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import abc
import math
import numpy as np
class ExpansionBasis1D(object):
"""
A family of one-dimensional functions that make up a basis.
"""
__metaclass__ = abc.ABCMeta
@abc.abstractmethod
def eval(self, idx, coord, diff_order = 0):
"""
Evaluate the diff_order-th derivative of the idx-th basis function at
the specified coordinates (where zeroth derivative means evaluating the
function itself).
"""
pass
@abc.abstractmethod
def colloc_grid(self, order):
"""
Get the coordinates of the optimal collocation grid of the specified
order (i.e. exactly order coordinates will be returned).
"""
pass
class CosBasis(ExpansionBasis1D):
PARITY_NONE = 0
PARITY_EVEN = 1
PARITY_ODD = 2
_diff_fact = [(1, np.cos), (-1, np.sin), (-1, np.cos), (1, np.sin)]
_parity = None
def __init__(self, parity = PARITY_NONE):
self._parity = parity
def eval(self, idx, coord, diff_order = 0):
fact, f = self._diff_fact[diff_order % 4]
if self._parity == self.PARITY_EVEN:
idx *= 2
elif self._parity == self.PARITY_ODD:
idx = 2 * idx + 1
fact *= idx ** diff_order
return fact * f(idx * coord)
def colloc_grid(self, order):
if self._parity == self.PARITY_NONE:
return (np.array(range(0, order)) + 1) * np.pi / (order + 1)
return (np.array(range(0, order))) * np.pi / (order - 1)
else:
return (np.array(range(0, order)) + 1) * np.pi / (2 * (order + 1))
class ChebBasis(ExpansionBasis1D):
_scale = None
def __init__(self, scale = 1.0):
self._scale = scale
def eval(self, idx, coord, diff_order = 0):
x = 2.0 * coord / self._scale - 1.0
t = np.arccos(x)
st = np.sin(t)
#idx += 1
if diff_order == 0:
return np.cos(idx * t);
elif diff_order == 1:
return (2.0 / self._scale) * np.where(np.fabs(np.fabs(x) - 1.0) < 1e-10, idx * idx, idx * np.sin(idx * t) / st)
elif diff_order == 2:
return ((2.0 / self._scale) ** 2) * np.where(np.fabs(np.fabs(x) - 1.0) < 1e-10, idx * idx * (idx * idx - 1.0) / 3., -(idx ** 2) * np.cos(idx * t) / (st * st) + idx * np.cos(t) * np.sin(idx * t) / (st * st * st))
else:
raise NotImplementedError
def colloc_grid(self, order):
if self._parity == self.PARITY_NONE:
return (np.array(range(0, order)) + 1) * 2 * np.pi / (order + 1)
else:
return (np.array(range(0, order)) + 1) * np.pi / (2 * (order + 1))
class ChebEvenBasis(ExpansionBasis1D):
_scale = None
def __init__(self, scale = 1.0):
self._scale = scale
def eval(self, idx, coord, diff_order = 0):
x = coord / self._scale
t = np.arccos(x)
st = np.sin(t)
idx *= 2
if diff_order == 0:
return np.cos(idx * t);
elif diff_order == 1:
return (1.0 / self._scale) * np.where(np.fabs(np.fabs(x) - 1.0) < 1e-10, idx * idx, idx * np.sin(idx * t) / st)
elif diff_order == 2:
return ((1.0 / self._scale) ** 2) * np.where(np.fabs(np.fabs(x) - 1.0) < 1e-10, idx * idx * (idx * idx - 1.0) / 3., -(idx ** 2) * np.cos(idx * t) / (st * st) + idx * np.cos(t) * np.sin(idx * t) / (st * st * st))
else:
raise NotImplementedError
def colloc_grid(self, order):
if self._parity == self.PARITY_NONE:
return (np.array(range(0, order)) + 1) * 2 * np.pi / (order + 1)
else:
return (np.array(range(0, order)) + 1) * np.pi / (2 * (order + 1))
class ChebTLBasis(ExpansionBasis1D):
_scale = None
def __init__(self, scale = 1.0):
self._scale = scale
def eval(self, idx, coord, diff_order = 0):
y = np.abs(coord)
y12 = np.sqrt(y)
t = 2 * np.arctan2(np.sqrt(self._scale), y12)
#idx += 1
if diff_order == 0:
return np.cos(idx * t)
elif diff_order == 1:
return np.where(np.abs(y) < 1e-10,
-2.0 * idx * idx * ((-1.0)**idx) / self._scale,
idx * np.sqrt(self._scale) * np.sin(idx * t) / (y12 * (y12 * y12 + self._scale)))
elif diff_order == 2:
return np.where(np.abs(y) < 1e-10,
4.0 * idx * idx * (idx * idx + 2) * ((-1.0) ** idx) / (3 * (self._scale**2)),
-((self._scale * idx * idx) * np.cos(idx * t) / (y * ((y + self._scale)**2)) + np.sqrt(self._scale) * idx * np.sin(idx * t) / (y12 * ((y + self._scale)**2)) + np.sqrt(self._scale) * idx * np.sin(idx * t) / (2 * y * y12 * (y +
self._scale))))
else:
raise NotImplementedError
def colloc_grid(self, order):
if self._parity == self.PARITY_NONE:
return (np.array(range(0, order)) + 1) * 2 * np.pi / (order + 1)
else:
return (np.array(range(0, order)) + 1) * np.pi / (2 * (order + 1))
class ChebSBBasis(ExpansionBasis1D):
_scale = None
def __init__(self, scale = 1.0):
self._scale = scale
def eval(self, idx, coord, diff_order = 0):
idx *= 2
val = np.arctan2(self._scale, np.abs(coord))
if diff_order == 0:
return np.sin((idx + 1) * val)
elif diff_order == 1:
return -self._scale * (idx + 1) * np.sign(coord) * np.cos((idx + 1) * val) / ((self._scale ** 2) + (coord ** 2))
elif diff_order == 2:
return self._scale * (idx + 1) * (2 * np.abs(coord) * np.cos((1 + idx) * val) - self._scale * (1 + idx) * np.sin((1 + idx) * val)) / (((self._scale ** 2) + (coord ** 2)) ** 2)
else:
raise NotImplementedError
def colloc_grid(self, order):
idx = np.array(range(0, order))
t = (idx + 2) * np.pi / (2 * order + 2)
return self._scale / np.tan(t)
class ChebTBBasis(ExpansionBasis1D):
_scale = None
def __init__(self, scale = 1.0):
self._scale = scale
def eval(self, idx, coord, diff_order = 0):
idx *= 2
val = np.arctan2(self._scale, np.abs(coord))
if diff_order == 0:
return np.cos(idx * val)
elif diff_order == 1:
return -self._scale * (idx + 1) * np.sign(coord) * np.cos((idx + 1) * val) / ((self._scale ** 2) + (coord ** 2))
elif diff_order == 2:
return self._scale * (idx + 1) * (2 * np.abs(coord) * np.cos((1 + idx) * val) - self._scale * (1 + idx) * np.sin((1 + idx) * val)) / (((self._scale ** 2) + (coord ** 2)) ** 2)
else:
raise NotImplementedError
def colloc_grid(self, order):
idx = np.array(range(0, order))
t = (idx + 2) * np.pi / (2 * order + 4)
return self._scale / np.tan(t)
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