"""Definition of some SPH kernel functions
"""
from math import pi, sqrt, exp
M_1_PI = 1.0/pi
M_2_SQRTPI = 2.0/sqrt(pi)
[docs]def get_correction(kernel, h0):
rij = kernel.deltap * h0
return kernel.kernel(rij=rij, h=h0)
[docs]def get_compiled_kernel(kernel):
"""Given a kernel, return a high performance wrapper kernel.
"""
import c_kernels
cls = getattr(c_kernels, kernel.__class__.__name__)
wrapper = getattr(c_kernels, kernel.__class__.__name__ + 'Wrapper')
kern = cls(**kernel.__dict__)
return wrapper(kern)
###############################################################################
# `CubicSpline` class.
###############################################################################
[docs]class CubicSpline(object):
r"""Cubic Spline Kernel: [Monaghan1992]_
.. math::
W(q) = \ &\sigma_3\left[ (2-q)^3 - 4(1-q)^3\right], \ & \textrm{for} \ 0 \leq q \leq 1,\\
= \ &\sigma_3(2-q)^3, & \textrm{for}\ 1 \leq q \leq 2,\\
= \ &0, & \textrm{for}\ q>2, \\
where :math:`\sigma_3` is a dimensional normalizing factor for the
cubic spline function given by:
.. math::
\sigma_3 = \ & \frac{2}{3h^1}, & \textrm{for dim=1}, \\
\sigma_3 = \ & \frac{10}{7\pi h^2}, \ & \textrm{for dim=2}, \\
\sigma_3 = \ & \frac{1}{\pi h^3}, & \textrm{for dim=3}. \\
References
----------
.. [Monaghan1992] `J. Monaghan, Smoothed Particle Hydrodynamics, "Annual
Review of Astronomy and Astrophysics", 30 (1992), pp. 543-574.
<http://adsabs.harvard.edu/abs/1992ARA&A..30..543M>`_
"""
def __init__(self, dim=1):
self.radius_scale = 2.0
self.dim = dim
if dim == 3:
self.fac = M_1_PI
elif dim == 2:
self.fac = 10*M_1_PI/7.0
else:
self.fac = 2.0/3.0
[docs] def get_deltap(self):
return 2./3
[docs] def kernel(self, xij=[0., 0, 0], rij=1.0, h=1.0):
h1 = 1./h
q = rij*h1
# get the kernel normalizing factor
if self.dim == 1:
fac = self.fac * h1
elif self.dim == 2:
fac = self.fac * h1 * h1
elif self.dim == 3:
fac = self.fac * h1 * h1 * h1
tmp2 = 2. - q
if ( q > 2.0 ):
val = 0.0
elif ( q > 1.0 ):
val = 0.25 * tmp2 * tmp2 * tmp2
else:
val = 1 - 1.5 * q * q * (1 - 0.5 * q)
return val * fac
[docs] def gradient(self, xij=[0., 0, 0], rij=1.0, h=1.0, grad=[0, 0, 0]):
h1 = 1./h
q = rij*h1
# get the kernel normalizing factor
if self.dim == 1:
fac = self.fac * h1
elif self.dim == 2:
fac = self.fac * h1 * h1
elif self.dim == 3:
