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code.py
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import numpy as np
import time
import matplotlib.pyplot as plt
from scipy import sparse
class ARC1D:
def __init__(self, domain, Ncells,
BoundaryConditions,
InitialConditions,
gamma, Courant, tol,
k2, k4):
"""
ARC1D for the subsonic channel flow
Inputs:
domain: [x_init,x_end]
Ncells: number of cells
Boundadry conditions: This is an input dictionary to construct the BoundConditions vector
BC = {'rho':[rho1,rho2],
'u':[u1,u2],
'p':[p1,p2],
'e':[e1,e2]
}
InitialConditions: This is the initial Q
gamma: constant
Courant: Courante number
tol: stopping criteria based on the residuals
Output:
Final Q
plots
"""
plt.rc('font', family='serif')
plt.rc('lines', linewidth=1.5)
plt.rc('font', size=9)
plt.rc('legend', **{'fontsize': 9})
self.x_init = domain[0]
self.x_end = domain[1]
self.Ncells = Ncells
self.tol = tol
self.k2 = k2
self.k4 = k4
self.C = Courant
self.N = int(self.Ncells - 1)
self.x = ARC1D.grid(self)
self.S = ARC1D.S_x(self)
self.BC = BoundaryConditions # This is an input dictionary
SS = np.array([self.S[1:-1],
self.S[1:-1],
self.S[1:-1]]).flatten()
self.Q = InitialConditions * SS
self.gamma = gamma
self.dS = ARC1D.S_derivative(self)
self.operator = ARC1D.difference_operator(self)
self.BCE = ARC1D.BoundConditions(self)
"""
Execution routine
"""
norm_residual = 1
p = ARC1D.GetPressure(self, self.Q[:self.N],
self.Q[self.N:2 * self.N],
self.Q[2 * self.N:])
a = ARC1D.GetSoundSpeed(self, p, self.Q[:self.N])
E = ARC1D.Evector(self)
self.fig = plt.figure(figsize = (9,3.5))
self.ax = self.fig.add_subplot(131)
self.ay = self.fig.add_subplot(132)
self.az = self.fig.add_subplot(133)
self.az.set_xlabel('Iterations')
self.az.set_ylabel('Residual norm')
self.fig.show()
self.residuals = []
self.norm_residuals = [1]
counter = 1
while norm_residual > tol:
#
self.ax.cla()
self.ax.set_xlabel('x [m]')
self.ax.set_ylabel('Pressure [Pa]')
self.ay.cla()
self.ay.set_xlabel('x [m]')
self.ay.set_ylabel('Mach')
self.ax.ticklabel_format(style='sci',axis='y',scilimits=(0,1))
#
h = self.C * (self.x[1] - self.x[0]) / (abs(self.Q[self.N: 2 * self.N] / self.Q[:self.N]) + a)
h = np.concatenate((h, h, h), axis=None)
#h = np.mean(h)
# Solve
A = np.identity(3 * self.N) + h * self.operator @ ARC1D.FluxA(self) \
- h * ARC1D.JacobianG(self) - h * ARC1D.FluxL(self, p, a)
B = h * (- self.operator @ ARC1D.Evector(self) -
self.BCE + ARC1D.Fvector(self) +
ARC1D.D_2(self, p, a) + ARC1D.D_4(self, p, a))
self.Q += np.linalg.solve(A, B)
# Get values
p = ARC1D.GetPressure(self, self.Q[:self.N], self.Q[self.N:2 * self.N], self.Q[2 * self.N:])
a = ARC1D.GetSoundSpeed(self, p, self.Q[:self.N])
Mach = self.Q[self.N: 2 * self.N] / self.Q[:self.N]/a
residual = np.sqrt(sum(B ** 2))
self.residuals.append(residual)
first_residual = self.residuals[0]
norm_residual = residual / first_residual
self.norm_residuals.append(norm_residual)
self.ax.plot(self.x, p / self.S[1:-1], 'k-',linewidth = 0.9)
