The normalization coefficient for 1D-QuinticSpline is $1/(60h)$ rather than $1/(120h)$.
Validation:
def quintic_spline(r, cut, dimension=3):
h = cut / 3
q = r / h
w = np.where(q < 3, (3 - q) ** 5, 0)
w -= np.where(q < 2, 6 * (2 - q) ** 5, 0)
w += np.where(q < 1, 15 * (1 - q) ** 5, 0)
if dimension == 1:
alphaD = 1 / (60 * h)
elif dimension == 2:
alphaD = 7 / (478 * np.pi * h ** 2)
else: # dimension == 3
alphaD = 1 / (120 * np.pi * h ** 3)
return w * alphaD
kernels = {'Quintic-Spline': quintic_spline}
def test_kernel(name, cutoff):
from scipy.integrate import quad, dblquad, tplquad
res1d, _ = quad(lambda r: kernels[name](r, cut=cutoff, dimension=1),
0, cutoff)
res2d, _ = dblquad(lambda r, theta: kernels[name](r, cut=cutoff, dimension=2) * r,
0, 2 * np.pi, 0, cutoff)
res3d, _ = tplquad(lambda r, phi, theta: kernels[name](r, cut=cutoff, dimension=3) * r * r * np.sin(phi),
0, 2 * np.pi, 0, np.pi, 0, cutoff)
flag_pass = True
for i, res in enumerate([res1d, res2d, res3d]):
print(f'Dimension {i+1}: result={res}.')
if abs(res - 1) > 1e-6:
print(f'Dimension {i + 1} does not satisfy the normalization condition!')
flag_pass = False
if flag_pass:
print(f'Kernel {name} has passed the normalization condition test!')
if __name__ == '__main__':
test_kernel('Quintic-Spline', 3)Output:
Dimension 1: result=0.9999999999999204.
Dimension 2: result=0.9999999999972096.
Dimension 3: result=0.9999999999634721.
Kernel Quintic-Spline has passed the normalization condition test!
The normalization coefficient for 1D-QuinticSpline is $1/(60h)$ rather than $1/(120h)$.
Validation:
Output: