"""
Transport Velocity Formulation
##############################
References
----------
.. [Adami2012] S. Adami et. al "A generalized wall boundary condition for
smoothed particle hydrodynamics", Journal of Computational Physics
(2012), pp. 7057--7075.
.. [Adami2013] S. Adami et. al "A transport-velocity formulation for
smoothed particle hydrodynamics", Journal of Computational Physics
(2013), pp. 292--307.
"""
from pysph.sph.equation import Equation
from math import sin, pi
# constants
M_PI = pi
[docs]class SummationDensity(Equation):
r"""**Summation density with volume summation**
In addition to the standard summation density, the number density
for the particle is also computed. The number density is important
for multi-phase flows to define a local particle volume
independent of the material density.
.. math::
\rho_a = \sum_b m_b W_{ab}\\
\mathcal{V}_a = \frac{1}{\sum_b W_{ab}}
Notes
-----
Note that in the pysph implementation, V is the inverse volume of a
particle, i.e. the equation computes V as follows:
.. math::
\mathcal{V}_a = \sum_b W_{ab}
For this equation, the destination particle array must define the
variable `V` for particle volume.
"""
[docs] def initialize(self, d_idx, d_V, d_rho):
d_V[d_idx] = 0.0
d_rho[d_idx] = 0.0
[docs] def loop(self, d_idx, d_V, d_rho, d_m, WIJ):
d_V[d_idx] += WIJ
d_rho[d_idx] += d_m[d_idx]*WIJ
[docs]class VolumeSummation(Equation):
r"""**Number density for volume computation**
See `SummationDensity`
Note that the quantity `V` is really :math:`\sigma` of the original paper,
i.e. inverse of the particle volume.
"""
[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 VolumeFromMassDensity(Equation):
"""**Set the inverse volume using mass density**"""
[docs] def loop(self, d_idx, d_V, d_rho, d_m):
d_V[d_idx] = d_rho[d_idx]/d_m[d_idx]
[docs]class SetWallVelocity(Equation):
r"""**Extrapolating the fluid velocity on to the wall**
Eq. (22) in [Adami2012]:
.. math::
\tilde{\boldsymbol{v}}_a = \frac{\sum_b\boldsymbol{v}_b W_{ab}}
{\sum_b W_{ab}}
Notes
-----
The destination particle array for this equation should define the
*filtered* velocity variables :math:`uf, vf, wf`.
"""
[docs] def initialize(self, d_idx, d_uf, d_vf, d_wf, d_wij):
d_uf[d_idx] = 0.0
d_vf[d_idx] = 0.0
d_wf[d_idx] = 0.0
d_wij[d_idx] = 0.0
[docs] def loop(self, d_idx, s_idx, d_uf, d_vf, d_wf,
s_u, s_v, s_w, d_wij, WIJ):
# normalisation factor is different from 'V' as the particles
# near the boundary do not have full kernel support
d_wij[d_idx] += WIJ
# sum in Eq. (22)
# this will be normalized in post loop
d_uf[d_idx] += s_u[s_idx] * WIJ
d_vf[d_idx] += s_v[s_idx] * WIJ
d_wf[d_idx] += s_w[s_idx] * WIJ
[docs] def post_loop(self, d_uf, d_vf, d_wf, d_wij, d_idx,
d_ug, d_vg, d_wg, d_u, d_v, d_w):
# calculation is done only for the relevant boundary particles.
# d_wij (and d_uf) is 0 for particles sufficiently away from the
# solid-fluid interface
if d_wij[d_idx] > 1e-12:
d_uf[d_idx] /= d_wij[d_idx]
d_vf[d_idx] /= d_wij[d_idx]
d_wf[d_idx] /= d_wij[d_idx]
# Dummy velocities at the ghost points using Eq. (23),
# d_u, d_v, d_w are the prescribed wall velocities.
d_ug[d_idx] = 2*d_u[d_idx] - d_uf[d_idx]
d_vg[d_idx] = 2*d_v[d_idx] - d_vf[d_idx]
d_wg[d_idx] = 2*d_w[d_idx] - d_wf[d_idx]
[docs]class ContinuityEquation(Equation):
r"""**Conservation of mass equation**
Eq (6) in [Adami2012]:
.. math::
\frac{d\rho_a}{dt} = \rho_a \sum_b \frac{m_b}{\rho_b}
\boldsymbol{v}_{ab} \cdot \nabla_a W_{ab}
"""
[docs] def initialize(self, d_idx, d_arho):
d_arho[d_idx] = 0.0
[docs] def loop(self, d_idx, s_idx, d_arho, s_m, s_rho, d_rho, VIJ, DWIJ):
vijdotdwij = VIJ[0] * DWIJ[0] + VIJ[1] * DWIJ[1] + VIJ[2] * DWIJ[2]
d_arho[d_idx] += d_rho[d_idx] * vijdotdwij * s_m[s_idx] / s_rho[s_idx]
[docs]class ContinuitySolid(Equation):
"""Continuity equation for the solid's ghost particles.
