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tools.py
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tools.py
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import numpy as np
import pyfftw.interfaces.numpy_fft as fft
# from numpy import fft
from pyfftw.interfaces import cache
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
cache.enable()
def N_boyd(M):
""" Boyd's rule of thumb. M is the number of wave lengths
given by M = L/l where L is the box size and l is the smallest
scale to be resolved """
return int(2**np.ceil(np.log2(4 * (M - 1) + 6)))
class ScalarTool(object):
"""
Description:
ScalarTool contains a collection of functions necessary to compute basic
operations such as gradients and norms on scalars defined on a 2D periodic
square domain of length L and discretized in each dimension into N
intervals.
Inputs:
N - number of discretized points in each dimension
L - length of side
"""
def __init__(self, N, L):
self.N = N
self.L = L
self.h = self.L / self.N
self.X = np.mgrid[:self.N, :self.N].astype(float) * self.h
self.Nf = self.N // 2 + 1
self.kx = np.fft.fftfreq(self.N, 1. / self.N).astype(int)
self.ky = self.kx[:self.Nf].copy()
self.ky[-1] *= -1
self.K = np.array(np.meshgrid(
self.kx, self.ky, indexing='ij'), dtype=int)
self.K2 = np.sum(self.K * self.K, 0, dtype=int)
self.KoverK2 = self.K.astype(
float) / np.where(self.K2 == 0, 1, self.K2).astype(float)
self.oneoverK2 = 1.0 / \
np.where(self.K2 == 0.0, 1.0, self.K2).astype(float)
self.mean_zero_array = self.K2 != 0.0
self.kmax_dealias = 2. / 3. * (self.N / 2 + 1)
self.dealias_array = np.array((abs(self.K[0]) < self.kmax_dealias) * (
abs(self.K[1]) < self.kmax_dealias), dtype=bool)
self.num_threads = 1
def l2norm(self, scalar):
self.scalar_input_test(scalar)
return np.sum(np.ravel(scalar)**2.0 * self.h**2.0)**0.5
def grad(self, scalar):
self.scalar_input_test(scalar)
scalar_hat = self.fft(scalar)
return fft.irfftn(1.0j * self.K * (2 * np.pi / self.L) * scalar_hat, axes=(1, 2), threads=self.num_threads)
def h1norm(self, scalar):
self.scalar_input_test(scalar)
grad_scalar = self.grad(scalar)
grad_scalar_sq = np.sum(grad_scalar * grad_scalar, 0)
integrand = np.ravel(grad_scalar_sq)
return np.sum(integrand * self.h**2.0)**0.5
def lap(self, scalar):
self.scalar_input_test(scalar)
scalar_hat = self.fft(scalar)
return self.ifft((-1.0) * self.K2 * (2 * np.pi / self.L)**2.0 * scalar_hat)
def invlap(self, scalar):
self.scalar_input_test(scalar)
scalar_hat = self.fft(scalar)
return np.real(self.ifft(-1.0 * (2.0 * np.pi / self.L)**(-2.0) *
self.oneoverK2 * self.mean_zero_array * scalar_hat))
def grad_invlap(self, scalar):
self.scalar_input_test(scalar)
scalar_hat = self.fft(scalar)
return fft.irfftn(-1.0j * self.KoverK2 * (2 * np.pi / self.L)**(-1.0) * scalar_hat, axes=(1, 2), threads=self.num_threads)
def hm1norm(self, scalar):
self.scalar_input_test(scalar)
grad_invlap_scalar = self.grad_invlap(scalar)
grad_invlap_scalar_sq = np.sum(grad_invlap_scalar *
grad_invlap_scalar, 0) # dot product
integrand = np.ravel(grad_invlap_scalar_sq)
return np.sum(integrand * self.h**2.0)**0.5
def plot(self, scalar, high_quality=False, fixed_vertical_axis=False):
if high_quality:
plt.rc('text', usetex=True)
plt.rc('font', family='serif', size=12)
else:
plt.rc('text', usetex=False)
plt.rc('font', family='sans-serif', size=12)
if fixed_vertical_axis:
vmin = -1
vmax = 1
else:
vmin = np.amin(scalar)
vmax = np.amax(scalar)
self.scalar_input_test(scalar)
im = plt.imshow(np.transpose(scalar),
cmap=plt.cm.gray,
extent=(0, self.L, 0, self.L),
origin="lower",
vmin=vmin,
vmax=vmax)
plt.xlabel(r'$x$')
plt.ylabel(r'$y$')
plt.colorbar(im)
def scalar_input_test(self, scalar):
if np.shape(scalar) != (self.N, self.N):
print(np.shape(scalar))
raise InputError("Scalar field array does not have correct shape.")