fac = self.fac * h1 * h1 * h1
# compute the gradient.
tmp2 = 2. - q
if (rij > 1e-12):
if (q > 2.0):
val = 0.0
elif ( q > 1.0 ):
val = -0.75 * tmp2*tmp2 * h1/rij
else:
val = -3.0*q * (1 - 0.75*q) * h1/rij
else:
val = 0.0
tmp = val * fac
grad[0] = tmp * xij[0]
grad[1] = tmp * xij[1]
grad[2] = tmp * xij[2]
[docs] def gradient_h(self, xij=[0., 0, 0], rij=1.0, h=1.0):
h1 = 1./h; q = rij * h1
# get the kernel normalizing factor
if self.dim == 1:
fac = self.fac * h1
elif self.dim == 2:
fac = self.fac * h1 * h1
elif self.dim == 3:
fac = self.fac * h1 * h1 * h1
# kernel and gradient evaluated at q
tmp2 = 2. - q
if ( q > 2.0 ):
w = 0.0
dw = 0.0
elif ( q > 1.0 ):
w = 0.25 * tmp2 * tmp2 * tmp2
dw = -0.75 * tmp2 * tmp2
else:
w = 1 - 1.5 * q * q * (1 - 0.5 * q)
dw = -3.0*q * (1 - 0.75*q)
return -fac * h1 * ( dw*q + w*self.dim )
[docs]class WendlandQuintic(object):
def __init__(self, dim=2):
self.radius_scale = 2.0
if dim == 1:
raise ValueError("WendlandQuintic: Dim %d not supported"%dim)
self.dim = dim
if dim == 3:
self.fac = M_1_PI * 21.0/16.0
if dim == 2:
self.fac = 7.0 * M_1_PI/4.0
[docs] def get_deltap(self):
return 0.5
[docs] def kernel(self, xij=[0., 0, 0], rij=1.0, h=1.0):
h1 = 1.0/h
q = rij*h1
# get the kernel normalizing factor
if self.dim == 1:
fac = self.fac * h1
elif self.dim == 2:
fac = self.fac * h1 * h1
elif self.dim == 3:
fac = self.fac * h1 * h1 * h1
val = 0.0
tmp = 1. - 0.5 * q
if ( q < 2.0 ):
val = tmp * tmp * tmp * tmp * (2.0*q + 1.0)
return val * fac
[docs] def gradient(self, xij=[0., 0, 0], rij=1.0, h=1.0, grad=[0, 0, 0]):
h1 = 1./h
q = rij*h1
# get the kernel normalizing factor
if self.dim == 1:
fac = self.fac * h1
elif self.dim == 2:
fac = self.fac * h1 * h1
elif self.dim == 3:
fac = self.fac * h1 * h1 * h1
# compute the gradient
val = 0.0
tmp = 1.0 - 0.5*q
if ( q < 2.0 ):
if (rij > 1e-12):
val = -5.0 * q * tmp * tmp * tmp * h1/rij
tmp = val * fac
grad[0] = tmp * xij[0]
grad[1] = tmp * xij[1]
grad[2] = tmp * xij[2]
[docs] def gradient_h(self, xij=[0., 0, 0], rij=1.0, h=1.0):
h1 = 1./h; q = rij*h1
# get the kernel normalizing factor
if self.dim == 1:
fac = self.fac * h1
elif self.dim == 2:
fac = self.fac * h1 * h1
elif self.dim == 3:
fac = self.fac * h1 * h1 * h1
# compute the kernel and gradient at q
w = 0.0; dw = 0.0
tmp = 1.0 - 0.5*q
if ( q < 2.0 ):
w = tmp * tmp * tmp * tmp * (2.0*q + 1.0)
dw = -5.0 * q * tmp * tmp * tmp
return -fac * h1 * ( dw*q + w*self.dim )
[docs]class Gaussian(object):
r"""Gaussian Kernel: [Liu2010]_
.. math::
W(q) = \ &\sigma_g e^{-q^2}, \ & \textrm{for} \ 0\leq q \leq 3,\\
= \ & 0, & \textrm{for} \ q>3,\\
where :math:`\sigma_g` is a dimensional normalizing factor for the gaussian function
given by:
.. math::
\sigma_g = \ & \frac{1}{\pi^{1/2} h^1}, \ & \textrm{for dim=1}, \\
\sigma_g = \ & \frac{1}{\pi h^2}, \ & \textrm{for dim=2}, \\
\sigma_g = \ & \frac{1}{\pi^{3/2} h^3}, & \textrm{for dim=3}. \\
References
----------
.. [Liu2010] `M. Liu, & G. Liu, Smoothed particle hydrodynamics (SPH):
an overview and recent developments, "Archives of computational
methods in engineering", 17.1 (2010), pp. 25-76.