The key difference is that we use the ghost velocity ug, and not the
particle velocity u.
"""
[docs] def loop(self, d_idx, s_idx, d_rho, d_u, d_v, d_w, d_arho,
s_m, s_rho, s_ug, s_vg, s_wg, DWIJ):
Vj = s_m[s_idx] / s_rho[s_idx]
rhoi = d_rho[d_idx]
uij = d_u[d_idx] - s_ug[s_idx]
vij = d_v[d_idx] - s_vg[s_idx]
wij = d_w[d_idx] - s_wg[s_idx]
vij_dot_dwij = uij*DWIJ[0] + vij*DWIJ[1] + wij*DWIJ[2]
d_arho[d_idx] += rhoi*Vj*vij_dot_dwij
[docs]class StateEquation(Equation):
r"""**Generalized Weakly Compressible Equation of State**
.. math::
p_a = p_0\left[ \left(\frac{\rho}{\rho_0}\right)^\gamma - b
\right] + \mathcal{X}
Notes
-----
This is the generalized Tait's equation of state and the suggested values
in [Adami2013] are :math:`\mathcal{X} = 0`, :math:`\gamma=1` and
:math:`b = 1`.
The reference pressure :math:`p_0` is calculated from the artificial
sound speed and reference density:
.. math::
p_0 = \frac{c^2\rho_0}{\gamma}
"""
def __init__(self, dest, sources, p0, rho0, b=1.0):
r"""
Parameters
----------
p0 : float
reference pressure
rho0 : float
reference density
b : float
constant (default 1.0).
"""
self.b = b
self.p0 = p0
self.rho0 = rho0
super(StateEquation, self).__init__(dest, sources)
[docs] def loop(self, d_idx, d_p, d_rho):
d_p[d_idx] = self.p0 * (d_rho[d_idx]/self.rho0 - self.b)
[docs]class MomentumEquationPressureGradient(Equation):
r"""**Momentum equation for the Transport Velocity Formulation: Pressure**
Eq. (8) in [Adami2013]:
.. math::
\frac{d \boldsymbol{v}_a}{dt} = \frac{1}{m_a}\sum_b (V_a^2 +
V_b^2)\left[-\bar{p}_{ab}\nabla_a W_{ab} \right]
where
.. math::
\bar{p}_{ab} = \frac{\rho_b p_a + \rho_a p_b}{\rho_a + \rho_b}
"""
def __init__(self, dest, sources, pb, gx=0., gy=0., gz=0.,
tdamp=0.0):
r"""
Parameters
----------
pb : float
background pressure
gx : float
Body force per unit mass along the x-axis
gy : float
Body force per unit mass along the y-axis
gz : float
Body force per unit mass along the z-axis
tdamp : float
damping time
Notes
-----
This equation should have the destination as fluid and sources as
fluid and boundary particles.
This function also computes the contribution to the background
pressure and accelerations due to a body force or gravity.
The body forces are damped according to Eq. (13) in [Adami2012] to
avoid instantaneous accelerations. By default, damping is neglected.
