"""Implementation of the equations used for surface tension modelling, for
example in KHI simulations. The references are [SY11], [JM00], [A10], [XA06].
References
----------
.. [SY11] M. Shadloo, M. Yildiz, "Numerical modelling of Kelvin-Helmholtz
instability using smoothed particle hydrodynamics", IJNME, 2011, 87, pp
988--1006
.. [JM00] Joseph P. Morris "Simulating surface tension with smoothed particle
hydrodynamics", JCP, 2000, 33, pp 333--353
.. [A10] Adami et al. "A new surface-tension formulation for multi-phase SPH
using a reproducing divergence approximation", JCP 2010, 229, pp 5011--5021
.. [XA06] X.Y.Hu, N.A. Adams. "A multi-phase SPH method for macroscopic and
mesoscopic flows", JCP 2006, 213, pp 844-861 [XA06]
"""
from math import sqrt
from pysph.sph.equation import Group, Equation
from pysph.sph.gas_dynamics.basic import ScaleSmoothingLength
from pysph.sph.wc.transport_velocity import SummationDensity, \
MomentumEquationPressureGradient, StateEquation,\
MomentumEquationArtificialStress, MomentumEquationViscosity, \
SolidWallNoSlipBC
from pysph.sph.wc.linalg import gj_solve, augmented_matrix
from pysph.sph.basic_equations import IsothermalEOS
from pysph.sph.wc.basic import TaitEOS
[docs]class SurfaceForceAdami(Equation):
[docs] def initialize(self, d_au, d_av, d_idx):
d_au[d_idx] = 0.0
d_av[d_idx] = 0.0
[docs] def loop(self, d_au, d_av, d_aw, d_idx, d_m, DWIJ, d_pi00, d_pi01, d_pi02,
d_pi10, d_pi11, d_pi12, d_pi20, d_pi21, d_pi22, s_pi00, s_pi01,
s_pi02, s_pi10, s_pi11, s_pi12, s_pi20, s_pi21, s_pi22, d_V, s_V,
s_idx):
s2 = s_V[s_idx]*s_V[s_idx]
f00 = (d_pi00[d_idx]/(d_V[d_idx]*d_V[d_idx]) + s_pi00[s_idx]/s2)
f01 = (d_pi01[d_idx]/(d_V[d_idx]*d_V[d_idx]) + s_pi01[s_idx]/s2)
f02 = (d_pi02[d_idx]/(d_V[d_idx]*d_V[d_idx]) + s_pi02[s_idx]/s2)
f10 = (d_pi10[d_idx]/(d_V[d_idx]*d_V[d_idx]) + s_pi10[s_idx]/s2)
f11 = (d_pi11[d_idx]/(d_V[d_idx]*d_V[d_idx]) + s_pi11[s_idx]/s2)
f12 = (d_pi12[d_idx]/(d_V[d_idx]*d_V[d_idx]) + s_pi12[s_idx]/s2)
f20 = (d_pi20[d_idx]/(d_V[d_idx]*d_V[d_idx]) + s_pi20[s_idx]/s2)
f21 = (d_pi21[d_idx]/(d_V[d_idx]*d_V[d_idx]) + s_pi21[s_idx]/s2)
f22 = (d_pi22[d_idx]/(d_V[d_idx]*d_V[d_idx]) + s_pi22[s_idx]/s2)
d_au[d_idx] += (DWIJ[0]*f00 + DWIJ[1]*f10 + DWIJ[2]*f20)/d_m[d_idx]
d_av[d_idx] += (DWIJ[0]*f01 + DWIJ[1]*f11 + DWIJ[2]*f21)/d_m[d_idx]
d_aw[d_idx] += (DWIJ[0]*f02 + DWIJ[1]*f12 + DWIJ[2]*f22)/d_m[d_idx]
[docs]class ConstructStressMatrix(Equation):
def __init__(self, dest, sources, sigma, d=2):
self.sigma = sigma
self.d = d
super(ConstructStressMatrix, self).__init__(dest, sources)
[docs] def initialize(self, d_pi00, d_pi01, d_pi02, d_pi10, d_pi11, d_pi12,
d_pi20, d_pi21, d_pi22, d_cx, d_cy, d_cz, d_idx, d_N):
mod_gradc2 = d_cx[d_idx]*d_cx[d_idx] + d_cy[d_idx]*d_cy[d_idx] + \
d_cz[d_idx]*d_cz[d_idx]
mod_gradc = sqrt(mod_gradc2)
d_N[d_idx] = 0.0
if mod_gradc > 1e-14:
factor = self.sigma/mod_gradc
d_pi00[d_idx] = (-d_cx[d_idx]*d_cx[d_idx] +
(mod_gradc2)/self.d)*factor
d_pi01[d_idx] = -factor*d_cx[d_idx]*d_cy[d_idx]
d_pi02[d_idx] = -factor*d_cx[d_idx]*d_cz[d_idx]
d_pi10[d_idx] = -factor*d_cx[d_idx]*d_cy[d_idx]
d_pi11[d_idx] = (-d_cy[d_idx]*d_cy[d_idx] +
(mod_gradc2)/self.d)*factor
d_pi12[d_idx] = -factor*d_cy[d_idx]*d_cz[d_idx]
d_pi20[d_idx] = -factor*d_cx[d_idx]*d_cz[d_idx]
d_pi21[d_idx] = -factor*d_cy[d_idx]*d_cz[d_idx]
d_pi22[d_idx] = (-d_cz[d_idx]*d_cz[d_idx] +
(mod_gradc2)/self.d)*factor
d_N[d_idx] = 1.0
else:
d_pi00[d_idx] = 0.0
d_pi01[d_idx] = 0.0
d_pi02[d_idx] = 0.0
d_pi10[d_idx] = 0.0
d_pi11[d_idx] = 0.0
d_pi12[d_idx] = 0.0
d_pi20[d_idx] = 0.0
d_pi21[d_idx] = 0.0
d_pi22[d_idx] = 0.0
[docs]class ColorGradientAdami(Equation):
[docs] def initialize(self, d_idx, d_cx, d_cy, d_cz):
d_cx[d_idx] = 0.0
d_cy[d_idx] = 0.0
d_cz[d_idx] = 0.0
[docs] def loop(self, d_idx, d_cx, d_cy, d_cz, d_V, s_V, d_color, s_color, DWIJ,
s_idx):
c_i = d_color[d_idx]/(d_V[d_idx]*d_V[d_idx])
