Zweite GHA numerisch
This commit is contained in:
Tammo.Weber
391
GHA_triaxial/panou_2013_2GHA_num.py
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391
GHA_triaxial/panou_2013_2GHA_num.py
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import numpy as np
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#from ellipsoide import EllipsoidTriaxial
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import Numerische_Integration.num_int_runge_kutta as rk
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# Panou 2013
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class EllipsoidTriaxial:
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def __init__(self, ax: float, ay: float, b: float):
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self.ax = ax
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self.ay = ay
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self.b = b
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self.ex = np.sqrt((self.ax**2 - self.b**2) / self.ax**2)
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self.ey = np.sqrt((self.ay**2 - self.b**2) / self.ay**2)
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self.ee = np.sqrt((self.ax**2 - self.ay**2) / self.ax**2)
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self.ex_ = np.sqrt((self.ax**2 - self.b**2) / self.b**2)
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self.ey_ = np.sqrt((self.ay**2 - self.b**2) / self.b**2)
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self.ee_ = np.sqrt((self.ax**2 - self.ay**2) / self.ay**2)
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self.Ex = np.sqrt(self.ax**2 - self.b**2)
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self.Ey = np.sqrt(self.ay**2 - self.b**2)
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self.Ee = np.sqrt(self.ax**2 - self.ay**2)
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@classmethod
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def init_name(cls, name: str):
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if name == "BursaFialova1993":
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ax = 6378171.36
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ay = 6378101.61
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b = 6356751.84
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return cls(ax, ay, b)
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elif name == "BursaSima1980":
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ax = 6378172
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ay = 6378102.7
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b = 6356752.6
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return cls(ax, ay, b)
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elif name == "BursaSima1980round":
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# Panou 2013
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ax = 6378172
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ay = 6378103
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b = 6356753
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return cls(ax, ay, b)
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elif name == "Eitschberger1978":
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ax = 6378173.435
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ay = 6378103.9
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b = 6356754.4
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return cls(ax, ay, b)
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elif name == "Bursa1972":
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ax = 6378173
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ay = 6378104
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b = 6356754
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return cls(ax, ay, b)
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elif name == "Bursa1970":
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ax = 6378173
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ay = 6378105
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b = 6356754
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return cls(ax, ay, b)
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elif name == "Bessel-biaxial":
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ax = 6377397.15509
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ay = 6377397.15508
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b = 6356078.96290
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return cls(ax, ay, b)
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def gha2_num(ell: EllipsoidTriaxial, beta_1, lamb_1, beta_2, lamb_2, n=16000, epsilon=10**-12, iter_max=30):
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"""
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:param ell: triaxiales Ellipsoid
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:param beta_1: reduzierte ellipsoidische Breite Punkt 1
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:param lamb_1: elllipsoidische Länge Punkt 1
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:param beta_2: reduzierte ellipsoidische Breite Punkt 2
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:param lamb_2: elllipsoidische Länge Punkt 2
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:param n: Anzahl Schritte
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:param epsilon:
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:param iter_max: Maximale Anzhal Iterationen
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:return:
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"""
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# h_x, h_y, h_e entsprechen E_x, E_y, E_e
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def arccot(x):
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return np.arctan2(1.0, x)
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def BETA_LAMBDA(beta, lamb):
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BETA = (ell.ay**2 * np.sin(beta)**2 + ell.b**2 * np.cos(beta)**2) / (ell.Ex**2 - ell.Ey**2 * np.sin(beta)**2)
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LAMBDA = (ell.ax**2 * np.sin(lamb)**2 + ell.ay**2 * np.cos(lamb)**2) / (ell.Ex**2 - ell.Ee**2 * np.cos(lamb)**2)
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# Erste Ableitungen von ΒETA und LAMBDA
