def dFdxi1(self, l, xi2):
"""
Calculates dF/dx_1 (see [1]).
[1] M. S. Olufsen. Modeling of the Arterial System with Reference to an Anesthesia Simulator. PhD thesis, University of Roskilde, Denmark, 1998.
:param l: Position, either M+1/2 or -1/2.
:param xi: Area.
:returns: Solution to dF/dx_1
"""
if l > self.L:
x_0 = self.L - self.dx
x_1 = self.L
R0_l = utils.extrapolate(
l, [x_0, x_1],
[np.sqrt(self.A0[-2] / np.pi),
np.sqrt(self.A0[-1] / np.pi)])
elif l < 0.0:
x_0 = self.dx
x_1 = 0.0
R0_l = utils.extrapolate(
l, [x_0, x_1],
[np.sqrt(self.A0[1] / np.pi),
np.sqrt(self.A0[0] / np.pi)])
elif l == self.L:
R0_l = np.sqrt(self.A0[-1] / np.pi)
else:
R0_l = np.sqrt(self.A0[0] / np.pi)
return -2 * np.pi * R0_l / (self.delta * self.Re * xi2)
def dpdx(self, l, xi):
"""
Calculates dp/dx (see [1]).
[1] M. S. Olufsen. Modeling of the Arterial System with Reference to an Anesthesia Simulator. PhD thesis, University of Roskilde, Denmark, 1998.
:param l: Position, either M+1/2 or -1/2.
:param xi: Area.
:returns: Solution to dp/dx
"""
if l > self.L:
x_0 = self.L - self.dx
x_1 = self.L
f_l = utils.extrapolate(l, [x_0, x_1], [self.f[-2], self.f[-1]])
A0_l = utils.extrapolate(l, [x_0, x_1], [self.A0[-2], self.A0[-1]])
elif l < 0.0:
x_0 = self.dx
x_1 = 0.0
f_l = utils.extrapolate(l, [x_0, x_1], [self.f[1], self.f[0]])
A0_l = utils.extrapolate(l, [x_0, x_1], [self.A0[1], self.A0[0]])
elif l == self.L:
f_l = self.f[-1]
A0_l = self.A0[-1]
else:
f_l = self.f[0]
A0_l = self.A0[0]
return f_l / 2 * np.sqrt(A0_l / xi**3)
def bifurcation(artery, d1, d2, dt):
x0 = np.zeros(18)
Dfr = np.zeros((18, 18)) # Jacobian
Dfr[0,0] = Dfr[1,3] = Dfr[2,6] = Dfr[3,9] = Dfr[4,12] = Dfr[5,15] = -1
Dfr[6,1] = Dfr[7,4] = Dfr[8,7] = Dfr[9,10] = Dfr[10,13] = Dfr[11,16] = -1
Dfr[12,1] = Dfr[13,0] = -1
Dfr[6,2] = Dfr[7,5] = Dfr[8,8] = Dfr[9,11] = Dfr[10,14] = Dfr[11,17] = 0.5
Dfr[12,4] = Dfr[12,7] = Dfr[13,3] = Dfr[13,6] = 1.0
Dfr[3,2] = -dt/artery.dx
R0_p_M12 = utils.extrapolate(artery.L+artery.dx/2,
[artery.L-artery.dx, artery.L],
[np.sqrt(artery.A0[-2]/np.pi), np.sqrt(artery.A0[-1]/np.pi)])
Dfr[2,0] = -2*dt/artery.dx * x[2]/x[11] -\
dt/2 * 2*np.pi*R0_p_M12/(delta*Re*x[11])
def dBdxdxi(self, l, xi):
"""
Calculates d^2B/dxdx_i (see [1]).
[1] M. S. Olufsen. Modeling of the Arterial System with Reference to an Anesthesia Simulator. PhD thesis, University of Roskilde, Denmark, 1998.
:param l: Position, either M+1/2 or -1/2.
:param xi: Area.
:returns: Solution to d^2B/dxdx_i
"""
if l > self.L:
x_0 = self.L - self.dx
x_1 = self.L
f_l = utils.extrapolate(l, [x_0, x_1], [self.f[-2], self.f[-1]])
df_l = utils.extrapolate(l, [x_0, x_1], [self.df[-2], self.df[-1]])
A0_l = utils.extrapolate(l, [x_0, x_1], [self.A0[-2], self.A0[-1]])
xgrad_l = utils.extrapolate(l, [x_0, x_1],
[self.xgrad[-2], self.xgrad[-1]])
elif l < 0.0:
x_0 = self.dx
x_1 = 0.0
f_l = utils.extrapolate(l, [x_0, x_1], [self.f[1], self.f[0]])
df_l = utils.extrapolate(l, [x_0, x_1], [self.df[1], self.df[0]])
A0_l = utils.extrapolate(l, [x_0, x_1], [self.A0[1], self.A0[0]])
xgrad_l = utils.extrapolate(l, [x_0, x_1],
[self.xgrad[1], self.xgrad[0]])
elif l == self.L:
f_l = self.f[-1]
df_l = self.df[-1]
A0_l = self.A0[-1]
xgrad_l = self.xgrad[-1]
else:
f_l = self.f[0]
df_l = self.df[0]
A0_l = self.A0[0]
xgrad_l = self.xgrad[0]
return (1/(2*np.sqrt(xi)) * (f_l*np.sqrt(np.pi) +\
df_l*np.sqrt(A0_l)) - df_l) * xgrad_l
def residuals(x, parent, d1, d2, theta, gamma, U_p_np, U_d1_np, U_d2_np):
"""
Calculates the residual equations for using Newton's method to solve bifurcation inlet and outlet boundary conditions [1].
