#! /usr/bin/env python
#
from fenics import *

def mitchell02 ( ):

#*****************************************************************************80
#
## mitchell02 sets up Mitchell test #2.
#
#  Reentrant corner
#  -Laplace(u) = f on [-1,+1]x[-1,+1] with a section removed from the clockwise
#  side of the positive X axis with angle omega,
#  u = u0 on the boundary.
#  u0 = r^alpha sin(alpha*theta) 
#  f = -(u0'')
#
#  Set omega = 3pi/2 for the L-domain problem.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    22 August 2014
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    William Mitchell,
#    A collection of 2D elliptic problems for testing adaptive
#    grid refinement algorithms,
#    Applied Mathematics and Computation,
#    Volume 220, 1 September 2013, pages 350-364.
#
  import matplotlib.pyplot as plt
#
#  Define parameters.
#  The symbol "pi" is predefined.
#
  omega = 3 * pi / 2
  alpha = pi / omega
#
#  The precomputed mesh should be read from the given file.
#
  mesh = Mesh ( "mitchell02_mesh.xml" )
#
#  Plot the mesh.
#
  plot ( mesh, title = 'Mitchell Test 02 Mesh' )
  filename = 'mitchell02_mesh.png'
  plt.savefig ( filename )
  print ( '  Graphics saved as "%s"' % ( filename ) )
  plt.close ( )
#
#  Define the function space.
#
  V = FunctionSpace ( mesh, "Lagrange", 1 )
#
#  Define the exact solution.
#
  u0 = Expression ( "pow ( sqrt(x[0]*x[0]+x[1]*x[1]), alpha )" 
                  "* sin ( alpha * atan2(x[1],x[0]) )", \
    alpha = alpha, degree = 10 )
#
#  Apply the boundary condition to points which are on the boundary of the mesh.
#
  def u0_boundary ( x, on_boundary ):
    return on_boundary
#
#  The value to be used at Dirichlet boundary points is U0.
#
  bc = DirichletBC ( V, u0, u0_boundary )
#
#  Define the variational problem.
#
  u = TrialFunction ( V )
  v = TestFunction ( V )
#
#  Got to figure out the correct value of F here!
#  Unless it's a miracle, f=0 is NOT the right value.
#
  f = Expression ( "0.0", degree = 0 )

  a = inner ( nabla_grad ( u ), nabla_grad ( v ) ) * dx
  L = f * v * dx
#
#  Compute the solution.
#
  u = Function ( V )
  solve ( a == L, u, bc )
#
#  Plot the solution.
#
  plot ( u, title = 'Mitchell Test 02 Solution' )
  filename = 'mitchell02_solution.png'
  plt.savefig ( filename )
  print ( '  Graphics saved as "%s"' % ( filename ) )
  plt.close ( )

  return

def mitchell02_test ( ):

#*****************************************************************************80
#
## mitchell02_test tests mitchell02.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    22 October 2018
#
#  Author:
#
#    John Burkardt
#

#
#  Report level = only warnings or higher.
#
  level = 30
  set_log_level ( level )

  print ( '' )
  print ( 'mitchell02_test:' )
  print ( '  FENICS/Python version' )
  print ( '  Mitchell test problem #2.' )

  mitchell02 ( )
#
#  Terminate.
#
  print ( '' )
  print ( 'mitchell02_test:' )
  print ( '  Normal end of execution.' )
  return

if ( __name__ == '__main__' ):

  mitchell02_test ( )