c_j = s_color[s_idx]/(s_V[s_idx]*s_V[s_idx])
factor = d_V[d_idx]*(c_i + c_j)
d_cx[d_idx] += factor*DWIJ[0]
d_cy[d_idx] += factor*DWIJ[1]
d_cz[d_idx] += factor*DWIJ[2]
[docs]class MomentumEquationViscosityAdami(Equation):
[docs] def initialize(self, d_au, d_av, d_aw, d_idx):
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_V, d_au, d_av, d_aw, s_V, d_p, s_p, DWIJ, s_idx,
d_m, R2IJ, XIJ, EPS, VIJ, d_nu, s_nu):
factor = 2.0*d_nu[d_idx]*s_nu[s_idx]/(d_nu[d_idx] + s_nu[s_idx])
V_i = 1/(d_V[d_idx]*d_V[d_idx])
V_j = 1/(s_V[s_idx]*s_V[s_idx])
dwijdotrij = (DWIJ[0]*XIJ[0] + DWIJ[1]*XIJ[1] + DWIJ[2]*XIJ[2])
dwijdotrij /= (R2IJ + EPS)
factor = factor*(V_i + V_j)*dwijdotrij/d_m[d_idx]
d_au[d_idx] += factor*VIJ[0]
d_av[d_idx] += factor*VIJ[1]
d_aw[d_idx] += factor*VIJ[2]
[docs]class MomentumEquationPressureGradientHuAdams(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(MomentumEquationPressureGradientHuAdams,
self).__init__(dest, sources)
[docs] def initialize(self, d_au, d_av, d_aw, d_idx):
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_V, d_au, d_av, d_aw, s_V, d_p, s_p, DWIJ, s_idx,
d_m):
p_i = d_p[d_idx]/(d_V[d_idx]*d_V[d_idx])
p_j = s_p[s_idx]/(s_V[s_idx]*s_V[s_idx])
d_au[d_idx] += -(p_i+p_j)*DWIJ[0]/d_m[d_idx]
d_av[d_idx] += -(p_i+p_j)*DWIJ[1]/d_m[d_idx]
d_aw[d_idx] += -(p_i+p_j)*DWIJ[2]/d_m[d_idx]
[docs] def post_loop(self, d_idx, d_au, d_av, d_aw):
d_au[d_idx] += self.gx
d_av[d_idx] += self.gy
d_aw[d_idx] += self.gz
[docs]class MomentumEquationPressureGradientAdami(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(MomentumEquationPressureGradientAdami,
self).__init__(dest, sources)
[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, s_idx, d_m, d_rho, s_rho,
d_au, d_av, d_aw, d_p, s_p,
d_V, s_V, DWIJ):
# averaged pressure Eq. (7)
rhoi = d_rho[d_idx]
rhoj = s_rho[s_idx]
p_i = d_p[d_idx]
p_j = s_p[s_idx]
pij = rhoj * p_i + rhoi * p_j
pij /= (rhoj + rhoi)
# particle volumes; d_V is inverse volume
Vi = 1./d_V[d_idx]
Vj = 1./s_V[s_idx]
Vi2 = Vi * Vi
Vj2 = Vj * Vj
# inverse mass of destination particle
mi1 = 1.0/d_m[d_idx]
# accelerations 1st term in Eq. (8)
tmp = -pij * mi1 * (Vi2 + Vj2)
d_au[d_idx] += tmp * DWIJ[0]
d_av[d_idx] += tmp * DWIJ[1]
d_aw[d_idx] += tmp * DWIJ[2]
[docs] def post_loop(self, d_idx, d_au, d_av, d_aw):
d_au[d_idx] += self.gx
d_av[d_idx] += self.gy
d_aw[d_idx] += self.gz
[docs]class MomentumEquationViscosityMorris(Equation):
def __init__(self, dest, sources, eta=0.01):
self.eta = eta*eta
super(MomentumEquationViscosityMorris, self).__init__(dest, sources)
[docs] def loop(self, d_idx, s_idx, d_au, d_av, d_aw, s_m, d_nu, s_nu, d_rho,
s_rho, DWIJ, R2IJ, VIJ, HIJ, XIJ):
r2 = R2IJ + self.eta*HIJ*HIJ
dw = (DWIJ[0]*XIJ[0] + DWIJ[1]*XIJ[1] + DWIJ[2]*XIJ[2])/(r2)
mult = s_m[s_idx]*(d_nu[d_idx] + s_nu[s_idx]) / \
(d_rho[d_idx]*s_rho[s_idx])
d_au[d_idx] += dw*mult*VIJ[0]
d_av[d_idx] += dw*mult*VIJ[1]
d_aw[d_idx] += dw*mult*VIJ[2]
[docs]class MomentumEquationPressureGradientMorris(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, s_idx, d_au, d_av, d_aw, s_m, d_p, s_p, DWIJ, d_rho,
s_rho):
factor = -s_m[s_idx]*(d_p[d_idx] + s_p[s_idx]) / \
(d_rho[d_idx]*s_rho[s_idx])
d_au[d_idx] += factor*DWIJ[0]
d_av[d_idx] += factor*DWIJ[1]
d_aw[d_idx] += factor*DWIJ[2]
[docs]class InterfaceCurvatureFromDensity(Equation):
def __init__(self, dest, sources, with_morris_correction=True):
self.with_morris_correction = with_morris_correction
super(InterfaceCurvatureFromDensity, self).__init__(dest, sources)
[docs] def initialize(self, d_idx, d_kappa, d_wij_sum):
d_kappa[d_idx] = 0.0
d_wij_sum[d_idx] = 0.0
[docs] def loop(self, d_idx, s_idx, d_kappa, d_nx, d_ny, d_nz, s_nx, s_ny, s_nz,
d_V, s_V, d_N, s_N, d_wij_sum, s_rho, s_m, WIJ, DWIJ):
nijdotdwij = (d_nx[d_idx] - s_nx[s_idx]) * DWIJ[0] + \
(d_ny[d_idx] - s_ny[s_idx]) * DWIJ[1] + \
(d_nz[d_idx] - s_nz[s_idx]) * DWIJ[2]
