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Commit 2d7485a8 authored by Michael Kuron's avatar Michael Kuron :mortar_board:
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FiniteDifferenceStaggeredStencilDerivation

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pystencils/fd/derivation.py
+ 113
− 0
View file @ 2d7485a8
import warnings import warnings
from collections import defaultdict from collections import defaultdict
import itertools
import numpy as np import numpy as np
import sympy as sp import sympy as sp
from pystencils.field import Field from pystencils.field import Field
from pystencils.stencil import direction_string_to_offset
from pystencils.sympyextensions import multidimensional_sum, prod from pystencils.sympyextensions import multidimensional_sum, prod
from pystencils.utils import LinearEquationSystem, fully_contains from pystencils.utils import LinearEquationSystem, fully_contains
...@@ -228,3 +230,114 @@ class FiniteDifferenceStencilDerivation: ...@@ -228,3 +230,114 @@ class FiniteDifferenceStencilDerivation:
def __repr__(self): def __repr__(self):
return "Finite difference stencil of accuracy {}, isotropic error: {}".format(self.accuracy, return "Finite difference stencil of accuracy {}, isotropic error: {}".format(self.accuracy,
self.is_isotropic) self.is_isotropic)
class FiniteDifferenceStaggeredStencilDerivation:
"""Derives a finite difference stencil for application at a staggered position
Args:
neighbor: the neighbor direction string or vector at whose staggered position to calculate the derivative
dim: how many dimensions (2 or 3)
derivative: a tuple of directions over which to perform derivatives
"""
def __init__(self, neighbor, dim, derivative=tuple()):
if type(neighbor) is str:
neighbor = direction_string_to_offset(neighbor)
if dim == 2:
assert neighbor[dim:] == 0
assert derivative is tuple() or max(derivative) < dim
neighbor = sp.Matrix(neighbor[:dim])
pos = neighbor / 2
def unitvec(i):
"""return the `i`-th unit vector in three dimensions"""
a = np.zeros(dim, dtype=int)
a[i] = 1
return a
def flipped(a, i):
"""return `a` with its `i`-th element's sign flipped"""
a = a.copy()
a[i] *= -1
return a
# determine the points to use, coordinates are relative to position
points = []
if np.linalg.norm(neighbor, 1) == 1:
main_points = [neighbor / 2, neighbor / -2]
elif np.linalg.norm(neighbor, 1) == 2:
nonzero_indices = [i for i, v in enumerate(neighbor) if v != 0 and i < dim]
main_points = [neighbor / 2, neighbor / -2, flipped(neighbor / 2, nonzero_indices[0]),
flipped(neighbor / -2, nonzero_indices[0])]
else:
main_points = [neighbor.multiply_elementwise(sp.Matrix(c) / 2)
for c in itertools.product([-1, 1], repeat=3)]
points += main_points
zero_indices = [i for i, v in enumerate(neighbor) if v == 0 and i < dim]
for i in zero_indices:
points += [point + sp.Matrix(unitvec(i)) for point in main_points]
points += [point - sp.Matrix(unitvec(i)) for point in main_points]
points_tuple = tuple([tuple(p) for p in points])
self._stencil = points_tuple
# determine the stencil weights
if len(derivative) == 0:
weights = None
else:
derivation = FiniteDifferenceStencilDerivation(derivative, points_tuple).get_stencil()
if not derivation.accuracy:
raise Exception('the requested derivative cannot be performed with the available neighbors')
weights = derivation.weights
# if the weights are underdefined, we can choose the free symbols to find the sparsest stencil
free_weights = set(itertools.chain(*[w.free_symbols for w in weights]))
if len(free_weights) > 0:
zero_counts = defaultdict(list)
for values in itertools.product([-1, -sp.Rational(1, 2), 0, 1, sp.Rational(1, 2)],
repeat=len(free_weights)):
subs = {free_weight: value for free_weight, value in zip(free_weights, values)}
weights = [w.subs(subs) for w in derivation.weights]
if not all(a == 0 for a in weights):