if not np.all(np.isrealobj(scalar)):
raise InputError("Scalar field array should be real.")
def scalar_hat_input_test(self, scalar_hat):
if np.shape(scalar_hat) != (self.N, self.Nf):
print(np.shape(scalar_hat))
raise InputError("Scalar field array does not have correct shape.")
def sint(self, scalar):
""" Performs spatial integration """
self.scalar_input_test(scalar)
return np.sum(np.ravel(scalar) * self.h**2.0)
def dealias(self, scalar):
""" Perform 1/3 dealias on scalar """
self.scalar_input_test(scalar)
temp_hat = self.fft(scalar) * self.dealias_array
return self.ifft(temp_hat)
def fft(self, scalar):
""" Performs fft of scalar field """
self.scalar_input_test(scalar)
return fft.rfftn(scalar, threads=self.num_threads)
def ifft(self, scalar_hat):
""" Performs inverse fft of scalar field """
self.scalar_hat_input_test(scalar_hat)
return fft.irfftn(scalar_hat, threads=self.num_threads)
def subtract_mean(self, scalar):
""" subtract off mean """
self.scalar_input_test(scalar)
scalar_hat = self.fft(scalar)
return np.real(self.ifft(scalar_hat * self.mean_zero_array))
def get_spectrum(self, scalar):
""" gets spectrum """
self.scalar_input_test(scalar)
scalar_hat = self.fft(scalar)
amp = 2.0 * np.absolute(scalar_hat) / \
self.N**2.0 # corrects normalization
k_list = np.arange(0, self.N // 2 + 1, 1) # beginning of bin intervals
K_inf = np.maximum(abs(self.K[0]), abs(self.K[1])) # infinity norm
amp_list = []
for k in k_list:
K_shell_bool = k == K_inf
max_amp_in_shell = np.amax(amp * K_shell_bool)
amp_list.append(max_amp_in_shell)
return [k_list, amp_list]
def isblocked(self, scalar, k_frac=0.85, amp_thres=10.**(-10)):
""" determines if spectral blocking is present """
self.scalar_input_test(scalar)
k_thres = int(k_frac * (self.N / 2))
[k_list, amp_list] = self.get_spectrum(scalar)
amp_beyond_k_thres = [amp_list[i]
for i in range(len(k_list)) if k_list[i] > k_thres]
return max(amp_beyond_k_thres) > amp_thres
class VectorTool(object):
"""
Description:
VectorTool contains a collection of functions necessary to compute basic
operations such as norms on scalars defined on a 2D periodic
square domain of length L and discretized in each dimension into N
intervals.
Inputs:
N - number of discretized points in each dimension
L - length of side
"""
def __init__(self, N, L):
self.N = N
self.L = L
self.h = self.L / self.N
self.X = np.mgrid[:self.N, :self.N].astype(float) * self.h
self.Nf = self.N // 2 + 1
self.kx = np.fft.fftfreq(self.N, 1. / self.N).astype(int)
self.ky = self.kx[:self.Nf].copy()
self.ky[-1] *= -1
self.K = np.array(np.meshgrid(
self.kx, self.ky, indexing='ij'), dtype=int)
self.K2 = np.sum(self.K * self.K, 0, dtype=int)
self.KoverK2 = self.K.astype(
float) / np.where(self.K2 == 0, 1, self.K2).astype(float)
self.oneoverK2 = 1.0 / \
np.where(self.K2 == 0.0, 1.0, self.K2).astype(float)
self.mean_zero_array = self.K2 != 0.0
self.kmax_dealias = 2. / 3. * (self.N / 2 + 1)
self.dealias_array = np.array((abs(self.K[0]) < self.kmax_dealias) * (
abs(self.K[1]) < self.kmax_dealias), dtype=bool)
self.num_threads = 1
def div(self, vector):
""" Take divergence of vector """
self.vector_input_test(vector)
vector_hat = self.fft(vector)
return fft.irfftn(np.sum(1j * self.K * (2 * np.pi / self.L) * vector_hat, 0), threads=self.num_threads)
def fft(self, vector):
""" Performs fft of vector field """
self.vector_input_test(vector)
return fft.rfftn(vector, axes=(1, 2), threads=self.num_threads)
def ifft(self, vector_hat):
""" Performs inverse fft of vector hat field """
