"""
Basic WCSPH Equations
#####################
"""
from pysph.sph.equation import Equation
[docs]class TaitEOS(Equation):
r"""**Tait equation of state for water-like fluids**
:math:`p_a = \frac{c_{0}^2\rho_0}{\gamma}\left(
\left(\frac{\rho_a}{\rho_0}\right)^{\gamma} -1\right)`
References
----------
.. [Cole1948] H. R. Cole, "Underwater Explosions", Princeton University
Press, 1948.
.. [Batchelor2002] G. Batchelor, "An Introduction to Fluid Dynamics",
Cambridge University Press, 2002.
.. [Monaghan2005] J. Monaghan, "Smoothed particle hydrodynamics",
Reports on Progress in Physics, 68 (2005), pp. 1703-1759.
"""
def __init__(self, dest, sources, rho0, c0, gamma, p0=0.0):
r"""
Parameters
----------
rho0 : float
reference density of fluid particles
c0 : float
maximum speed of sound expected in the system
gamma : float
constant
p0 : float
reference pressure in the system (defaults to zero).
Notes
-----
The reference speed of sound, c0, is to be taken approximately as
10 times the maximum expected velocity in the system. The particle
sound speed is given by the usual expression:
:math:`c_a = \sqrt{\frac{\partial p}{\partial \rho}}`
"""
self.rho0 = rho0
self.rho01 = 1.0/rho0
self.c0 = c0
self.gamma = gamma
self.gamma1 = 0.5*(gamma - 1.0)
self.B = rho0*c0*c0/gamma
self.p0 = p0
super(TaitEOS, self).__init__(dest, sources)
[docs] def loop(self, d_idx, d_rho, d_p, d_cs):
ratio = d_rho[d_idx] * self.rho01
tmp = pow(ratio, self.gamma)
d_p[d_idx] = self.p0 + self.B * (tmp - 1.0)
d_cs[d_idx] = self.c0 * pow(ratio, self.gamma1)
[docs]class TaitEOSHGCorrection(Equation):
r"""**Tait Equation of State with Hughes and Graham Correction**
.. math::
p_a = \frac{c_{0}^2\rho_0}{\gamma}\left(
\left(\frac{\rho_a}{\rho_0}\right)^{\gamma} -1\right)
where
.. math::
\rho_{a}=\begin{cases}\rho_{a} & \rho_{a}\geq\rho_{0}\\
\rho_{0} & \rho_{a}<\rho_{0}\end{cases}`
References
----------
.. [Hughes2010] J. P. Hughes and D. I. Graham, "Comparison of
incompressible and weakly-compressible SPH models for free-surface
water flows", Journal of Hydraulic Research, 48 (2010), pp. 105-117.
"""
def __init__(self, dest, sources, rho0, c0, gamma):
r"""
Parameters
----------
rho0 : float
reference density
c0 : float
reference speed of sound
gamma : float
constant
Notes
-----
The correction is to be applied on boundary particles and imposes
a minimum value of the density (rho0) which is set upon
instantiation. This correction avoids particle sticking behaviour
at walls.
"""
self.rho0 = rho0
self.rho01 = 1.0/rho0
self.c0 = c0
self.gamma = gamma
self.gamma1 = 0.5*(gamma - 1.0)
self.B = rho0*c0*c0/gamma
super(TaitEOSHGCorrection, self).__init__(dest, sources)
[docs] def loop(self, d_idx, d_rho, d_p, d_cs):
if d_rho[d_idx] < self.rho0:
d_rho[d_idx] = self.rho0
ratio = d_rho[d_idx] * self.rho01
tmp = pow(ratio, self.gamma)
d_p[d_idx] = self.B * (tmp - 1.0)
d_cs[d_idx] = self.c0 * pow(ratio, self.gamma1)
[docs]class MomentumEquation(Equation):
r"""**Classic Monaghan Style Momentum Equation with Artificial Viscosity**
.. math::
\frac{d\mathbf{v}_{a}}{dt}=-\sum_{b}m_{b}\left(\frac{p_{b}}
{\rho_{b}^{2}}+\frac{p_{a}}{\rho_{a}^{2}}+\Pi_{ab}\right)
\nabla_{a}W_{ab}
where
.. math::
\Pi_{ab}=\begin{cases}
\frac{-\alpha\bar{c}_{ab}\mu_{ab}+\beta\mu_{ab}^{2}}{\bar{\rho}_{ab}} &
\mathbf{v}_{ab}\cdot\mathbf{r}_{ab}<0;\\
0 & \mathbf{v}_{ab}\cdot\mathbf{r}_{ab}\geq0;
\end{cases}
with
.. math::
\mu_{ab}=\frac{h\mathbf{v}_{ab}\cdot\mathbf{r}_{ab}}
{\mathbf{r}_{ab}^{2}+\eta^{2}}\\
\bar{c}_{ab} = \frac{c_a + c_b}{2}\\
\bar{\rho}_{ab} = \frac{\rho_a + \rho_b}{2}
References
----------
.. [Monaghan1992] J. Monaghan, Smoothed Particle Hydrodynamics, "Annual
Review of Astronomy and Astrophysics", 30 (1992), pp. 543-574.
