#!/usr/bin/env python
# coding: utf-8

# # Combining Dirichlet and Neumann conditions
# Author: Jørgen S. Dokken
# 
# Let's return to the Poisson problem from the [Fundamentals chapter](./../chapter1/fundamentals.md) and see how to extend the mathematics and the implementation to handle Dirichlet condition in combination with a Neumann condition.
# The domain is still the unit square, but now we set the Dirichlet condition $u=u_D$ at the left and right sides, while the Neumann condition
# 
# $$
# -\frac{\partial u}{\partial n}=g
# $$
# 
# is applied to the  remaining sides $y=0$ and $y=1$.
# 
# ## The PDE problem
# Let $\Lambda_D$ and $\Lambda_N$ denote parts of the boundary $\partial \Omega$ where the Dirichlet and Neumann conditions apply, respectively.
# The complete boundary-value problem can be written as
# 
# $$
# -\nabla^2 u =f \qquad \text{in } \Omega,
# $$
# $$
# u=u_D \qquad\text{on } \Lambda_D,
# $$
# $$
# -\frac{\partial u}{\partial n}=g \qquad \text{on }\Lambda_N
# $$
# 
# Again, we choose $u=1+x^2+2y^2$ as the exact solution and adjust $f, g,$ and $u_D$ accordingly
# 
# $$
# f(x,y)=-6,
# $$
# $$
# g(x,y)=\begin{cases}
# 0, & y=0,\\
# -4, & y=1,
# \end{cases}
# $$
# $$
# u_D(x,y)=1+x^2+2y^2.
# $$
# 
# For the ease of programming, we define $g$ as a function over the whole domain $\Omega$ such that $g$ takes on the correct values at $y=0$ and $y=1$. One possible extension is
# 
# $$
#  g(x,y)=-4y.
# $$
# 
# ## The variational formulation
# The first task is to derive the variational formulatin. This time we cannot omit the boundary term arising from integration by parts, because $v$ is only zero on $\Lambda_D$. We have
# 
# $$
# -\int_\Omega (\nabla^2u)v~\mathrm{d} x = \int_\Omega \nabla u \cdot \nabla v ~\mathrm{d} x - \int_{\partial\Omega}\frac{\partial u}{\partial n}v~\mathrm{d}s,
# $$
# 
# and since $v=0$ on $\Lambda_D$,
# 
# $$
# - \int_{\partial\Omega}\frac{\partial u}{\partial n}v~\mathrm{d}s= - \int_{\Lambda_N}\frac{\partial u}{\partial n}v~\mathrm{d}s =\int_{\Lambda_N} gv~\mathrm{d}s,
# $$
# 
# by applying the boundary condition on $\Lambda_N$.
# The resulting weak from reads
# 
# $$
#     \int_\Omega \nabla u \cdot \nabla v~\mathrm{d} x = \int_\Omega fv~\mathrm{d} x - \int_{\Lambda_N}gv~\mathrm{d}s.
# $$
# Expressing this equation in the standard notation $a(u,v)=L(v)$ is straight-forward with
# 
# $$
#     a(u,v) = \int_{\Omega} \nabla u \cdot \nabla v ~\mathrm{d} x,\\
# $$
# $$
# L(v) = \int_{\Omega} fv ~\mathrm{d} x - \int_{\Lambda_N} gv~\mathrm{d} s.
# $$
# 
# ## Implementation
# As in the previous example, we define our mesh,function space and bilinear form $a(u,v)$.

# In[ ]:


from dolfinx import default_scalar_type
from dolfinx.fem import (Constant, Function, functionspace,
                         assemble_scalar, dirichletbc, form, locate_dofs_geometrical)
from dolfinx.fem.petsc import LinearProblem
from dolfinx.mesh import create_unit_square
from dolfinx.plot import vtk_mesh

from mpi4py import MPI
from ufl import SpatialCoordinate, TestFunction, TrialFunction, dot, ds, dx, grad

import numpy as np
import pyvista

mesh = create_unit_square(MPI.COMM_WORLD, 10, 10)
V = functionspace(mesh, ("Lagrange", 1))
u = TrialFunction(V)
v = TestFunction(V)
a = dot(grad(u), grad(v)) * dx


# Now we get to the Neumann and Dirichlet boundary condition. As previously, we use a Python-function to define the boundary where we should have a Dirichlet condition. Then, with this function, we locate degrees of freedom that fulfill this condition.

# In[ ]:


def u_exact(x):
    return 1 + x[0]**2 + 2 * x[1]**2


def boundary_D(x):
    return np.logical_or(np.isclose(x[0], 0), np.isclose(x[0], 1))


dofs_D = locate_dofs_geometrical(V, boundary_D)
u_bc = Function(V)
u_bc.interpolate(u_exact)
bc = dirichletbc(u_bc, dofs_D)


# The next step is to define the Neumann condition. We first define $g$ uses `UFL`s `SpatialCoordinate`-function, and then in turn create a boundary integration measure `ds`. As the test function $v$ is zero on the boundary integrals over the Dirichlet boundary disappears, and we can integrate `g*v*ds` over the entire boundary.

# In[ ]:


x = SpatialCoordinate(mesh)
g = -4 * x[1]
f = Constant(mesh, default_scalar_type(-6))
L = f * v * dx - g * v * ds


# We can now assemble and solve the linear system of equations

# In[ ]:


problem = LinearProblem(a, L, bcs=[bc], petsc_options={"ksp_type": "preonly", "pc_type": "lu"})
uh = problem.solve()

V2 = functionspace(mesh, ("Lagrange", 2))
uex = Function(V2)
uex.interpolate(u_exact)
error_L2 = assemble_scalar(form((uh - uex)**2 * dx))
error_L2 = np.sqrt(MPI.COMM_WORLD.allreduce(error_L2, op=MPI.SUM))

u_vertex_values = uh.x.array
uex_1 = Function(V)
uex_1.interpolate(uex)
u_ex_vertex_values = uex_1.x.array
error_max = np.max(np.abs(u_vertex_values - u_ex_vertex_values))
error_max = MPI.COMM_WORLD.allreduce(error_max, op=MPI.MAX)
print(f"Error_L2 : {error_L2:.2e}")
print(f"Error_max : {error_max:.2e}")


# ## Visualization
# To look at the actual solution, run the script as a python script with `off_screen=True` or as a Jupyter notebook with `off_screen=False`

# In[ ]:


pyvista.start_xvfb()

pyvista_cells, cell_types, geometry = vtk_mesh(V)
grid = pyvista.UnstructuredGrid(pyvista_cells, cell_types, geometry)
grid.point_data["u"] = uh.x.array
grid.set_active_scalars("u")

plotter = pyvista.Plotter()
plotter.add_text("uh", position="upper_edge", font_size=14, color="black")
plotter.add_mesh(grid, show_edges=True)
plotter.view_xy()

if not pyvista.OFF_SCREEN:
    plotter.show()
else:
    figure = plotter.screenshot("neumann_dirichlet.png")