tmp = 1.0
if self.with_morris_correction:
tmp = min(d_N[d_idx], s_N[s_idx])
d_wij_sum[d_idx] += tmp * s_m[s_idx]/s_rho[s_idx] * WIJ
d_kappa[d_idx] += tmp*nijdotdwij*s_m[s_idx]/s_rho[s_idx]
[docs] def post_loop(self, d_idx, d_wij_sum, d_nx, d_kappa):
if self.with_morris_correction:
if d_wij_sum[d_idx] > 1e-12:
d_kappa[d_idx] /= d_wij_sum[d_idx]
[docs]class SolidWallPressureBCnoDensity(Equation):
[docs] def initialize(self, d_idx, d_p, d_wij):
d_p[d_idx] = 0.0
d_wij[d_idx] = 0.0
[docs] def loop(self, d_idx, s_idx, d_p, s_p, d_wij, s_rho,
d_au, d_av, d_aw, WIJ, XIJ):
d_p[d_idx] += s_p[s_idx]*WIJ
d_wij[d_idx] += WIJ
[docs] def post_loop(self, d_idx, d_wij, d_p, d_rho):
if d_wij[d_idx] > 1e-14:
d_p[d_idx] /= d_wij[d_idx]
[docs]class SummationDensitySourceMass(Equation):
[docs] def initialize(self, d_idx, d_V, d_rho):
d_rho[d_idx] = 0.0
[docs] def loop(self, d_idx, d_V, d_rho, d_m, WIJ, s_m, s_idx):
d_rho[d_idx] += d_m[d_idx]*WIJ
[docs] def post_loop(self, d_idx, d_V, d_rho, d_m):
d_V[d_idx] = d_rho[d_idx]/d_m[d_idx]
[docs]class SmoothedColor(Equation):
r"""Smoothed color function. Eq. (17) in [JM00]
.. math::
c_a = \sum_b \frac{m_b}{\rho_b} c_b^i \nabla_a W_{ab}\,,
where, :math:`c_b^i` is the color index associated with a
particle.
"""
[docs] def initialize(self, d_idx, d_scolor):
d_scolor[d_idx] = 0.0
[docs] def loop(self, d_idx, s_idx, d_rho, s_rho, s_m,
s_color, d_scolor, WIJ):
# Smoothed color Eq. (17) in [JM00]
d_scolor[d_idx] += s_m[s_idx]/s_rho[s_idx] * s_color[s_idx] * WIJ
[docs]class ColorGradientUsingNumberDensity(Equation):
r"""Gradient of the color function using Eq. (13) of [SY11]:
.. math::
\nabla C_a = \sum_b \frac{2 C_b - C_a}{\psi_a + \psi_a}
\nabla_{a} W_{ab}
Using the gradient of the color function, the normal and
discretized dirac delta is calculated in the post
loop.
Singularities are avoided as per the recommendation by [JM00] (see eqs
20 & 21) using the parameter :math:`\epsilon`
"""
def __init__(self, dest, sources, epsilon=1e-6):
self.epsilon2 = epsilon*epsilon
super(ColorGradientUsingNumberDensity, self).__init__(dest, sources)
[docs] def initialize(self, d_idx, d_cx, d_cy, d_cz, d_nx, d_ny, d_nz,
d_ddelta, d_N):
# color gradient
d_cx[d_idx] = 0.0
d_cy[d_idx] = 0.0
d_cz[d_idx] = 0.0
# interface normals
d_nx[d_idx] = 0.0
d_ny[d_idx] = 0.0
d_nz[d_idx] = 0.0
# discretized dirac delta
d_ddelta[d_idx] = 0.0
# reliability indicator for normals
d_N[d_idx] = 0.0
[docs] def loop(self, d_idx, s_idx, d_scolor, s_scolor, d_cx, d_cy, d_cz,
d_V, s_V, DWIJ):
# average particle volume
psiab1 = 2.0/(d_V[d_idx] + s_V[s_idx])
# difference in color divided by psiab. Eq. (13) in [SY11]
Cba = (s_scolor[s_idx] - d_scolor[d_idx]) * psiab1
# color gradient
d_cx[d_idx] += Cba * DWIJ[0]
d_cy[d_idx] += Cba * DWIJ[1]
d_cz[d_idx] += Cba * DWIJ[2]
[docs] def post_loop(self, d_idx, d_cx, d_cy, d_cz,
d_nx, d_ny, d_nz, d_N, d_ddelta):
# absolute value of the color gradient
mod_gradc2 = d_cx[d_idx]*d_cx[d_idx] + \
d_cy[d_idx]*d_cy[d_idx] + \
d_cz[d_idx]*d_cz[d_idx]
# avoid sqrt computations on non-interface particles
# (particles for which the color gradient is zero) Eq. (19,
# 20) in [JM00]
if mod_gradc2 > self.epsilon2:
# this normal is reliable in the sense of [JM00]
d_N[d_idx] = 1.0
# compute the normals
mod_gradc = 1./sqrt(mod_gradc2)
d_nx[d_idx] = d_cx[d_idx] * mod_gradc
d_ny[d_idx] = d_cy[d_idx] * mod_gradc
d_nz[d_idx] = d_cz[d_idx] * mod_gradc
# discretized Dirac Delta function
d_ddelta[d_idx] = 1./mod_gradc
[docs]class MorrisColorGradient(Equation):
r"""Gradient of the color function using Eq. (17) of [JM00]:
.. math::
\nabla c_a = \sum_b \frac{m_b}{\rho_b}(c_b - c_a) \nabla_{a}
W_{ab}\,,
where a smoothed representation of the color is used in the
equation. Using the gradient of the color function, the normal and
discretized dirac delta is calculated in the post loop.
