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

def mitchell09 ( ):

#*****************************************************************************80
#
## mitchell09 sets up Mitchell test #9.
#
#  Wave front
#  -Laplace(u) = f on the unit square.
#  u = u0 on the boundary.
#  u0 = atan ( alpha * (sqrt(x-xc)**2+(y-yc)**2)-r0) )
#  f = -Laplace(u0)
#
#  Discussion:
#
#    Suggested parameter values:
#    * alpha =   20, xc = -0.05, yc = -0.05, r0 = 0.7
#    * alpha = 1000, xc = -0.05, yc = -0.05, r0 = 0.7
#    * alpha = 1000, xc =  1.5,  yc =  0.25, r0 = 0.92
#    * alpha =   50, xc =  0.5,  yc =  0.5,  r0 = 0.25
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    23 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
#
#  Set parameters.
#
  alpha = 20.0
  xc = -0.05
  yc = -0.05
  r0 = 0.7
#
#  Divide the unit square into a 4x4 mesh of quadrilaterals, and
#  split each into triangles.
#
  mesh = UnitSquareMesh ( 4, 4 )
#
#  Plot the mesh.
#
  plot ( mesh, title = 'Mitchell Test 09 Mesh' )
  filename = 'mitchell09_mesh.png'
  plt.savefig ( filename )
  print ( '  Graphics saved as "%s"' % ( filename ) )
  plt.close ( )
#
#  Define the function space.
#
  V = FunctionSpace ( mesh, "Lagrange", 1 )
#
#  Define the radial coordinate R.
#
  r = Expression ( "sqrt ( pow ( x[0] - xc, 2 ) + pow ( x[1] - yc, 2 ) - r0 )" ,
                 xc = xc, \
                 yc = yc, \
                 r0 = r0, degree = 10 )
#
#  Define the exact solution.
#  Can we use r in this way?  Do we need "r = r"?
#
  u0 = Expression ( " atan ( alpha * r )" ,
                    alpha = alpha, degee = 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 )
#
#  Define F.
#
  f = Expression ( "?" )
  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 )

  return

def mitchell09_test ( ):

#*****************************************************************************80
#
## mitchell09_test tests mitchell09.
#
#  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 ( 'mitchell09_test:' )
  print ( '  FENICS/Python version' )
  print ( '  Mitchell test problem #9.' )

  mitchell09 ( )
#
#  Terminate.
#
  print ( '' )
  print ( 'mitchell09_test:' )
  print ( '  Normal end of execution.' )
  return

if ( __name__ == '__main__' ):

  mitchell09_test ( )