Time-dependent demo (bubble shear)

demos/monge_ampere_bubble_shear.py

@@ -0,0 +1,266 @@
+# Mesh Movement in a time-dependent problem
+# =========================================
+#
+# In the `previous demo <./monge_ampere_helmholtz.py.html>`__, we demonstrated mesh
+# movement with the Monge-Ampère method in the context of a steady-state PDE. In this
+# demo, we progress to the time-dependent case.
+#
+# The demo solves an advection equation,
+#
+# ..math::
+#    \frac{\partial c}{\partial t}+\mathbf{u}\cdot\nabla c=0,
+#
+# where the solution field :math:`c` is approximated in :math:`\mathbb{P}1` space, and
+# the background velocity field is denoted by :math:`\mathbf{u}`. The velocity field is
+# approximated in vector :math:`\mathbb{P}1` space.
+#
+# As in previous examples, import from Firedrake and the :class:`~.MongeAmpereMover`. In
+# this demo, we also import :class:`~.RiemannianMetric` from the Animate metric-based
+# mesh adaptation package, as it comes with the functionality to recover Hessians. ::
+
+from animate.metric import RiemannianMetric
+from firedrake import *
+
+from movement.monge_ampere import MongeAmpereMover
+from movement.monitor import RingMonitorBuilder
+
+# As in most demos, we define a uniform mesh of the unit square. ::
+
+n = 20
+mesh = UnitSquareMesh(n, n)
+
+# The initial condition for the problem is unity in a circle shape of radius 0.15,
+# centred at :math:`(0.5,0.65)` and zero elsewhere. ::
+
+ball_r, ball_x0, ball_y0 = 0.15, 0.5, 0.65
+
+
+def initial_condition(mesh2d):
+    x, y = SpatialCoordinate(mesh2d)
+    r = sqrt(pow(x - ball_x0, 2) + pow(y - ball_y0, 2))
+    return conditional(r < ball_r, 1.0, 0.0)
+
+
+# Given this initial condition, it makes sense to move the initial mesh so that it has
+# mesh resolution aligned with the edge of the circle. (Recall the earlier
+# `ring demo <./monge_ampere_ring.py.html>`__.) We keep the relative tolerance used in
+# the Monge-Ampère solver to :math:`10^{-3}` for computational efficiency. ::
+
+mb = RingMonitorBuilder(centre=(0.5, 0.65), radius=0.15, amplitude=10.0, width=200.0)
+ring_monitor = mb.get_monitor()
+rtol = 1.0e-03
+mover = MongeAmpereMover(mesh, ring_monitor, method="quasi_newton", rtol=rtol)
+mover.move()
+mesh = mover.mesh
+
+# .. code-block:: none
+#
+#       0   Volume ratio  7.19   Variation (σ/μ) 8.77e-01   Residual 8.55e-01
+#       1   Volume ratio  7.60   Variation (σ/μ) 5.80e-01   Residual 5.39e-01
+#       2   Volume ratio  7.39   Variation (σ/μ) 5.27e-01   Residual 4.54e-01
+#       3   Volume ratio  4.67   Variation (σ/μ) 4.31e-01   Residual 3.49e-01
+#       4   Volume ratio  4.96   Variation (σ/μ) 3.95e-01   Residual 2.90e-01
+#       5   Volume ratio  4.44   Variation (σ/μ) 3.36e-01   Residual 2.25e-01
+#       6   Volume ratio  4.03   Variation (σ/μ) 2.98e-01   Residual 1.70e-01
+#       7   Volume ratio  3.51   Variation (σ/μ) 2.56e-01   Residual 1.14e-01
+#       8   Volume ratio  3.52   Variation (σ/μ) 2.24e-01   Residual 7.84e-02
+#       9   Volume ratio  3.43   Variation (σ/μ) 1.98e-01   Residual 4.80e-02
+#      10   Volume ratio  3.27   Variation (σ/μ) 1.83e-01   Residual 2.91e-02
+#      11   Volume ratio  3.11   Variation (σ/μ) 1.72e-01   Residual 1.50e-02
+#
+# Plotting the adapted mesh gives a similar result as in the `earlier demo
+# <./monge_ampere_ring.py.html>`__. ::
+
+import matplotlib.pyplot as plt
+
+fig, axes = plt.subplots()
+triplot(mesh, axes=axes)
+axes.set_aspect(1)
+plt.savefig("monge_ampere_bubble_shear-initial_adapted_mesh.jpg")
+
+# .. figure:: monge_ampere_bubble_shear-initial_adapted_mesh.jpg