Singularities are avoided as per the recommendation by [JM00] (see eqs
20 & 21) using the parameter :math:`\epsilon`
"""
def __init__(self, dest, sources, epsilon=1e-6):
self.epsilon2 = epsilon*epsilon
super(MorrisColorGradient, self).__init__(dest, sources)
[docs] def initialize(self, d_idx, d_cx, d_cy, d_cz, d_nx, d_ny, d_nz,
d_ddelta, d_N):
# color gradient
d_cx[d_idx] = 0.0
d_cy[d_idx] = 0.0
d_cz[d_idx] = 0.0
# interface normals
d_nx[d_idx] = 0.0
d_ny[d_idx] = 0.0
d_nz[d_idx] = 0.0
# reliability indicator for normals and dirac delta
d_N[d_idx] = 0.0
d_ddelta[d_idx] = 0.0
[docs] def loop(self, d_idx, s_idx, d_scolor, s_scolor, d_cx, d_cy, d_cz,
s_m, s_rho, DWIJ):
# Eq. (17) in [JM00]
Cba = (s_scolor[s_idx] - d_scolor[d_idx]) * s_m[s_idx]/s_rho[s_idx]
# color gradient
d_cx[d_idx] += Cba * DWIJ[0]
d_cy[d_idx] += Cba * DWIJ[1]
d_cz[d_idx] += Cba * DWIJ[2]
[docs] def post_loop(self, d_idx, d_cx, d_cy, d_cz,
d_nx, d_ny, d_nz, d_N, d_ddelta):
# absolute value of the color gradient
mod_gradc2 = d_cx[d_idx]*d_cx[d_idx] + \
d_cy[d_idx]*d_cy[d_idx] + \
d_cz[d_idx]*d_cz[d_idx]
# avoid sqrt computations on non-interface particles
# (particles for which the color gradient is zero) Eq. (19,
# 20) in [JM00]
if mod_gradc2 > self.epsilon2:
# this normal is reliable in the sense of [JM00]
d_N[d_idx] = 1.0
# compute the normals
mod_gradc = 1./sqrt(mod_gradc2)
d_nx[d_idx] = d_cx[d_idx] * mod_gradc
d_ny[d_idx] = d_cy[d_idx] * mod_gradc
d_nz[d_idx] = d_cz[d_idx] * mod_gradc
# discretized Dirac Delta function
d_ddelta[d_idx] = 1./mod_gradc
[docs]class SY11ColorGradient(Equation):
r"""Gradient of the color function using Eq. (13) of [SY11]:
.. math::
\nabla C_a = \sum_b \frac{2 C_b - C_a}{\psi_a + \psi_a}
\nabla_{a} W_{ab}
Using the gradient of the color function, the normal and
discretized dirac delta is calculated in the post
loop.
"""
def __init__(self, dest, sources, epsilon=1e-6):
self.epsilon2 = epsilon*epsilon
super(SY11ColorGradient, self).__init__(dest, sources)
[docs] def initialize(self, d_idx, d_cx, d_cy, d_cz, d_nx, d_ny, d_nz,
d_ddelta, d_N):
# color gradient
d_cx[d_idx] = 0.0
d_cy[d_idx] = 0.0
d_cz[d_idx] = 0.0
# interface normals
d_nx[d_idx] = 0.0
d_ny[d_idx] = 0.0
d_nz[d_idx] = 0.0
# discretized dirac delta
d_ddelta[d_idx] = 0.0
# reliability indicator for normals
d_N[d_idx] = 0.0
[docs] def loop(self, d_idx, s_idx, d_color, s_color, d_cx, d_cy, d_cz,
d_V, s_V, DWIJ):
# average particle volume
psiab1 = 2.0/(d_V[d_idx] + s_V[s_idx])
# difference in color divided by psiab. Eq. (13) in [SY11]
Cba = (s_color[s_idx] - d_color[d_idx]) * psiab1
# color gradient
d_cx[d_idx] += Cba * DWIJ[0]
d_cy[d_idx] += Cba * DWIJ[1]
d_cz[d_idx] += Cba * DWIJ[2]
[docs] def post_loop(self, d_idx, d_cx, d_cy, d_cz,
d_nx, d_ny, d_nz, d_N, d_ddelta):
# absolute value of the color gradient
mod_gradc2 = d_cx[d_idx]*d_cx[d_idx] + \
d_cy[d_idx]*d_cy[d_idx] + \
d_cz[d_idx]*d_cz[d_idx]
# avoid sqrt computations on non-interface particles
if mod_gradc2 > self.epsilon2:
# this normal is reliable in the sense of [JM00]
d_N[d_idx] = 1.0
# compute the normals
mod_gradc = 1./sqrt(mod_gradc2)
d_nx[d_idx] = d_cx[d_idx] * mod_gradc
d_ny[d_idx] = d_cy[d_idx] * mod_gradc
d_nz[d_idx] = d_cz[d_idx] * mod_gradc
# discretized Dirac Delta function
d_ddelta[d_idx] = 1./mod_gradc
[docs]class SY11DiracDelta(Equation):
r"""Discretized dirac-delta for the SY11 formulation Eq. (14) in [SY11]
This is essentially the same as computing the color gradient, the
only difference being that this might be called with a reduced
smoothing length.
