221 lines
5.1 KiB
Python
221 lines
5.1 KiB
Python
import sys
|
|
|
|
import numpy as np
|
|
|
|
from functools import wraps
|
|
import itertools
|
|
|
|
import interp
|
|
import logging
|
|
log = logging.getLogger('interp')
|
|
|
|
def get_phis(X, R):
|
|
"""
|
|
The get_phis function is used to get barycentric coordonites for a
|
|
point on a triangle or tetrahedron. This is equation (*\ref{eq:qlinarea}*)
|
|
|
|
in 2D:
|
|
|
|
X - the destination point (2D)
|
|
X = [0,0]
|
|
R - the three points that make up the 2-D triangular simplex
|
|
R = [[-1, -1], [0, 2], [1, -1]]
|
|
|
|
this will return [0.333, 0.333, 0.333]
|
|
|
|
|
|
in 3D:
|
|
|
|
X - the destination point (3D)
|
|
X = [0,0,0]
|
|
R - the four points that make up the 3-D simplex (tetrahedron)
|
|
R = [
|
|
[ 0.0000, 0.0000, 1.0000],
|
|
[ 0.9428, 0.0000, -0.3333],
|
|
[-0.4714, 0.8165, -0.3333],
|
|
[-0.4714, -0.8165, -0.3333],
|
|
]
|
|
|
|
this will return [0.25, 0.25, 0.25, 0.25]
|
|
"""
|
|
|
|
# equations (*\ref{eq:lin3d}*) and (*\ref{eq:lin2d}*)
|
|
if len(X) == 2:
|
|
log.debug("running 2D")
|
|
A = np.array([
|
|
[ 1, 1, 1],
|
|
[R[0][0], R[1][0], R[2][0]],
|
|
[R[0][1], R[1][1], R[2][1]],
|
|
])
|
|
b = np.array([ 1,
|
|
X[0],
|
|
X[1]
|
|
])
|
|
elif len(X) == 3:
|
|
log.debug("running 3D")
|
|
A = np.array([
|
|
[ 1, 1, 1, 1 ],
|
|
[R[0][0], R[1][0], R[2][0], R[3][0]],
|
|
[R[0][1], R[1][1], R[2][1], R[3][1]],
|
|
[R[0][2], R[1][2], R[2][2], R[3][2]],
|
|
])
|
|
b = np.array([ 1,
|
|
X[0],
|
|
X[1],
|
|
X[2]
|
|
])
|
|
else:
|
|
raise Exception("inapropriate demension on X")
|
|
|
|
try:
|
|
phi = np.linalg.solve(A,b)
|
|
except np.linalg.LinAlgError as e:
|
|
msg = "calculation of phis yielded a linearly dependant system (%s)" % e
|
|
log.error(msg)
|
|
# raise Exception(msg)
|
|
phi = np.dot(np.linalg.pinv(A), b)
|
|
|
|
log.debug("phi: %s", phi)
|
|
|
|
return phi
|
|
|
|
def qlinear(X, R):
|
|
"""
|
|
this calculates the linear portion of q from R to X
|
|
|
|
This is equation (*\ref{eq:qlinbasis}*)
|
|
|
|
X = destination point
|
|
R = a inter.grid object; must have R.points and R.q
|
|
"""
|
|
|
|
phis = get_phis(X, R.verts)
|
|
qlin = np.sum([q_i * phi_i for q_i, phi_i in zip(R.q, phis)])
|
|
|
|
log.debug("phis: %s", phis)
|
|
log.debug("qlin: %s", qlin)
|
|
|
|
return phis, qlin
|
|
|
|
def get_error(phi, R, S, order = 2):
|
|
"""
|
|
Calculate the error approximation terms, returning the unknowns
|
|
a,b, and c in equation (*\ref{eq:quadratic2d}*).
