"""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.
"""
from pysph.base 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[ 1 - \frac{3}{2}q^2\left( 1 -
\frac{q}{2} \right) \right],
\ & \textrm{for} \ 0 \leq q \leq 1,\\
= \ &\frac{\sigma_3}{4}(2-q)^3, & \textrm{for}
\ 1 < 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 dwdq(self, rij=1.0, h=1.0):
"""Gradient of a kernel is given by
.. math::
\nabla W = normalization \frac{dW}{dq} \frac{dq}{dx}
\nabla W = w_dash \frac{dq}{dx}
Here we get `w_dash` by using `dwdq` method
"""
h1 = 1. / h
q = rij * h1
# get the kernel normalizing factor ( sigma )
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 sigma * dw_dq
tmp2 = 2. - q
if (rij > 1e-12):
if (q > 2.0):
val = 0.0
elif (q > 1.0):
val = -0.75 * tmp2 * tmp2
else:
val = -3.0 * q * (1 - 0.75 * q)
else:
val = 0.0
return val * fac
[docs] def gradient(self, xij=[0., 0, 0], rij=1.0, h=1.0, grad=[0, 0, 0]):
h1 = 1. / h
# compute the gradient.
if (rij > 1e-12):
wdash = self.dwdq(rij, h)
tmp = wdash * h1 / rij
else:
tmp = 0.0
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 WendlandQuinticC2_1D(object):
r"""The following is the WendlandQuintic kernel (Wendland C2) kernel for 1D.
.. math::
W(q) = \ & \alpha_d (1-q/2)^3 (1.5q +1))),
\ & \textrm{for} \ 0\leq q \leq 2,\\
= \ & 0, & \textrm{for} \ q>2,\\
where :math:`d` is the number of dimensions and
.. math::
\alpha_d = \frac{5}{8h}, \textrm{for dim=1}
"""
def __init__(self, dim=1):
self.radius_scale = 2.0
self.dim = dim
if dim == 1:
self.fac = 5.0 / 8.0
elif dim == 2:
raise ValueError(
"WendlandQuinticC2_1D: Dim %d not supported" % dim)
elif dim == 3:
raise ValueError(
"WendlandQuinticC2_1D: Dim %d not supported" % dim)
[docs] def get_deltap(self):
return 2.0/3
[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 * (1.5 * q + 1.0)
return val * fac
[docs] def dwdq(self, 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 gradient
val = 0.0
tmp = 1.0 - 0.5 * q
if (q < 2.0):
if (rij > 1e-12):
val = -3.0 * q * tmp * tmp
return val * fac
[docs] def gradient(self, xij=[0., 0, 0], rij=1.0, h=1.0, grad=[0, 0, 0]):
h1 = 1. / h
# compute the gradient.
if (rij > 1e-12):
wdash = self.dwdq(rij, h)
tmp = wdash * h1 / rij
else:
tmp = 0.0
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 * (1.5 * q + 1.0)
dw = -3.0 * q * tmp * tmp
return -fac * h1 * (dw * q + w * self.dim)
[docs]class WendlandQuintic(object):
r"""The following is the WendlandQuintic kernel(C2) kernel for 2D and 3D.
.. math::
W(q) = \ & \alpha_d (1-q/2)^4(2q +1))),
\ & \textrm{for} \ 0\leq q \leq 2,\\
= \ & 0, & \textrm{for} \ q>2,\\
where :math:`d` is the number of dimensions and
.. math::
\alpha_d = \ & \frac{7}{4\pi h^2}, \ & \textrm{for dim=2}, \\
\alpha_d = \ & \frac{21}{16\pi h^3}, \ & \textrm{for dim=3}
"""
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 == 2:
self.fac = 7.0 * M_1_PI / 4.0
elif dim == 3:
self.fac = M_1_PI * 21.0 / 16.0
[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 dwdq(self, 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 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
return val * fac
[docs] def gradient(self, xij=[0., 0, 0], rij=1.0, h=1.0,
grad=[0, 0, 0]):
h1 = 1. / h
# compute the gradient.
if (rij > 1e-12):
wdash = self.dwdq(rij, h)
tmp = wdash * h1 / rij
else:
tmp = 0.0
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 WendlandQuinticC4_1D(object):
r"""The following is the WendlandQuintic kernel (Wendland C4) kernel for 1D.
