smbinterp/interp/baker/__init__.py

import sys

import numpy as np

import itertools
from   interp.tools import log

def get_phis(X, R):
  """
    The get_phis function is used to get barycentric coordonites for a point on a triangle.

    X -- the destination point (2D)
         X = [0,0]
    r -- the three points that make up the containing triangular simplex (2D)
         r = [[-1, -1], [0, 2], [1, -1]]

    this will return [0.333, 0.333, 0.333]
  """

  # baker: eq 7
  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]
              ])
  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)

  return phi

def get_phis_3D(X, R):
  """
    The get_phis function is used to get barycentric coordonites for a point on a tetrahedron.

    X -- the destination point (3D)
         X = [0,0,0]
    R -- the four points that make up the containing simplex, tetrahedron (3D)
         R = [
               [0.0, 0.0, 1.0],
               [0.94280904333606508, 0.0, -0.3333333283722672],
               [-0.47140452166803232, 0.81649658244673617, -0.3333333283722672],
               [-0.47140452166803298, -0.81649658244673584, -0.3333333283722672],
             ]

    this (should) will return [0.25, 0.25, 0.25, 0.25]
  """

  # baker: eq 7
  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]
              ])
  try:
    phi = np.linalg.solve(A,b)
  except np.linalg.LinAlgError as e:
    log.error("calculation of phis yielded a linearly dependant system: %s" % e)
    phi = np.dot(np.linalg.pinv(A), b)

  return phi


def qlinear(X, R):
  """
    this calculates the linear portion of q from X to R

    also, this is baker eq 3

    X = destination point
    R = simplex points
    q = CFD quantities of interest at the simplex points
  """

  phis = get_phis(X, R.verts)
  qlin = np.sum([q_i * phi_i for q_i, phi_i in zip(R.q, phis)])
  return phis, qlin

def qlinear_3D(X, R):
  """
    this calculates the linear portion of q from X to R

    X = destination point
    R = simplex points
    q = CFD quantities of interest at the simplex points(R)
  """

  phis = get_phis_3D(X, R.verts)
  qlin = sum([q_i * phi_i for q_i, phi_i in zip(R.q, phis)])
  return phis, qlin

def get_error(phi, R, S, order = 2):
  log.debug("len(phi): %d"% len(phi))
  B = [] # baker eq 9
  w = [] # baker eq 11

  p  = pattern(order, len(phi), offset = -1)
  log.debug("pattern: %s" % p)

  for (s,q) in zip(S.verts, S.q):
    cur_phi, cur_qlin = qlinear(s, R)
    l = []
    for i in p:
      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.debug("B: %s" % B)
  log.debug("w: %s" % w)


  B = np.array(B)
  w = np.array(w)

  A = np.dot(B.T, B)
  b = np.dot(B.T, w)

  # baker solve eq 10
  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)

  log.debug(len(abc) == len(p))

  error_term = 0.0
  for (a, i) in zip(abc, p):
    cur_sum = a
    for j in i:
      cur_sum *= phi[j]
    error_term += cur_sum

  log.debug("error_term smb: %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 (2D)
         X = [0,0]

    R = Simplex
    S = extra points
  """
  log.debug("order = %d" % order)

  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
    return answer
  elif order in (2,3,4,5,6):
    error_term, abc = get_error(phi, R, S, order)
  else:
    raise Exception('unsupported order for baker method')

  q_final = qlin + error_term

  answer['qlin' ] =  qlin
  answer['error'] =  error_term
  answer['final'] =  q_final
  answer['abc'  ] =  abc

  return answer

def run_baker_3D(X, R, S):
  """
    This is the main function to call to get an interpolation to X from the input meshes

    X -- the destination point (3D)
         X = [0,0,0]

    R = Simplex (4 points, contains X)
    S = extra points (surrounding, in some manner, R and X, but not in R)
  """

  # calculate values only for the triangle
  phi, qlin = qlinear_3D(X, R)

  if len(S.verts) == 0:
    answer = {
               'a': None,
               'b': None,
               'c': None,
               'd': None,
               'e': None,
               'f': None,
               'qlin': qlin,
               'error': None,
               'final': None,
              }
    return answer

  B = [] # baker eq 9
  w = [] # baker eq 11

  for (s, q) in zip(S.verts, S.q):
    cur_phi, cur_qlin = qlinear_3D(s, R)
    (phi1, phi2, phi3, phi4) = cur_phi

    B.append(
              [
                phi1 * phi2,
                phi1 * phi3,
                phi1 * phi4,
                phi2 * phi3,
                phi2 * phi4,
                phi3 * phi4,
              ]
            )

    w.append(q - cur_qlin)

  B = np.array(B)
  w = np.array(w)

  A = np.dot(B.T, B)
  b = np.dot(B.T, w)

  # baker solve eq 10
  try:
    (a, b, c, d, e, f) = np.linalg.solve(A,b)
  except np.linalg.LinAlgError as e:
    log.error("linear calculation went bad, resorting to np.linalg.pinv: %s", e)
    (a, b, c, d, e, f) = np.dot(np.linalg.pinv(A), b)

  error_term =  a * phi[0] * phi[1]\
              + b * phi[0] * phi[2]\
              + c * phi[0] * phi[3]\
              + d * phi[1] * phi[2]\
              + e * phi[1] * phi[3]\
              + f * phi[2] * phi[3]

  q_final = qlin + error_term

  answer = {
             'a': a,
             'b': b,
             'c': c,
             'd': d,
             'e': e,
             'f': f,
             'qlin': qlin,
             'error': error_term,
             'final': q_final,
            }

  return answer

def _boxings(n, k):
  """\
    source for this function:
    http://old.nabble.com/Simple-combinatorics-with-Numpy-td20086915.html
    http://old.nabble.com/Re:-Simple-combinatorics-with-Numpy-p20099736.html
  """
  seq, i = [n] * k + [0], k
  while i:
    yield tuple(seq[i] - seq[i+1] for i in xrange(k))
    i = seq.index(0) - 1
    seq[i:k] = [seq[i] - 1] * (k-i)

def _samples_ur(items, k, offset = 0):
  """Returns k unordered samples (with replacement) from items."""
  n = len(items)
  for sample in _boxings(k, n):
    selections = [[items[i]]*count for i,count in enumerate(sample)]
    yield tuple([x + offset  for sel in selections for x in sel])

def pattern(power, phicount, offset = 0):
  log.debug("(power = %s, phicount = %s)" % (power, phicount))
  r = []
  for i in _samples_ur(range(1, phicount + 1), power, offset):
    if not len(set(i)) == 1:
      r.append(i)
  return r