"""
def __init__(self, dest, sources, c0,
alpha=1.0, beta=1.0, gx=0.0, gy=0.0, gz=0.0,
tensile_correction=False):
r"""
Parameters
----------
c0 : float
reference speed of sound
alpha : float
produces a shear and bulk viscosity
beta : float
used to handle high Mach number shocks
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
tensilte_correction : bool
switch for tensile instability correction (Default: False)
"""
self.alpha = alpha
self.beta = beta
self.gx = gx
self.gy = gy
self.gz = gz
self.c0 = c0
self.tensile_correction = tensile_correction
super(MomentumEquation, self).__init__(dest, sources)
[docs] def initialize(self, d_idx, d_au, d_av, d_aw, d_dt_cfl):
d_au[d_idx] = 0.0
d_av[d_idx] = 0.0
d_aw[d_idx] = 0.0
d_dt_cfl[d_idx] = 0.0
[docs] def loop(self, d_idx, s_idx, d_rho, d_cs,
d_p, d_au, d_av, d_aw, s_m,
s_rho, s_cs, s_p, VIJ,
XIJ, HIJ, R2IJ, RHOIJ1, EPS,
DWIJ, WIJ, WDP, d_dt_cfl):
rhoi21 = 1.0/(d_rho[d_idx]*d_rho[d_idx])
rhoj21 = 1.0/(s_rho[s_idx]*s_rho[s_idx])
vijdotxij = VIJ[0]*XIJ[0] + VIJ[1]*XIJ[1] + VIJ[2]*XIJ[2]
piij = 0.0
if vijdotxij < 0:
cij = 0.5 * (d_cs[d_idx] + s_cs[s_idx])
muij = (HIJ * vijdotxij)/(R2IJ + EPS)
piij = -self.alpha*cij*muij + self.beta*muij*muij
piij = piij*RHOIJ1
# compute the CFL time step factor
_dt_cfl = 0.0
if R2IJ > 1e-12:
_dt_cfl = abs(HIJ * vijdotxij/R2IJ) + self.c0
d_dt_cfl[d_idx] = max(_dt_cfl, d_dt_cfl[d_idx])
tmpi = d_p[d_idx]*rhoi21
tmpj = s_p[s_idx]*rhoj21
fij = WIJ/WDP
Ri = 0.0
Rj = 0.0
# tensile instability correction
if self.tensile_correction:
fij = fij*fij
fij = fij*fij
if d_p[d_idx] > 0:
Ri = 0.01 * tmpi
else:
Ri = 0.2*abs(tmpi)
if s_p[s_idx] > 0:
Rj = 0.01 * tmpj
else:
Rj = 0.2 * abs(tmpj)
# gradient and correction terms
tmp = (tmpi + tmpj) + (Ri + Rj)*fij
d_au[d_idx] += -s_m[s_idx] * (tmp + piij) * DWIJ[0]
d_av[d_idx] += -s_m[s_idx] * (tmp + piij) * DWIJ[1]
d_aw[d_idx] += -s_m[s_idx] * (tmp + piij) * DWIJ[2]
[docs] def post_loop(self, d_idx, d_au, d_av, d_aw, d_dt_force):
d_au[d_idx] += self.gx
d_av[d_idx] += self.gy
d_aw[d_idx] += self.gz
acc2 = (d_au[d_idx]*d_au[d_idx] +
d_av[d_idx]*d_av[d_idx] +
d_aw[d_idx]*d_aw[d_idx])
# store the square of the max acceleration
d_dt_force[d_idx] = acc2
[docs]class MomentumEquationDeltaSPH(Equation):
r"""**Momentum equation defined in JOSEPHINE and the delta-SPH model**
.. math::
\frac{du_{i}}{dt}=-\frac{1}{\rho_{i}}\sum_{j}\left(p_{j}+p_{i}\right)
\nabla_{i}W_{ij}V_{j}+\mathbf{g}_{i}+\alpha hc_{0}\rho_{0}\sum_{j}
\pi_{ij}\nabla_{i}W_{ij}V_{j}
where
.. math::
\pi_{ij}=\frac{\mathbf{u}_{ij}\cdot\mathbf{r}_{ij}}
{|\mathbf{r}_{ij}|^{2}}
References
----------
.. [Marrone2011] S. Marrone et al., "delta-SPH model for simulating
violent impact flows", Computer Methods in Applied Mechanics and
Engineering, 200 (2011), pp 1526--1542.
