def star_stepper(QL, QR, dt, MPL, MPR, SL=None, SR=None):
d = 0
PL = State(QL, MPL)
PR = State(QR, MPR)
LL, RL = riemann_constraints(PL, 1, MPL)
LR, RR = riemann_constraints(PR, -1, MPR)
YL = RL[11:15, :4]
YR = RR[11:15, :4]
xL = concatenate([PL.Σ()[d], [PL.T()]])
xR = concatenate([PR.Σ()[d], [PR.T()]])
Ξ1L = Xi1(PL, d, MPL)
Ξ2L = Xi2(PL, d, MPL)
OL = dot(Ξ1L, Ξ2L)
Ξ1R = Xi1(PR, d, MPR)
Ξ2R = Xi2(PR, d, MPR)
OR = dot(Ξ1R, Ξ2R)
_, QL_1 = eig(OL)
_, QR_1 = eig(OR)
if SL is not None:
cL = dot(LL, reorder(SL, order='atypical'))
cR = dot(LR, reorder(SR, order='atypical'))
XL = dot(QL_1, cL[4:8])
XR = dot(QR_1, cR[4:8])
yL = concatenate([PL.v, [PL.J[d]]])
yR = concatenate([PR.v, [PR.J[d]]])
x_ = solve(YL - YR,
yR - yL - dt * (XL + XR) + dot(YL, xL) - dot(YR, xR))
else:
yL = concatenate([PL.v, [PL.J[d]]])
yR = concatenate([PR.v, [PR.J[d]]])
x_ = solve(YL - YR, yR - yL + dot(YL, xL) - dot(YR, xR))
cL[:4] = x_ - xL
cR[:4] = x_ - xR
PLvec = reorder(Pvec(PL, MPL.THERMAL), order='atypical')
PRvec = reorder(Pvec(PR, MPR.THERMAL), order='atypical')
PL_vec = dot(RL, cL) + PLvec
PR_vec = dot(RR, cR) + PRvec
QL_ = Pvec_to_Cvec(reorder(PL_vec), MPL)
QR_ = Pvec_to_Cvec(reorder(PR_vec), MPR)
return QL_, QR_
def riemann_constraints(P, sgn, MP):
""" K=R: sgn = -1
K=L: sgn = 1
NOTE: Uses atypical ordering
Extra constraints are:
dΣ = dΣ/dρ * dρ + dΣ/dp * dp + dΣ/dA * dA
dT = dT/dρ * dρ + dT/dp * dp
v*L = v*R
J*L = J*R
"""
_, Lhat, Rhat = eigen(P, 0, MP, False)
Lhat = reorder(Lhat.T, order='atypical').T
Rhat = reorder(Rhat, order='atypical')
σA = P.dσdA()
σρ = P.dσdρ()
Tρ = P.dTdρ()
Tp = P.dTdp()
Lhat[:3, 0] = -σρ[0]
Lhat[:3, 1] = array([1, 0, 0])
for i in range(3):
Lhat[:3, 2 + 3 * i:5 + 3 * i] = -σA[0, :, :, i]
Lhat[:3, 11:] = 0
if MP.THERMAL:
Lhat[3, 0] = Tρ
Lhat[3, 1] = Tp
Lhat[3, 2:] = 0
Lhat[4:8, 11:15] *= -sgn
Ξ1 = Xi1(P, 0, MP)
Ξ2 = Xi2(P, 0, MP)
O = dot(Ξ1, Ξ2)
w, vl, vr = eig(O, left=1)
D_1 = diag(1 / sqrt(w.real))
Q = vl.T
Q_1 = vr
I = dot(Q, Q_1)
Q = solve(I, Q)
tmp = zeros([5, 5])
tmp[:4] = Lhat[:4, :5]
tmp[4] = Lhat[8, :5]
b = zeros([5, 4])
b[:4, :4] = eye(4)
X = solve(tmp, b)
Rhat[:5, :4] = X
Y0 = dot(Q_1, dot(D_1, Q))
Y = -sgn * dot(Y0, dot(Ξ1, X))
Rhat[11:15, :4] = Y
Rhat[:, 4:8] = 0
Rhat[11:15, 4:8] = sgn * Q_1
return Lhat, Rhat
def Y_matrix(P, MP, sgn):
ρ = P.ρ
A = P.A
σρ = P.dσdρ()
σA = P.dσdA()