"""
self.pb = pb
self.gx = gx
self.gy = gy
self.gz = gz
self.tdamp = tdamp
super(MomentumEquationPressureGradient, self).__init__(dest, sources)
[docs] def initialize(self, d_idx, d_au, d_av, d_aw, d_auhat, d_avhat, d_awhat):
d_au[d_idx] = 0.0
d_av[d_idx] = 0.0
d_aw[d_idx] = 0.0
d_auhat[d_idx] = 0.0
d_avhat[d_idx] = 0.0
d_awhat[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_auhat, d_avhat, d_awhat, 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]
# contribution due to the background pressure Eq. (13)
tmp = -self.pb * mi1 * (Vi2 + Vj2)
d_auhat[d_idx] += tmp * DWIJ[0]
d_avhat[d_idx] += tmp * DWIJ[1]
d_awhat[d_idx] += tmp * DWIJ[2]
[docs] def post_loop(self, d_idx, d_au, d_av, d_aw, t):
# damped accelerations due to body or external force
damping_factor = 1.0
if t < self.tdamp:
damping_factor = 0.5 * (sin((-0.5 + t/self.tdamp)*M_PI) + 1.0)
d_au[d_idx] += self.gx * damping_factor
d_av[d_idx] += self.gy * damping_factor
d_aw[d_idx] += self.gz * damping_factor
[docs]class MomentumEquationViscosity(Equation):
r"""**Momentum equation for the Transport Velocity Formulation: Viscosity**
Eq. (8) in [Adami2013]:
.. math::
\frac{d \boldsymbol{v}_a}{dt} = \frac{1}{m_a}\sum_b (V_a^2 +
V_b^2)\left[ \bar{\eta}_{ab}\hat{r}_{ab}\cdot \nabla_a W_{ab}
\frac{\boldsymbol{v}_{ab}}{|\boldsymbol{r}_{ab}|}\right]
where
.. math::
\bar{\eta}_{ab} = \frac{2\eta_a \eta_b}{\eta_a + \eta_b}
"""
def __init__(self, dest, sources, nu):
r"""
Parameters
----------
nu : float
kinematic viscosity
"""
self.nu = nu
super(MomentumEquationViscosity, 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_rho, s_rho, d_m, d_V, s_V,
d_au, d_av, d_aw,
R2IJ, EPS, DWIJ, VIJ, XIJ):
# averaged shear viscosity Eq. (6)
etai = self.nu * d_rho[d_idx]
etaj = self.nu * s_rho[s_idx]
etaij = 2 * (etai * etaj)/(etai + etaj)
# scalar part of the kernel gradient
Fij = DWIJ[0]*XIJ[0] + DWIJ[1]*XIJ[1] + DWIJ[2]*XIJ[2]
# 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
# accelerations 3rd term in Eq. (8)
tmp = 1./d_m[d_idx] * (Vi2 + Vj2) * etaij * Fij/(R2IJ + EPS)
d_au[d_idx] += tmp * VIJ[0]
d_av[d_idx] += tmp * VIJ[1]
d_aw[d_idx] += tmp * VIJ[2]
[docs]class MomentumEquationArtificialViscosity(Equation):
r"""**Artificial viscosity for the momentum equation**
Eq. (11) in [Adami2012]:
.. math::
\frac{d \boldsymbol{v}_a}{dt} = -\sum_b m_b \alpha h_{ab}
c_{ab} \frac{\boldsymbol{v}_{ab}\cdot
\boldsymbol{r}_{ab}}{\rho_{ab}\left(|r_{ab}|^2 + \epsilon
\right)}\nabla_a W_{ab}
where
.. math::
\rho_{ab} = \frac{\rho_a + \rho_b}{2}\\
c_{ab} = \frac{c_a + c_b}{2}\\
h_{ab} = \frac{h_a + h_b}{2}
"""
def __init__(self, dest, sources, c0, alpha=0.1):
r"""
Parameters
----------
alpha : float
constant
c0 : float
speed of sound
"""
self.alpha = alpha
self.c0 = c0
super(MomentumEquationArtificialViscosity, 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, s_m, d_au, d_av, d_aw,
RHOIJ1, R2IJ, EPS, DWIJ, VIJ, XIJ, HIJ):
# v_{ab} \cdot r_{ab}
vijdotrij = VIJ[0]*XIJ[0] + VIJ[1]*XIJ[1] + VIJ[2]*XIJ[2]
# scalar part of the accelerations Eq. (11)
piij = 0.0
if vijdotrij < 0:
muij = (HIJ * vijdotrij)/(R2IJ + EPS)
piij = -self.alpha*self.c0*muij
piij = s_m[s_idx] * piij*RHOIJ1
d_au[d_idx] += -piij * DWIJ[0]