self.ay.plot(self.x, Mach, 'b-',linewidth = 0.9)
self.az.semilogy([counter - 1, counter],
[self.norm_residuals[counter-1],
self.norm_residuals[counter]],
'g-', linewidth = 0.9)
self.fig.tight_layout()
self.fig.canvas.draw()
time.sleep(0.001)
counter = counter + 1
print('Convergence reached')
plt.close('all')
self.counter = counter
def grid(self):
"""
Construct the computational grid given the
domain boundaries: xinit and xend
"""
dx = (self.x_end - self.x_init) / self.Ncells
x = np.linspace(self.x_init + dx,
self.x_end, self.N,
endpoint=False)
return x
def S_x(self):
"""
Variable area as function of x: S(x) for the subsonic case
"""
x = np.concatenate(([self.x_init], self.x, [self.x_end]))
S = np.ones_like(x)
for i, j in enumerate(x):
if j < 5:
S[i] = 1 + 1.5 * (1 - (j / 5)) ** 2
elif j >= 5:
S[i] = 1 + 0.5 * (1 - (j / 5)) ** 2
return S
def difference_operator(self):
"""
Contruct the difference operator for the spatial derivative
"""
center_diagonal = np.zeros((self.N, 1)).ravel()
diagonal_left = -1 * np.ones((self.N, 1)).ravel()
diagonal_right = 1 * np.ones((self.N, 1)).ravel()
a = diagonal_left.shape[0]
diagonals = [center_diagonal, diagonal_left, diagonal_right]
A = sparse.diags(diagonals, [0, -1, 1], shape=(a, a)).toarray()
A[0, 0] = 0
A[0, 1] = 1
A[-1, -1] = 0
A[-1, -2] = -1
A = 1 / (2 * (self.x[1] - self.x[0])) * A
operator = np.block([[A, np.zeros((self.N, self.N)), np.zeros((self.N, self.N))],
[np.zeros((self.N, self.N)), A, np.zeros((self.N, self.N))],
[np.zeros((self.N, self.N)), np.zeros((self.N, self.N)), A]])
return operator
def BoundConditions(self):
"""
Create the boundary conditions vector based on the given BoundaryConditions input dictionary
"""
bcE = np.zeros(3 * self.N)
bcE[0] = -self.BC['rho'][0] * self.BC['u'][0] / 2 / (self.x[1] - self.x[0]) * self.S[0]
bcE[self.N - 1] = self.BC['rho'][-1] * self.BC['u'][-1] / 2 / (self.x[1] - self.x[0]) * self.S[-1]
bcE[self.N] = - ((self.BC['rho'][0] * self.BC['u'][0] ** 2 +
self.BC['p'][0]) / 2 / (self.x[1] - self.x[0])) * self.S[0]
bcE[2 * self.N - 1] = (self.BC['rho'][-1] * self.BC['u'][-1] ** 2 + self.BC['p'][-1]) / 2 / (
self.x[1] - self.x[0]) * self.S[-1]
bcE[2 * self.N] = -(self.BC['u'][0] * (self.BC['e'][0] +
self.BC['p'][0]) / 2 / (self.x[1] - self.x[0])) * self.S[0]
bcE[3 * self.N - 1] = self.BC['u'][-1] * (self.BC['e'][-1] +
self.BC['p'][-1]) / 2 / (self.x[1] - self.x[0]) * self.S[-1]
return bcE
def S_derivative(self):
"""
Compute numerically the first derivative of S(x)
"""
center_diagonal = np.zeros((self.N, 1)).ravel()
diagonal_left = -1 * np.ones((self.N, 1)).ravel()
diagonal_right = 1 * np.ones((self.N, 1)).ravel()
a = diagonal_left.shape[0]
diagonals = [center_diagonal, diagonal_left, diagonal_right]
A = sparse.diags(diagonals, [0, -1, 1], shape=(a, a)).toarray()
A[0, 0] = 0
A[0, 1] = 1
A[-1, -1] = 0
A[-1, -2] = -1
A = 1 / (2 * (self.x[1] - self.x[0])) * A
ds = np.zeros(self.N)
ds[0], ds[-1] = - self.S[0] / 2 / (self.x[1] - self.x[0]), self.S[-1] / 2 / (self.x[1] - self.x[0])
DS = A @ self.S[1:-1] + ds