<http://link.springer.com/article/10.1007/s11831-010-9040-7>`_
"""
def __init__(self, dim=2):
self.radius_scale = 3.0
self.dim = dim
self.fac = 0.5*M_2_SQRTPI
if dim > 1:
self.fac *= 0.5*M_2_SQRTPI
if dim > 2:
self.fac *= 0.5*M_2_SQRTPI
[docs] def get_deltap(self):
# The inflection point is at q=1/sqrt(2)
# the deltap values for some standard kernels
# have been tabulated in sec 3.2 of
# http://cfd.mace.manchester.ac.uk/sph/SPH_PhDs/2008/crespo_thesis.pdf
return 0.70710678118654746
[docs] def kernel(self, xij=[0., 0, 0], rij=1.0, h=1.0):
h1 = 1./h
q = rij*h1
# get the kernel normalizing factor
if self.dim == 1:
fac = self.fac * h1
elif self.dim == 2:
fac = self.fac * h1 * h1
elif self.dim == 3:
fac = self.fac * h1 * h1 * h1
val = 0.0
if ( q < 3.0 ):
val = exp(-q*q) * fac
return val
[docs] def gradient(self, xij=[0., 0, 0], rij=1.0, h=1.0, grad=[0., 0, 0]):
h1 = 1./h
q = rij*h1
# get the kernel normalizing factor
if self.dim == 1:
fac = self.fac * h1
elif self.dim == 2:
fac = self.fac * h1 * h1
elif self.dim == 3:
fac = self.fac * h1 * h1 * h1
# compute the gradient
val = 0.0
if (q < 3.0):
if (rij > 1e-12):
val = -2.0* q * exp(-q*q) * h1/rij
tmp = val * fac
grad[0] = tmp * xij[0]
grad[1] = tmp * xij[1]
grad[2] = tmp * xij[2]
[docs] def gradient_h(self, xij=[0., 0., 0.], rij=1.0, h=1.0):
h1 = 1./h; q= rij*h1
# get the kernel normalizing factor
if self.dim == 1:
fac = self.fac * h1
elif self.dim == 2:
fac = self.fac * h1 * h1
elif self.dim == 3:
fac = self.fac * h1 * h1 * h1
# kernel and gradient evaluated at q
w = 0.0; dw = 0.0
if ( q < 3.0 ):
w = exp(-q*q)
dw = -2.0 * q * w
return -fac * h1 * ( dw*q + w*self.dim )
[docs]class QuinticSpline(object):
r"""Quintic Spline SPH kernel: [Liu2010]_
.. math::
W(q) = \ &\sigma_5\left[ (3-q)^5 - 6(2-q)^5 + 15(1-q)^5 \right], \ & \textrm{for} \ 0\leq q \leq 1,\\
= \ &\sigma_5\left[ (3-q)^5 - 6(2-q)^5 \right], & \textrm{for} \ 1 \leq q \leq 2,\\
= \ &\sigma_5 \ (3-5)^5 , & \textrm{for} \ 2 \leq q \leq 3,\\
= \ & 0, & \textrm{for} \ q>3,\\
where :math:`\sigma_5` is a dimensional normalizing factor for the
quintic spline function given by:
.. math::
\sigma_5 = \ & \frac{120}{h^1}, & \textrm{for dim=1}, \\
\sigma_5 = \ & \frac{7}{478\pi h^2}, \ & \textrm{for dim=2}, \\
\sigma_5 = \ & \frac{3}{359\pi h^3}, & \textrm{for dim=3}. \\
Raises
------
NotImplementedError
Quintic spline currently only supports 2D kernels.
"""
def __init__(self, dim=2):
self.radius_scale = 3.0
if dim != 2:
raise NotImplementedError('Quintic spline currently only supports 2D kernels.')
