uBoundary=dirichletBC)
"""
Note that on boundary 4 (right) no BC is explicitly applied leading to default or natural BC that are of homogeneous Neumann type
:math:`\frac{\partial u}{\partial n}=0`
"""
ax = showMesh(grid, data=u, filled=True, colorBar=True,
orientation='vertical', label='Solution $u$',
levels=np.linspace(min(u), max(u), 14), showLater=True)[0]
"""
Instead of the grid we now want to add streamlines to the plot to show the gradients of the solution (i.e., the flow direction).
"""
drawStreamLines(ax, grid, u, nx=25, ny=25, color='Black')
ax.text(0.0, 1.02, '$u=1$',
horizontalalignment='center' )
ax.text(-1.08, 0.0, '$\partial u/\partial n=-0.5$',
verticalalignment='center', rotation='vertical')
ax.text(0.0, -1.08, '$\partial u/\partial n=2.5$',
horizontalalignment='center')
ax.text(1.02, 0.0, '$\partial u/\partial n=0$',
verticalalignment='center', rotation='vertical')
ax.set_title('$\\nabla\cdot(1\\nabla u)=0$')
ax.set_xlim([-1.1, 1.1])
ax.set_ylim([-1.1, 1.1])
while len(downwind) > 0:
fastMarch(mesh, downwind, times, upTags, downTags)
print(time.time() - tic, "s")
# compare with analytical solution along the x axis
x = np.arange(0., 100., 0.5)
t = pg.interpolate(mesh, times, pg.asvector(x), x * 0., x * 0.)
tdirect = x / v[0] # direct wave
alfa = asin(v[0] / v[1]) # critically refracted wave angle
xreflec = tan(alfa) * zlay * 2. # first critically refracted
trefrac = (x - xreflec) / v[1] + xreflec * v[1] / v[0]**2
tana = np.where(trefrac < tdirect, trefrac, tdirect) # minimum of both
print("min(dt)=",
min(t - tana) * 1000,
"ms max(dt)=",
max(t - tana) * 1000,
"ms")
# plot traveltime field, a few lines
fig = plt.figure()
a = fig.add_subplot(211)
drawMesh(a, mesh)
drawField(a, mesh, times, True, 'Spectral')
drawStreamLines(a, mesh, times, nx=50, ny=50)
# plot calculated and measured travel times
a2 = fig.add_subplot(212)
plt.plot(x, t, 'b.-', x, tdirect, 'r-', x, trefrac, 'g-')
plt.show()
###############################################################################
# Then we start marching until all fields are filled.
tic = time.time()
while len(downwind) > 0:
fastMarch(mesh, downwind, times, upTags, downTags)
print(time.time() - tic, "s")
###############################################################################
# First, we plot the traveltime field and streamlines
fig, ax = plt.subplots(figsize=(10, 5))
drawMesh(ax, mesh)
ax.set_xlabel('x [m]')
ax.set_ylabel('y [m]')
drawField(ax, mesh, times, cmap='Spectral', fillContour=True)
drawStreamLines(ax, mesh, -times, nx=50, ny=50)
###############################################################################
# We compare the result with the analytical solution along the x axis
x = np.arange(0., 140., 0.5)
tFMM = pg.interpolate(mesh, times, x, x * 0., x * 0.)
tAna = analyticalSolution2Layer(x)
print("min(dt)={} ms max(dt)={} ms".format(
min(tFMM - tAna) * 1000,
max(tFMM - tAna) * 1000))
###############################################################################
# In order to use the Dijkstra, we extract the surface positions >0
mx = pg.x(mesh.positions())
my = pg.y(mesh.positions())
fi = pg.find((my == 0.0) & (mx >= 0))
###############################################################################
# Then we start marching until all fields are filled.
tic = time.time()
while len(downwind) > 0:
fastMarch(mesh, downwind, times, upTags, downTags)
print(time.time() - tic, "s")
###############################################################################
# First, we plot the traveltime field and streamlines
fig, ax = plt.subplots(figsize=(10, 5))
drawMesh(ax, mesh)
ax.set_xlabel('x [m]')
ax.set_ylabel('y [m]')
pg.show(mesh, times, cMap='Spectral', fillContour=True, ax=ax)
drawStreamLines(ax, mesh, -times, nx=50, ny=50)
###############################################################################
# We compare the result with the analytical solution along the x axis
x = np.arange(0., 140., 0.5)
tFMM = pg.interpolate(mesh, times, x, x * 0., x * 0.)
tAna = analyticalSolution2Layer(x, zlay, v[0], v[1])
print("min(dt)={} ms max(dt)={} ms".format(min(tFMM - tAna) * 1000,
max(tFMM - tAna) * 1000))
###############################################################################
# In order to use the Dijkstra, we extract the surface positions >0
mx = pg.x(mesh.positions())
my = pg.y(mesh.positions())
fi = pg.find((my == 0.0) & (mx >= 0))
px = np.sort(mx(fi))
# start fast marching
tic = time.time()
while len(downwind) > 0:
fastMarch(mesh, downwind, times, upTags, downTags)
print(time.time() - tic, "s")
# compare with analytical solution along the x axis
x = np.arange(0., 100., 0.5)
t = pg.interpolate(mesh, times, x, x * 0., x * 0.)
tdirect = x / v[0] # direct wave
alfa = asin(v[0] / v[1]) # critically refracted wave angle
xreflec = tan(alfa) * zlay * 2. # first critically refracted
trefrac = (x - xreflec) / v[1] + xreflec * v[1] / v[0]**2
tana = np.where(trefrac < tdirect, trefrac, tdirect) # minimum of both
print("min(dt)=",
min(t - tana) * 1000, "ms max(dt)=",
max(t - tana) * 1000, "ms")
# plot traveltime field, a few lines
fig = plt.figure()
a = fig.add_subplot(211)
drawMesh(a, mesh)
drawField(a, mesh, times, True, 'Spectral')
drawStreamLines(a, mesh, times, nx=50, ny=50)
# plot calculated and measured travel times
a2 = fig.add_subplot(212)
plt.plot(x, t, 'b.-', x, tdirect, 'r-', x, trefrac, 'g-')
plt.show()
for nn in tmpNodes:
if not upTags[nn.id()] and not downTags[nn.id()]:
downwind.add(nn)
downTags[nn.id()] = 1
# start fast marching
tic = time.time()
while len(downwind) > 0:
fastMarch(mesh, downwind, times, upTags, downTags)
print(time.time() - tic, "s")
# compare with analytical solution along the x axis
x = np.arange(-20., 150., 0.5)
t = pg.interpolate(mesh, times, pg.asvector(x), x * 0., x * 0.)
tdirect = np.abs(x) / v[0] # direct wave
alfa = asin(v[0] / v[1]) # critically refracted wave angle
xreflec = tan(alfa) * zlay * 2. # first critically refracted
trefrac = (np.abs(x) - xreflec) / v[1] + xreflec * v[1] / v[0]**2
tana = np.where(trefrac < tdirect, trefrac, tdirect) # minimum of both
print("min(dt)=", min(t-tana)*1e3, "ms max(dt)=", max(t-tana)*1e3, "ms")
# %% plot traveltime field, a few lines
fig, ax = plt.subplots(nrows=2, sharex=True)
drawMesh(ax[0], mesh)
drawField(ax[0], mesh, times, True)
drawStreamLines(ax[0], mesh, times, color='white')
# plot calculated and measured travel times
ax[1].plot(x, t, 'b.-', x, tdirect, 'r-', x, trefrac, 'g-')
plt.show()