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

"""
***********************************************************************************
tutorial_cv_10.py
DAE Tools: pyDAE module, www.daetools.com
***********************************************************************************
DAE Tools is free software; you can redistribute it and/or modify it under the
Foundation. DAE Tools is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A
PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with the
DAE Tools software; if not, see <http://www.gnu.org/licenses/>.
************************************************************************************
"""
__doc__ = """
Code verification using the Method of Exact Solutions (Rotating Gaussian Hill problem).

Reference (section 4.4.6.3 Convection-Diffusion):

- D. Kuzmin (2010). A Guide to Numerical Methods for Transport Equations.
`PDF <http://www.mathematik.uni-dortmund.de/~kuzmin/Transport.pdf>`_

Here, a 2D transient convection-diffusion problem in a rectangular (-1,1)x(-1,1) domain
is solved using the FE method:

.. code-block:: none

dc/dt + div(u*c) - eps*nabla(c) = 0, in Omega = (-1,1)x(-1,1)

The exact solution is given by the following function:

.. code-block:: none

(x0, y0) = (0.0, 0.5)
x_bar(t) = x0*cos(t) - y0*sin(t)
y_bar(t) = -x0*sin(t) + y0*cos(t)
r2(x,y,t) = (x-x_bar(t))**2 + (y-y_bar(t))**2

c_exact(x,y,t) = 1.0 / (4*pi*eps*t) * exp(-r2(x,y,t) / (4*eps*t))

The initial conditions define a Gaussian hill which is rotated counterclockwise around
the point (0.0, 0.0) using the velocity field u = (-y, x). Since at t = 0 the
value of c_exact is the Dirac delta function it is better to start the simulation at t > 0.
Therefore, the simulation is started and t = pi/2 by reinitialising variable c to:

.. code-block:: none

c(x,y,pi/2) = c_exact(x,y,pi/2)

At t = 5/2 pi the peak smeared by the diffusion should arrive at the starting position.

Homogeneous Dirichlet boundary conditions are prescribed at all four edges:

.. code-block:: none

c(x,y,t) = 0.0

The mesh is a rectangle (-1,1)x(-1,1):

.. image:: _static/square(-1,1)x(-1,1)-64x64.png
:width: 300 px

The solution plots at t = pi/2 (the initial peak) and t = 5/2pi (96x96 grid):

.. image:: _static/tutorial_cv_10-results1.png
:height: 400 px

.. image:: _static/tutorial_cv_10-results2.png
:height: 400 px

Animations for 32x32 and 96x96 grids:

.. image:: _static/tutorial_cv_10-animation-32x32.gif
:height: 400 px

.. image:: _static/tutorial_cv_10-animation-96x96.gif
:height: 400 px

Again, some low-magnitude oscillations in the solution appear, which are more pronounced
for coarser grids.
In the original example this problem was resolved using the flux linearisation technique.

The normalised global errors and the order of accuracy plots
(no. elements = [32x32, 64x64, 96x96, 128x128], t = 5/2pi):

.. image:: _static/tutorial_cv_10-results3.png
:width: 800 px
"""

import os, sys, numpy, json, tempfile
from time import localtime, strftime
import matplotlib.pyplot as plt
from daetools.pyDAE import *
from daetools.solvers.deal_II import *
from daetools.solvers.superlu import pySuperLU

# Standard variable types are defined in variable_types.py
from pyUnits import m, kg, s, K, Pa, mol, J, W

eps = 1E-3
(x0, y0) = (0.0, 0.5)
x_bar = lambda t: x0*numpy.cos(t) - y0*numpy.sin(t)
y_bar = lambda t: -x0*numpy.sin(t) + y0*numpy.cos(t)
r2 = lambda x,y,t: (x-x_bar(t))**2 + (y-y_bar(t))**2
ct = lambda x,y,t: 1.0 / (4*numpy.pi*eps*t) * numpy.exp(-r2(x,y,t) / (4*eps*t))

class VelocityFunction_2D(Function_2D):
def __init__(self, n_components = 1):
Function_2D.__init__(self, n_components)
self.m_velocity = Tensor_1_2D()

def gradient(self, point, component = 0):
self.m_velocity[0] = -point.y
self.m_velocity[1] = point.x
return self.m_velocity

