
p = 0.0
if p != 0.0:
d_compression[d_idx] = fabs(p*dnr - tmp) + self.rho0
else:
d_compression[d_idx] = self.rho0
d_piter[d_idx] = p
d_p[d_idx] = p
[docs] def reduce(self, dst, t, dt):
dst.tmp_comp[0] = serial_reduce_array(dst.compression > 0.0, 'sum')
dst.tmp_comp[1] = serial_reduce_array(dst.compression, 'sum')
dst.tmp_comp[:] = parallel_reduce_array(dst.tmp_comp, 'sum')
if dst.tmp_comp[0] > 0:
avg_rho = dst.tmp_comp[1]/dst.tmp_comp[0]
else:
avg_rho = self.rho0
self.compression = fabs(avg_rho - self.rho0)/self.rho0
[docs] def converged(self):
debug = self.debug
compression = self.compression
if compression > self.tolerance:
if debug:
print("Not converged:", compression)
return -1.0
else:
if debug:
print("Converged:", compression)
return 1.0
[docs]class PressureSolveBoundary(Equation):
def __init__(self, dest, sources, rho0):
self.rho0 = rho0
super(PressureSolveBoundary, self).__init__(dest, sources)
[docs] def loop(self, d_idx, d_p, d_rho, d_dijpj0, d_dijpj1, d_dijpj2,
s_idx, s_V, DWIJ):
phi_b = self.rho0/s_V[s_idx]
dijdotwij = (d_dijpj0[d_idx]*DWIJ[0] +
d_dijpj1[d_idx]*DWIJ[1] +
d_dijpj2[d_idx]*DWIJ[2])
d_p[d_idx] += phi_b*dijdotwij
[docs]class UpdateGhostPressure(Equation):
[docs] def initialize(self, d_idx, d_tag, d_orig_idx, d_p, d_piter):
idx = declare('int')
if d_tag[d_idx] == 2:
idx = d_orig_idx[d_idx]
d_piter[d_idx] = d_piter[idx]
d_p[d_idx] = d_p[idx]
[docs]class PressureForce(Equation):
[docs] def initialize(self, d_idx, d_au, d_av, d_aw):
d_au[d_idx] = 0.0
d_av[d_idx] = 0.0
d_aw[d_idx] = 0.0
[docs] def loop(self, d_idx, d_rho, d_p, d_au, d_av, d_aw,
s_idx, s_m, s_rho, s_p, DWIJ):
rhoi1 = 1.0/d_rho[d_idx]
rhoj1 = 1.0/s_rho[s_idx]
fac = -s_m[s_idx]*(d_p[d_idx]*rhoi1*rhoi1 + s_p[s_idx]*rhoj1*rhoj1)
d_au[d_idx] += fac*DWIJ[0]
d_av[d_idx] += fac*DWIJ[1]
d_aw[d_idx] += fac*DWIJ[2]
[docs] def post_loop(self, d_idx, d_au, d_av, d_aw,
d_uadv, d_vadv, d_wadv, d_dt_cfl, d_dt_force):
fac = d_au[d_idx]*d_au[d_idx] + d_av[d_idx]*d_av[d_idx] +\
d_aw[d_idx]*d_aw[d_idx]
vmag = sqrt(d_uadv[d_idx]*d_uadv[d_idx] + d_vadv[d_idx]*d_vadv[d_idx] +
d_wadv[d_idx]*d_wadv[d_idx])
d_dt_cfl[d_idx] = 2.0*vmag
d_dt_force[d_idx] = 2.0*fac
[docs]class PressureForceBoundary(Equation):
def __init__(self, dest, sources, rho0):
self.rho0 = rho0
super(PressureForceBoundary, self).__init__(dest, sources)
[docs] def loop(self, d_idx, d_rho, d_au, d_av, d_aw, d_p, s_idx, s_V, DWIJ):
rho1 = 1.0/d_rho[d_idx]
fac = -d_p[d_idx]*rho1*rho1*self.rho0/s_V[s_idx]
d_au[d_idx] += fac*DWIJ[0]
d_av[d_idx] += fac*DWIJ[1]
d_aw[d_idx] += fac*DWIJ[2]
[docs]class IISPHScheme(Scheme):
def __init__(self, fluids, solids, dim, rho0, nu=0.0,
gx=0.0, gy=0.0, gz=0.0, omega=0.5, tolerance=1e-2,
debug=False, has_ghosts=False):
'''The IISPH scheme
Parameters
----------
fluids : list(str)
List of names of fluid particle arrays.
solids : list(str)
List of names of solid particle arrays.
dim: int
Dimensionality of the problem.
rho0 : float
Density of fluid.
nu : float
Kinematic viscosity.
gx, gy, gz : float
Componenents of body acceleration (gravity, external forcing etc.)
omega : float
Relaxation parameter for relaxed-Jacobi iterations.
tolerance: float
Tolerance for the convergence of pressure iterations as a fraction.
debug: bool
Produce some debugging output on iterations.
has_ghosts: bool
The problem has ghost particles so add equations for those.