d_av[d_idx] += -piij * DWIJ[1]
d_aw[d_idx] += -piij * DWIJ[2]
[docs]class MomentumEquationArtificialStress(Equation):
r"""**Artificial stress contribution to the Momentum Equation**
.. math::
\frac{d\boldsymbol{v}_a}{dt} = \frac{1}{m_a}\sum_b (V_a^2 +
V_b^2)\left[ \frac{1}{2}(\boldsymbol{A}_a +
\boldsymbol{A}_b) : \nabla_a W_{ab}\right]
where the artificial stress terms are given by:
.. math::
\boldsymbol{A} = \rho \boldsymbol{v} (\tilde{\boldsymbol{v}}
- \boldsymbol{v})
"""
[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_rho, d_u, d_v, d_w, d_V,
d_uhat, d_vhat, d_what, d_au, d_av, d_aw, d_m,
s_rho, s_u, s_v, s_w, s_V, s_uhat, s_vhat, s_what, DWIJ):
rhoi = d_rho[d_idx]
rhoj = s_rho[s_idx]
# physical and advection velocities
ui = d_u[d_idx]
uhati = d_uhat[d_idx]
vi = d_v[d_idx]
vhati = d_vhat[d_idx]
wi = d_w[d_idx]
whati = d_what[d_idx]
uj = s_u[s_idx]
uhatj = s_uhat[s_idx]
vj = s_v[s_idx]
vhatj = s_vhat[s_idx]
wj = s_w[s_idx]
whatj = s_what[s_idx]
# 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
# artificial stress tensor
Axxi = rhoi*ui*(uhati - ui)
Axyi = rhoi*ui*(vhati - vi)
Axzi = rhoi*ui*(whati - wi)
Ayxi = rhoi*vi*(uhati - ui)
Ayyi = rhoi*vi*(vhati - vi)
Ayzi = rhoi*vi*(whati - wi)
Azxi = rhoi*wi*(uhati - ui)
Azyi = rhoi*wi*(vhati - vi)
Azzi = rhoi*wi*(whati - wi)
Axxj = rhoj*uj*(uhatj - uj)
Axyj = rhoj*uj*(vhatj - vj)
Axzj = rhoj*uj*(whatj - wj)
Ayxj = rhoj*vj*(uhatj - uj)
Ayyj = rhoj*vj*(vhatj - vj)
Ayzj = rhoj*vj*(whatj - wj)
Azxj = rhoj*wj*(uhatj - uj)
Azyj = rhoj*wj*(vhatj - vj)
Azzj = rhoj*wj*(whatj - wj)
# contraction of stress tensor with kernel gradient
Ax = 0.5*(
(Axxi + Axxj)*DWIJ[0] +
(Axyi + Axyj)*DWIJ[1] +
(Axzi + Axzj)*DWIJ[2]
)
Ay = 0.5*(
(Ayxi + Ayxj)*DWIJ[0] +
(Ayyi + Ayyj)*DWIJ[1] +
(Ayzi + Ayzj)*DWIJ[2]
)
Az = 0.5*(
(Azxi + Azxj)*DWIJ[0] +
(Azyi + Azyj)*DWIJ[1] +
(Azzi + Azzj)*DWIJ[2]
)
# accelerations 2nd part of Eq. (8)
tmp = 1./d_m[d_idx] * (Vi2 + Vj2)
d_au[d_idx] += tmp * Ax
d_av[d_idx] += tmp * Ay
d_aw[d_idx] += tmp * Az
[docs]class SolidWallNoSlipBC(Equation):
r"""**Solid wall boundary condition** [Adami2012]_
This boundary condition is to be used with fixed ghost particles
in SPH simulations and is formulated for the general case of
moving boundaries.
The velocity and pressure of the fluid particles is extrapolated
to the ghost particles and these values are used in the equations
of motion.
No-penetration:
Ghost particles participate in the continuity and state equations
with fluid particles. This means as fluid particles approach the
wall, the pressure of the ghost particles increases to generate a
repulsion force that prevents particle penetration.
No-slip:
Extrapolation is used to set the `dummy` velocity of the ghost
particles for viscous interaction. First, the smoothed velocity
field of the fluid phase is extrapolated to the wall particles:
.. math::
\tilde{v}_a = \frac{\sum_b v_b W_{ab}}{\sum_b W_{ab}}
In the second step, for the viscous interaction in Eqs. (10) in [Adami2012]
and Eq. (8) in [Adami2013], the velocity of the ghost particles is
assigned as:
.. math::
v_b = 2v_w -\tilde{v}_a,
where :math:`v_w` is the prescribed wall velocity and :math:`v_b`
is the ghost particle in the interaction.
