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_quadratic(phi, R, S): B = [] # baker eq 9 w = [] # baker eq 11 for (s, q) in zip(S.verts, S.q): cur_phi, cur_qlin = qlinear(s, R) (phi1, phi2, phi3) = cur_phi B.append( [ phi1 * phi2, phi2 * phi3, phi3 * phi1, ] ) 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) = 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) = np.dot(np.linalg.pinv(A), b) error_term = a * phi[0] * phi[1]\ + b * phi[1] * phi[2]\ + c * phi[2] * phi[0] return error_term def get_error_cubic(phi, R, S): B = [] # baker eq 9 w = [] # baker eq 11 for (s, q) in zip(S.verts, S.q): cur_phi, cur_qlin = qlinear(s, R) (phi1, phi2, phi3) = cur_phi # basing this on eq 17 B.append( [ phi1 * phi2 * phi2, # a phi1 * phi3 * phi3, # b phi2 * phi1 * phi1, # c phi2 * phi3 * phi3, # d phi3 * phi1 * phi1, # e phi3 * phi2 * phi2, # f phi1 * phi2 * phi3, # g ] ) 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, g) = 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, g) = np.dot(np.linalg.pinv(A), b) error_term = a * phi[0] * phi[1] * phi[1]\ + b * phi[0] * phi[2] * phi[2]\ + c * phi[1] * phi[0] * phi[0]\ + d * phi[1] * phi[2] * phi[2]\ + e * phi[2] * phi[0] * phi[0]\ + f * phi[2] * phi[1] * phi[1]\ + g * phi[0] * phi[1] * phi[2]\ return error_term def get_error_sauron(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): error_term, abc = get_error_sauron(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 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