'''
self.fluids = fluids
self.solids = solids
self.dim = dim
self.rho0 = rho0
self.nu = nu
self.gx = gx
self.gy = gy
self.gz = gz
self.omega = omega
self.tolerance = tolerance
self.debug = debug
self.has_ghosts = has_ghosts
[docs] def add_user_options(self, group):
group.add_argument(
'--omega', action="store", type=float, dest="omega",
default=None, help="Relaxation parameter for Jacobi iterations."
)
group.add_argument(
'--tolerance', action='store', type=float, dest='tolerance',
default=None,
help='Tolerance for convergence of iterations as a fraction'
)
add_bool_argument(
group, 'iisph-debug', dest='debug', default=None,
help="Produce some debugging output on convergence of iterations."
)
[docs] def consume_user_options(self, options):
vars = ['omega', 'tolerance', 'debug']
data = dict((var, self._smart_getattr(options, var))
for var in vars)
self.configure(**data)
[docs] def get_equations(self):
from pysph.sph.equation import Group
has_ghosts = self.has_ghosts
equations = []
if self.solids:
g1 = Group(
equations=[NumberDensity(dest=x, sources=[x])
for x in self.solids]
)
equations.append(g1)
g2 = Group(
equations=[SummationDensity(dest=x, sources=self.fluids)
for x in self.fluids],
real=False
)
equations.append(g2)
if self.solids:
g3 = Group(
equations=[
SummationDensityBoundary(
dest=x, sources=self.solids, rho0=self.rho0
)
for x in self.fluids],
real=False
)
equations.append(g3)
eq = []
for fluid in self.fluids:
eq.extend([
AdvectionAcceleration(
dest=fluid, sources=None,
gx=self.gx, gy=self.gy, gz=self.gz
),
ComputeDII(dest=fluid, sources=self.fluids)
])
if self.nu > 0.0:
eq.append(
ViscosityAcceleration(
dest=fluid, sources=self.fluids, nu=self.nu
)
)
if self.solids:
if self.nu > 0.0:
eq.append(
ViscosityAccelerationBoundary(
dest=fluid, sources=self.solids, nu=self.nu,
rho0=self.rho0,
)
)
eq.append(
ComputeDIIBoundary(dest=fluid, sources=self.solids,
rho0=self.rho0)
)
g4 = Group(equations=eq, real=False)
equations.append(g4)
eq = []
for fluid in self.fluids:
eq.extend([
ComputeRhoAdvection(dest=fluid, sources=self.fluids),
ComputeAII(dest=fluid, sources=self.fluids),
])
if self.solids:
eq.extend([
ComputeRhoBoundary(dest=fluid, sources=self.solids,
rho0=self.rho0),
ComputeAIIBoundary(dest=fluid, sources=self.solids,
rho0=self.rho0),
])
g5 = Group(equations=eq)
equations.append(g5)
sg1 = Group(
equations=[
ComputeDIJPJ(dest=x, sources=self.fluids)
for x in self.fluids
]
)
eq = []
for fluid in self.fluids:
eq.append(
PressureSolve(dest=fluid, sources=self.fluids,
rho0=self.rho0, tolerance=self.tolerance,
debug=self.debug)
)
if self.solids:
eq.append(
PressureSolveBoundary(
dest=fluid, sources=self.solids, rho0=self.rho0,
)
)
sg2 = Group(equations=eq)
if has_ghosts:
ghost1 = Group(
equations=[UpdateGhostProps(dest=x, sources=None)
for x in self.fluids],
real=False
)
ghost2 = Group(
equations=[UpdateGhostPressure(dest=x, sources=None)
for x in self.fluids],
real=False
)
solver_eqs = [sg1, ghost1, sg2, ghost2]
else:
solver_eqs = [sg1, sg2]
g6 = Group(
equations=solver_eqs,
iterate=True, max_iterations=30, min_iterations=2
)
equations.append(g6)
eq = []
for fluid in self.fluids:
eq.append(
PressureForce(dest=fluid, sources=self.fluids)
)
if self.solids:
eq.append(
PressureForceBoundary(
dest=fluid, sources=self.solids, rho0=self.rho0
)
)
g7 = Group(equations=eq)
equations.append(g7)
return equations
[docs] def setup_properties(self, particles, clean=True):
"""Setup the particle arrays so they have the right set of properties
for this scheme.
Parameters
----------
particles : list
List of particle arrays.
clean : bool
If True, removes any unnecessary properties.
"""
from pysph.base.utils import get_particle_array_iisph
dummy = get_particle_array_iisph()
props = set(dummy.properties.keys())
for pa in particles:
self._ensure_properties(pa, props, clean)
for c, v in dummy.constants.items():
if c not in pa.constants:
pa.add_constant(c, v)
pa.set_output_arrays(dummy.output_property_arrays)
if self.has_ghosts:
particle_arrays = dict([(p.name, p) for p in particles])
for fluid in self.fluids:
pa = particle_arrays[fluid]
pa.add_property('orig_idx', type='int')