+#    :figwidth: 60%
+#    :align: center
+#
+# With a suitable initial mesh, we can now define the PDE problem.
+#
+# We denote the solution at the current timestep by `c` and the lagged solution (at the
+# previous timestep) by `c_`. Similarly, we use `u` to denote the velocity field at the
+# current timestep and `u_` to denote the lagged velocity field. ::
+
+Q = FunctionSpace(mesh, "CG", 1)
+V = VectorFunctionSpace(mesh, "CG", 1)
+c = Function(Q, name="concentration").interpolate(initial_condition(mesh))
+c_ = Function(Q).assign(c)
+u = Function(V, name="velocity")
+u_ = Function(V)
+
+# The velocity field for the problem is an analyically prescribed rotational field, set
+# up such that it is periodic in time, returning to its initial condition after half of
+# the full period :math:`T`. Defining the current time as a variable in
+# :math:`\mathbb{R}`-space, we can define the velocity expression in UFL once and then
+# update the time as the solution progresses. ::
+
+x, y = SpatialCoordinate(mesh)
+T = Constant(6.0, name="period")
+R = FunctionSpace(mesh, "R", 0)
+dt = Function(R, name="timestep").assign(0.01)
+t = Function(R, name="current time").assign(dt)
+u_expression = as_vector(
+    [
+        2 * sin(pi * x) ** 2 * sin(2 * pi * y) * cos(2 * pi * t / T),
+        -sin(2 * pi * x) * sin(pi * y) ** 2 * cos(2 * pi * t / T),
+    ]
+)
+
+# To stabilise the :math:`\mathbb{P}1` discretisation, we make use of SUPG with an
+# artificial diffusivity :math:`D = 0.1`. ::
+
+D = Function(R).assign(0.1)
+h = CellSize(mesh)
+U = sqrt(dot(u, u))
+tau = min_value(0.5 * h / U, U * h / (6 * D))
+phi = TestFunction(Q)
+phi += tau * dot(u, grad(phi))
+
+# We are now able to define the vriational form of the advection equation. ::
+
+trial = TrialFunction(Q)
+theta = Function(R).assign(0.5)  # Crank-Nicolson implicitness
+a = inner(trial, phi) * dx + dt * theta * inner(dot(u, grad(trial)), phi) * dx
+L = inner(c_, phi) * dx - dt * (1 - theta) * inner(dot(u_, grad(c_)), phi) * dx
+lvp = LinearVariationalProblem(a, L, c, bcs=DirichletBC(Q, 0.0, "on_boundary"))
+
+# For the linear solver, it should be sufficient to use GMRES preconditioned with SOR.
+# ::
+
+solver_parameters = {
+    "snes_type": "ksponly",
+    "ksp_type": "gmres",
+    "pc_type": "sor",
+}
+lvs = LinearVariationalSolver(lvp, solver_parameters=solver_parameters)
+
+# To properly transfer the solution field when the mesh is moved within the time
+# integration loop, we need to stash a separate mesh and solution variable to hold the
+# data. ::
+
+
+class Stash:
+    """
+    Class for holding temporary mesh and solution data.
+    """
+
+    mesh = Mesh(mesh.coordinates.copy(deepcopy=True))
+    solution = None
+
+
+# For the subsequent mesh movement, we switch to a similar Hessian-based monitor
+# function as considered in the `previous demo
+# <./monge_ampere_helmholtz.py.html>`__. For efficiency, define the Hessian, its norm,
+# and the monitor scaling once, rather than inside the monitor function. ::
+
+P1_ten = TensorFunctionSpace(mesh, "CG", 1)
+H = RiemannianMetric(P1_ten)
+Hnorm = Function(Q, name="Hnorm")
+alpha = Constant(10.0)
+
+
+def monitor(mesh):
+    # Project the stashed solution data onto the current mesh
+    Q_tmp = FunctionSpace(Stash.mesh, "CG", 1)
+    Stash.solution = Function(Q_tmp)
+    Stash.solution.dat.data[:] = c.dat.data
+    c.project(Stash.solution)
+
+    # Compute the monitor expression using the Hessian
+    H.compute_hessian(c)
+    Hnorm.interpolate(sqrt(inner(H, H)))
+    Hnorm_max = Hnorm.dat.data.max()
+    m = 1 + alpha * Hnorm / Hnorm_max
+
