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461 lines (376 loc) · 14.3 KB
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from matplotlib import cm
import taichi as ti
# A compressible euler equation solver using two methods
# 1: 2nd order muscl
# 2: thinc BVD, ref: "Limiter-free discontinuity-capturing scheme
# for compressible gas dynamics with reactive fronts"
real = ti.f32
ti.init(arch=ti.gpu, default_fp=real)
N = 512 # grid resolution
CFL = 0.9 # keep below 1
method = 1 # 0:muscl, 1:thinc
IC_type = 0 # 0:sod
BC_type = 0 # 0:walls
img_field = 0 # 0:density, 1: schlieren, 2:vorticity, 3: velocity mag
res = 512 # gui resolution
cmap_name = "magma_r" # python colormap
use_fixed_caxis = 0 # 1: use fixed caxis limits, 0: automatic caxis limits
fixed_caxis = [0.0, 5.0] # fixed caxis limits
Q = ti.Vector.field(4, dtype=real, shape=(N, N)) # [rho, rho*u, rho*v, rho*e] consv vars
Q_old = ti.Vector.field(4, dtype=real, shape=(N, N))
W = ti.Vector.field(4, dtype=real, shape=(N, N)) # [rho, u, v, p] cell avg
W_xl = ti.Vector.field(4, dtype=real, shape=(N, N, 3)) # left side of x-face
W_xr = ti.Vector.field(4, dtype=real, shape=(N, N, 3)) # right side of x-face
W_yl = ti.Vector.field(4, dtype=real, shape=(N, N, 3)) # left side of y-face
W_yr = ti.Vector.field(4, dtype=real, shape=(N, N, 3)) # right side of y-face
F_x = ti.Vector.field(4, dtype=real, shape=(N, N)) # x-face flux
F_y = ti.Vector.field(4, dtype=real, shape=(N, N)) # y-face flux
dt = ti.field(dtype=real, shape=())
img = ti.field(dtype=ti.f32, shape=(res, res))
beta_smooth = 1.2
beta_sharp = 2.0
gamma = 1.4 # ratio of specific heats
h = 1.0 / (N - 2) # cell size
vol = h * h # cell volume
@ti.func
def is_interior_cell(i, j):
return 0 < i < N - 1 and 0 < j < N - 1
@ti.func
def is_interior_x_face(i, j):
return 1 < i < N - 1 and 0 < j < N - 1
@ti.func
def is_boundary_x_face(i, j):
return (i == 1 or i == N - 1) and 0 < j < N - 1
@ti.func
def is_interior_y_face(i, j):
return 0 < i < N - 1 and 1 < j < N - 1
@ti.func
def is_boundary_y_face(i, j):
return 0 < i < N - 1 and (j == 1 or j == N - 1)
@ti.func
def get_cell_pos(i, j):
return ti.Vector([i * h - h / 2.0, j * h - h / 2.0])
@ti.kernel
def compute_W():
# conversion from conservative variables to primitive variables
for i, j in Q:
W[i, j] = q_to_w(Q[i, j])
@ti.kernel
def copy_to_old():
for i, j in Q:
Q_old[i, j] = Q[i, j]
@ti.kernel
def set_ic():
for i, j in Q:
if IC_type == 0:
# primitive variable initial conditions
w_in = ti.Vector([10.0, 0.0, 0.0, 10.0])
w_out = ti.Vector([0.125, 0.0, 0.0, 0.1])
pos = get_cell_pos(i, j)
center = ti.Vector([0.5, 0.5])
if (pos - center).norm() < 0.25:
Q[i, j] = w_to_q(w_in)
else:
Q[i, j] = w_to_q(w_out)
# implement more ic's later
@ti.kernel
def set_bc():
# enforce boundary conditions by setting ghost cells
for i, j in Q:
if not is_interior_cell(i, j):
if BC_type == 0: # walls
# enforce neumann=0 and zero normal velocity on face
if i == 0:
