def set_qoi_kernel(self, prob, i):
r"""
Assuming the QoI is power output,
..math::
J(\mathbf q) = \int_\Omega C_t \|\mathbf u\|^3 \;\mathrm dx,
this can be represented as an inner product with :math:`mathbf q`
using the kernel function
..math::
\mathbf k(\mathbf q) = (\frac13 C_t \|\mathbf u\|\mathbf u, 0).
That is,
..math::
J(\mathbf q) = \int_\Omega \mathbf k(\mathbf q) \cdot \mathbf q \;\mathrm dx.
"""
prob.kernels[i] = Function(prob.V[i])
u, eta = prob.fwd_solutions[i].split()
k_u, k_eta = prob.kernels[i].split()
# k_u.interpolate(Constant(1/3)*prob.turbine_densities[i]*speed(u)*u)
k_u.interpolate(prob.turbine_densities[i]*speed(u)*u)
with open(os.path.join(ramp_dir, "log"), "w+") as logfile:
logfile.write(msg + "\n")
op.plot_pvd = True
swp.export_state(0, ramp_dir)
else:
swp.load_state(0, ramp_dir)
# --- Plot
plot_dir = create_directory(os.path.join(os.path.dirname(__file__), 'plots'))
# Get fluid speed and elevation in P1 space
q = swp.fwd_solutions[0]
u, eta = q.split()
speed_proj = interpolate(speed(q), swp.P1[0])
eta_proj = project(eta, swp.P1[0])
# Plot fluid speed
fig, axes = plt.subplots(figsize=(14, 6))
cbar = fig.colorbar(tricontourf(speed_proj, axes=axes, levels=50, cmap='coolwarm'), ax=axes)
cbar.set_label(r"Fluid speed [$\mathrm{m\,s}^{-1}$]")
axes.set_xlim([-L/2, L/2])
axes.set_ylim([-W/2, W/2])
axes.set_xlabel(r"$x$-coordinate $[\mathrm m]$")
axes.set_ylabel(r"$y$-coordinate $[\mathrm m]$")
axes.set_xticks(np.linspace(-20000, 20000, 3))
axes.set_yticks(np.linspace(-20000, 20000, 5))
plt.tight_layout()
for ext in ("png", "pdf"):
plt.savefig(os.path.join(plot_dir, ".".join(["speed", ext])))
# --- Solve forward problem
tp = AdaptiveSteadyTurbineProblem(op,
discrete_turbines=discrete_turbines,
callback_dir=op.di)
tp.solve_forward()
op.print_debug("Power output: {:.4e}MW".format(tp.quantity_of_interest() *
1.030e-03))
# --- Plot fluid speed
if plot_any:
# Compute fluid speed
spd = interpolate(speed(tp.fwd_solution), tp.P1[0])
# Plot
fig, axes = plt.subplots(figsize=(12, 6.5))
tc = tricontourf(spd, axes=axes, **tricontourf_kwargs)
cbar = fig.colorbar(tc, ax=axes, orientation="horizontal", pad=0.1)
cbar.ax.set_yticklabels(cbar.ax.get_yticklabels(),
fontsize=fontsizes['cbar'])
cbar.set_label(r'Fluid speed [$m\,s^{-1}$]', fontsize=fontsizes['cbar'])
axes.set_xlim([0, op.domain_length])
axes.set_ylim([0, op.domain_width])
# axes.xaxis.tick_top()
# for axis in (axes.xaxis, axes.yaxis):
# for tick in axis.get_major_ticks():
# tick.label.set_fontsize(fontsizes['tick'])
axes.set_xticks([])
def set_qoi_kernel(self, prob, i):
prob.kernels[i] = Function(prob.V[i])
u, eta = prob.fwd_solutions[i].split()
k_u, k_eta = prob.kernels[i].split()
# k_u.interpolate(Constant(1/3)*prob.turbine_density*speed(u)*u)
k_u.interpolate(prob.turbine_density*speed(u)*u)
fig, axes = plt.subplots(figsize=(8, 3))
hmin = np.floor(h.vector().gather().min())
hmax = np.ceil(h.vector().gather().max())
levels = np.linspace(hmin, hmax, 40)
tc = tricontourf(h, axes=axes, levels=levels, cmap='coolwarm')
cbar = fig.colorbar(tc, ax=axes, orientation="horizontal", pad=0.1)
cbar.set_ticks(np.linspace(hmin, hmax, 5))
axes.axis(False)
axes.set_xlim([0, 50])
axes.set_ylim([0, 10])
fname = 'anisotropic_cell_size' if op.anisotropic_stabilisation else 'isotropic_cell_size'
savefig(fname, op.di, extensions=["jpg"])
# Calculate stabilisation parameter
uv = Constant(as_vector(op.base_velocity))
unorm = speed(uv)
tau = 0.5 * h / unorm
Pe = 0.5 * h * unorm / op.base_diffusivity
tau *= min_value(1, Pe / 3)
tau = interpolate(tau, h.function_space())
# Plot stabilisation parameter
fig, axes = plt.subplots(figsize=(8, 3))
taumin = np.floor(tau.vector().gather().min())
taumax = np.ceil(tau.vector().gather().max())
levels = np.linspace(taumin, taumax, 40)
tc = tricontourf(tau, axes=axes, levels=levels, cmap='coolwarm')
cbar = fig.colorbar(tc, ax=axes, orientation="horizontal", pad=0.1)
cbar.set_ticks(np.linspace(taumin, taumax, 5))
axes.axis(False)
axes.set_xlim([0, 50])