.. [Cherfils2012] J. M. Cherfils et al., "JOSEPHINE: A parallel SPH code
for free-surface flows", Computer Physics Communications, 183 (2012),
pp 1468--1480.
"""
def __init__(self, dest, sources, rho0, c0, alpha=1.0):
r"""
Parameters
----------
rho0 : float
reference density
c0 : float
reference speed of sound
alpha : float
coefficient used to control the intensity of the
diffusion of velocity
Notes
-----
Artificial viscosity is used in this momentum equation and is
controlled by the parameter :math:`\alpha`. This form of the
artificial viscosity is similar but not identical to the
Monaghan-style artificial viscosity.
"""
self.alpha = alpha
self.c0 = c0
self.rho0 = rho0
super(MomentumEquationDeltaSPH, self).__init__(dest, sources)
[docs] def loop(self, d_idx, s_idx, d_rho, d_cs, d_p, d_au, d_av, d_aw, s_m,
s_rho, s_cs, s_p, VIJ, XIJ, HIJ, R2IJ, RHOIJ1, EPS, WIJ, DWIJ):
# src paricle volume mj/rhoj
Vj = s_m[s_idx]/s_rho[s_idx]
# viscous contribution second part of eqn (5b) in [Maronne2011]
vijdotxij = VIJ[0]*XIJ[0] + VIJ[1]*XIJ[1] + VIJ[2]*XIJ[2]
fac = self.alpha * HIJ * self.c0 * self.rho0
piij = vijdotxij/(R2IJ + EPS)
# gradient and viscous terms eqn 5b in [Maronne2011]
tmp = fac * piij * Vj/d_rho[d_idx]
# accelerations
d_au[d_idx] += tmp * DWIJ[0]
d_av[d_idx] += tmp * DWIJ[1]
d_aw[d_idx] += tmp * DWIJ[2]
[docs]class ContinuityEquationDeltaSPHPreStep(Equation):
r"""**Continuity equation with dissipative terms**
See :class:`pysph.sph.wc.basic.ContinuityEquationDeltaSPH`
The matrix :math:`L_a` is multiplied to :math:`\nabla W_{ij}` in the
:class:`pysph.sph.scheme.WCSPHScheme` class by using
:class:`pysph.sph.wc.kernel_correction.GradientCorrectionPreStep` and
:class:`pysph.sph.wc.kernel_correction.GradientCorrection`.
"""
[docs] def initialize(self, d_idx, d_gradrho):
d_gradrho[d_idx*3 + 0] = 0.0
d_gradrho[d_idx*3 + 1] = 0.0
d_gradrho[d_idx*3 + 2] = 0.0
[docs] def loop(self, d_idx, d_arho, s_idx, d_rho, s_rho, s_m, d_gradrho, DWIJ):
rhoi = d_rho[d_idx]
rhoj = s_rho[s_idx]
Vj = s_m[s_idx]/rhoj
# renormalized density of eqn (5a)
d_gradrho[d_idx*3 + 0] += (rhoj - rhoi) * DWIJ[0] * Vj
d_gradrho[d_idx*3 + 1] += (rhoj - rhoi) * DWIJ[1] * Vj
d_gradrho[d_idx*3 + 2] += (rhoj - rhoi) * DWIJ[2] * Vj
[docs]class ContinuityEquationDeltaSPH(Equation):
r"""**Continuity equation with dissipative terms**
:math:`\frac{d\rho_a}{dt} = \sum_b \rho_a \frac{m_b}{\rho_b}
\left( \boldsymbol{v}_{ab}\cdot \nabla_a W_{ab} + \delta \eta_{ab}
\cdot \nabla_{a} W_{ab} (h_{ab}\frac{c_{ab}}{\rho_a}(\rho_b -
\rho_a)) \right)`
References
----------
.. [Marrone2011] S. Marrone et al., "delta-SPH model for simulating
violent impact flows", Computer Methods in Applied Mechanics and
Engineering, 200 (2011), pp 1526--1542.