Q[i, j] = Q[i + 1, j]
Q[i, j][1] = -Q[i + 1, j][1]
if i == N - 1:
Q[i, j] = Q[i - 1, j] # neumann 0 bc
Q[i, j][1] = -Q[i - 1, j][1] # enforce 0 normal velocty at face
if j == 0:
Q[i, j] = Q[i, j + 1]
Q[i, j][2] = -Q[i, j + 1][2]
if j == N - 1:
Q[i, j] = Q[i, j - 1]
Q[i, j][2] = -Q[i, j - 1][2]
# implement more bc's later
@ti.func
def mc_lim(r):
# MC flux limiter
return max(0.0, min(2.0 * r, min(0.5 * (r + 1.0), 2.0)))
@ti.func
def w_to_q(w):
# convert primitive variables to conserved variables
q = ti.Vector([0.0, 0.0, 0.0, 0.0])
q[0] = w[0] # rho
q[1] = w[0] * w[1] # rho*u
q[2] = w[0] * w[2] # rho*v
q[3] = w[0] * (w[3] / ((gamma - 1) * w[0]) + 0.5 * (w[1] ** 2 + w[2] ** 2))
# rho*e
return q
@ti.func
def q_to_w(q):
# convert conserved variables to primitive variables
w = ti.Vector([0.0, 0.0, 0.0, 0.0])
w[0] = q[0] # rho
w[1] = q[1] / q[0] # u
w[2] = q[2] / q[0] # v
w[3] = (gamma - 1) * (q[3] - 0.5 * (q[1] ** 2 + q[2] ** 2) / q[0])
# p
return w
@ti.func
def HLLC_flux(qL, qR, n):
# normal vector
nx = n[0]
ny = n[1]
# Left state
rL = qL[0] # rho
uL = qL[1] / qL[0] # u
vL = qL[2] / qL[0] # v
pL = (gamma - 1.0) * (qL[3] - 0.5 * (qL[1] ** 2 + qL[2] ** 2) / qL[0])
# p
vnL = uL * nx + vL * ny
vtL = -uL * ny + vL * nx
aL = ti.sqrt(gamma * pL / rL)
HL = (qL[3] + pL) / rL
# Right state
rR = qR[0] # rho
uR = qR[1] / qR[0] # u
vR = qR[2] / qR[0] # v
pR = (gamma - 1.0) * (qR[3] - 0.5 * (qR[1] ** 2 + qR[2] ** 2) / qR[0])
# p
vnR = uR * nx + vR * ny
vtR = -uR * ny + vR * nx
aR = ti.sqrt(gamma * pR / rR)
HR = (qR[3] + pR) / rR
# Left and Right fluxes
fL = ti.Vector([rL * vnL, rL * vnL * uL + pL * nx, rL * vnL * vL + pL * ny, rL * vnL * HL])
fR = ti.Vector([rR * vnR, rR * vnR * uR + pR * nx, rR * vnR * vR + pR * ny, rR * vnR * HR])
# Roe Averages
rt = ti.sqrt(rR / rL)
u = (uL + rt * uR) / (1.0 + rt)
v = (vL + rt * vR) / (1.0 + rt)
H = (HL + rt * HR) / (1.0 + rt)
a = ti.sqrt((gamma - 1.0) * (H - (u**2 + v**2) / 2.0))
vn = u * nx + v * ny
# wavespeeds
sL = min(vnL - aL, vn - a)
sR = max(vnR + aR, vn + a)
sM = (pL - pR + rR * vnR * (sR - vnR) - rL * vnL * (sL - vnL)) / (rR * (sR - vnR) - rL * (sL - vnL))
# HLLC flux.
HLLC = ti.Vector([0.0, 0.0, 0.0, 0.0])
if 0 <= sL:
HLLC = fL
elif 0 <= sM:
qsL = (
rL
* (sL - vnL)
/ (sL - sM)
* ti.Vector(
[
1.0,
sM * nx - vtL * ny,
sM * ny + vtL * nx,
qL[3] / rL + (sM - vnL) * (sM + pL / (rL * (sL - vnL))),
]
)
)
HLLC = fL + sL * (qsL - qL)
elif 0 <= sR:
qsR = (
rR
* (sR - vnR)
/ (sR - sM)
* ti.Vector(
[
1.0,
sM * nx - vtR * ny,
sM * ny + vtR * nx,
qR[3] / rR + (sM - vnR) * (sM + pR / (rR * (sR - vnR))),
]
)
)
HLLC = fR + sR * (qsR - qR)
elif 0 >= sR:
HLLC = fR
return HLLC
@ti.kernel
def compute_F_muscl():
for i, j in Q:
if is_interior_x_face(i, j):
# muscl reconstrucion of left and right states with HLLC flux
wL = ti.Vector([0.0, 0.0, 0.0, 0.0])
wR = ti.Vector([0.0, 0.0, 0.0, 0.0])
for f in ti.static(range(4)):
ratio_l = (W[i, j][f] - W[i - 1, j][f]) / (W[i - 1, j][f] - W[i - 2, j][f])
