Oedometric loading

#-------------------------------------------------------------------------------
#Librairies
#-------------------------------------------------------------------------------

from yade import pack, plot, export
import numpy as np
import matplotlib.pyplot as plt
import os
import shutil
import time
import math
import random
import pickle
from pathlib import Path

#-------------------------------------------------------------------------------
#User
#-------------------------------------------------------------------------------

# PSD
n_grains = 3000
L_r = []

# Particles
rMean = 0.000100  # m
rRelFuzz = .25

# Box
Dz_on_Dx = 1 # ratio Dz / Dxy
Dz = 0.0028 # m
Dx = Dz/Dz_on_Dx
Dy = Dx

# IC
n_steps_ic = 100

# top wall
P_load = 1e5 # Pa
kp = 1e-9 # m.N-1
k_v_max = 0.00002 #-

# cementation
P_cementation = P_load*0.1 # Pa
# 2T   : f_cemented (0.13), m_log (6.79), s_log (0.70), E ( 300MPa), Ab_mean (148-12)
# 2MB  : f_cemented (0.88), m_log (7.69), s_log (0.60), E ( 320MPa), Ab_mean (2303e-12)
# 11BB : f_cemented (0.98), m_log (8.01), s_log (0.88), E ( 760MPa), Ab_mean (4267e-12)
# 13BT : f_cemented (1.00), m_log (8.44), s_log (0.92), E ( 860MPa), Ab_mean (6577e-12)
# 13MB : f_cemented (1.00), m_log (8.77), s_log (0.73), E (1000MPa), Ab_mean (8137e-12)
type_cementation = '13MB' # only for the report
f_cemented = 1. # -
m_log = 8.77 # -
s_log = 0.73 # -
YoungModulus = 1000e6
Ab_mean = 8137e-12 # m2
# mechanical rupture
factor_strength = 10
tensileCohesion = 2.75e6*factor_strength # Pa
shearCohesion = 6.6e6*factor_strength # Pa

# Dissolution
mass_removal = 0 # 0-1 maximum bond surface (4e4 µm2)

# additional loading to determine properties
n_load = 20
d_P = 0.1*1e5

# time step
factor_dt_crit_1 = 0.6
factor_dt_crit_2 = 0.2

# steady-state detection
unbalancedForce_criteria = 0.01

# Report
simulation_report_name = O.tags['d.id']+'_report.txt'
simulation_report = open(simulation_report_name, 'w')
simulation_report.write('Oedometric Loading test\n')
simulation_report.write('Type of sample: Rock\n')
simulation_report.write('Cementation at '+str(int(P_cementation))+' Pa\n')
simulation_report.write('Type of cementation: '+type_cementation+'\n')
simulation_report.write('Confinement at '+str(int(P_load))+' Pa\n')
simulation_report.write('Mass removal: '+str(round(mass_removal,2))+' x 4e4 µm2\n\n')
simulation_report.close()

#-------------------------------------------------------------------------------
#Initialisation
#-------------------------------------------------------------------------------

# clock to show performances
tic = time.perf_counter()
tic_0 = tic
iter_0 = 0

# plan simulation
if Path('plot').exists():
    shutil.rmtree('plot')
os.mkdir('plot')
if Path('data').exists():
    shutil.rmtree('data')
os.mkdir('data')
if Path('vtk').exists():
    shutil.rmtree('vtk')
os.mkdir('vtk')
if Path('save').exists():
    shutil.rmtree('save')
os.mkdir('save')

# define wall material (no friction)
O.materials.append(CohFrictMat(young=80e6, poisson=0.25, frictionAngle=0, density=2650, isCohesive=False, momentRotationLaw=False))