zero_count = sum([1 for w in weights if w == 0])
zero_counts[zero_count].append(weights)
best = zero_counts[max(zero_counts.keys())]
if len(best) > 1: # if there are multiple, pick the one that contains a nonzero center weight
center = [tuple(p + pos) for p in points].index((0, 0, 0))
best = [b for b in best if b[center] != 0]
if len(best) > 1:
raise NotImplementedError("more than one suitable set of weights found, don't know how to proceed")
weights = best[0]
assert weights
points_tuple = tuple([tuple(p + pos) for p in points])
self._points = points_tuple
self._weights = weights
@property
def points(self):
"""return the points of the stencil"""
return self._points
@property
def stencil(self):
"""return the points of the stencil relative to the staggered position specified by neighbor"""
return self._stencil
@property
def weights(self):
"""return the weights of the stencil"""
assert self._weights is not None
return self._weights
def visualize(self):
if self._weights is None:
ws = None
else:
ws = np.array([w for w in self.weights if w != 0], dtype=float)
pts = np.array([p for i, p in enumerate(self.points) if self.weights[i] != 0], dtype=int)
from pystencils.stencil import plot
plot(pts, data=ws)
def apply(self, field):
return sum([field.__getitem__(point) * weight for point, weight in zip(self.points, self.weights)])
pystencils_tests/test_fd_derivation.ipynb
+ 213
− 26
View file @ 2d7485a8
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
from pystencils.session import * from pystencils.session import *
from pystencils.fd.derivation import * from pystencils.fd.derivation import *
``` ```
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
# 2D standard stencils # 2D standard stencils
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
stencil = [(-1, 0), (1, 0), (0, -1), (0, 1), (0, 0)] stencil = [(-1, 0), (1, 0), (0, -1), (0, 1), (0, 0)]
standard_2d_00 = FiniteDifferenceStencilDerivation((0,0), stencil) standard_2d_00 = FiniteDifferenceStencilDerivation((0,0), stencil)
f = ps.fields("f: [2D]") f = ps.fields("f: [2D]")
standard_2d_00_res = standard_2d_00.get_stencil() standard_2d_00_res = standard_2d_00.get_stencil()
res = standard_2d_00_res.apply(f.center) res = standard_2d_00_res.apply(f.center)
expected = f[-1, 0] - 2 * f[0, 0] + f[1, 0] expected = f[-1, 0] - 2 * f[0, 0] + f[1, 0]
assert res == expected assert res == expected
``` ```
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
assert standard_2d_00_res.accuracy == 2 assert standard_2d_00_res.accuracy == 2
assert not standard_2d_00_res.is_isotropic assert not standard_2d_00_res.is_isotropic
standard_2d_00_res standard_2d_00_res
``` ```
%% Output %% Output
Finite difference stencil of accuracy 2, isotropic error: False Finite difference stencil of accuracy 2, isotropic error: False
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
standard_2d_00.get_stencil().as_matrix() standard_2d_00.get_stencil().as_matrix()
``` ```
%% Output %% Output
---------------------------------------------------------------------------
AttributeError Traceback (most recent call last) $$\left[\begin{matrix}0 & 0 & 0\\1 & -2 & 1\\0 & 0 & 0\end{matrix}\right]$$
<ipython-input-4-ea41cd8e50a0> in <module> ⎡0 0 0⎤
----> 1 standard_2d_00.get_stencil().as_matrix() ⎢ ⎥
⎢1 -2 1⎥
~/pystencils/pystencils/pystencils/fd/derivation.py in as_matrix(self) ⎢ ⎥
185 for direction, weight in zip(self.stencil, self.weights): ⎣0 0 0⎦
186 result[max_offset - direction[1], max_offset + direction[0]] = weight
--> 187 result.tomatrix().tomatrix()
188 print("test")
189 if dim == 3:
~/anaconda3/envs/pystencils/lib/python3.7/site-packages/sympy/matrices/matrices.py in __getattr__(self, attr)
2139 else:
2140 raise AttributeError(
-> 2141 "%s has no attribute %s." % (self.__class__.__name__, attr))
2142
2143 def __len__(self):
AttributeError: MutableDenseMatrix has no attribute tomatrix.