ratio_r = (W[i, j][f] - W[i - 1, j][f]) / (W[i + 1, j][f] - W[i, j][f])
wL[f] = W[i - 1, j][f] + 0.5 * mc_lim(ratio_l) * (W[i - 1, j][f] - W[i - 2, j][f])
wR[f] = W[i, j][f] - 0.5 * mc_lim(ratio_r) * (W[i + 1, j][f] - W[i, j][f])
F_x[i, j] = HLLC_flux(w_to_q(wL), w_to_q(wR), ti.Vector([1.0, 0.0]))
elif is_boundary_x_face(i, j):
F_x[i, j] = HLLC_flux(Q[i - 1, j], Q[i, j], ti.Vector([1.0, 0.0]))
if is_interior_y_face(i, j):
# muscl reconstrucion of left and right states with HLLC flux
wL = ti.Vector([0.0, 0.0, 0.0, 0.0])
wR = ti.Vector([0.0, 0.0, 0.0, 0.0])
for f in ti.static(range(4)):
ratio_l = (W[i, j][f] - W[i, j - 1][f]) / (W[i, j - 1][f] - W[i, j - 2][f])
ratio_r = (W[i, j][f] - W[i, j - 1][f]) / (W[i, j + 1][f] - W[i, j][f])
wL[f] = W[i, j - 1][f] + 0.5 * mc_lim(ratio_l) * (W[i, j - 1][f] - W[i, j - 2][f])
wR[f] = W[i, j][f] - 0.5 * mc_lim(ratio_r) * (W[i, j + 1][f] - W[i, j][f])
F_y[i, j] = HLLC_flux(w_to_q(wL), w_to_q(wR), ti.Vector([0.0, 1.0]))
elif is_boundary_y_face(i, j):
F_y[i, j] = HLLC_flux(Q[i, j - 1], Q[i, j], ti.Vector([0.0, 1.0]))
@ti.func
def sign(a):
sgn = 0.0
if a > 0.0:
sgn = 1.0
elif a < 0.0:
sgn = -1.0
return sgn
@ti.func
def cosh(a):
return (ti.exp(a) + ti.exp(-a)) / 2.0
@ti.func
def thinc(wl, wc, wr, beta):
w0 = wc
w1 = wc
if (wr - wc) * (wc - wl) > 0.0:
# use thinc reconstruction
eps = 1.0e-15
wmin = ti.min(wr, wl)
wmax = ti.max(wr, wl)
wdelta = wmax - wmin
theta = sign(wr - wl)
C = (wc - wmin + eps) / (wdelta + eps)
B = ti.exp(theta * beta * (2 * C - 1))
A = (B / cosh(beta) - 1) / ti.tanh(beta)
# reconstructed value on right side of left face
w0 = wmin + wdelta / 2.0 * (1.0 + theta * A)
# reconstructed value on left side of right face
w1 = wmin + wdelta / 2.0 * (1.0 + theta * (ti.tanh(beta) + A) / (1.0 + A * ti.tanh(beta)))
return w0, w1
@ti.kernel
def compute_F_thinc():
# reconstruct primitive variables on interior faces of each cell using
# multiple candidate thinc reconstructions
for i, j in Q:
if is_interior_cell(i, j):
for f in ti.static(range(4)):
# smooth x-dir reconstruction
w0, w1 = thinc(W[i - 1, j][f], W[i, j][f], W[i + 1, j][f], beta_smooth)
W_xr[i, j, 0][f] = w0
W_xl[i + 1, j, 0][f] = w1
# sharp x-dir reconstruction
w0, w1 = thinc(W[i - 1, j][f], W[i, j][f], W[i + 1, j][f], beta_sharp)
W_xr[i, j, 1][f] = w0
W_xl[i + 1, j, 1][f] = w1
# smooth y-dir reconstruction
w0, w1 = thinc(W[i, j - 1][f], W[i, j][f], W[i, j + 1][f], beta_smooth)
W_yr[i, j, 0][f] = w0
W_yl[i, j + 1, 0][f] = w1
# sharp y-dir reconstruction
w0, w1 = thinc(W[i, j - 1][f], W[i, j][f], W[i, j + 1][f], beta_sharp)
W_yr[i, j, 1][f] = w0
W_yl[i, j + 1, 1][f] = w1
for i, j in Q:
# choose the final reconstruction for each cell using the BVD algorithm
if is_interior_cell(i, j):
for f in ti.static(range(4)):
# x-dir
TBV_smooth = abs(W_xl[i, j, 0][f] - W_xr[i, j, 0][f]) + abs(W_xl[i + 1, j, 0][f] - W_xr[i + 1, j, 0][f])
TBV_sharp = abs(W_xl[i, j, 1][f] - W_xr[i, j, 1][f]) + abs(W_xl[i + 1, j, 1][f] - W_xr[i + 1, j, 1][f])