# create box and grains
O.bodies.append(aabbWalls([Vector3(0,0,0),Vector3(Dx,Dy,Dz)], thickness=0.,oversizeFactor=1))
# a list of 6 boxes Bodies enclosing the packing, in the order minX, maxX, minY, maxY, minZ, maxZ
# extent the plates
O.bodies[0].shape.extents = Vector3(0,1.5*Dy/2,1.5*Dz/2)
O.bodies[1].shape.extents = Vector3(0,1.5*Dy/2,1.5*Dz/2)
O.bodies[2].shape.extents = Vector3(1.5*Dx/2,0,1.5*Dz/2)
O.bodies[3].shape.extents = Vector3(1.5*Dx/2,0,1.5*Dz/2)
O.bodies[4].shape.extents = Vector3(1.5*Dx/2,1.5*Dy/2,0)
O.bodies[5].shape.extents = Vector3(1.5*Dx/2,1.5*Dy/2,0)
# global names
plate_x = O.bodies[1]
plate_y = O.bodies[3]
plate_z = O.bodies[5]

# define grain material
O.materials.append(CohFrictMat(young=80e6, poisson=0.25, frictionAngle=atan(0.05), density=2650,\
                               isCohesive=True, normalCohesion=tensileCohesion, shearCohesion=shearCohesion,\
                               momentRotationLaw=True, alphaKr=0, alphaKtw=0))
# frictionAngle, alphaKr, alphaKtw are set to 0 during IC. The real value is set after IC.
frictionAngleReal = radians(20)
alphaKrReal = 0.5
alphaKtwReal = 0.5

# generate grain
for i in range(n_grains):
    radius = random.uniform(rMean*(1-rRelFuzz),rMean*(1+rRelFuzz))
    center_x = random.uniform(0+radius/n_steps_ic, Dx-radius/n_steps_ic)
    center_y = random.uniform(0+radius/n_steps_ic, Dy-radius/n_steps_ic)
    center_z = random.uniform(0+radius/n_steps_ic, Dz-radius/n_steps_ic)
    O.bodies.append(sphere(center=[center_x, center_y, center_z], radius=radius/n_steps_ic))
    # can use b.state.blockedDOFs = 'xyzXYZ' to block translation of rotation of a body
    L_r.append(radius)
O.tags['Step ic'] = '1'

# yade algorithm
O.engines = [
        PyRunner(command='grain_in_box()', iterPeriod = 1000),
        ForceResetter(),
        # sphere, wall
        InsertionSortCollider([Bo1_Sphere_Aabb(), Bo1_Box_Aabb()]),
        InteractionLoop(
                # need to handle sphere+sphere and sphere+wall
                # Ig : compute contact point. Ig2_Sphere (3DOF) or Ig2_Sphere6D (6DOF)
                # Ip : compute parameters needed
                # Law : compute contact law with parameters from Ip
                [Ig2_Sphere_Sphere_ScGeom6D(), Ig2_Box_Sphere_ScGeom6D()],
                [Ip2_CohFrictMat_CohFrictMat_CohFrictPhys()],
                [Law2_ScGeom6D_CohFrictPhys_CohesionMoment(always_use_moment_law=True)]
        ),
        NewtonIntegrator(gravity=(0, 0, 0), damping=0.001, label = 'Newton'),
        PyRunner(command='checkUnbalanced_ir_ic()', iterPeriod = 200, label='checker')
]
# time step
O.dt = factor_dt_crit_1 * PWaveTimeStep()

#-------------------------------------------------------------------------------

def grain_in_box():
    '''
    Delete grains outside the box.
    '''
    #detect grain outside the box
    L_id_to_delete = []
    for b in O.bodies :
        if isinstance(b.shape, Sphere):
            #limit x \ limit y \ limit z
            if b.state.pos[0] < O.bodies[0].state.pos[0] or O.bodies[1].state.pos[0] < b.state.pos[0] or \
            b.state.pos[1] < O.bodies[2].state.pos[1] or O.bodies[3].state.pos[1] < b.state.pos[1] or \
            b.state.pos[2] < O.bodies[4].state.pos[2] or O.bodies[5].state.pos[2] < b.state.pos[2] :
                L_id_to_delete.append(b.id)
    if L_id_to_delete != []:
        #delete grain detected
        for id in L_id_to_delete:
            O.bodies.erase(id)
        #print and report
        simulation_report = open(simulation_report_name, 'a')
        simulation_report.write(str(len(L_id_to_delete))+" grains erased (outside of the box)\n")
        simulation_report.close()
        print("\n"+str(len(L_id_to_delete))+" grains erased (outside of the box)\n")