return [self.value(point, c) for c in range(self.n_components)]

c_t = daeVariableType("c_t", unit(),  0.0, 1E20, 0, 1e-07)

class modTutorial(daeModel):
def __init__(self, Name, Nx, Parent = None, Description = ""):
daeModel.__init__(self, Name, Parent, Description)

dofs = [dealiiFiniteElementDOF_2D(name='c',
description='Something',
fe = FE_Q_2D(1),
multiplicity=1,
variableType=c_t)]
self.n_components = int(numpy.sum([dof.Multiplicity for dof in dofs]))

meshes_dir = os.path.join(os.path.dirname(os.path.abspath(__file__)), 'meshes')
mesh_file  = os.path.join(meshes_dir, 'square(-1,1)x(-1,1)-%dx%d.msh' % (Nx, Nx))

# Store the object so it does not go out of scope while still in use by daetools
self.fe_system = dealiiFiniteElementSystem_2D(meshFilename    = mesh_file,
dofs            = dofs)

self.fe_model = daeFiniteElementModel('GaussianHill', self, 'GaussianHill problem', self.fe_system)

def DeclareEquations(self):
daeModel.DeclareEquations(self)

# Create some auxiliary objects for readability
phi_i  =  phi_2D('c', fe_i, fe_q)
phi_j  =  phi_2D('c', fe_j, fe_q)
dphi_i = dphi_2D('c', fe_i, fe_q)
dphi_j = dphi_2D('c', fe_j, fe_q)
xyz    = xyz_2D(fe_q)
JxW    = JxW_2D(fe_q)

# The counterclockwise velocity field (0.5-y, x-0.5) Function<dim>::gradient wrapper.
self.fun_u = VelocityFunction_2D()

# Boundary IDs
left_edge   = 0
top_edge    = 1
right_edge  = 2
bottom_edge = 3

dirichletBC = {}
dirichletBC[left_edge]   = [
]
dirichletBC[top_edge]    = [
]
dirichletBC[right_edge]  = [
]
dirichletBC[bottom_edge] = [
]

# FE weak form terms
accumulation = (phi_i * phi_j) * JxW
diffusion    = (dphi_i * dphi_j) * eps * JxW
convection   = phi_i * (u_grad * dphi_j) * JxW
source       = 0.0 * JxW

weakForm = dealiiFiniteElementWeakForm_2D(Aij = diffusion + convection,
Mij = accumulation,
Fi  = source,
functionsDirichletBC = dirichletBC)

# Setting the weak form of the FE system will declare a set of equations:
# [Mij]{dx/dt} + [Aij]{x} = {Fi} and boundary integral equations
self.fe_system.WeakForm = weakForm

class simTutorial(daeSimulation):
def __init__(self, Nx):
daeSimulation.__init__(self)
self.m = modTutorial("tutorial_cv_10", Nx)
self.m.Description = __doc__
self.m.fe_model.Description = __doc__

def SetUpParametersAndDomains(self):
pass

def SetUpVariables(self):
setFEInitialConditions(self.m.fe_model, self.m.fe_system, 'c', 0.0)

def Run(self):
# Get coordinates for every DOF
sp = self.m.fe_system.GetDOFSupportPoints()

# Define a peak
def ic(internal_index, overall_index):
p = sp[overall_index]
return ct(p.x, p.y, numpy.pi/2)

# Integrate for pi/2, c(x,y) = 0 everywhere
self.Log.Message("Integrating for pi/2 seconds ... ", 0)
time = self.IntegrateForTimeInterval(numpy.pi/2, eDoNotStopAtDiscontinuity)
self.ReportData(self.CurrentTime)
self.Log.SetProgress(int(100.0 * self.CurrentTime/self.TimeHorizon));

# Set the initial peak at t = pi/2
self.Log.Message("Setting an initial peak at t = pi/2 seconds ... ", 0)
setFEInitialConditions(self.m.fe_model, self.m.fe_system, 'c', ic)
self.Reinitialize()
self.ReportData(self.CurrentTime)
daeSimulation.Run(self)