if TBV_smooth < TBV_sharp:
W_xr[i, j, 2][f] = W_xr[i, j, 0][f]
W_xl[i + 1, j, 2][f] = W_xl[i + 1, j, 0][f]
else:
W_xr[i, j, 2][f] = W_xr[i, j, 1][f]
W_xl[i + 1, j, 2][f] = W_xl[i + 1, j, 1][f]
# y-dir
TBV_smooth = abs(W_yl[i, j, 0][f] - W_yr[i, j, 0][f]) + abs(W_yl[i, j + 1, 0][f] - W_yr[i, j + 1, 0][f])
TBV_sharp = abs(W_yl[i, j, 1][f] - W_yr[i, j, 1][f]) + abs(W_yl[i, j + 1, 1][f] - W_yr[i, j + 1, 1][f])
if TBV_smooth < TBV_sharp:
W_yr[i, j, 2][f] = W_yr[i, j, 0][f]
W_yl[i, j + 1, 2][f] = W_yl[i, j + 1, 0][f]
else:
W_yr[i, j, 2][f] = W_yr[i, j, 1][f]
W_yl[i, j + 1, 2][f] = W_yl[i, j + 1, 1][f]
for i, j in Q:
# compute numerical fluxes of with Riemann solver
if is_interior_x_face(i, j):
# muscl reconstrucion of left and right states with HLLC flux
F_x[i, j] = HLLC_flux(w_to_q(W_xl[i, j, 2]), w_to_q(W_xr[i, j, 2]), ti.Vector([1.0, 0.0]))
elif is_boundary_x_face(i, j):
F_x[i, j] = HLLC_flux(Q[i - 1, j], Q[i, j], ti.Vector([1.0, 0.0]))
if is_interior_y_face(i, j):
F_y[i, j] = HLLC_flux(w_to_q(W_yl[i, j, 2]), w_to_q(W_yr[i, j, 2]), ti.Vector([0.0, 1.0]))
elif is_boundary_y_face(i, j):
F_y[i, j] = HLLC_flux(Q[i, j - 1], Q[i, j], ti.Vector([0.0, 1.0]))
@ti.kernel
def calc_dt():
dt[None] = 1.0e5
for i, j in Q:
w = q_to_w(Q[i, j])
a = ti.sqrt(gamma * w[3] / w[0])
vel = ti.sqrt(w[1] ** 2 + w[2] ** 2)
ws = a + vel
ti.atomic_min(dt[None], CFL * h / ws / 2.0)
@ti.kernel
def update_Q(rk_step: ti.template()):
for i, j in Q:
if is_interior_cell(i, j):
if ti.static(rk_step == 0):
Q[i, j] = Q[i, j] + dt[None] * (F_x[i, j] - F_x[i + 1, j] + F_y[i, j] - F_y[i, j + 1]) / h
if ti.static(rk_step == 1):
Q[i, j] = (Q[i, j] + Q_old[i, j]) / 2.0 + dt[None] * (
F_x[i, j] - F_x[i + 1, j] + F_y[i, j] - F_y[i, j + 1]
) / h
@ti.kernel
def paint():
for i, j in img:
ii = min(max(1, i * N // res), N - 2)
jj = min(max(1, j * N // res), N - 2)
if img_field == 0: # density
img[i, j] = Q[ii, jj][0]
elif img_field == 1: # numerical schlieren
img[i, j] = ti.sqrt(
((Q[ii + 1, jj][0] - Q[ii - 1, jj][0]) / h) ** 2 + ((Q[ii, jj + 1][0] - Q[ii, jj - 1][0]) / h) ** 2
)
elif img_field == 2: # vorticity
img[i, j] = (Q[ii + 1, jj][2] - Q[ii - 1, jj][2]) / h - (Q[ii, jj + 1][1] - Q[ii, jj - 1][1]) / h
elif img_field == 3: # velocity magnitude
img[i, j] = ti.sqrt(Q[ii, jj][1] ** 2 + Q[ii, jj][2] ** 2)
max_ = -1.0e10
min_ = 1.0e10
for i, j in img:
ti.atomic_max(max_, img[i, j])
ti.atomic_min(min_, img[i, j])
for i, j in img:
if use_fixed_caxis:
min_ = fixed_caxis[0]
max_ = fixed_caxis[1]
img[i, j] = (img[i, j] - min_) / (max_ - min_)
def main():
gui = ti.GUI("Euler Equations", (res, res))
cmap = cm.get_cmap(cmap_name)
set_ic()
set_bc()
n = 0
while gui.running:
calc_dt()
copy_to_old()
for rk_step in range(2):
compute_W()
if method == 0:
compute_F_muscl()
else:
compute_F_thinc()
update_Q(rk_step)
set_bc()
if n % 10 == 0:
paint()
gui.set_image(cmap(img.to_numpy()))
gui.show()
n += 1
if __name__ == "__main__":
main()