Note that the normals should be computed using the
SY11ColorGradient equation. This function will effectively
overwrite the color gradient.
"""
def __init__(self, dest, sources, epsilon=1e-6):
self.epsilon2 = epsilon*epsilon
super(SY11DiracDelta, self).__init__(dest, sources)
[docs] def initialize(self, d_idx, d_cx, d_cy, d_cz, d_ddelta):
# color gradient
d_cx[d_idx] = 0.0
d_cy[d_idx] = 0.0
d_cz[d_idx] = 0.0
# discretized dirac delta
d_ddelta[d_idx] = 0.0
[docs] def loop(self, d_idx, s_idx, d_color, s_color, d_cx, d_cy, d_cz,
d_V, s_V, DWIJ):
# average particle volume
psiab1 = 2.0/(d_V[d_idx] + s_V[s_idx])
# difference in color divided by psiab. Eq. (13) in [SY11]
Cba = (s_color[s_idx] - d_color[d_idx]) * psiab1
# color gradient
d_cx[d_idx] += Cba * DWIJ[0]
d_cy[d_idx] += Cba * DWIJ[1]
d_cz[d_idx] += Cba * DWIJ[2]
[docs] def post_loop(self, d_idx, d_cx, d_cy, d_cz,
d_nx, d_ny, d_nz, d_N, d_ddelta):
# absolute value of the color gradient
mod_gradc2 = d_cx[d_idx]*d_cx[d_idx] + \
d_cy[d_idx]*d_cy[d_idx] + \
d_cz[d_idx]*d_cz[d_idx]
# avoid sqrt computations on non-interface particles
if mod_gradc2 > self.epsilon2:
mod_gradc = sqrt(mod_gradc2)
# discretized Dirac Delta function
d_ddelta[d_idx] = mod_gradc
[docs]class InterfaceCurvatureFromNumberDensity(Equation):
r"""Interface curvature using number density. Eq. (15) in [SY11]:
.. math::
\kappa_a = \sum_b \frac{2.0}{\psi_a + \psi_b}
\left(\boldsymbol{n_a} - \boldsymbol{n_b}\right) \cdot
\nabla_a W_{ab}
"""
def __init__(self, dest, sources, with_morris_correction=True):
self.with_morris_correction = with_morris_correction
super(InterfaceCurvatureFromNumberDensity, self).__init__(dest,
sources)
[docs] def initialize(self, d_idx, d_kappa, d_wij_sum):
d_kappa[d_idx] = 0.0
d_wij_sum[d_idx] = 0.0
[docs] def loop(self, d_idx, s_idx, d_kappa, d_nx, d_ny, d_nz, s_nx, s_ny, s_nz,
d_V, s_V, d_N, s_N, d_wij_sum, s_rho, s_m, WIJ, DWIJ):
nijdotdwij = (d_nx[d_idx] - s_nx[s_idx]) * DWIJ[0] + \
(d_ny[d_idx] - s_ny[s_idx]) * DWIJ[1] + \
(d_nz[d_idx] - s_nz[s_idx]) * DWIJ[2]
# averaged particle number density
psiij1 = 2.0/(d_V[d_idx] + s_V[s_idx])
# local number density with reliable normals Eq. (24) in [JM00]
tmp = 1.0
if self.with_morris_correction:
tmp = min(d_N[d_idx], s_N[s_idx])
d_wij_sum[d_idx] += tmp * s_m[s_idx]/s_rho[s_idx] * WIJ
# Eq. (15) in [SY11] with correction Eq. (22) in [JM00]
d_kappa[d_idx] += tmp * psiij1 * nijdotdwij
[docs] def post_loop(self, d_idx, d_wij_sum, d_nx, d_kappa):
# correct the curvature estimate. Eq. (23) in [JM00]
if self.with_morris_correction:
if d_wij_sum[d_idx] > 1e-12:
d_kappa[d_idx] /= d_wij_sum[d_idx]
[docs]class ShadlooYildizSurfaceTensionForce(Equation):
r"""Acceleration due to surface tension force Eq. (7,9) in [SY11]:
.. math:
\frac{d\boldsymbol{v}_a} = \frac{1}{m_a} \sigma \kappa_a
\boldsymbol{n}_a \delta_a^s\,,
where, :math:`\delta^s` is the discretized dirac delta function,
:math:`\boldsymbol{n}` is the interface normal, :math:`\kappa` is
the discretized interface curvature and :math:`\sigma` is the
surface tension force constant.