#-------------------------------------------------------------------------------

def checkUnbalanced_ir_ic():
    '''
    Increase particle radius until a steady-state is found.
    '''
    # the rest will be run only if unbalanced is < .1 (stabilized packing)
    # Compute the ratio of mean summary force on bodies and mean force magnitude on interactions.
    if unbalancedForce() > .1:
        return
    # increase the radius of particles
    if int(O.tags['Step ic']) < n_steps_ic :
        print('IC step '+O.tags['Step ic']+'/'+str(n_steps_ic)+' done')
        O.tags['Step ic'] = str(int(O.tags['Step ic'])+1)
        i_L_r = 0
        for b in O.bodies :
            if isinstance(b.shape, Sphere):
                growParticle(b.id, int(O.tags['Step ic'])/n_steps_ic*L_r[i_L_r]/b.shape.radius)
                i_L_r = i_L_r + 1
        # update the dt as the radii change
        O.dt = factor_dt_crit_1 * PWaveTimeStep()
        return
    # plot the psd
    global L_L_psd_binsSizes, L_L_psd_binsProc
    L_L_psd_binsSizes = []
    L_L_psd_binsProc = []
    binsSizes, binsProc, binsSumCum = psd(bins=10)
    L_L_psd_binsSizes.append(binsSizes)
    L_L_psd_binsProc.append(binsProc)
    # characterize the ic algorithm
    global tic
    global iter_0
    tac = time.perf_counter()
    hours = (tac-tic)//(60*60)
    minutes = (tac-tic -hours*60*60)//(60)
    seconds = int(tac-tic -hours*60*60 -minutes*60)
    tic = tac
    # report
    simulation_report = open(simulation_report_name, 'a')
    simulation_report.write("IC Generated : "+str(hours)+" hours "+str(minutes)+" minutes "+str(seconds)+" seconds\n")
    simulation_report.write(str(O.iter-iter_0)+' Iterations\n')
    simulation_report.write(str(n_grains)+' grains\n\n')
    simulation_report.close()
    print("\nIC Generated : "+str(hours)+" hours "+str(minutes)+" minutes "+str(seconds)+" seconds\n")
    # save
    #O.save('save/simu_ic.yade.bz2')
    # next time, do not call this function anymore, but the next one instead
    iter_0 = O.iter
    checker.command = 'checkUnbalanced_load_cementation_ic()'
    checker.iterPeriod = 500
    # control top wall
    O.engines = O.engines + [PyRunner(command='controlWalls_ic()', iterPeriod = 1)]
    # switch on the gravity
    Newton.gravity = [0, 0, -9.81]

#-------------------------------------------------------------------------------

def controlWalls_ic():
    '''
    Control the walls to applied a defined confinement force.

    The displacement of the wall depends on the force difference. A maximum value is defined.
    '''
    Fz = O.forces.f(plate_z.id)[2]
    if Fz == 0:
        plate_z.state.pos =  (plate_x.state.pos[0]/2, plate_y.state.pos[1]/2, max([b.state.pos[2]+0.99*b.shape.radius for b in O.bodies if isinstance(b.shape, Sphere)]))
    else :
        dF = Fz - P_cementation*plate_x.state.pos[0]*plate_y.state.pos[1]
        v_plate_max = rMean*k_v_max/O.dt
        v_try_abs = abs(kp*dF)/O.dt
        # maximal speed is applied to top wall
        if v_try_abs < v_plate_max :
            plate_z.state.vel = (0, 0, np.sign(dF)*v_try_abs)
        else :
            plate_z.state.vel = (0, 0, np.sign(dF)*v_plate_max)

