Source code for pysph.sph.surface_tension

"""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