add poisson example

examples/poisson.py

@@ -0,0 +1,53 @@
+#! /usr/bin/env python3
+#
+# In this script we solve Poisson's equation Δu = 1 subject to boundary
+# constraints, using the fact that the solution to the strong form minimizes
+# the functional ∫ .5 ‖∇u‖² - u. The domain is a unit square, and the solution
+# is constrained to zero along the entire boundary.
+
+from nutils import mesh, function, solver, export, cli, testing
+
+def main(nelems: int):
+  '''
+  Poisson's equation on a unit square.
+
+  .. arguments::
+
+     nelems [10]
+       Number of elements along edge.
+  '''
+
+  domain, x = mesh.unitsquare(nelems, etype='square')
+  u = function.dotarg('udofs', domain.basis('std', degree=1))
+  g = u.grad(x)
+  J = function.J(x)
+  cons = solver.optimize('udofs',
+    domain.boundary.integral(u**2 * J, degree=2), droptol=1e-12)
+  udofs = solver.optimize('udofs',
+    domain.integral((g @ g / 2 - u) * J, degree=1), constrain=cons)
+  bezier = domain.sample('bezier', 3)
+  x, u = bezier.eval([x, u], udofs=udofs)
+  export.triplot('u.png', x, u, tri=bezier.tri, hull=bezier.hull)
+
+  return udofs
+
+# If the script is executed (as opposed to imported), :func:`nutils.cli.run`
+# calls the main function with arguments provided from the command line. To
+# keep with the default arguments simply run :sh:`python3 poisson.py`.
+
+if __name__ == '__main__':
+  cli.run(main)
+
+# Once a simulation is developed and tested, it is good practice to save a few
+# strategic return values for regression testing. The :mod:`nutils.testing`
+# module, which builds on the standard :mod:`unittest` framework, facilitates
+# this by providing :func:`nutils.testing.TestCase.assertAlmostEqual64` for the
+# embedding of desired results as compressed base64 data.
+
+class test(testing.TestCase):
+
+  def test_default(self):
+    udofs = main(nelems=10)
+    self.assertAlmostEqual64(udofs, '''
+      eNp9zrENwCAMBEBGYQJ444o2ozAAYgFmYhLEFqxAmye1FUtf+PSy7Jw9J6yoKGiMYsUTrq44kaVKZ7JM
+      +lWlDdlymEFXXC2o3H1C8mmzXz5t6OwhPfTDO+2na9+1f7D/teYFdsk5vQ==''')