"""
def __init__(self, dest, sources, nu):
r"""
Parameters
----------
nu : float
kinematic viscosity
Notes
-----
For this equation the destination particle array should be the
fluid and the source should be ghost or boundary particles. The
boundary particles must define a prescribed velocity :math:`u_0,
v_0, w_0`
"""
self.nu = nu
super(SolidWallNoSlipBC, 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_V, s_V,
d_u, d_v, d_w,
d_au, d_av, d_aw,
s_ug, s_vg, s_wg,
DWIJ, R2IJ, EPS, XIJ):
# averaged shear viscosity Eq. (6).
etai = self.nu * d_rho[d_idx]
etaj = self.nu * s_rho[s_idx]
etaij = 2 * (etai * etaj)/(etai + etaj)
# particle volumes; d_V inverse volume.
Vi = 1./d_V[d_idx]
Vj = 1./s_V[s_idx]
Vi2 = Vi * Vi
Vj2 = Vj * Vj
# scalar part of the kernel gradient
Fij = XIJ[0]*DWIJ[0] + XIJ[1]*DWIJ[1] + XIJ[2]*DWIJ[2]
# viscous contribution (third term) from Eq. (8), with VIJ
# defined appropriately using the ghost values
tmp = 1./d_m[d_idx] * (Vi2 + Vj2) * (etaij * Fij/(R2IJ + EPS))
d_au[d_idx] += tmp * (d_u[d_idx] - s_ug[s_idx])
d_av[d_idx] += tmp * (d_v[d_idx] - s_vg[s_idx])
d_aw[d_idx] += tmp * (d_w[d_idx] - s_wg[s_idx])
[docs]class SolidWallPressureBC(Equation):
r"""**Solid wall pressure boundary condition** [Adami2012]_
This boundary condition is to be used with fixed ghost particles
in SPH simulations and is formulated for the general case of
moving boundaries.
The velocity and pressure of the fluid particles is extrapolated
to the ghost particles and these values are used in the equations
of motion.
Pressure boundary condition:
The pressure of the ghost particle is also calculated from the
fluid particle by interpolation using:
.. math::
p_g = \frac{\sum_f p_f W_{gf} + \boldsymbol{g - a_g} \cdot
\sum_f \rho_f \boldsymbol{r}_{gf}W_{gf}}{\sum_f W_{gf}},
where the subscripts `g` and `f` relate to the ghost and fluid
particles respectively.
Density of the wall particle is then set using this pressure
.. math::
\rho_w=\rho_0\left(\frac{p_w - \mathcal{X}}{p_0} +
1\right)^{\frac{1}{\gamma}}
"""
def __init__(self, dest, sources, rho0, p0, b=1.0, gx=0.0, gy=0.0, gz=0.0):
r"""
Parameters
----------
rho0 : float
reference density
p0 : float
reference pressure
b : float
constant (default 1.0)
gx : float
Body force per unit mass along the x-axis
gy : float
Body force per unit mass along the y-axis
gz : float
Body force per unit mass along the z-axis
Notes
-----
For a two fluid system (boundary, fluid), this equation must be
instantiated with boundary as the destination and fluid as the
source.
The boundary particle array must additionally define a property
:math:`wij` for the denominator in Eq. (27) from [Adami2012]. This
array sums the kernel terms from the ghost particle to the fluid
particle.
"""
self.rho0 = rho0
self.p0 = p0
self.b = b
self.gx = gx
self.gy = gy
self.gz = gz
super(SolidWallPressureBC, self).__init__(dest, sources)
[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):
# numerator of Eq. (27) ax, ay and az are the prescribed wall
# accelerations which must be defined for the wall boundary
# particle
gdotxij = (self.gx - d_au[d_idx])*XIJ[0] + \
(self.gy - d_av[d_idx])*XIJ[1] + \
(self.gz - d_aw[d_idx])*XIJ[2]
d_p[d_idx] += s_p[s_idx]*WIJ + s_rho[s_idx]*gdotxij*WIJ
# denominator of Eq. (27)
d_wij[d_idx] += WIJ
[docs] def post_loop(self, d_idx, d_wij, d_p, d_rho):
# extrapolated pressure at the ghost particle
if d_wij[d_idx] > 1e-14:
d_p[d_idx] /= d_wij[d_idx]
# update the density from the pressure Eq. (28)
d_rho[d_idx] = self.rho0 * (d_p[d_idx]/self.p0 + self.b)