"""
def __init__(self, dest, sources, c0, delta=0.1):
r"""
Parameters
----------
c0 : float
reference speed of sound
delta : float
coefficient used to control the intensity of diffusion of density
"""
self.c0 = c0
self.delta = delta
super(ContinuityEquationDeltaSPH, self).__init__(dest, sources)
[docs] def loop(self, d_idx, d_arho, s_idx, s_m, d_rho, s_rho, DWIJ, XIJ, R2IJ,
HIJ, EPS, d_gradrho, s_gradrho):
rhoi = d_rho[d_idx]
rhoj = s_rho[s_idx]
Vj = s_m[s_idx]/rhoj
fac = -2.0 * (rhoj - rhoi)/(R2IJ + EPS)
psix = fac*XIJ[0] - d_gradrho[d_idx*3 + 0] - s_gradrho[s_idx*3 + 0]
psiy = fac*XIJ[1] - d_gradrho[d_idx*3 + 1] - s_gradrho[s_idx*3 + 1]
psiz = fac*XIJ[2] - d_gradrho[d_idx*3 + 2] - s_gradrho[s_idx*3 + 2]
psidotdwij = psix*DWIJ[0] + psiy*DWIJ[1] + psiz*DWIJ[2]
# standard term with dissipative penalization eqn (5a)
d_arho[d_idx] += self.delta * HIJ * self.c0 * psidotdwij * Vj
[docs]class UpdateSmoothingLengthFerrari(Equation):
r"""**Update the particle smoothing lengths**
:math:`h_a = hdx \left(\frac{m_a}{\rho_a}\right)^{\frac{1}{d}}`
References
----------
.. [Ferrari2009] A. Ferrari et al., "A new 3D parallel SPH scheme for free
surface flows", Computers and Fluids, 38 (2009), pp. 1203--1217.
"""
def __init__(self, dest, sources, dim, hdx):
r"""
Parameters
----------
dim : float
number of dimensions
hdx : float
scaling factor
Notes
-----
Ideally, the kernel scaling factor should be determined from the
kernel used based on a linear stability analysis. The default
value of (hdx=1) reduces to the formulation suggested by Ferrari
et al. who used a Cubic Spline kernel.
Typically, a change in the smoothing length should mean the
neighbors are re-computed which in PySPH means the NNPS must be
updated. This equation should therefore be placed as the last
equation so that after the final corrector stage, the smoothing
lengths are updated and the new NNPS data structure is computed.
Note however that since this is to be used with incompressible flow
equations, the density variations are small and hence the smoothing
lengths should also not vary too much.
"""
self.dim1 = 1./dim
self.hdx = hdx
super(UpdateSmoothingLengthFerrari, self).__init__(dest, sources)
[docs] def loop(self, d_idx, d_rho, d_h, d_m):
# naive estimate of particle volume
Vj = d_m[d_idx]/d_rho[d_idx]
d_h[d_idx] = self.hdx * pow(Vj, self.dim1)
[docs]class PressureGradientUsingNumberDensity(Equation):
r"""Pressure gradient discretized using number density:
.. math::
\frac{d \boldsymbol{v}_a}{dt} = -\frac{1}{m_a}\sum_b
(\frac{p_a}{V_a^2} + \frac{p_b}{V_b^2})\nabla_a W_{ab}
"""
[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):
# particle volumes
Vi = 1./d_V[d_idx]
Vj = 1./s_V[s_idx]
Vi2 = Vi * Vi
Vj2 = Vj * Vj
# pressure gradient term
p_i = d_p[d_idx]
pj = s_p[s_idx]
pij = p_i*Vi2 + pj*Vj2
# accelerations
tmp = -pij * 1.0/(d_m[d_idx])
d_au[d_idx] += tmp * DWIJ[0]
d_av[d_idx] += tmp * DWIJ[1]
d_aw[d_idx] += tmp * DWIJ[2]