e0 = array([1, 0, 0])
Π1 = σA[0, :, :, 0]
tmp = zeros([5, 5])
tmp[:3, 0] = -σρ[0]
tmp[0, 1] = 1
tmp[:3, 2:5] = -Π1
if THERMAL:
tmp[3, 0] = P.dTdρ()
tmp[3, 1] = P.dTdp()
tmp[4, 0] = -1 / ρ
tmp[4, 2:5] = inv(A)[0]
else:
p = P.p()
σ = P.σ()
c0 = c_0(ρ, p, A, MP)
B = zeros([2, 3])
B[0, 0] = ρ
B[1] = σ[0] - ρ * σρ[0]
B[1, 0] += ρ * c0**2
rhs = column_stack((-σρ[0], e0))
C = solve(Π1, rhs)
BA_1 = dot(B, inv(A))
Z = eye(2) - dot(BA_1, C)
W = concatenate([eye(2), -BA_1], axis=1)
tmp[3:5] = solve(Z, W)
b = zeros([5, n1])
b[:n1, :n1] = eye(n1)
X = solve(tmp, b)
Ξ1 = Xi1(P, 0, MP)
Ξ2 = Xi2(P, 0, MP)
Ξ = dot(Ξ1, Ξ2)
w, vr = eig(Ξ)
D_1 = diag(1 / sqrt(w.real))
Q_1 = vr
Q = inv(Q_1)
Y0 = dot(Q_1, dot(D_1, Q))
return sgn * dot(Y0, dot(Ξ1, X))
def TAT_test(d, MP):
Q = generate_vector(MP)
P = State(Q, MP)
Ξ1 = Xi1(P, d, MP)
Ξ2 = Xi2(P, d, MP)
TAT_py = dot(Ξ1, Ξ2)
TAT_cp = GPRpy.system.thermo_acoustic_tensor(Q, d, MP)
print("TAT ", check(TAT_cp, TAT_py))
return TAT_cp, TAT_py
def eigenvalues(P, d, MP):
v = P.v
vd = v[d]
n1, n2, n3, n4, n5, n6 = get_indexes()
Ξ1 = Xi1(P, d, MP)
Ξ2 = Xi2(P, d, MP)
Ξ = dot(Ξ1, Ξ2)
w, vl, vr = eig(Ξ, left=1)
sw = sqrt(w.real)
D = diag(sw)
l = array([vd + λ for λ in diag(D)] +
[vd - λ for λ in diag(D)] + [vd] * n3)
return l
def riemann_constraints(P, sgn, MP, left=False):
""" K=R: sgn = -1
K=L: sgn = 1
NOTE: Uses atypical ordering
Extra constraints are:
dΣ = dΣ/dρ * dρ + dΣ/dp * dp + dΣ/dA * dA
dT = dT/dρ * dρ + dT/dp * dp
v*L = v*R
J*L = J*R
"""
_, Lhat, Rhat = eigen(P, 0, False, MP, values=False, right=True, left=left,
typical_order=False)
if left:
Lhat = left_riemann_constraints(P, Lhat, sgn)
ρ = P.ρ
A = P.A
σρ = P.dσdρ()
σA = P.dσdA()
e0 = array([1, 0, 0])
Π1 = σA[0, :, :, 0]
tmp = zeros([5, 5])
tmp[:3, 0] = -σρ[0]
tmp[0, 1] = 1
tmp[:3, 2:5] = -Π1
if THERMAL:
tmp[3, 0] = P.dTdρ()
tmp[3, 1] = P.dTdp()
tmp[4, 0] = -1 / ρ
tmp[4, 2:5] = inv(A)[0]
else:
p = P.p()
σ = P.σ()
c0 = c_0(ρ, p, A, MP)
B = zeros([2, 3])
B[0, 0] = ρ
B[1] = σ[0] - ρ * σρ[0]
B[1, 0] += ρ * c0**2
rhs = column_stack((-σρ[0], e0))
C = solve(Π1, rhs)
BA_1 = dot(B, inv(A))
Z = eye(2) - dot(BA_1, C)
W = concatenate([eye(2), -BA_1], axis=1)
tmp[3:5] = solve(Z, W)
b = zeros([5, n1])
b[:n1, :n1] = eye(n1)
X = solve(tmp, b)
Rhat[:5, :n1] = X
Ξ1 = Xi1(P, 0, MP)
Ξ2 = Xi2(P, 0, MP)
Q, Q_1, _, D_1 = decompose_Ξ(Ξ1, Ξ2)