return DS
def Evector(self):
"""
Compute E given Q
"""
P = ARC1D.GetPressure(self, self.Q[:self.N],
self.Q[self.N: 2 * self.N],
self.Q[2 * self.N:])
E = np.empty(np.shape(self.Q))
E[:self.N] = self.Q[self.N:2 * self.N]
E[self.N:2 * self.N] = self.Q[self.N: 2 * self.N] ** 2 / self.Q[:self.N] + P
E[2 * self.N:] = self.Q[self.N: 2 * self.N] / self.Q[:self.N] * (self.Q[2 * self.N:] + P)
return E
def Fvector(self):
"""
Compute F given Q
"""
P = ARC1D.GetPressure(self, self.Q[:self.N],
self.Q[self.N: 2 * self.N],
self.Q[2 * self.N:]) / self.S[1:-1]
F = np.zeros(3 * self.N)
F[self.N: 2 * self.N] = P * self.dS
return F
def FluxA(self):
"""
Compute flux jacobian matrix of E
"""
B_1 = np.zeros((self.N, self.N))
B_2 = np.identity(self.N)
B_3 = np.zeros((self.N, self.N))
B_4 = 0.5 * (self.gamma - 1) * self.Q[self.N:2 * self.N] ** 2 / self.Q[:self.N] ** 2
B_5 = (3 - self.gamma) * self.Q[self.N: 2 * self.N] / self.Q[:self.N]
B_6 = np.identity(self.N) * (self.gamma - 1)
B_7 = (self.gamma - 1) * self.Q[self.N: 2 * self.N] ** 3 / self.Q[:self.N] ** 3 \
- self.gamma * self.Q[self.N: 2 * self.N] * self.Q[2 * self.N:] / self.Q[: self.N] ** 2
B_8 = self.gamma * self.Q[2 * self.N:] / self.Q[:self.N] \
- 1.5 * (self.gamma - 1) * self.Q[self.N: 2 * self.N] ** 2 / self.Q[: self.N] ** 2
B_9 = self.gamma * self.Q[self.N: 2 * self.N] / self.Q[: self.N]
A = np.block([[B_1, B_2, B_3],
[np.diag(B_4), np.diag(B_5), B_6],
[np.diag(B_7), np.diag(B_8), np.diag(B_9)]])
return A
def FluxL(self, P, a):
"""
Compute flux Jacobian matrix of D
"""
rho = np.pad(self.Q[:self.N], (1, 1), 'constant',
constant_values=(self.BC['rho'][0] * self.S[0],
self.BC['rho'][-1] * self.S[-1])
)
rhou = np.pad(self.Q[self.N: 2 * self.N], (1, 1), 'constant',
constant_values=(self.BC['rho'][0] * self.BC['u'][0] * self.S[0],
self.BC['rho'][-1] * self.BC['u'][0] * self.S[-1])
)
e = np.pad(self.Q[2 * self.N:], (1, 1), 'constant',
constant_values=(self.BC['e'][0] * self.S[0],
self.BC['e'][-1] * self.S[-1])
)
a = np.pad(a, (1, 1), 'constant',
constant_values=(ARC1D.GetSoundSpeed(self, self.BC['p'][0], self.BC['rho'][0]),
ARC1D.GetSoundSpeed(self, self.BC['p'][-1], self.BC['rho'][-1]))
)
dx = self.x[1] - self.x[0]
C2 = ARC1D.Eps2(self, P) * (abs(rhou / rho) + a)
C2_1 = 1 / 2 / dx * (C2[1:-1] + C2[2:])
C2_2 = 1 / 2 / dx * (C2[1:-1] + C2[:-2])
C2_3 = -1 / 2 / dx * (C2[2:] + 2 * C2[1:-1] + C2[:-2])
matrix = np.diag(C2_3) + np.diag(C2_2[1:], -1) + np.diag(C2_1[:-1], 1)
D2 = np.block([[matrix, np.zeros((self.N, self.N)),
np.zeros((self.N, self.N))],
[np.zeros((self.N, self.N)),
matrix, np.zeros((self.N, self.N))],
[np.zeros((self.N, self.N)),
np.zeros((self.N, self.N)), matrix]]
)
C4 = ARC1D.Eps4(self, P) * (abs(rhou / rho) + a)
C4_1 = -1 / 2 / dx * (C4[2:-2] + C4[3:-1])
C4_2 = -1 / 2 / dx * (C4[2:-2] + C4[1:-3])
C4_3 = 1 / 2 / dx * (4 * C4[2:-2] + C4[1:-3] + 3 * C4[3:-1])
C4_4 = 1 / 2 / dx * (4 * C4[2:-2] + 3 * C4[1:-3] + C4[3:-1])
C4_5 = -1 / 2 / dx * (6 * C4[2:-2] + 3 * C4[1:-3] + 3 * C4[3:-1])
C4_5 = np.pad(C4_5, (1, 1), 'constant',