self.dim = dim
self.fac = M_1_PI * 7.0/478.0
[docs] def get_deltap(self):
# The inflection points for the polynomial are obtained as
# http://www.wolframalpha.com/input/?i=%28%283-x%29%5E5+-+6*%282-x%29%5E5+%2B+15*%281-x%29%5E5%29%27%27
# the only permissible value is taken
return 0.759298480738450
[docs] def kernel(self, xij=[0., 0, 0], rij=1.0, h=1.0):
h1 = 1./h
q = rij*h1
# get the kernel normalizing factor
if self.dim == 1:
fac = self.fac * h1
elif self.dim == 2:
fac = self.fac * h1 * h1
elif self.dim == 3:
fac = self.fac * h1 * h1 * h1
tmp3 = 3. - q
tmp2 = 2. - q
tmp1 = 1. - q
if ( q > 3.0 ):
val = 0.0
elif ( q > 2.0 ):
val = tmp3 * tmp3 * tmp3 * tmp3 * tmp3
elif ( q > 1.0 ):
val = tmp3 * tmp3 * tmp3 * tmp3 * tmp3
val -= 6.0 * tmp2 * tmp2 * tmp2 * tmp2 * tmp2
else:
val = tmp3 * tmp3 * tmp3 * tmp3 * tmp3
val -= 6.0 * tmp2 * tmp2 * tmp2 * tmp2 * tmp2
val += 15. * tmp1 * tmp1 * tmp1 * tmp1 * tmp1
return val * fac
[docs] def gradient(self, xij=[0., 0, 0], rij=1.0, h=1.0, grad=[0., 0, 0]):
h1 = 1./h
q = rij*h1
# get the kernel normalizing factor
if self.dim == 1:
fac = self.fac * h1
elif self.dim == 2:
fac = self.fac * h1 * h1
elif self.dim == 3:
fac = self.fac * h1 * h1 * h1
tmp3 = 3. - q
tmp2 = 2. - q
tmp1 = 1. - q
# compute the gradient
if (rij > 1e-12):
if ( q > 3.0 ):
val = 0.0
elif ( q > 2.0 ):
val = -5.0 * tmp3 * tmp3 * tmp3 * tmp3
val *= h1/rij
elif ( q > 1.0 ):
val = -5.0 * tmp3 * tmp3 * tmp3 * tmp3
val += 30.0 * tmp2 * tmp2 * tmp2 * tmp2
val *= h1/rij
else:
val = -5.0 * tmp3 * tmp3 * tmp3 * tmp3
val += 30.0 * tmp2 * tmp2 * tmp2 * tmp2
val -= 75.0 * tmp1 * tmp1 * tmp1 * tmp1
val *= h1/rij
else:
val = 0.0
tmp = val * fac
grad[0] = tmp * xij[0]
grad[1] = tmp * xij[1]
grad[2] = tmp * xij[2]
[docs] def gradient_h(self, xij=[0., 0, 0], rij=1.0, h=1.0):
h1 = 1./h; q = rij*h1
# get the kernel normalizing factor
if self.dim == 1:
fac = self.fac * h1
elif self.dim == 2:
fac = self.fac * h1 * h1
elif self.dim == 3:
fac = self.fac * h1 * h1 * h1
tmp3 = 3. - q
tmp2 = 2. - q
tmp1 = 1. - q
# compute the kernel & gradient at q
if ( q > 3.0 ):
w = 0.0
dw = 0.0
elif ( q > 2.0 ):
w = tmp3 * tmp3 * tmp3 * tmp3 * tmp3
dw = -5.0 * tmp3 * tmp3 * tmp3 * tmp3
elif ( q > 1.0 ):
w = tmp3 * tmp3 * tmp3 * tmp3 * tmp3
w -= 6.0 * tmp2 * tmp2 * tmp2 * tmp2 * tmp2
dw = -5.0 * tmp3 * tmp3 * tmp3 * tmp3
dw += 30.0 * tmp2 * tmp2 * tmp2 * tmp2
else:
w = tmp3 * tmp3 * tmp3 * tmp3 * tmp3
w -= 6.0 * tmp2 * tmp2 * tmp2 * tmp2 * tmp2
w += 15. * tmp1 * tmp1 * tmp1 * tmp1 * tmp1
dw = -5.0 * tmp3 * tmp3 * tmp3 * tmp3
dw += 30.0 * tmp2 * tmp2 * tmp2 * tmp2
dw -= 75.0 * tmp1 * tmp1 * tmp1 * tmp1
return -fac * h1 * ( dw*q + w*self.dim )