from baker import * import numpy as np import sys from tools import smberror 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 = "warning: get_phis: calculation of phis yielded a linearly dependant system (%s)" % e # TODO: log this -- > print >> sys.stderr, msg raise smberror(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: print >> sys.stderr, "warning: get_phis_3D: calculation of phis yielded a linearly dependant system", 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.points) qlin = 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.points) 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.points, 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: print >> sys.stderr, "warning: run_baker: linear calculation went bad, resorting to np.linalg.pinv", 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, a, b, c def get_error_cubic(phi, R, S): B = [] # baker eq 9 w = [] # baker eq 11 for (s, q) in zip(S.points, S.q): cur_phi, cur_qlin = qlinear(s, R) (phi1, phi2, phi3) = cur_phi # basing this on eq 17 B.append( [ phi1 * phi2, # a phi1 * phi3, # b phi2 * phi1, # c phi2 * phi3, # d phi3 * phi1, # e phi3 * 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: print >> sys.stderr, "warning: run_baker: linear calculation went bad, resorting to np.linalg.pinv", e (a, b, c, d, e, f, g) = np.dot(np.linalg.pinv(A), b) error_term = a * phi[0] * phi[1]\ + b * phi[0] * phi[2]\ + c * phi[1] * phi[0]\ + d * phi[1] * phi[2]\ + e * phi[2] * phi[0]\ + f * phi[2] * phi[1]\ + g * phi[0] * phi[1] * phi[2]\ return error_term, a, b, c 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 """ # calculate values only for the simplex triangle phi, qlin = qlinear(X, R) if [i for i in phi if i <= 0.0]: s = "this is not a containing simplex:\n" s += " X: %s\n" % X s += " R: %s\n" % R s += " phi: %s, sum(%0.4e)\n" % (phi, sum(phi)) print >> sys.stderr, s raise smberror("simplex does not contain point") if len(S.points) == 0: answer = { 'a': None, 'b': None, 'c': None, 'qlin': qlin, 'error': None, 'final': None, } return answer if order == 2: error_term, a, b, c = get_error_quadratic(phi, R, S) elif order == 3: error_term, a, b, c = get_error_cubic(phi, R, S) else: raise smberror('unacceptable order for baker method') q_final = qlin + error_term answer = { 'a': a, 'b': b, 'c': c, 'qlin': qlin, 'error': error_term, '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 [i for i in phi if i <= 0.0]: s = "this is not a containing simplex:\n" s += " X: %s\n" % X s += " R: %s\n" % R s += " phi: %s, sum(%0.4e)\n" % (phi, sum(phi)) print >> sys.stderr, s raise smberror("not containing simplex") if len(S.points) == 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.points, 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: print >> sys.stderr, "warning: run_baker: linear calculation went bad, resorting to np.linalg.pinv", 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