r"""
.. _ref_1714:

Three point bending test
^^^^^^^^^^^^^^^^^^^^^^^^

A well-known benchmark simulation in fracture mechanics is performed, relying on the simulation conducted by [Miehe]_. This simulation considers an anisotropic formulation with spectral energy decomposition.

A rectangular plate with an initial notch positioned at the bottom center is shown in the figure below. This beam is supported at its ends, as shown in the figure. The bottom left part is fixed in all directions, while the bottom right part is fixed in the vertical direction. A vertical displacement is applied at the top. The geometry and boundary conditions are depicted in the figure. We discretize the model with triangular elements, refining the areas (element size h) where crack evolution is expected. The element size h must be sufficiently small to avoid mesh dependencies.

.. code-block::
      
   #                            u||||      
   #                             \/\/               
   #            ------------------------------------* 
   #            |                                   | 
   #            |                                   |
   #            |                                   |
   #            |                 /\                | 
   #            *-----------------  ----------------*
   #            /_\/_\        (0,0,0)          /_\/_\       
   #    |Y     ///////                         oo  oo
   #    |
   #    ---X
   # Z /

The Young's modulus, Poisson's ratio, and the critical energy release rate are given in the table :ref:`Properties <table_properties_label>`. Young's modulus $E$ and Poisson's ratio $\nu$ can be represented with the Lamé parameters as: $\lambda=\frac{E\nu}{(1+\nu)(1-2\nu)}$; $\mu=\frac{E}{2(1+\nu)}$.

.. _table_properties_label:

+----+---------+--------+
|    | VALUE   | UNITS  |
+====+=========+========+
| E  | 20.8    | kN/mm2 |
+----+---------+--------+
| nu | 0.3     | [-]    |
+----+---------+--------+
| Gc | 0.0005  | kN/mm  |
+----+---------+--------+
| l  | 0.006   | mm     |
+----+---------+--------+

.. [Miehe] A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits, https://doi.org/10.1016/j.cma.2010.04.011.

"""

###############################################################################
# Import necessary libraries
# --------------------------
import numpy as np
import matplotlib.pyplot as plt
import pyvista as pv
import dolfinx
import mpi4py
import petsc4py
import os


###############################################################################
# Import from phasefieldx package
# -------------------------------
from phasefieldx.Element.Phase_Field_Fracture.Input import Input
from phasefieldx.Element.Phase_Field_Fracture.solver.solver import solve
from phasefieldx.Boundary.boundary_conditions import bc_xy, bc_y, get_ds_bound_from_marker
from phasefieldx.PostProcessing.ReferenceResult import AllResults


###############################################################################
# Parameters Definition
# ---------------------
# `Data` is an input object containing essential parameters for simulation setup
# and result storage:
#
# - `E`: Young's modulus, set to 20.8 $kN/mm^2$.
# - `nu`: Poisson's ratio, set to 0.3.
# - `Gc`: Critical energy release rate, set to 0.0005 $kN/mm$.
# - `l`: Length scale parameter, set to 0.06 $mm$.
# - `degradation`: Specifies the degradation type. Options are "isotropic" or "anisotropic".
# - `split_energy`: Controls how the energy is split; options include "no" (default), "spectral," or "deviatoric."
#   In this case an anisotropic model with spectral decomposition is considered.
# - `degradation_function`: Specifies the degradation function; here, it is "quadratic."
# - `irreversibility`: Determines the irreversibility criterion; in this case, set to "miehe."
# - `fatigue`: Enables fatigue simulation when set to `True`.
# - `fatigue_degradation_function`: Defines the function for fatigue degradation, set to "asymptotic."
# - `fatigue_val`: Fatigue parameter value (used only in fatigue simulations, not in this one).
# - `k`: Stiffness penalty parameter, set to 0.0.
# - `min_stagger_iter`: Minimum number of staggered iterations, set to 2.
# - `max_stagger_iter`: Maximum number of staggered iterations, set to 500.
# - `stagger_error_tol`: Error tolerance for staggered iterations, set to 1e-8.
# - `save_solution_xdmf` and `save_solution_vtu`: Specify the file formats to save displacement results.
#   In this case, results are saved as `.vtu` files.
# - `results_folder_name`: Name of the folder for saving results. If it exists,
#   it will be replaced with a new empty folder.
Data = Input(E=20.8,     # young modulus
             nu=0.3,     # poisson
             Gc=0.0005,  # critical energy release rate
             l=0.06,     # lenght scale parameter
             degradation="anisotropic",  # "isotropic" "anisotropic"
             split_energy="spectral",   # "spectral" "deviatoric"
             degradation_function="quadratic",
             irreversibility="miehe",  # "miehe"
             fatigue=False,
             fatigue_degradation_function="asymptotic",
             fatigue_val=0.05625,
             k=0.0,
             save_solution_xdmf=False,
             save_solution_vtu=True,
             results_folder_name="1714_Three_point_bending")


