Demo created at Dartington Hall - Shape Optimization

demos/shape_optimization/shape_optimization.py.rst

@@ -0,0 +1,109 @@
+Shape optimization
+==================
+
+Shape optimization is about modifying the shape of a domain :math:`\Omega` so
+that an objective function :math:`J(\Omega)` is minimized. In this demo, we
+consider an objective function constrained to a boundary value problem and
+implement a simple mesh-moving shape optimization strategy using Firedrake and
+pyadjoint.  This tutorial was contributed by `Alberto Paganini
+<mailto:[email protected]>`__ with support from `Ado Farsi
+<mailto:[email protected]>`__ and `Mirko Ciceri
+<mailto:[email protected]>`__, and was written during the `ccp-dcm hackaton
+<https://ccp-dcm.github.io/exeter_hackathon>`__ at Dartington Hall.
+
+Let
+
+.. math::
+
+   J(\Omega) = \int_\Omega \big(u(\mathbf{x}) - u_t(\mathbf{x})\big)^2 \,\mathrm{d}\mathbf{x}\,.
+
+measure the difference between a steady-state temperature profile
+:math:`u:\mathbb{R}^2\to\mathbb{R}` and a target steady-state temperature
+profile :math:`u_t:\mathbb{R}^2\to\mathbb{R}`. Specifically, the function
+:math:`u` is the solution to the steady-state heat equation
+
+.. math::
+
+    -\Delta u = 4 \quad \text{in }\Omega\,, \qquad u = 0 \quad \text{on } \partial\Omega
+
+
+and the target temperature profile :math:`u_t` is
+
+.. math::
+
+    u_t(x,y) = 1.21 - (x - 0.5)^2 - (y - 0.5)^2\,.
+
+Beside the empty set, the domain that minimizes :math:`J(\Omega)` is a disc of
+radius :math:`1.1` centered at :math:`(0.5,0.5)`.
+
+We can now proceed to set up the problem. We import firedrake and pyadjoint and
+choose an initial guess (in this case, a unit disc centred at the origin)::
+
+  from firedrake import *
+  from firedrake.adjoint import *
+  mesh = UnitDiskMesh(refinement_level=3)
+
+Then, we :ref:`start annotating <adjoint-taping>` and turn the mesh coordinates into a control variable::
+
+  continue_annotation()
+  Q = mesh.coordinates.function_space()
+  dT = Function(Q)
+  mesh.coordinates.assign(mesh.coordinates + dT)
+
+We can now implement the target function::
+
+  x, y = SpatialCoordinate(mesh)
+  u_t = Constant(1.21) - (x - Constant(0.5))**2 - (y - Constant(0.5))**2
+
+solve the weak form of the boundary value problem::
+
+  V = FunctionSpace(mesh, "CG", 1)
+  u = Function(V, name='state')
+  v = TestFunction(V)
+  F = (dot(grad(u), grad(v)) - 4 * v) * dx
+  bcs = DirichletBC(V, Constant(0.), "on_boundary")
+  solve(F == 0, u, bcs=bcs)
+
+and evaluate the objective function::
+
+  J = assemble((u - u_t)**2*dx)
+
+We now turn the objective function into a reduced function so that pyadjoint
+(and UFL shape differentiation capability) can automatically compute shape
+gradients, that is, directions of steepest ascent::
+
+  Jred = ReducedFunctional(J, Control(dT))
+  stop_annotating()
+
+We now have all the ingredients to implement a basic steepest descent shape
+optimization algorithm with fixed step size.::
+
+  File = VTKFile("shape_iterates.pvd")
+  for ii in range(30):
+    print("J(ii =", ii, ") =", Jred(dT))
+    File.write(mesh.coordinates)
+
+    # compute the gradient (steepest ascent)
+    opts = {"riesz_representation": "H1"}
+    gradJ = Jred.derivative(options=opts)
+
+    # update domain
+    dT -= 0.2*gradJ
+
+  File.write(mesh.coordinates)
+  print("J(final) =", Jred(dT))
+
+.. only:: html
+
+  .. container:: youtube
+
+    .. vimeo:: 1083822714?loop=1
+       :width: 600px
+
+
+**Remark:** mesh-moving shape optimization can lead to mesh tangling, which
+invalidates finite element computations. For faster and more robust shape
+optimization, we recommend using Firedrake's shape optimization toolbox
+`Fireshape <https://github.com/fireshape/fireshape>`__.
+
+A python script version of this demo can be found :demo:`here <shape_optimization.py>`.

docs/source/adjoint.rst

@@ -159,6 +159,8 @@ The essence of this process is:
    derived automatically by applying UFL's :func:`~ufl.derivative`,
    :func:`~ufl.adjoint`, and :func:`~ufl.action` operators.
 
+.. _adjoint-taping:
+
 Taping an example calculation
 -----------------------------
 

docs/source/advanced_tut.rst

@@ -24,3 +24,4 @@ element systems.
    Rayleigh-Benard convection.<demos/rayleigh-benard.py>
    Netgen support.<demos/netgen_mesh.py>
    Full-waveform inversion: spatial and wave sources parallelism.<demos/full_waveform_inversion.py>
+   Shape optimisation.<demos/shape_optimization.py>