[1] M. S. Olufsen. Modeling of the Arterial System with Reference to an Anesthesia Simulator. PhD thesis, University of Roskilde, Denmark, 1998.
[2] R. J. LeVeque. Numerical Methods for Conservation Laws. Birkhauser Verlag, Basel, Switzerland, 2nd edition, 1992.
:param x: Solution of the system of equations.
:param parent: Artery object of the parent vessel.
:param d1: Artery object of the first daughter vessel.
:param d2: Artery object of the second daughter vessel.
:param theta: dt/dx
:param gamma: dt/2
:param U_p_np: U_(M-1/2)^(n+1/2) [2]
:param U_p_np: U_(M-1/2)^(n+1/2) [2]
:returns: The residual equations for Newton's method.
"""
f_p_mp = utils.extrapolate(parent.L+parent.dx/2,
[parent.L-parent.dx, parent.L], [parent.f[-2], parent.f[-1]])
f_d1_mp = utils.extrapolate(-d1.dx/2, [d1.dx, 0.0],
[d1.f[1], d1.f[0]])
f_d2_mp = utils.extrapolate(-d2.dx/2, [d2.dx, 0.0],
[d2.f[1], d2.f[0]])
A0_p_mp = utils.extrapolate(parent.L+parent.dx/2,
[parent.L-parent.dx, parent.L], [parent.A0[-2], parent.A0[-1]])
A0_d1_mp = utils.extrapolate(-d1.dx/2, [d1.dx, 0.0],
[d1.A0[1], d1.A0[0]])
A0_d2_mp = utils.extrapolate(-d2.dx/2, [d2.dx, 0.0],
[d2.A0[1], d2.A0[0]])
R0_p_mp = np.sqrt(A0_p_mp/np.pi)
R0_d1_mp = np.sqrt(A0_d1_mp/np.pi)
R0_d2_mp = np.sqrt(A0_d2_mp/np.pi)
B_p_mp = f_p_mp * np.sqrt(x[11]*A0_p_mp)
B_d1_mp = f_d1_mp * np.sqrt(x[14]*A0_d1_mp)
B_d2_mp = f_d2_mp * np.sqrt(x[17]*A0_d2_mp)
k1 = parent.U0[1,-1] + theta * (parent.F(U_p_np, j=-1)[1]) +\
gamma * (parent.S(U_p_np, j=-1)[1])
k2 = d1.U0[1,0] - theta * (d1.F(U_d1_np, j=0)[1]) +\
gamma * (d1.S(U_d1_np, j=0)[1])
k3 = d2.U0[1,0] - theta * (d2.F(U_d2_np, j=0)[1]) +\
gamma * (d2.S(U_d2_np, j=0)[1])
k4 = parent.U0[0,-1] + theta*parent.F(U_p_np, j=-1)[0]
k5 = d1.U0[0,0] - theta*d1.F(U_d1_np, j=0)[0]
k6 = d2.U0[0,0] - theta*d2.F(U_d2_np, j=0)[0]
k7 = U_p_np[1]/2
k8 = U_d1_np[1]/2
k9 = U_d2_np[1]/2
k10 = U_p_np[0]/2
k11 = U_d1_np[0]/2
k12 = U_d2_np[0]/2
k15a = -parent.f[-1] + d1.f[0]
k15b = d1.f[0] * np.sqrt(d1.A0[0])
k16a = -parent.f[-1] + d2.f[0]
k16b = d2.f[0] * np.sqrt(d2.A0[0])
k156 = parent.f[-1] * np.sqrt(parent.A0[-1])
fr1 = k1 - x[0] - theta*(x[2]**2/x[11] + B_p_mp) +\
gamma*(-2*np.pi*R0_p_mp*x[2]/(parent.delta*parent.Re*x[11]) +\
parent.dBdx(parent.L+parent.dx/2, x[11]))
fr2 = k2 - x[3] + theta*(x[5]**2/x[14] + B_d1_mp) +\
gamma*(-2*np.pi*R0_d1_mp*x[5]/(d1.delta*d1.Re*x[14]) +\
d1.dBdx(-d1.dx/2, x[14]))
fr3 = k3 - x[6] + theta*(x[8]**2/x[17] + B_d2_mp) +\
gamma*(-2*np.pi*R0_d2_mp*x[8]/(d2.delta*d2.Re*x[17]) +\
d2.dBdx(-d2.dx/2, x[17]))
fr4 = -x[9] - theta*x[2] + k4
fr5 = -x[12] + theta*x[5] + k5
fr6 = -x[15] + theta*x[8] + k6
fr7 = -x[1] + x[2]/2 + k7
fr8 = -x[4] + x[5]/2 + k8
fr9 = -x[7] + x[8]/2 + k9
fr10 = -x[10] + x[11]/2 + k10
fr11 = -x[13] + x[14]/2 + k11
fr12 = -x[16] + x[17]/2 + k12
fr13 = -x[1] + x[4] + x[7]
fr14 = -x[0] + x[3] + x[6]
fr15 = k156/np.sqrt(x[10]) - k15b/np.sqrt(x[13]) + k15a
fr16 = k156/np.sqrt(x[10]) - k16b/np.sqrt(x[16]) + k16a
fr17 = k156/np.sqrt(x[9]) - k15b/np.sqrt(x[12]) + k15a
fr18 = k156/np.sqrt(x[9]) - k16b/np.sqrt(x[15]) + k16a
return np.array([fr1, fr2, fr3, fr4, fr5, fr6, fr7, fr8, fr9, fr10,
fr11, fr12, fr13, fr14, fr15, fr16, fr17, fr18])