###############################################################################
# Mesh Definition
# ---------------
# The mesh is generated using Gmsh and saved as a 'mesh.msh' file. For more details 
# on how to create the mesh, refer to the :ref:`ref_examples_91` examples.
# The following lines 
msh_file = os.path.join("mesh", "three_point_bending_test.msh") # Path to the mesh file
gdim = 2                                     # Geometric dimension of the mesh
gmsh_model_rank = 0                          # Rank of the Gmsh model in a parallel setting
mesh_comm = mpi4py.MPI.COMM_WORLD            # MPI communicator for parallel computation

# %%
# The mesh, cell markers, and facet markers are extracted from the 'mesh.msh' file
# using the `read_from_msh` function.
mesh_data = dolfinx.io.gmsh.read_from_msh(msh_file, mesh_comm, gmsh_model_rank, gdim)
msh = mesh_data.mesh
cell_markers = mesh_data.cell_tags
facet_markers = mesh_data.facet_tags

fdim = msh.topology.dim - 1 # Dimension of the mesh facets


###############################################################################
# Boundary Identification Functions
# ---------------------------------
# These functions identify points on the specific boundaries of the domain
# where boundary conditions will be applied. The `bottom_left` function checks 
# if a point lies on the bottom left boundary, returning `True` for points where 
# `y=0` and `x` is less than `-3.9`, and `False` otherwise. Similarly, the `bottom_right` 
# function identifies points on the bottom right boundary, returning `True` for points 
# where `y=0` and `x` is greater than `3.9`, and `False` otherwise. 
# The `top` function identifies points on the top boundary, returning `True` for points 
# where `y=2`, and `x` is between `-0.25` and `0.25`, and `False` otherwise.
#
# This approach ensures that boundary conditions are applied to specific parts of 
# the mesh, which helps in defining the simulation's physical constraints.
def bottom_left(x):
    return np.logical_and(np.isclose(x[1], 0), np.less(x[0], -3.9))

def bottom_right(x):
    return np.logical_and(np.isclose(x[1], 0), np.greater(x[0], 3.9))

def top(x):
    return np.logical_and(np.logical_and(np.isclose(x[1], 2), np.greater(x[0], -0.25)), np.less(x[0], 0.25))


# %%
# Using the `bottom` and `top` functions, we locate the facets on the bottom and top sides of the mesh,
# where $y = 0$ and $y = 1$, respectively. The `locate_entities_boundary` function returns an array of facet
# indices representing these identified boundary entities.
bottom_left_facet_marker = dolfinx.mesh.locate_entities_boundary(msh, fdim, bottom_left)
bottom_right_facet_marker = dolfinx.mesh.locate_entities_boundary(msh, fdim, bottom_right)
top_facet_marker = dolfinx.mesh.locate_entities_boundary(msh, fdim, top)

# %%
# The `get_ds_bound_from_marker` function generates a measure for applying boundary conditions 
# specifically to the facets identified by `top_facet_marker` and `bottom_facet_marker`, respectively. 
# This measure is then assigned to `ds_bottom` and `ds_top`.
ds_bottom_left = get_ds_bound_from_marker(bottom_left_facet_marker, msh, fdim)
ds_bottom_right = get_ds_bound_from_marker(bottom_right_facet_marker, msh, fdim)
ds_top = get_ds_bound_from_marker(top_facet_marker, msh, fdim)

# %%
# `ds_list` is an array that stores boundary condition measures along with names 
# for each boundary, simplifying result-saving processes. Each entry in `ds_list` 
# is formatted as `[ds_, "name"]`, where `ds_` represents the boundary condition measure, 
# and `"name"` is a label used for saving. Here, `ds_bottom` and `ds_top` are labeled 
# as `"bottom"` and `"top"`, respectively, to ensure clarity when saving results.
ds_list = np.array([
                   [ds_top, "top"],
                   [ds_bottom_left, "bottom_left"],
                   [ds_bottom_right, "bottom_right"],
                   ])


###############################################################################
# Function Space Definition
# -------------------------
# Define function spaces for displacement and phase-field using Lagrange elements.
V_u = dolfinx.fem.functionspace(msh, ("Lagrange", 1, (msh.geometry.dim, )))
V_phi = dolfinx.fem.functionspace(msh, ("Lagrange", 1))