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
# 2D isotropic stencils # 2D isotropic stencils
## second x-derivative ## second x-derivative
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
stencil = [(i, j) for i in (-1, 0, 1) for j in (-1, 0, 1)] stencil = [(i, j) for i in (-1, 0, 1) for j in (-1, 0, 1)]
isotropic_2d_00 = FiniteDifferenceStencilDerivation((0,0), stencil) isotropic_2d_00 = FiniteDifferenceStencilDerivation((0,0), stencil)
isotropic_2d_00_res = isotropic_2d_00.get_stencil(isotropic=True) isotropic_2d_00_res = isotropic_2d_00.get_stencil(isotropic=True)
assert isotropic_2d_00_res.is_isotropic assert isotropic_2d_00_res.is_isotropic
assert isotropic_2d_00_res.accuracy == 2 assert isotropic_2d_00_res.accuracy == 2
isotropic_2d_00_res isotropic_2d_00_res
``` ```
%% Output
Finite difference stencil of accuracy 2, isotropic error: True
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
isotropic_2d_00_res.as_matrix() isotropic_2d_00_res.as_matrix()
``` ```
%% Output
$$\left[\begin{matrix}\frac{1}{12} & - \frac{1}{6} & \frac{1}{12}\\\frac{5}{6} & - \frac{5}{3} & \frac{5}{6}\\\frac{1}{12} & - \frac{1}{6} & \frac{1}{12}\end{matrix}\right]$$
⎡1/12 -1/6 1/12⎤
⎢ ⎥
⎢5/6 -5/3 5/6 ⎥
⎢ ⎥
⎣1/12 -1/6 1/12⎦
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
plt.figure(figsize=(2,2)) plt.figure(figsize=(2,2))
isotropic_2d_00_res.visualize() isotropic_2d_00_res.visualize()
``` ```
%% Output
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
expected_result = sp.Matrix([[1, -2, 1], [10, -20, 10], [1, -2, 1]]) / 12 expected_result = sp.Matrix([[1, -2, 1], [10, -20, 10], [1, -2, 1]]) / 12
assert expected_result == isotropic_2d_00_res.as_matrix() assert expected_result == isotropic_2d_00_res.as_matrix()
``` ```
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
type(isotropic_2d_00_res.as_matrix()) type(isotropic_2d_00_res.as_matrix())
``` ```
%% Output
sympy.matrices.dense.MutableDenseMatrix
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
type(expected_result) type(expected_result)
``` ```
%% Output
sympy.matrices.dense.MutableDenseMatrix
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
## Isotropic laplacian ## Isotropic laplacian
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
isotropic_2d_11 = FiniteDifferenceStencilDerivation((1,1), stencil) isotropic_2d_11 = FiniteDifferenceStencilDerivation((1,1), stencil)
isotropic_2d_11_res = isotropic_2d_11.get_stencil(isotropic=True) isotropic_2d_11_res = isotropic_2d_11.get_stencil(isotropic=True)
iso_laplacian = isotropic_2d_00_res.as_matrix() + isotropic_2d_11_res.as_matrix() iso_laplacian = isotropic_2d_00_res.as_matrix() + isotropic_2d_11_res.as_matrix()
iso_laplacian iso_laplacian
``` ```
%% Output
$$\left[\begin{matrix}\frac{1}{6} & \frac{2}{3} & \frac{1}{6}\\\frac{2}{3} & - \frac{10}{3} & \frac{2}{3}\\\frac{1}{6} & \frac{2}{3} & \frac{1}{6}\end{matrix}\right]$$