|
|
"""
|
|
B = [] # equation ((*\ref{eq:B2d}*)
|
|
w = [] # equation ((*\ref{eq:w}*)
|
|
|
|
cur_pattern = pattern(len(phi), order)
|
|
log.info("pattern: %s" % cur_pattern)
|
|
|
|
for (s,q) in zip(S.verts, S.q):
|
|
cur_phi, cur_qlin = qlinear(s, R)
|
|
l = []
|
|
for i in cur_pattern:
|
|
cur_sum = cur_phi[i[0]]
|
|
for j in i[1:]:
|
|
cur_sum *= cur_phi[j]
|
|
l.append(cur_sum)
|
|
|
|
B.append(l)
|
|
w.append(q - cur_qlin)
|
|
|
|
log.info("B: %s" % B)
|
|
log.info("w: %s" % w)
|
|
|
|
|
|
B = np.array(B)
|
|
w = np.array(w)
|
|
|
|
A = np.dot(B.T, B)
|
|
b = np.dot(B.T, w)
|
|
|
|
try:
|
|
abc = np.linalg.solve(A,b)
|
|
except np.linalg.LinAlgError as e:
|
|
log.error("linear calculation went bad, resorting to np.linalg.pinv: %s" % e)
|
|
abc = np.dot(np.linalg.pinv(A), b)
|
|
|
|
error_term = 0.0
|
|
for (a, i) in zip(abc, cur_pattern):
|
|
cur_sum = a
|
|
for j in i:
|
|
cur_sum *= phi[j]
|
|
error_term += cur_sum
|
|
|
|
log.debug("error_term: %s" % error_term)
|
|
return error_term, abc
|
|
|
|
def run_baker(X, R, S, order=2):
|
|
"""
|
|
This is the main function to call to get an interpolation to X from the
|
|
input meshes
|
|
|
|
X -- the destination point
|
|
|
|
R = Simplex
|
|
S = extra points
|
|
"""
|
|
log.debug("order = %d" % order)
|
|
log.debug("extra points = %d" % len(S.verts))
|
|
|
|
answer = {
|
|
'qlin': None,
|
|
'error': None,
|
|
'final': None,
|
|
}
|
|
# calculate values only for the simplex triangle
|
|
phi, qlin = qlinear(X, R)
|
|
|
|
if order == 1:
|
|
answer['qlin'] = qlin
|
|
answer['final'] = qlin
|
|
return answer
|
|
elif order in xrange(2,11):
|
|
error_term, abc = get_error(phi, R, S, order)
|
|
else:
|
|
raise Exception('unsupported order "%d" for baker method' % order)
|
|
|
|
q_final = qlin + error_term
|
|
|
|
answer['qlin' ] = qlin
|
|
answer['error'] = error_term
|
|
answer['final'] = q_final
|
|
answer['abc' ] = abc
|
|
|
|
log.debug(answer)
|
|
|
|
return answer
|
|
|
|
|
|
def memoize(f):
|
|
"""
|
|
for more information on what I'm doing here, please read:
|
|
http://en.wikipedia.org/wiki/Memoize
|
|
"""
|
|
cache = {}
|
|
@wraps(f)
|
|
def memf(simplex_size, nu):
|
|
x = (simplex_size, nu)
|
|
if x not in cache:
|
|
log.debug("adding to cache: %s", x)
|
|
cache[x] = f(simplex_size, nu)
|
|
return cache[x]
|
|
return memf
|
|
|
|
|
|
@memoize
|
|
def pattern(simplex_size, nu):
|
|
"""
|
|
This function returns the pattern requisite to compose the error
|
|
approximation function, and the matrix B.
|
|
"""
|
|
log.debug("pattern: simplex: %d, order: %d" % (simplex_size, nu))
|
|
|
|
r = []
|
|
for i in itertools.product(xrange(simplex_size), repeat = nu):
|
|
if len(set(i)) !=1:
|
|
r.append(tuple(sorted(i)))
|
|
unique_r = list(set(r))
|
|
return unique_r
|