.. math::
W(q) = \ & \alpha_d (1-q/2)^5 (2q^2 + 2.5q +1))),
\ & \textrm{for} \ 0\leq q \leq 2,\\
= \ & 0, & \textrm{for} \ q>2,\\
where :math:`d` is the number of dimensions and
.. math::
\alpha_d = \frac{3}{4h}, \ \textrm{for dim=1}
"""
def __init__(self, dim=1):
self.radius_scale = 2.0
self.dim = dim
if dim == 1:
self.fac = 0.75
if dim == 2:
raise ValueError(
"WendlandQuinticC4_1D: Dim %d not supported" % dim)
elif dim == 3:
raise ValueError(
"WendlandQuinticC4_1D: Dim %d not supported" % dim)
[docs] def get_deltap(self):
return 0.55195628
[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 * tmp * (2 * q * q + 2.5 * q + 1.0)
return val * fac
[docs] def dwdq(self, 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 gradient
val = 0.0
tmp = 1.0 - 0.5 * q
if (q < 2.0):
if (rij > 1e-12):
val = -3.5 * q * (2 * q + 1) * tmp * tmp * tmp * tmp
return val * fac
[docs] def gradient(self, xij=[0., 0., 0.], rij=1.0, h=1.0, grad=[0, 0, 0]):
h1 = 1. / h
# compute the gradient.
if (rij > 1e-12):
wdash = self.dwdq(rij, h)
tmp = wdash * h1 / rij
else:
tmp = 0.0
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 * tmp * (2 * q * q + 2.5 * q + 1.0)
dw = -3.5 * q * (2 * q + 1) * tmp * tmp * tmp * tmp
return -fac * h1 * (dw * q + w * self.dim)
[docs]class WendlandQuinticC4(object):
r"""The following is the WendlandQuintic kernel (Wendland C4) kernel for
2D and 3D.
.. math::
W(q) = \ & \alpha_d (1-q/2)^6(\frac{35}{12} q^2 + 3q +1))),
\ & \textrm{for} \ 0\leq q \leq 2,\\
= \ & 0, & \textrm{for} \ q>2,\\
where :math:`d` is the number of dimensions and
.. math::
\alpha_d = \ & \frac{9}{4\pi h^2}, \ & \textrm{for dim=2}, \\
\alpha_d = \ & \frac{495}{256\pi h^3}, \ & \textrm{for dim=3}
"""
def __init__(self, dim=2):
self.radius_scale = 2.0
self.dim = dim
if dim == 1:
raise ValueError("WendlandQuinticC4: Dim %d not supported" % dim)
if dim == 2:
self.fac = 9.0 * M_1_PI / 4.0
elif dim == 3:
self.fac = M_1_PI * 495.0 / 256.0
[docs] def get_deltap(self):
return 0.47114274
[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 * tmp * tmp * \
((35.0 / 12.0) * q * q + 3.0 * q + 1.0)
return val * fac
[docs] def dwdq(self, 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 gradient
val = 0.0
tmp = 1.0 - 0.5 * q
if (q < 2.0):
if (rij > 1e-12):
val = (-14.0 / 3.0) * q * (1 + 2.5 * q) * \
tmp * tmp * tmp * tmp * tmp
return val * fac
[docs] def gradient(self, xij=[0., 0., 0.], rij=1.0, h=1.0, grad=[0, 0, 0]):
h1 = 1. / h
# compute the gradient.
if (rij > 1e-12):
wdash = self.dwdq(rij, h)
tmp = wdash * h1 / rij
else:
tmp = 0.0
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 * tmp * tmp * \
((35.0 / 12.0) * q * q + 3.0 * q + 1.0)
dw = (-14.0 / 3.0) * q * (1 + 2.5 * q) * \
tmp * tmp * tmp * tmp * tmp
return -fac * h1 * (dw * q + w * self.dim)
[docs]class WendlandQuinticC6_1D(object):
r"""The following is the WendlandQuintic kernel (Wendland C6) kernel for 1D.