⎡1/6 2/3 1/6⎤
⎢ ⎥
⎢2/3 -10/3 2/3⎥
⎢ ⎥
⎣1/6 2/3 1/6⎦
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
expected_result = sp.Matrix([[1, 4, 1], [4, -20, 4], [1, 4, 1]]) / 6 expected_result = sp.Matrix([[1, 4, 1], [4, -20, 4], [1, 4, 1]]) / 6
assert iso_laplacian == expected_result assert iso_laplacian == expected_result
``` ```
%% Cell type:markdown id: tags:
# stencils for staggered fields
%% Cell type:code id: tags:
``` python
half = sp.Rational(1, 2)
fd_points_ex = (
(half, 0),
(-half, 0),
(half, 1),
(half, -1),
(-half, 1),
(-half, -1)
)
assert set(fd_points_ex) == set(FiniteDifferenceStaggeredStencilDerivation("E", 2).stencil)
fd_points_ey = (
(0, half),
(0, -half),
(-1,-half),
(-1, half),
(1, -half),
(1, half)
)
assert set(fd_points_ey) == set(FiniteDifferenceStaggeredStencilDerivation("N",2).stencil)
fd_points_c = (
(half, half),
(-half, -half),
(half, -half),
(-half, half)
)
assert set(fd_points_c) == set(FiniteDifferenceStaggeredStencilDerivation("NE",2).stencil)
assert len(FiniteDifferenceStaggeredStencilDerivation("E",3).points) == 10
assert len(FiniteDifferenceStaggeredStencilDerivation("NE",3).points) == 12
assert len(FiniteDifferenceStaggeredStencilDerivation("TNE",3).points) == 8
```
%% Cell type:code id: tags:
``` python
c = ps.fields("c: [2D]")
c3 = ps.fields("c3: [3D]")
assert FiniteDifferenceStaggeredStencilDerivation("E", 2, (0,)).apply(c) == c[1, 0] - c[0, 0]
assert FiniteDifferenceStaggeredStencilDerivation("W", 2, (0,)).apply(c) == c[0, 0] - c[-1, 0]
assert FiniteDifferenceStaggeredStencilDerivation("N", 2, (1,)).apply(c) == c[0, 1] - c[0, 0]
assert FiniteDifferenceStaggeredStencilDerivation("S", 2, (1,)).apply(c) == c[0, 0] - c[0, -1]
assert FiniteDifferenceStaggeredStencilDerivation("E", 3, (0,)).apply(c3) == c3[1, 0, 0] - c3[0, 0, 0]
assert FiniteDifferenceStaggeredStencilDerivation("W", 3, (0,)).apply(c3) == c3[0, 0, 0] - c3[-1, 0, 0]
assert FiniteDifferenceStaggeredStencilDerivation("N", 3, (1,)).apply(c3) == c3[0, 1, 0] - c3[0, 0, 0]
assert FiniteDifferenceStaggeredStencilDerivation("S", 3, (1,)).apply(c3) == c3[0, 0, 0] - c3[0, -1, 0]
assert FiniteDifferenceStaggeredStencilDerivation("T", 3, (2,)).apply(c3) == c3[0, 0, 1] - c3[0, 0, 0]
assert FiniteDifferenceStaggeredStencilDerivation("B", 3, (2,)).apply(c3) == c3[0, 0, 0] - c3[0, 0, -1]
assert FiniteDifferenceStaggeredStencilDerivation("NE", 2, (0,)).apply(c) + FiniteDifferenceStaggeredStencilDerivation("NE", 2, (1,)).apply(c) == c[1, 1] - c[0, 0]
assert FiniteDifferenceStaggeredStencilDerivation("NE", 3, (0,)).apply(c3) + FiniteDifferenceStaggeredStencilDerivation("NE", 3, (1,)).apply(c3) == c3[1, 1, 0] - c3[0, 0, 0]
```
%% Cell type:code id: tags:
``` python
FiniteDifferenceStaggeredStencilDerivation("NE", 2, (0, 1)).visualize()
```
%% Output
%% Cell type:code id: tags:
``` python
```
......
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