+    # Stash the mesh before it's adapted
+    Stash.mesh = Mesh(mesh.coordinates.copy(deepcopy=True))
+    return m
+
+
+# Switch the monitor function used by the :class:`~.MongeAmpereMover`. ::
+
+mover.monitor_function = monitor
+
+# Now we're ready to solve the PDE. Let's just solve for a quarter period, which is
+# sufficient to see the bubble of tracer concentration stretch out considerably. ::
+
+t_end = 1.5
+c_.interpolate(initial_condition(mesh))
+c.assign(c_)
+outfile = VTKFile("bubble_shear.pvd")
+outfile.write(c)
+while float(t) < t_end + 0.5 * float(dt):
+    # Update the background velocity field at the current timestep
+    u.interpolate(u_expression)
+
+    # Solve the advection equation
+    lvs.solve()
+
+    # Move the mesh
+    mover.move()
+
+    # Export the solution
+    outfile.write(c, time=float(t))
+
+    # Update the solution at the previous timestep
+    c_.assign(c)
+    u_.assign(u)
+    t.assign(t + dt)
+
+# The outputs of the mesh movement steps are not shown here, for brevity. However,
+# looking at a small sample, we observe that very few iterations are required once the
+# mesh movement has become established:
+#
+# .. code-block:: none
+#
+#    ...
+#       0   Volume ratio  2.24   Variation (σ/μ) 7.24e-02   Residual 3.08e-02
+#       1   Volume ratio  2.09   Variation (σ/μ) 7.02e-02   Residual 1.32e-02
+#    Solver converged in 1 iteration.
+#       0   Volume ratio  2.23   Variation (σ/μ) 6.73e-02   Residual 2.95e-02
+#       1   Volume ratio  1.98   Variation (σ/μ) 6.35e-02   Residual 1.24e-02
+#    Solver converged in 1 iteration.
+#    ...
+
+
+fig, axes = plt.subplots()
+tricontourf(c, axes=axes)
+axes.set_aspect(1)
+plt.savefig("monge_ampere_bubble_shear-final_solution.jpg")
+
+# .. figure:: monge_ampere_bubble_shear-final_solution.jpg
+#    :figwidth: 60%
+#    :align: center
+
+fig, axes = plt.subplots()
+triplot(mesh, axes=axes)
+axes.set_aspect(1)
+plt.savefig("monge_ampere_bubble_shear-final_adapted_mesh.jpg")
+
+# .. figure:: monge_ampere_bubble_shear-final_adapted_mesh.jpg
+#    :figwidth: 60%
+#    :align: center
+#
+# As you might have already realised, the approach documented here is a completely
+# over-the-top way to solve a simple linear advection problem. However, it shows how the
+# Monge-Ampère approach might be used in other problems for which it brings a greater
+# advantage.
+#
+# Given that the PDE represents linear advection, this example is a prime candidate for
+# Lagrangian-type mesh movement approaches, which are likely to have far lower
+# computational costs than a full Monge-Ampère approach as used here.
+#
+# This tutorial can be dowloaded as a `Python script <monge_ampere_bubble_shear.py>`__.
+#
+#
+# .. rubric:: References
+#
+# .. bibliography::
+#    :filter: docname in docnames

demos/monge_ampere_helmholtz.py

@@ -449,6 +449,9 @@ def monitor_interp_Hessian(mesh):
 # function that computes the solution less frequently, or even only once per iteration.
 # Movement allows and encourages such experimentation with different monitor functions.
 #
+# In the `next demo <./monge_ampere_bubble_shear.py.html>`__, we will demonstrate
+# an application of Monge-Ampère mesh movement to a time-dependent problem.
+#
 # This tutorial can be dowloaded as a `Python script <monge_ampere_helmholtz.py>`__.
 #
 # .. rubric:: Footnotes

test/test_demos.py

@@ -25,6 +25,7 @@
     "monge_ampere_helmholtz.py": changes_dict,
     "monge_ampere_ring.py": changes_dict,
     "monge_ampere_periodic.py": changes_dict,
+    "monge_ampere_bubble_shear.py": {"n = 20": "n = 10", "t_end = 1.5": "t_end = 0.15"},
 }