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BETA_ = (ell.ax**2 * ell.Ey**2 * np.sin(2*beta)) / (ell.Ex**2 - ell.Ey**2 * np.sin(beta)**2)**2
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LAMBDA_ = - (ell.b**2 * ell.Ee**2 * np.sin(2*lamb)) / (ell.Ex**2 - ell.Ee**2 * np.cos(lamb)**2)**2
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# Zweite Ableitungen von ΒETA und LAMBDA
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BETA__ = ((2 * ell.ax**2 * ell.Ey**4 * np.sin(2*beta)**2) / (ell.Ex**2 - ell.Ey**2 * np.sin(beta)**2)**3) + ((2 * ell.ax**2 * ell.Ey**2 * np.cos(2*beta)) / (ell.Ex**2 - ell.Ey**2 * np.sin(beta)**2)**2)
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LAMBDA__ = (((2 * ell.b**2 * ell.Ee**4 * np.sin(2*lamb)**2) / (ell.Ex**2 - ell.Ee**2 * np.cos(lamb)**2)**3) -
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((2 * ell.b**2 * ell.Ee**2 * np.sin(2*lamb)) / (ell.Ex**2 - ell.Ee**2 * np.cos(lamb)**2)**2))
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E = BETA * (ell.Ey ** 2 * np.cos(beta) ** 2 + ell.Ee ** 2 * np.sin(lamb) ** 2)
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F = 0
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G = LAMBDA * (ell.Ey ** 2 * np.cos(beta) ** 2 + ell.Ee ** 2 * np.sin(lamb) ** 2)
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# Erste Ableitungen von E und G
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E_beta = BETA_ * (ell.Ey**2 * np.cos(beta)**2 + ell.Ee**2 * np.sin(lamb)**2) - BETA * ell.Ey**2 * np.sin(2*beta)
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E_lamb = BETA * ell.Ee**2 * np.sin(2*lamb)
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G_beta = - LAMBDA * ell.Ey**2 * np.sin(2*beta)
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G_lamb = LAMBDA_ * (ell.Ey**2 * np.cos(beta)**2 + ell.Ee**2 * np.sin(lamb)**2) + LAMBDA * ell.Ee**2 * np.sin(2*lamb)
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# Zweite Ableitungen von E und G
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E_beta_beta = BETA__ * (ell.Ey**2 * np.cos(beta)**2 + ell.Ee**2 * np.sin(lamb)**2) - 2 * BETA_ * ell.Ey**2 * np.sin(2*beta) - 2 * BETA * ell.Ey**2 * np.cos(2*beta)
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E_beta_lamb = BETA_ * ell.Ee**2 * np.sin(2*lamb)
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E_lamb_lamb = 2 * BETA * ell.Ee**2 * np.cos(2*lamb)
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G_beta_beta = - 2 * LAMBDA * ell.Ey**2 * np.cos(2*beta)
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G_beta_lamb = - LAMBDA_ * ell.Ey**2 * np.sin(2*beta)
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G_lamb_lamb = LAMBDA__ * (ell.Ey**2 * np.cos(beta)**2 + ell.Ee**2 * np.sin(lamb)**2) + 2 * LAMBDA_ * ell.Ee**2 * np.sin(2*lamb) + 2 * LAMBDA * ell.Ee**2 * np.cos(2*lamb)
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return (BETA, LAMBDA, E, G,
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BETA_, LAMBDA_, BETA__, LAMBDA__,
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E_beta, E_lamb, G_beta, G_lamb,
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E_beta_beta, E_beta_lamb, E_lamb_lamb,
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G_beta_beta, G_beta_lamb, G_lamb_lamb)
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def p_coef(beta, lamb):
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(BETA, LAMBDA, E, G,
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BETA_, LAMBDA_, BETA__, LAMBDA__,
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E_beta, E_lamb, G_beta, G_lamb,
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E_beta_beta, E_beta_lamb, E_lamb_lamb,
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G_beta_beta, G_beta_lamb, G_lamb_lamb) = BETA_LAMBDA(beta, lamb)
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p_3 = - 0.5 * (E_lamb / G)
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p_2 = (G_beta / G) - 0.5 * (E_beta / E)
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p_1 = 0.5 * (G_lamb / G) - (E_lamb / E)
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p_0 = 0.5 * (G_beta / E)
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p_33 = - 0.5 * ((E_beta_lamb * G - E_lamb * G_beta) / (G ** 2))
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p_22 = ((G * G_beta_beta - G_beta * G_beta) / (G ** 2)) - 0.5 * ((E * E_beta_beta - E_beta * E_beta) / (E ** 2))
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p_11 = 0.5 * ((G * G_beta_lamb - G_beta * G_lamb) / (G ** 2)) - ((E * E_beta_lamb - E_beta * E_lamb) / (E ** 2))
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p_00 = 0.5 * ((E * G_beta_beta - E_beta * G_beta) / (E ** 2))
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return (BETA, LAMBDA, E, G,
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p_3, p_2, p_1, p_0,
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p_33, p_22, p_11, p_00)
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def q_coef(beta, lamb):
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(BETA, LAMBDA, E, G,
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BETA_, LAMBDA_, BETA__, LAMBDA__,
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E_beta, E_lamb, G_beta, G_lamb,
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E_beta_beta, E_beta_lamb, E_lamb_lamb,
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G_beta_beta, G_beta_lamb, G_lamb_lamb) = BETA_LAMBDA(beta, lamb)
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q_3 = - 0.5 * (G_beta / E)
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q_2 = (E_lamb / E) - 0.5 * (G_lamb / G)
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q_1 = 0.5 * (E_beta / E) - (G_beta / G)
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q_0 = 0.5 * (E_lamb / G)
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q_33 = - 0.5 * ((E * G_beta_lamb - E_lamb * G_lamb) / (E ** 2))
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q_22 = ((E * E_lamb_lamb - E_lamb * E_lamb) / (E ** 2)) - 0.5 * ((G * G_lamb_lamb - G_lamb * G_lamb) / (G ** 2))
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q_11 = 0.5 * ((E * E_beta_lamb - E_beta * E_lamb) / (E ** 2)) - ((G * G_beta_lamb - G_beta * G_lamb) / (G ** 2))
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q_00 = 0.5 * ((E_lamb_lamb * G - E_lamb * G_lamb) / (G ** 2))
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return (BETA, LAMBDA, E, G,