"""
def __init__(self, dest, sources, sigma=0.1):
self.sigma = sigma
# base class initialization
super(ShadlooYildizSurfaceTensionForce, self).__init__(dest, sources)
[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_au, d_av, d_aw, d_kappa,
d_nx, d_ny, d_nz, d_m, d_rho, d_ddelta):
mi = 1./d_m[d_idx]
rhoi = 1./d_rho[d_idx]
# acceleration per uint mass term Eq. (7) in [SY11]
tmp = self.sigma * d_kappa[d_idx] * d_ddelta[d_idx] * rhoi
d_au[d_idx] += tmp * d_nx[d_idx]
d_av[d_idx] += tmp * d_ny[d_idx]
d_aw[d_idx] += tmp * d_nz[d_idx]
[docs]class CSFSurfaceTensionForce(Equation):
r"""Acceleration due to surface tension force Eq. (25) in [JM00]:
.. math:
\frac{d\boldsymbol{v}_a}{dt} = \frac{1}{rho_a} \sigma_b \kappa_a
\boldsymbol{n}_a
Note that as per Eq. (17) in [JM00], the un-normalized normal is
basically the gradient of the color function. The acceleration
term therefore depends on the gradient of the color field.
"""
def __init__(self, dest, sources, sigma=0.1):
self.sigma = sigma
# base class initialization
super(CSFSurfaceTensionForce, self).__init__(dest, sources)
[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_au, d_av, d_aw, d_kappa,
d_cx, d_cy, d_cz, d_rho):
rhoi = 1./d_rho[d_idx]
# acceleration per uint mass term Eq. (25) in [JM00]
tmp = self.sigma * d_kappa[d_idx] * rhoi
d_au[d_idx] += tmp * d_cx[d_idx]
d_av[d_idx] += tmp * d_cy[d_idx]
d_aw[d_idx] += tmp * d_cz[d_idx]
[docs]class AdamiReproducingDivergence(Equation):
r"""Reproducing divergence approximation Eq. (20) in [A10] to
compute the curvature
.. math::
\nabla \cdot \boldsymbol{\phi}_a = d\frac{\sum_b
\boldsymbol{\phi}_{ab}\cdot \nabla_a
W_{ab}V_b}{\sum_b\boldsymbol{x}_{ab}\cdot \nabla_a W_{ab} V_b}
"""
def __init__(self, dest, sources, dim):
self.dim = dim
super(AdamiReproducingDivergence, self).__init__(dest, sources)
[docs] def initialize(self, d_idx, d_kappa, d_wij_sum):
d_kappa[d_idx] = 0.0
d_wij_sum[d_idx] = 0.0
[docs] def loop(self, d_idx, s_idx, d_kappa, d_wij_sum,
d_nx, d_ny, d_nz, s_nx, s_ny, s_nz, d_V, s_V,
DWIJ, XIJ, RIJ, EPS, d_N, s_N, d_color, s_color):
# particle volumes
Vi = 1./d_V[d_idx]
Vj = 1./s_V[s_idx]
color_diff = abs(d_color[d_idx] - s_color[s_idx])
if color_diff == 1.0:
phi_ij = -1.0
else:
phi_ij = 1.0
# dot product in the numerator of Eq. (20)
nijdotdwij = (d_nx[d_idx] - phi_ij * s_nx[s_idx]) * DWIJ[0] + \
(d_ny[d_idx] - phi_ij * s_ny[s_idx]) * DWIJ[1] + \
(d_nz[d_idx] - phi_ij * s_nz[s_idx]) * DWIJ[2]
# dot product in the denominator of Eq. (20)
xijdotdwij = XIJ[0]*DWIJ[0] + XIJ[1]*DWIJ[1] + XIJ[2]*DWIJ[2]
tmp = min(d_N[d_idx], s_N[s_idx])
# accumulate the contributions
d_kappa[d_idx] += tmp * nijdotdwij * Vj
d_wij_sum[d_idx] += tmp * xijdotdwij * Vj
[docs] def post_loop(self, d_idx, d_kappa, d_wij_sum):
# normalize the curvature estimate
if abs(d_wij_sum[d_idx]) > 1e-12:
d_kappa[d_idx] /= d_wij_sum[d_idx]
d_kappa[d_idx] *= self.dim
[docs]class CSFSurfaceTensionForceAdami(Equation):
def __init__(self, dest, sources, sigma):
self.sigma = sigma
super(CSFSurfaceTensionForceAdami, self).__init__(dest, sources)
[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 post_loop(self, d_idx, d_au, d_av, d_aw, d_kappa, d_cx, d_cy, d_cz,
d_m, d_alpha, d_rho):
d_au[d_idx] += -self.sigma * d_kappa[d_idx] * d_cx[d_idx]/d_rho[d_idx]
d_av[d_idx] += -self.sigma * d_kappa[d_idx] * d_cy[d_idx]/d_rho[d_idx]
d_aw[d_idx] += -self.sigma * d_kappa[d_idx] * d_cz[d_idx]/d_rho[d_idx]
[docs]class ShadlooViscosity(Equation):
def __init__(self, dest, sources, alpha):
self.alpha = alpha