self.vector_hat_input_test(vector_hat)
return fft.irfftn(vector_hat, axes=(1, 2), threads=self.num_threads)
def plot(self, vector, high_quality=False):
""" Plots a quiver plot of the vector field """
self.vector_input_test(vector)
if high_quality:
plt.rc('text', usetex=True)
plt.rc('font', family='serif', size=12)
else:
plt.rc('text', usetex=False)
plt.rc('font', family='sans-serif', size=12)
m = max(round(self.N / 25), 1)
Q = plt.quiver(self.X[0][1:-1:m, 1:-1:m],
self.X[1][1:-1:m, 1:-1:m],
vector[0][1:-1:m, 1:-1:m],
vector[1][1:-1:m, 1:-1:m], linewidths=2.0)
plt.quiverkey(
Q, 0.8, 1.03, 2, r'%.2f $\frac{m}{s}$' % np.amax(vector), labelpos='E',)
plt.xlabel(r'$x$')
plt.ylabel(r'$y$')
plt.title('')
plt.xlim(0.0, self.L)
plt.ylim(0.0, self.L)
plt.axis('scaled')
def dealias(self, vector):
""" Dealias vector """
self.vector_input_test(vector)
vector_hat = self.fft(vector)
vector_hat = vector_hat * self.dealias_array
return np.real(self.ifft(vector_hat))
def l2norm(self, vector):
""" L2 norm of a vector field """
self.vector_input_test(vector)
integrand = np.sum(vector * vector, 0)
return np.sum(np.ravel(integrand) * self.h**2)**0.5
def h1norm(self, vector):
""" L2 norm of a vector field """
self.vector_input_test(vector)
vector_hat = self.fft(vector)
grad_vx = fft.irfftn(1.0j * self.K * (2 * np.pi / self.L)
* vector_hat[0], axes=(1, 2), threads=self.num_threads)
grad_vy = fft.irfftn(1.0j * self.K * (2 * np.pi / self.L)
* vector_hat[1], axes=(1, 2), threads=self.num_threads)
integrand = (grad_vx[0]**2.0 + grad_vx[1]**2.0
+ grad_vy[0]**2.0 + grad_vy[1]**2.0)
return np.sum(np.ravel(integrand) * self.h**2)**0.5
def vector_input_test(self, vector):
""" Determines if vector is correct size """
if np.shape(vector) != (2, self.N, self.N):
print(np.shape(vector))
raise InputError("Vector field array does not have correct shape")
if not np.all(np.isrealobj(vector)):
raise InputError("Scalar field array should be real.")
def vector_hat_input_test(self, vector_hat):
""" Determines if vector is correct size """
if np.shape(vector_hat) != (2, self.N, self.Nf):
print(np.shape(vector_hat))
raise InputError("Vector field array does not have correct shape")
def is_incompressible(self, vector):
self.vector_input_test(vector)
return np.allclose(self.div(vector), 0)
def div_free_proj(self, vector):
""" performs leray divergence-free projection """
self.vector_input_test(vector)
vector_hat = self.fft(vector)
return self.ifft(vector_hat - self.KoverK2 * np.sum(self.K * vector_hat, 0))
def curl(self, vector):
""" Perform curl of vector """
self.vector_input_test(vector)
vector_hat = self.fft(vector)
w = fft.irfftn(
1j * self.K[0] * (2.0 * np.pi / self.L) * vector_hat[1]
- 1j * self.K[1] * (2.0 * np.pi / self.L) * vector_hat[0], threads=self.num_threads)
return w
def invlap(self, vector):
""" Inverse laplacian of vector """
self.vector_input_test(vector)
vector_hat = self.fft(vector)
return np.real(self.ifft(-1.0 * (2.0 * np.pi / self.L)**(-2.0) *
self.oneoverK2 * self.mean_zero_array * vector_hat))
def lap(self, vector):
""" Laplacian of vector """
self.vector_input_test(vector)
vector_hat = self.fft(vector)
return np.real(self.ifft(-1.0 * (2.0 * np.pi / self.L)**(2.0) * (self.K2) * vector_hat))
def subtract_mean(self, vector):
""" subtract off mean """
self.vector_input_test(vector)
vector_hat = self.fft(vector)
return np.real(self.ifft(vector_hat * self.mean_zero_array))
class InputError(Exception):
""" Input Error """
def __init__(self, message):
self.message = message