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q_3, q_2, q_1, q_0,
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q_33, q_22, q_11, q_00)
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if lamb_1 != lamb_2:
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def functions():
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def f_beta(lamb, beta, beta_p, X3, X4):
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return beta_p
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def f_beta_p(lamb, beta, beta_p, X3, X4):
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(BETA, LAMBDA, E, G,
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p_3, p_2, p_1, p_0,
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p_33, p_22, p_11, p_00) = p_coef(beta, lamb)
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return p_3 * beta_p ** 3 + p_2 * beta_p ** 2 + p_1 * beta_p + p_0
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def f_X3(lamb, beta, beta_p, X3, X4):
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return X4
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def f_X4(lamb, beta, beta_p, X3, X4):
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(BETA, LAMBDA, E, G,
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p_3, p_2, p_1, p_0,
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p_33, p_22, p_11, p_00) = p_coef(beta, lamb)
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return (p_33 * beta_p ** 3 + p_22 * beta_p ** 2 + p_11 * beta_p + p_00) * X3 + \
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(3 * p_3 * beta_p ** 2 + 2 * p_2 * beta_p + p_1) * X4
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return [f_beta, f_beta_p, f_X3, f_X4]
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N = n
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dlamb = lamb_2 - lamb_1
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if abs(dlamb) < 1e-15:
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beta_0 = 0.0
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else:
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beta_0 = (beta_2 - beta_1) / (lamb_2 - lamb_1)
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converged = False
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iterations = 0
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funcs = functions()
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for i in range(iter_max):
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iterations = i + 1
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startwerte = [lamb_1, beta_1, beta_0, 0.0, 1.0]
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werte = rk.verfahren(funcs, startwerte, dlamb, N)
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lamb_end, beta_end, beta_p_end, X3_end, X4_end = werte[-1]
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d_beta_end_d_beta0 = X3_end
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delta = beta_end - beta_2
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if abs(delta) < epsilon:
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converged = True
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break
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if abs(d_beta_end_d_beta0) < 1e-20:
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raise RuntimeError("Abbruch.")
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max_step = 0.5
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step = delta / d_beta_end_d_beta0
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if abs(step) > max_step:
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step = np.sign(step) * max_step
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beta_0 = beta_0 - step
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if not converged:
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raise RuntimeError("konvergiert nicht.")
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# Z
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werte = rk.verfahren(funcs, [lamb_1, beta_1, beta_0, 0.0, 1.0], dlamb, N)
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beta_arr = np.zeros(N + 1)
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lamb_arr = np.zeros(N + 1)
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beta_p_arr = np.zeros(N + 1)
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for i, state in enumerate(werte):
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lamb_arr[i] = state[0]
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beta_arr[i] = state[1]
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beta_p_arr[i] = state[2]
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(_, _, E1, G1,
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*_) = BETA_LAMBDA(beta_arr[0], lamb_arr[0])
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(_, _, E2, G2,
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*_) = BETA_LAMBDA(beta_arr[-1], lamb_arr[-1])
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alpha_1 = arccot(np.sqrt(E1 / G1) * beta_p_arr[0])
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alpha_2 = arccot(np.sqrt(E2 / G2) * beta_p_arr[-1])
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integrand = np.zeros(N + 1)
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for i in range(N + 1):
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(_, _, Ei, Gi,
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*_) = BETA_LAMBDA(beta_arr[i], lamb_arr[i])
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integrand[i] = np.sqrt(Ei * beta_p_arr[i] ** 2 + Gi)
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h = abs(dlamb) / N
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if N % 2 == 0:
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S = integrand[0] + integrand[-1] \
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+ 4.0 * np.sum(integrand[1:-1:2]) \
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+ 2.0 * np.sum(integrand[2:-1:2])
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s = h / 3.0 * S
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else:
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s = np.trapz(integrand, dx=h)
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beta0 = beta_arr[0]
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lamb0 = lamb_arr[0]
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c = np.sqrt(