super(ShadlooViscosity, self).__init__(dest, sources)
[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_au, d_av, d_aw, d_h, s_idx, s_h, d_cs, s_cs, d_rho,
s_rho, VIJ, XIJ, d_V, s_V, R2IJ, EPS, DWIJ):
mu1 = 0.125*self.alpha*d_h[d_idx]*d_cs[d_idx]*d_rho[d_idx]
mu2 = 0.125*self.alpha*s_h[s_idx]*s_cs[s_idx]*s_rho[s_idx]
mu12 = 2.0*mu1*mu2/(mu1 + mu2)
vijdotxij = VIJ[0]*XIJ[0] + VIJ[1]*XIJ[1] + VIJ[2]*XIJ[2]
denominator = d_V[d_idx]*s_V[s_idx]*(R2IJ + EPS)
piij = 8.0*mu12*vijdotxij/denominator
d_au[d_idx] += -piij*DWIJ[0]
d_av[d_idx] += -piij*DWIJ[1]
d_aw[d_idx] += -piij*DWIJ[2]
[docs]class AdamiColorGradient(Equation):
r"""Gradient of color Eq. (14) in [A10]
.. math::
\nabla c_a = \frac{1}{V_a}\sum_b \left[V_a^2 + V_b^2
\right]\tilde{c}_{ab}\nabla_a W_{ab}\,,
where, the average :math:`\tilde{c}_{ab}` is defined as
.. math::
\tilde{c}_{ab} = \frac{\rho_b}{\rho_a + \rho_b}c_a +
\frac{\rho_a}{\rho_a + \rho_b}c_b
"""
[docs] def initialize(self, d_idx, d_cx, d_cy, d_cz, d_nx, d_ny, d_nz, d_ddelta,
d_N):
d_cx[d_idx] = 0.0
d_cy[d_idx] = 0.0
d_cz[d_idx] = 0.0
d_nx[d_idx] = 0.0
d_ny[d_idx] = 0.0
d_nz[d_idx] = 0.0
# reliability indicator for normals
d_N[d_idx] = 0.0
# Discretized dirac-delta
d_ddelta[d_idx] = 0.0
[docs] def loop(self, d_idx, s_idx, d_V, s_V, d_rho, s_rho,
d_cx, d_cy, d_cz, d_color, s_color, DWIJ):
# particle volumes
Vi = 1./d_V[d_idx]
Vj = 1./s_V[s_idx]
Vi2 = Vi * Vi
Vj2 = Vj * Vj
# averaged particle color
rhoi = d_rho[d_idx]
rhoj = s_rho[s_idx]
rhoij1 = 1./(rhoi + rhoj)
color_diff = abs((d_color[d_idx] - s_color[s_idx]))
# Eq. (15) in [A10]
if color_diff == 0.0:
cij = 0.0
else:
cij = rhoj * rhoij1 * 0.0 + rhoi * rhoij1 * 1.0
# comute the gradient
tmp = cij * (Vi2 + Vj2)/Vi
d_cx[d_idx] += tmp * DWIJ[0]
d_cy[d_idx] += tmp * DWIJ[1]
d_cz[d_idx] += tmp * DWIJ[2]
[docs] def post_loop(self, d_idx, d_cx, d_cy, d_cz, d_h,
d_nx, d_ny, d_nz, d_ddelta, d_N):
# absolute value of the color gradient
mod_gradc2 = d_cx[d_idx]*d_cx[d_idx] + \
d_cy[d_idx]*d_cy[d_idx] + \
d_cz[d_idx]*d_cz[d_idx]
# avoid sqrt computations on non-interface particles
h2 = d_h[d_idx]*d_h[d_idx]
if mod_gradc2 > 0.0:
# this normal is reliable in the sense of [JM00]
d_N[d_idx] = 1.0
# compute the normals
one_mod_gradc = 1./sqrt(mod_gradc2)
d_nx[d_idx] = d_cx[d_idx] * one_mod_gradc
d_ny[d_idx] = d_cy[d_idx] * one_mod_gradc
d_nz[d_idx] = d_cz[d_idx] * one_mod_gradc
# discretized dirac delta
d_ddelta[d_idx] = 1./one_mod_gradc
[docs]def get_surface_tension_equations(fluids, solids, scheme, rho0, p0, c0, b,
factor1, factor2, nu, sigma, d, epsilon,
gamma, real=False):
"""This function returns the required equations for the multiphase formulation
taking inputs of the fluid particles array, solid particles array, the
scheme to be used and other physical parameters
Parameters
-----------
fluids: list
List of names of fluid particle arrays
solids: list
List of names of solid particle arrays
scheme: string
The scheme with which the equations are to be setup.
Supported Schemes:
1. TVF scheme with Morris' surface tension.
String to be used: "tvf"
2. Adami's surface tension implementation which doesn't involve
calculation of curvature. String to be used: "adami_stress"
3. Adami's surface tension implementation which involves
calculation of curvature. String to be used: "adami"
4. Shadloo Yildiz surface tension formulation.
String to be used: "shadloo"
5. Morris' surface tension formulation. This is the default scheme
which will be used if none of the above strings are input as
scheme.