###############################################################################
# Boundary Conditions
# -------------------
# Dirichlet boundary conditions are defined as follows:
#
# - `bc_bottom_left`: Constrains both x and y displacements on the bottom left boundary, 
#   ensuring that the leftmost bottom edge remains fixed.
# - `bc_bottom_right`: Constrains only the vertical displacement (y-displacement) on the 
#   bottom right boundary, while allowing horizontal movement.
# - `bc_top`: Constrains the vertical displacement (y-displacement) on the top boundary, 
#   while allowing horizontal movement.
#
# These boundary conditions ensure that the relevant portions of the mesh are correctly 
# fixed or allowed to move according to the simulation requirements.

bc_bottom_left = bc_xy(bottom_left_facet_marker, V_u, fdim)
bc_bottom_right = bc_y(bottom_right_facet_marker, V_u, fdim)
bc_top = bc_y(top_facet_marker, V_u, fdim)

# %%
# The bcs_list_u variable is a list that stores all boundary conditions for the displacement
# field $\boldsymbol u$. This list facilitates easy management of multiple boundary
# conditions and can be expanded if additional conditions are needed.
bcs_list_u = [bc_top, bc_bottom_left, bc_bottom_right]
bcs_list_u_names = ["top", "bottom_left", "bottom_right"]

###############################################################################
# Function: `update_boundary_conditions`
# --------------------------------------
# The `update_boundary_conditions` function dynamically updates the displacement boundary conditions at each time step.
# This allows for quasi-static analysis by incrementally adjusting the displacements applied to specific degrees of freedom.
#
# Parameters:
#
# - `bcs`: A list of boundary conditions, where each element corresponds to a boundary condition applied to a specific facet of the mesh.
# - `time`: A scalar representing the current time step in the analysis.
#
# Function Details:
#
# - The displacement value `val` is computed based on the current `time`:
# - For `time <= 36`, `val` increases linearly as `val = dt0 * time`, where `dt0` is a small time step factor (`10^-3`), simulating gradual displacement along the y-axis.
# - For `time > 36`, `val` increases more gradually as `val = 36 * dt0 + (dt0 / 10) * (time - 36)`, which represents a slower displacement rate after the initial phase.
# - This calculated value is assigned to the y-component of the displacement field on the top boundary by modifying `bcs[0].g.value[1]`, where `bcs[0]` represents the top boundary condition.
#   The displacement is negated to represent a displacement in the opposite direction.
#
# Return Value:
#
# - A tuple `(0, val, 0)` is returned, representing the incremental displacement vector:
# - The first element (0) corresponds to no update for the x-displacement.
# - The second element (`val`) is the calculated y-displacement.
# - The third element (0) corresponds to no update for the z-displacement, applicable in 2D simulations.
#
# Purpose:
#
# - This function facilitates quasi-static analysis by applying controlled, time-dependent boundary displacements. It is essential for simulations that involve gradual loading or unloading, with a slower displacement evolution after the initial phase.
def update_boundary_conditions(bcs, time):
    dt0 = 10**-3
    if time <= 36:
        val = dt0 * time
    else:
        val = 36 * dt0 + dt0 / 10 * (time - 36)
    bcs[0].g.value[...] = petsc4py.PETSc.ScalarType(-val)
    return 0, val, 0


T_list_u = None
update_loading = None
f = None
T = dolfinx.fem.Constant(msh, petsc4py.PETSc.ScalarType((0.0, 0.0)))


###############################################################################
# Boundary Conditions for phase field
bcs_list_phi = []


###############################################################################
# Solver Call for a Phase-Field Fracture Problem
# ----------------------------------------------
# This section sets up and calls the solver for a phase-field fracture problem.
#
# **Key Points:**
#
# - The simulation is run for a final time of 150, with a time step of 1.0.
# - The solver will manage the mesh, boundary conditions, and update the solution
#   over the specified time steps.
#
# **Parameters:**
#
# - `dt`: The time step for the simulation, set to 1.0.
# - `final_time`: The total simulation time, set to 200.0, which determines how 
#   long the problem will be solved.
# - `path`: Optional parameter for specifying the folder where results will be saved; 
#   here it is set to `None`, meaning results will be saved to the default location.
#
# **Function Call:**
#
# The `solve` function is invoked with the following arguments:
#
# - `Data`: Contains the simulation parameters and configurations.
# - `msh`: The mesh representing the domain for the problem.
# - `final_time`: The total duration of the simulation (200.0).
# - `V_u`: Function space for the displacement field, $\boldsymbol{u}$.
# - `V_phi`: Function space for the phase field, $\phi$.
# - `bcs_list_u`: List of Dirichlet boundary conditions for the displacement field.
# - `bcs_list_phi`: List of boundary conditions for the phase field (empty in this case).
# - `update_boundary_conditions`: Function to update boundary conditions for the displacement field.
# - `f`: The body force applied to the domain (if any).
# - `T_list_u`: Time-dependent loading parameters for the displacement field.
# - `update_loading`: Function to update loading parameters for the quasi-static analysis.
# - `ds_list`: Boundary measures for integration over the domain boundaries.
# - `dt`: The time step for the simulation.
# - `path`: Directory for saving results (if specified).
#
# This setup provides a framework for solving static problems with specified boundary 
# conditions and loading parameters.