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(np.cos(beta0) ** 2 + (ell.Ee**2 / ell.Ex**2) * np.sin(beta0) ** 2) * np.sin(alpha_1) ** 2
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+ (ell.Ee**2 / ell.Ex**2) * np.cos(lamb0) ** 2 * np.cos(alpha_1) ** 2
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)
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return alpha_1, alpha_2, s
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if lamb_1 == lamb_2:
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N = n
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dbeta = beta_2 - beta_1
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if abs(dbeta) < 10**-15:
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return 0, 0, 0
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lamb_0 = 0
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converged = False
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iterations = 0
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def functions_beta():
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def g_lamb(beta, lamb, lamb_p, Y3, Y4):
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return lamb_p
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def g_lamb_p(beta, lamb, lamb_p, Y3, Y4):
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(BETA, LAMBDA, E, G,
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q_3, q_2, q_1, q_0,
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q_33, q_22, q_11, q_00) = q_coef(beta, lamb)
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return q_3 * lamb_p ** 3 + q_2 * lamb_p ** 2 + q_1 * lamb_p + q_0
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def g_Y3(beta, lamb, lamb_p, Y3, Y4):
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return Y4
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def g_Y4(beta, lamb, lamb_p, Y3, Y4):
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(BETA, LAMBDA, E, G,
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q_3, q_2, q_1, q_0,
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q_33, q_22, q_11, q_00) = q_coef(beta, lamb)
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return (q_33 * lamb_p ** 3 + q_22 * lamb_p ** 2 + q_11 * lamb_p + q_00) * Y3 + \
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(3 * q_3 * lamb_p ** 2 + 2 * q_2 * lamb_p + q_1) * Y4
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return [g_lamb, g_lamb_p, g_Y3, g_Y4]
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funcs_beta = functions_beta()
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for i in range(iter_max):
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iterations = i + 1
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startwerte = [beta_1, lamb_1, lamb_0, 0.0, 1.0]
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werte = rk.verfahren(funcs_beta, startwerte, dbeta, N)
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beta_end, lamb_end, lamb_p_end, Y3_end, Y4_end = werte[-1]
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d_lamb_end_d_lambda0 = Y3_end
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delta = lamb_end - lamb_2
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if abs(delta) < epsilon:
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converged = True
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break
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if abs(d_lamb_end_d_lambda0) < 1e-20:
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raise RuntimeError("Abbruch (Ableitung ~ 0).")
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max_step = 1.0
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step = delta / d_lamb_end_d_lambda0
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if abs(step) > max_step:
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step = np.sign(step) * max_step
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lamb_0 = lamb_0 - step
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werte = rk.verfahren(funcs_beta, [beta_1, lamb_1, lamb_0, 0.0, 1.0], dbeta, N)
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beta_arr = np.zeros(N + 1)
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lamb_arr = np.zeros(N + 1)
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lambda_p_arr = np.zeros(N + 1)
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for i, state in enumerate(werte):
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beta_arr[i] = state[0]
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lamb_arr[i] = state[1]
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lambda_p_arr[i] = state[2]
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# Azimute
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(BETA1, LAMBDA1, E1, G1,
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*_) = BETA_LAMBDA(beta_arr[0], lamb_arr[0])
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(BETA2, LAMBDA2, E2, G2,
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*_) = BETA_LAMBDA(beta_arr[-1], lamb_arr[-1])
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alpha_1 = (np.pi / 2.0) - arccot(np.sqrt(LAMBDA1 / BETA1) * lambda_p_arr[0])
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alpha_2 = (np.pi / 2.0) - arccot(np.sqrt(LAMBDA2 / BETA2) * lambda_p_arr[-1])
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integrand = np.zeros(N + 1)
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for i in range(N + 1):
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(_, _, Ei, Gi,
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*_) = BETA_LAMBDA(beta_arr[i], lamb_arr[i])
|
||||
integrand[i] = np.sqrt(Ei + Gi * lambda_p_arr[i] ** 2)
|
||||
|
||||
h = abs(dbeta) / N
|
||||
if N % 2 == 0:
|
||||
S = integrand[0] + integrand[-1] \
|
||||
+ 4.0 * np.sum(integrand[1:-1:2]) \
|
||||
+ 2.0 * np.sum(integrand[2:-1:2])
|
||||
s = h / 3.0 * S
|
||||
else:
|
||||
s = np.trapz(integrand, dx=h)
|
||||
|
||||
return alpha_1, alpha_2, s
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
ell = EllipsoidTriaxial.init_name("BursaSima1980round")
|
||||
beta1 = np.deg2rad(75)
|
||||
lamb1 = np.deg2rad(-90)
|
||||
beta2 = np.deg2rad(75)
|
||||
lamb2 = np.deg2rad(66)
|
||||
a1, a2, s = gha2_num(ell, beta1, lamb1, beta2, lamb2)
|
||||
print(np.rad2deg(a1))
|
||||
print(np.rad2deg(a2))
|
||||
print(s)
|
||||
|
||||
|
||||
|
||||
Reference in New Issue
Block a user