constant_values=(-1 / 2 / dx * (2 * C4[0] + 5 * C4[1] + 3 * C4[2]),
-1 / 2 / dx * (2 * C4[-1] + 5 * C4[-2] + 3 * C4[-3])
)
)
C4_3 = np.insert(C4_3, 0, 1 / 2 / dx * (C4[0] + 4 * C4[1] + 3 * C4[2]))
C4_1 = np.insert(C4_1, 0, -1 / 2 / dx * (C4[1] + C4[2]))
C4_4 = np.insert(C4_4, self.N - 2, 1 / 2 / dx * (C4[-1] + 4 * C4[-2] + 3 * C4[-3]))
C4_2 = np.insert(C4_2, self.N - 2, -1 / 2 / dx * (C4[-1] + C4[-2]))
matrix = np.diag(C4_5) + np.diag(C4_3, 1) + np.diag(C4_1[:-1], 2) \
+ np.diag(C4_4, -1) + np.diag(C4_2[1:], -2)
D4 = np.block([[matrix, np.zeros((self.N, self.N)), np.zeros((self.N, self.N))],
[np.zeros((self.N, self.N)), matrix, np.zeros((self.N, self.N))],
[np.zeros((self.N, self.N)), np.zeros((self.N, self.N)), matrix]])
return D2 + D4
def JacobianG(self):
"""
Compute the jacobian of the source term given Q
"""
A = (self.gamma - 1) * self.dS * self.Q[self.N: 2 * self.N] ** 2 / self.Q[:self.N] ** 2 / 2 / self.S[1:-1]
B = -(self.gamma - 1) * self.dS * self.Q[self.N: 2 * self.N] / self.Q[:self.N] ** 2 / self.S[1:-1]
G = np.block([[np.zeros((self.N, self.N)), np.zeros((self.N, self.N)), np.zeros((self.N, self.N))],
[np.diag(A), np.diag(B), np.diag(np.ones(self.N) * (self.gamma - 1) * self.dS)],
[np.zeros((self.N, self.N)), np.zeros((self.N, self.N)), np.zeros((self.N, self.N))]])
return G
# Below, dissipation terms
def Eps2(self, P):
"""
Activate second-dorder artificial dissipation
"""
P = np.pad(P, (1, 1), 'constant',
constant_values=(self.BC['p'][0] * self.S[0],
self.BC['p'][-1] * self.S[-1])
)
Y = np.pad(abs((P[2:] - 2 * P[1:-1] + P[:-2]) / (P[2:] + 2 * P[1:-1] + P[:-2])),
(2, 2), 'constant', constant_values=(0, 0)
)
return self.k2 * np.maximum.reduce([Y[2:], Y[1:-1], Y[:-2]])
def Eps4(self, P):
"""
Activate fourth-order artificial dissipation
"""
return np.maximum(np.zeros(self.N + 2), self.k4 - ARC1D.Eps2(self, P))
def D_2(self, P, a):
"""
Second order artificial dissipation term.
"""
rho = np.pad(self.Q[:self.N], (1, 1), 'constant',
constant_values=(self.BC['rho'][0] * self.S[0],
self.BC['rho'][-1] * self.S[-1])
)
rhou = np.pad(self.Q[self.N: 2 * self.N], (1, 1), 'constant',
constant_values=(self.BC['rho'][0] * self.BC['u'][0] * self.S[0],
self.BC['rho'][-1] * self.BC['u'][-1] * self.S[-1])
)
e = np.pad(self.Q[2 * self.N:], (1, 1), 'constant',
constant_values=(self.BC['e'][0] * self.S[0],
self.BC['e'][-1] * self.S[-1])
)
a = np.pad(a, (1, 1), 'constant',
constant_values=(ARC1D.GetSoundSpeed(self, self.BC['p'][0], self.BC['rho'][0]),
ARC1D.GetSoundSpeed(self, self.BC['p'][-1], self.BC['rho'][-1]))
)
C = ARC1D.Eps2(self, P) * (abs(rhou / rho) + a)
dx = self.x[1] - self.x[0]
C1 = C[1:-1] + C[2:]
C2 = C[1:-1] + C[:-2]
C3 = C[2:] + 2 * C[1:-1] + C[:-2]
drho = 1 / 2 / dx * (C1 * rho[2:] + C2 * rho[:-2] - C3 * rho[1:-1])
drhou = 1 / 2 / dx * (C1 * rhou[2:] + C2 * rhou[:-2] - C3 * rhou[1:-1])
de = 1 / 2 / dx * (C1 * e[2:] + C2 * e[:-2] - C3 * e[1:-1])
return np.array([drho, drhou, de]).flatten()
def D_4(self, P, a):
"""
Fourth order artificial dissipation term.