final_time = 150
dt = 1

# %%
# Uncomment the following lines to run the solver with the specified parameters.

# solve(Data,
#       msh,
#       final_time,
#       V_u,
#       V_phi,
#       bcs_list_u,
#       bcs_list_phi,
#       update_boundary_conditions,
#       f,
#       T_list_u,
#       update_loading,
#       ds_list,
#       dt,
#       path=None,
#       bcs_list_u_names=bcs_list_u_names,
#       min_stagger_iter=2,
#       max_stagger_iter=1000,
#       stagger_error_tol=1e-6)


###############################################################################
# Load results
# ------------
# Once the simulation finishes, the results are loaded from the results folder.
# The AllResults class takes the folder path as an argument and stores all
# the results, including logs, energy, convergence, and DOF files.
# Note that it is possible to load results from other results folders to compare results.
# It is also possible to define a custom label and color to automate plot labels.
S = AllResults(Data.results_folder_name)
S.set_label('Simulation')
S.set_color('b')


###############################################################################
# Plot: phase-field $\phi$
# ------------------------
# The phase-field result saved in the .vtu file is shown.
# For this, the file is loaded using PyVista.
file_vtu = pv.read(os.path.join(Data.results_folder_name, "paraview-solutions_vtu", "phasefieldx_p0_000065.vtu"))
file_vtu.plot(scalars='phi', cpos='xy', show_scalar_bar=True, show_edges=False)


###############################################################################
# Plot: displacement $\boldsymbol u$
# ----------------------------------
# The displacements results saved in the .vtu file are shown.
# For this, the file is loaded using PyVista.
file_vtu.plot(scalars='u', cpos='xy', show_scalar_bar=True, show_edges=False)

plt.show()


###############################################################################
# Plot: Displacement vs Fracture Energy
# -------------------------------------
# The vertical displacement is saved in S.dof_files["top.dof"]["Uy"].
displacement = S.dof_files["top.dof"]["Uy"]

fig, energyW = plt.subplots()

energyW.plot(displacement, S.energy_files['total.energy']["W"], 'b-', linewidth=2.0, label=r'$W$')
energyW.plot(displacement, S.energy_files['total.energy']["W_phi"], 'y-', linewidth=2.0, label=r'$W_{\phi}$')
energyW.plot(displacement, S.energy_files['total.energy']["W_gradphi"], 'g-', linewidth=2.0, label=r'$W_{\nabla \phi}$')

energyW.grid(color='k', linestyle='-', linewidth=0.3)
energyW.set_xlabel('displacement - u $[mm]$')
energyW.set_ylabel('Energy')
energyW.legend()


###############################################################################
# Plot: Force vs Vertical Displacement
# ------------------------------------
Miehe = np.loadtxt(os.path.join("reference_solutions", "miehe_three_point.csv"))

fig, ax_reaction = plt.subplots()

ax_reaction.plot(Miehe[:, 0], Miehe[:, 1], 'g-', linewidth=2.0, label='Miehe')
ax_reaction.plot(displacement, -S.reaction_files['top.reaction']["Ry"], 'k.', linewidth=2.0, label=S.label)
ax_reaction.grid(color='k', linestyle='-', linewidth=0.3)
ax_reaction.set_xlabel('displacement - u $[mm]$')
ax_reaction.set_ylabel('reaction force - F $[kN]$')
ax_reaction.set_title('Reaction Force vs Vertical Displacement')
ax_reaction.legend()


###############################################################################
# Plot: Staggered Iterations vs Vertical Displacement
# ---------------------------------------------------
fig, ax_convergence = plt.subplots()

ax_convergence.plot(displacement, S.convergence_files["phasefieldx.conv"]["stagger"], 'k.', linewidth=2.0, label='Stagger iterations')

ax_convergence.grid(color='k', linestyle='-', linewidth=0.3)
ax_convergence.set_xlabel('displacement - u $[mm]$')
ax_convergence.set_ylabel('stagger iterations - []')
ax_convergence.set_title('Stagger iterations vs vertical displacement')
ax_convergence.legend()

plt.show()