.. math::
W(q) = \ & \alpha_d (1-q/2)^7 (\frac{21}{8} q^3
+ \frac{19}{4} q^2 + 3.5q +1))),
\ & \textrm{for} \ 0\leq q \leq 2,\\
= \ & 0, & \textrm{for} \ q>2,\\
where :math:`d` is the number of dimensions and
.. math::
\alpha_d = \ \frac{55}{64h}, \textrm{for dim=1}
"""
def __init__(self, dim=1):
self.radius_scale = 2.0
self.dim = dim
if dim == 1:
self.fac = 55.0 / 64.0
if dim == 2:
raise ValueError(
"WendlandQuinticC6_1D: Dim %d not supported" % dim)
elif dim == 3:
raise ValueError(
"WendlandQuinticC6_1D: Dim %d not supported" % dim)
[docs] def get_deltap(self):
return 0.47996698
[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 * tmp * tmp * tmp * \
(2.625 * q * q * q + 4.75 * q * q + 3.5 * q + 1.0)
return val * fac
[docs] def dwdq(self, 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 gradient
val = 0.0
tmp = 1.0 - 0.5 * q
if (q < 2.0):
if (rij > 1e-12):
val = -0.5 * q * (26.25 * q * q + 27 * q + 9.0) * \
tmp * tmp * tmp * tmp * tmp * tmp
return val * fac
[docs] def gradient(self, xij=[0., 0., 0.], rij=1.0, h=1.0, grad=[0, 0, 0]):
h1 = 1. / h
# compute the gradient.
if (rij > 1e-12):
wdash = self.dwdq(rij, h)
tmp = wdash * h1 / rij
else:
tmp = 0.0
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 * tmp * tmp * tmp * \
(2.625 * q * q * q + 4.75 * q * q + 3.5 * q + 1.0)
dw = -0.5 * q * (26.25 * q * q + 27 * q + 9.0) * \
tmp * tmp * tmp * tmp * tmp * tmp
return -fac * h1 * (dw * q + w * self.dim)
[docs]class WendlandQuinticC6(object):
r"""The following is the WendlandQuintic kernel(C6) kernel for 2D and 3D.
.. math::
W(q) = \ & \alpha_d (1-q/2)^8 (4 q^3 + 6.25 q^2 + 4q +1))),
\ & \textrm{for} \ 0\leq q \leq 2,\\
= \ & 0, & \textrm{for} \ q>2,\\
where :math:`d` is the number of dimensions and
.. math::
\alpha_d = \ & \frac{78}{28\pi h^2}, \ & \textrm{for dim=2}, \\
\alpha_d = \ & \frac{1365}{512\pi h^3}, \ & \textrm{for dim=3}
"""
def __init__(self, dim=2):
self.radius_scale = 2.0
self.dim = dim
if dim == 1:
raise ValueError("WendlandQuinticC6: Dim %d not supported" % dim)
if dim == 2:
self.fac = 78.0 * M_1_PI / 28.0
elif dim == 3:
self.fac = M_1_PI * 1365.0 / 512.0
[docs] def get_deltap(self):
return 0.4305720757
[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 * tmp * tmp * tmp * tmp * \
(4.0 * q * q * q + 6.25 * q * q + 4.0 * q + 1.0)
return val * fac
[docs] def dwdq(self, 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 gradient
val = 0.0
tmp = 1.0 - 0.5 * q
if (q < 2.0):
if (rij > 1e-12):
val = -5.50 * q * tmp * tmp * tmp * tmp * tmp * \
tmp * tmp * (1.0 + 3.5 * q + 4 * q * 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