# Setup everything manually and run in a console
def simulate(Nx):
# Create Log, Solver, DataReporter and Simulation object
log          = daePythonStdOutLog()
daesolver    = daeIDAS()
datareporter = daeDelegateDataReporter()
simulation   = simTutorial(Nx)

daesolver.RelativeTolerance = 1E-6

# Do no print progress
log.PrintProgress = False

lasolver = pySuperLU.daeCreateSuperLUSolver()
daesolver.SetLASolver(lasolver)

simName = simulation.m.Name + 'Nx=%d'%Nx + strftime(" [%d.%m.%Y %H:%M:%S]", localtime())
results_folder = os.path.join(os.path.dirname(os.path.abspath(__file__)), 'tutorial_cv_10-results(Nx=%d)' % Nx)

# Create three data reporters:
# 1. DealII
feDataReporter = simulation.m.fe_system.CreateDataReporter()
if not feDataReporter.Connect(results_folder, simName):
sys.exit()

# 2. TCP/IP
tcpipDataReporter = daeTCPIPDataReporter()
if not tcpipDataReporter.Connect("", simName):
sys.exit()

# 3. Data
dr = daeNoOpDataReporter()

# Enable reporting of all variables
simulation.m.SetReportingOn(True)

# Set the time horizon and the reporting interval
simulation.ReportingInterval = 5.0/2.0*numpy.pi / 100
simulation.TimeHorizon = 5.0/2.0*numpy.pi

# Initialize the simulation
simulation.Initialize(daesolver, datareporter, log)

# Save the model report and the runtime model report
#simulation.m.fe_model.SaveModelReport(simulation.m.Name + ".xml")
#simulation.m.fe_model.SaveRuntimeModelReport(simulation.m.Name + "-rt.xml")

# Solve at time=0 (initialization)
simulation.SolveInitial()

# Run
simulation.Run()
simulation.Finalize()

###########################################
#  Plots and data                         #
###########################################
results = dr.Process.dictVariables
cvar = results[simulation.m.Name + '.GaussianHill.c']
points = cvar.Domains[0].Points
c      = cvar.Values[-1,:] # 2D array [t,omega]

sp = simulation.m.fe_system.GetDOFSupportPoints()
Nsp = len(sp)
c_exact = numpy.zeros(Nsp)
for i, p in enumerate(sp):
c_exact[i] = ct(p.x, p.y, 5.0/2.0*numpy.pi)

return points, c, c_exact

def run(**kwargs):
Nxs = numpy.array([32, 64, 96, 128])
n = len(Nxs)
L = 1.0
hs = L / Nxs
E = numpy.zeros(n)
C = numpy.zeros(n)
p = numpy.zeros(n)
E2 = numpy.zeros(n)

# The normalised global errors
for i,Nx in enumerate(Nxs):
points, numerical_sol, manufactured_sol = simulate(int(Nx))
E[i] = numpy.sqrt((1.0/Nx) * numpy.sum((numerical_sol-manufactured_sol)**2))

# Order of accuracy
for i,Nx in enumerate(Nxs):
p[i] = numpy.log(E[i]/E[i-1]) / numpy.log(hs[i]/hs[i-1])
C[i] = E[i] / hs[i]**p[i]

C2 = 400.0 # constant for the second order slope line (to get close to the actual line)
E2 = C2 * hs**2 # E for the second order slope

fontsize = 14
fontsize_legend = 11
fig = plt.figure(figsize=(10,4), facecolor='white')
fig.canvas.set_window_title('The Normalised global errors and the Orders of accuracy (Nelems = %s) (cv_10)' % Nxs.tolist())

ax = plt.subplot(121)
plt.figure(1, facecolor='white')
plt.loglog(hs, E,  'ro', label='E(h)')
plt.loglog(hs, E2, 'b-', label='2nd order slope')
plt.xlabel('h', fontsize=fontsize)
plt.ylabel('||E||', fontsize=fontsize)
plt.legend(fontsize=fontsize_legend)
#plt.xlim((0.04, 0.11))

ax = plt.subplot(122)
plt.figure(1, facecolor='white')
plt.semilogx(hs[1:], p[1:],  'rs-', label='Order of Accuracy (p)')
plt.xlabel('h', fontsize=fontsize)
plt.ylabel('p', fontsize=fontsize)
plt.legend(fontsize=fontsize_legend)
#plt.xlim((0.04, 0.075))
#plt.ylim((2.0, 2.04))

plt.tight_layout()
plt.show()

if __name__ == "__main__":
run()

```