Y0 = dot(Q_1, dot(D_1, Q))
Rhat[11:n5, :n1] = -sgn * dot(Y0, dot(Ξ1, X))
Rhat[:11, n1:n2] = 0
Rhat[11:n5, n1:n2] = sgn * dot(Q_1, D_1)
return Lhat, Rhat
def right_eigenvectors(P, d, MP, typical_order):
σA = P.dσdA()
σρ = P.dσdρ()
Π1 = σA[d, :, :, 0]
Π2 = σA[d, :, :, 1]
Π3 = σA[d, :, :, 2]
Ξ1 = Xi1(P, d, MP)
Ξ2 = Xi2(P, d, MP)
Q, Q_1, D, D_1 = decompose_Ξ(Ξ1, Ξ2)
n1, n2, n3, n4, n5, n6 = get_indexes()
e0 = array([1, 0, 0])
R = zeros([17, 17])
tmp1 = 0.5 * dot(Ξ2, dot(Q_1, D_1**2))
tmp2 = 0.5 * dot(Q_1, D_1)
R[:5, :n1] = tmp1
R[:5, n1:n2] = tmp1
R[11:n5, :n1] = tmp2
R[11:n5, n1:n2] = -tmp2
if THERMAL:
ρ = P.ρ
A = P.A
Tρ = P.dTdρ()
Tp = P.dTdp()
b = Tp * σρ[d] + Tρ * e0
c = 1 / (solve(dot(Π1, A), b)[d] + Tp / ρ)
R[0, 8] = -c * Tp
R[1, 8] = c * Tρ
R[2:5, 8] = c * solve(Π1, b)
R[15, 15] = 1
R[16, 16] = 1
else:
R[0, 6] = 1
R[1, 7] = 1
R[2:5, 6] = -solve(Π1, σρ[0])
R[2:5, 7] = solve(Π1, e0)
R[2:5, n3:n4] = -solve(Π1, Π2)
R[2:5, n4:n5] = -solve(Π1, Π3)
for i in range(6):
R[5 + i, n3 + i] = 1
if typical_order:
R = reorder(R)
if not THERMAL:
R = R[:14, :14]
return R
def left_eigenvectors(P, d, MP, typical_order):
ρ = P.ρ
A = P.A
σA = P.dσdA()
σρ = P.dσdρ()
Π1 = σA[d, :, :, 0]
Π2 = σA[d, :, :, 1]
Π3 = σA[d, :, :, 2]
Ξ1 = Xi1(P, d, MP)
Ξ2 = Xi2(P, d, MP)
Q, Q_1, D, D_1 = decompose_Ξ(Ξ1, Ξ2)
n1, n2, n3, n4, n5, n6 = get_indexes()
e0 = array([1, 0, 0])
L = zeros([17, 17])
tmp1 = dot(Q, Ξ1)
tmp2 = -dot(Q[:, :3], Π2) / ρ
tmp3 = -dot(Q[:, :3], Π3) / ρ
tmp4 = dot(D, Q)
L[:n1, :5] = tmp1
L[n1:n2, :5] = tmp1
L[:n1, 5:8] = tmp2
L[n1:n2, 5:8] = tmp2
L[:n1, 8:11] = tmp3
L[n1:n2, 8:11] = tmp3
L[:n1, 11:n5] = tmp4
L[n1:n2, 11:n5] = -tmp4
if THERMAL:
tmp = inv(A)[0]
L[8, 0] = -1 / ρ
L[8, 2:5] = tmp
L[8, 5:8] = dot(tmp, solve(Π1, Π2))
L[8, 8:11] = dot(tmp, solve(Π1, Π3))
L[15, 15] = 1
L[16, 16] = 1
else:
σ = P.σ()
p = P.p()
c0 = c_0(ρ, p, A, MP)
B = zeros([2, 3])
B[0, 0] = ρ
B[1] = σ[0] - ρ * σρ[0]
B[1, 0] += ρ * c0**2
C = zeros([3, 2])
C[:, 0] = -solve(Π1, σρ[0])
C[:, 1] = solve(Π1, e0)
BA_1 = dot(B, inv(A))
Z = eye(2) - dot(BA_1, C)
X = zeros([2, 14])
X[:, :2] = eye(2)
X[:, 2:5] = -BA_1
X[:, 5:8] = -dot(BA_1, solve(Π1, Π2))
X[:, 8:11] = -dot(BA_1, solve(Π1, Π3))
L[6:8, :14] = solve(Z, X)
for i in range(6):
L[n3 + i, 5 + i] = 1
if typical_order:
L = reorder(L.T).T
if not THERMAL:
L = L[:14, :14]
return L