"""
rho = np.pad(self.Q[:self.N], (1, 1), 'constant',
constant_values=(self.BC['rho'][0] * self.S[0],
self.BC['rho'][-1] * self.S[-1])
)
rhou = np.pad(self.Q[self.N: 2 * self.N], (1, 1), 'constant',
constant_values=(self.BC['rho'][0] * self.BC['u'][0] * self.S[0],
self.BC['rho'][-1] * self.BC['u'][-1] * self.S[-1])
)
e = np.pad(self.Q[2 * self.N:], (1, 1), 'constant',
constant_values=(self.BC['e'][0] * self.S[0],
self.BC['e'][-1] * self.S[-1])
)
a = np.pad(a, (1, 1), 'constant',
constant_values=(ARC1D.GetSoundSpeed(self, self.BC['p'][0], self.BC['rho'][0]),
ARC1D.GetSoundSpeed(self, self.BC['p'][-1], self.BC['rho'][-1]))
)
C = ARC1D.Eps4(self, P) * (abs(rhou / rho) + a)
dx = self.x[1] - self.x[0]
C1 = C[2:-2] + C[3:-1]
C2 = C[2:-2] + C[1:-3]
C3 = 4 * C[2:-2] + C[1:-3] + 3 * C[3:-1]
C4 = 4 * C[2:-2] + 3 * C[1:-3] + C[3:-1]
C5 = 6 * C[2:-2] + 3 * C[1:-3] + 3 * C[3:-1]
drho = np.empty(self.N)
drhou = np.empty(self.N)
de = np.empty(self.N)
drho[1:-1] = 1 / 2 / dx * (-C1 * rho[4:] - C2 * rho[:-4]
+ C3 * rho[3:-1] + C4 * rho[1:-3]
- C5 * rho[2:-2])
drhou[1:-1] = 1 / 2 / dx * (-C1 * rhou[4:] - C2 * rhou[:-4]
+ C3 * rhou[3:-1] + C4 * rhou[1:-3]
- C5 * rhou[2:-2])
de[1:-1] = 1 / 2 / dx * (-C1 * e[4:] - C2 * e[:-4] + C3 * e[3:-1]
+ C4 * e[1:-3] - C5 * e[2:-2])
drho[0] = 1 / 2 / dx * ((C[0] + 2 * C[1] + C[2]) * rho[0]
- (2 * C[0] + 5 * C[1] + 3 * C[2]) * rho[1]
+ (C[0] + 4 * C[1] + 3 * C[2]) * rho[2]
- (C[1] + C[2]) * rho[3])
drhou[0] = 1 / 2 / dx * ((C[0] + 2 * C[1] + C[2]) * rhou[0]
- (2 * C[0] + 5 * C[1] + 3 * C[2]) * rhou[1]
+ (C[0] + 4 * C[1] + 3 * C[2]) * rhou[2]
- (C[1] + C[2]) * rhou[3])
de[0] = 1 / 2 / dx * ((C[0] + 2 * C[1] + C[2]) * e[0]
- (2 * C[0] + 5 * C[1] + 3 * C[2]) * e[1]
+ (C[0] + 4 * C[1] + 3 * C[2]) * e[2]
- (C[1] + C[2]) * e[3])
drho[-1] = 1 / 2 / dx * ((C[-1] + 2 * C[-2] + C[-3]) * rho[-1]
- (2 * C[-1] + 5 * C[-2] + 3 * C[-3]) * rho[-2]
+ (C[-1] + 4 * C[-2] + 3 * C[-3]) * rho[-3]
- (C[-2] + C[-3]) * rho[-4])
drhou[-1] = 1 / 2 / dx * ((C[-1] + 2 * C[-2] + C[-3]) * rhou[-1]
- (2 * C[-1] + 5 * C[-2] + 3 * C[-3]) * rhou[-2]
+ (C[-1] + 4 * C[-2] + 3 * C[-3]) * rhou[-3]
- (C[-2] + C[-3]) * rhou[-4])
de[-1] = 1 / 2 / dx * ((C[-1] + 2 * C[-2] + C[-3]) * e[-1]
- (2 * C[-1] + 5 * C[-2] + 3 * C[-3]) * e[-2]
+ (C[-1] + 4 * C[-2] + 3 * C[-3]) * e[-3]
- (C[-2] + C[-3]) * e[-4])
return np.array([drho, drhou, de]).flatten()
def GetSoundSpeed(self, P, rho):
"""
Compute the sound speed
"""
return np.sqrt(self.gamma * P / rho)
def GetPressure(self, rho, rhou, e):
"""
Compute the pressure
"""
return (self.gamma - 1) * (e - (rhou ** 2 / 2 / rho))
def ARC1Dexceptions():
if ZeroDivisionError:
raise Exception('Warning: denominator is equal to zero')
if ValueError:
raise Exception('Warning: Invalid value encountered in sqrt')
if RuntimeWarning:
raise Exception('Warning: Invalid value encountered in sqrt')