# compute the gradient.
if (rij > 1e-12):
wdash = self.dwdq(rij, h)
tmp = wdash * h1 / rij
else:
tmp = 0.0
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 * tmp * tmp * tmp * tmp * \
(4.0 * q * q * q + 6.25 * q * q + 4.0 * q + 1.0)
dw = -5.50 * q * tmp * tmp * tmp * tmp * tmp * \
tmp * tmp * (1.0 + 3.5 * q + 4 * q * q)
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}, \ & \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 dwdq(self, 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 gradient
val = 0.0
if (q < 3.0):
if (rij > 1e-12):
val = -2.0 * q * exp(-q * 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
# compute the gradient.
if (rij > 1e-12):
wdash = self.dwdq(rij, h)
tmp = wdash * h1 / rij
else:
tmp = 0.0
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 SuperGaussian(object):
r"""Super Gaussian Kernel: [Monaghan1992]_
.. math::
W(q) = \ &\frac{1}{h^{d}\pi^{d/2}} e^{-q^2} (d/2 + 1 - q^2),
\ & \textrm{for} \ 0\leq q \leq 3,\\
= \ & 0, & \textrm{for} \ q>3,\\
where :math:`d` is the number of dimensions.
"""
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):
# Found inflection point using sympy.
if self.dim == 1:
return 0.584540507426389
elif self.dim == 2:
return 0.6021141014644256
else:
return 0.615369528365158
[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):
q2 = q * q
val = exp(-q2) * (1.0 + self.dim * 0.5 - q2) * fac
return val
[docs] def dwdq(self, 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 gradient
val = 0.0
if (q < 3.0):
if (rij > 1e-12):
q2 = q * q
val = q * (2.0 * q2 - self.dim - 4) * exp(-q2)
return val * fac
[docs] def gradient(self, xij=[0., 0., 0.], rij=1.0, h=1.0, grad=[0, 0, 0]):
h1 = 1. / h
# compute the gradient.
if (rij > 1e-12):
wdash = self.dwdq(rij, h)
tmp = wdash * h1 / rij
else:
tmp = 0.0
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
d = self.dim
# get the kernel normalizing factor
if d == 1:
fac = self.fac * h1
elif d == 2:
fac = self.fac * h1 * h1
elif d == 3:
fac = self.fac * h1 * h1 * h1
# kernel and gradient evaluated at q
val = 0.0
if (q < 3.0):
q2 = q * q
val = (-d * d * 0.5 + 2.0 * d * q2 - d -
2.0 * q2 * q2 + 4 * q2) * exp(-q2)
return -fac * h1 * val
[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 < q \leq 2,\\
= \ &\sigma_5 \ (3-q)^5 , & \textrm{for} \ 2 < 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{1}{120 h^1}, & \textrm{for dim=1}, \\
\sigma_5 = \ & \frac{7}{478\pi h^2}, \ & \textrm{for dim=2}, \\
\sigma_5 = \ & \frac{1}{120\pi h^3}, & \textrm{for dim=3}. \\
"""
def __init__(self, dim=2):
self.radius_scale = 3.0
self.dim = dim
if dim == 1:
self.fac = 1.0 / 120.0
elif dim == 2:
self.fac = M_1_PI * 7.0 / 478.0
elif dim == 3:
self.fac = M_1_PI * 1.0 / 120.0
[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 dwdq(self, 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 gradient
if (rij > 1e-12):
if (q > 3.0):
val = 0.0
elif (q > 2.0):
val = -5.0 * tmp3 * tmp3 * tmp3 * tmp3
elif (q > 1.0):
val = -5.0 * tmp3 * tmp3 * tmp3 * tmp3
val += 30.0 * tmp2 * tmp2 * tmp2 * tmp2
else:
val = -5.0 * tmp3 * tmp3 * tmp3 * tmp3
val += 30.0 * tmp2 * tmp2 * tmp2 * tmp2
val -= 75.0 * tmp1 * tmp1 * tmp1 * tmp1
else:
val = 0.0
return val * fac
[docs] def gradient(self, xij=[0., 0., 0.], rij=1.0, h=1.0, grad=[0, 0, 0]):
h1 = 1. / h
# compute the gradient.
if (rij > 1e-12):
wdash = self.dwdq(rij, h)
tmp = wdash * h1 / rij
else:
tmp = 0.0
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)