rho0 : float
The reference density of the medium (Currently multiple reference
densities for different particles is not supported)
p0 : float
The background pressure of the medium(Currently multiple background
pressures for different particles is not supported)
c0 : float
The speed of sound of the medium(Currently multiple speeds of sounds
for different particles is not supported)
b : float
The b parameter of the generalized Tait Equation of State. Refer to
the Tait Equation's documentation for reference
factor1 : float
The factor for scaling of smoothing length for calculation of
interface curvature number for shadloo's scheme
factor2 : float
The factor for scaling back of smoothing length for calculation of
forces after calculating the interface curvature number in shadloo's
scheme
nu : float
The kinematic viscosity of the medium
sigma : float
The surface tension of the system
d : int
The number of dimensions of the problem in the cartesian space
epsilon: float
Put this option false if the equations are supposed to be evaluated
for the ghost particles, else keep it True
"""
if scheme == 'tvf':
result = []
equations = []
for i in fluids+solids:
equations.append(SummationDensity(dest=i, sources=fluids+solids))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(StateEquation(dest=i, sources=None, rho0=rho0,
p0=p0))
equations.append(SmoothedColor(dest=i, sources=fluids+solids))
for i in solids:
equations.append(SolidWallPressureBCnoDensity(dest=i,
sources=fluids))
equations.append(SmoothedColor(dest=i, sources=fluids+solids))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(MorrisColorGradient(dest=i, sources=fluids+solids,
epsilon=epsilon))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(InterfaceCurvatureFromNumberDensity(
dest=i, sources=fluids+solids, with_morris_correction=True))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(MomentumEquationPressureGradient(dest=i,
sources=fluids+solids, pb=p0))
equations.append(MomentumEquationViscosity(dest=i, sources=fluids,
nu=nu))
equations.append(CSFSurfaceTensionForce(dest=i, sources=None,
sigma=sigma))
equations.append(MomentumEquationArtificialStress(dest=i,
sources=fluids))
if len(solids) != 0:
equations.append(SolidWallNoSlipBC(dest=i, sources=solids,
nu=nu))
result.append(Group(equations))
elif scheme == 'adami_stress':
result = []
equations = []
for i in fluids+solids:
equations.append(SummationDensitySourceMass(dest=i,
sources=fluids+solids))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(TaitEOS(dest=i, sources=None, c0=c0,
gamma=gamma, p0=p0, rho0=rho0))
for i in solids:
equations.append(SolidWallPressureBCnoDensity(dest=i,
sources=fluids))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(ColorGradientAdami(dest=i, sources=fluids+solids))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(ConstructStressMatrix(dest=i, sources=None,
sigma=sigma, d=d))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(MomentumEquationPressureGradientHuAdams(dest=i,
sources=fluids+solids))
equations.append(MomentumEquationViscosityAdami(dest=i,
sources=fluids))
equations.append(SurfaceForceAdami(dest=i, sources=fluids+solids))
if len(solids) != 0:
equations.append(SolidWallNoSlipBC(dest=i, sources=solids,
nu=nu))
result.append(Group(equations))
elif scheme == 'adami':
result = []
equations = []
for i in fluids+solids:
equations.append(SummationDensitySourceMass(dest=i,
sources=fluids+solids))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(TaitEOS(dest=i, sources=None, c0=c0,
gamma=gamma, p0=p0, rho0=rho0))
for i in solids:
equations.append(SolidWallPressureBCnoDensity(dest=i,
sources=fluids))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(AdamiColorGradient(dest=i, sources=fluids+solids))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(AdamiReproducingDivergence(dest=i,
sources=fluids+solids,
dim=d))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(MomentumEquationPressureGradientAdami(
dest=i, sources=fluids+solids))
equations.append(MomentumEquationViscosityAdami(dest=i,
sources=fluids))
equations.append(CSFSurfaceTensionForceAdami(dest=i,
sources=None,
sigma=sigma))
if len(solids) != 0:
equations.append(SolidWallNoSlipBC(dest=i, sources=solids,
nu=nu))
result.append(Group(equations))
elif scheme == 'shadloo':
result = []
equations = []
for i in fluids+solids:
equations.append(SummationDensitySourceMass(dest=i,
sources=fluids+solids))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(IsothermalEOS(dest=i, sources=None,
p0=p0, c0=c0, rho0=rho0))
equations.append(SY11ColorGradient(dest=i, sources=fluids+solids))
for i in solids:
equations.append(SolidWallPressureBCnoDensity(dest=i,
sources=fluids))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(ScaleSmoothingLength(dest=i, sources=None,
factor=factor1))
result.append(Group(equations, real=real, update_nnps=True))
equations = []
for i in fluids:
equations.append(SY11DiracDelta(dest=i, sources=fluids+solids))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(InterfaceCurvatureFromNumberDensity(
dest=i, sources=fluids+solids, with_morris_correction=True))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(ScaleSmoothingLength(dest=i, sources=None,
factor=factor2))
result.append(Group(equations, real=real, update_nnps=True))
equations = []
for i in fluids:
equations.append(MomentumEquationPressureGradient(
dest=i, sources=fluids+solids, pb=0.0))
equations.append(MomentumEquationViscosity(dest=i, sources=fluids,
nu=nu))
equations.append(ShadlooYildizSurfaceTensionForce(dest=i,
sources=None,
sigma=sigma))
if len(solids) != 0:
equations.append(SolidWallNoSlipBC(dest=i, sources=solids,
nu=nu))
result.append(Group(equations))
else:
result = []
equations = []
for i in fluids+solids:
equations.append(SummationDensitySourceMass(dest=i,
sources=fluids+solids))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(TaitEOS(dest=i, sources=None, rho0=rho0, c0=c0,
gamma=gamma, p0=p0))
equations.append(SmoothedColor(dest=i, sources=fluids+solids))
for i in solids:
equations.append(SolidWallPressureBCnoDensity(dest=i,
sources=fluids))
equations.append(SmoothedColor(dest=i, sources=fluids+solids))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(MorrisColorGradient(dest=i, sources=fluids+solids,
epsilon=epsilon))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(InterfaceCurvatureFromDensity(
dest=i, sources=fluids+solids, with_morris_correction=True))
result.append(Group(equations, real=real))
equations = []
for i in fluids:
equations.append(MomentumEquationPressureGradientMorris(dest=i,
sources=fluids+solids))
equations.append(MomentumEquationViscosityMorris(dest=i,
sources=fluids))
equations.append(CSFSurfaceTensionForce(dest=i, sources=None,
sigma=sigma))
if len(solids) != 0:
equations.append(SolidWallNoSlipBC(dest=i, sources=solids,
nu=nu))
result.append(Group(equations))
return result