#-------------------------------------------------------------------------------

def checkUnbalanced_load_cementation_ic():
    '''
    Wait to reach the confining pressure targetted for cementation.
    '''
    addPlotData_cementation_ic()
    saveData_ic()
    # check the force applied
    if abs(O.forces.f(plate_z.id)[2] - P_cementation*plate_x.state.pos[0]*plate_y.state.pos[1])/(P_cementation*plate_x.state.pos[0]*plate_y.state.pos[1]) > 0.005 :
        return
    if unbalancedForce() > unbalancedForce_criteria :
        return
    # characterize the ic algorithm
    global tic
    tac = time.perf_counter()
    hours = (tac-tic)//(60*60)
    minutes = (tac-tic -hours*60*60)//(60)
    seconds = int(tac-tic -hours*60*60 -minutes*60)
    tic = tac
    # report
    simulation_report = open(simulation_report_name, 'a')
    simulation_report.write("Pressure (Cementation) applied : "+str(hours)+" hours "+str(minutes)+" minutes "+str(seconds)+" seconds\n")
    simulation_report.write(str(n_grains)+' grains\n\n')
    simulation_report.close()
    print("\nPressure (Cementation) applied : "+str(hours)+" hours "+str(minutes)+" minutes "+str(seconds)+" seconds\n")
    # switch on friction, bending resistance and twisting resistance between particles
    O.materials[-1].frictionAngle = frictionAngleReal
    O.materials[-1].alphaKr = alphaKrReal
    O.materials[-1].alphaKtw = alphaKtwReal
    # for existing contacts, clear them
    O.interactions.clear()
    # calm down particles
    for b in O.bodies:
        if isinstance(b.shape,Sphere):
            b.state.angVel = Vector3(0,0,0)
            b.state.vel = Vector3(0,0,0)
    # switch off damping
    #Newton.damping = 0
    # next time, do not call this function anymore, but the next one instead
    checker.command = 'checkUnbalanced_param_ic()'

#-------------------------------------------------------------------------------

def checkUnbalanced_param_ic():
    '''
    Wait to reach the equilibrium after switching on the friction and the rolling resistances.
    '''
    addPlotData_cementation_ic()
    saveData_ic()
    # check the force applied
    if abs(O.forces.f(plate_z.id)[2] - P_cementation*plate_x.state.pos[0]*plate_y.state.pos[1])/(P_cementation*plate_x.state.pos[0]*plate_y.state.pos[1]) > 0.005 :
        return
    if unbalancedForce() > unbalancedForce_criteria :
        return
    # characterize the ic algorithm
    global tic
    tac = time.perf_counter()
    hours = (tac-tic)//(60*60)
    minutes = (tac-tic -hours*60*60)//(60)
    seconds = int(tac-tic -hours*60*60 -minutes*60)
    tic = tac
    # report
    simulation_report = open(simulation_report_name, 'a')
    simulation_report.write("Parameters applied : "+str(hours)+" hours "+str(minutes)+" minutes "+str(seconds)+" seconds\n\n")
    simulation_report.close()
    print("\nParameters applied : "+str(hours)+" hours "+str(minutes)+" minutes "+str(seconds)+" seconds\n")
    # save
    #O.save('save/'+O.tags['d.id']+'_ic.yade.bz2')
    # next time, do not call this function anymore, but the next one instead
    checker.command = 'cementation()'
    checker.iterPeriod = 10

#------------------------------------------------------------------------------

def pdf_lognormal(x,m_log,s_log):
    '''
    Return the probability of a value x with a log normal function defined by the mean m_log and the variance s_log.
    '''
    p = np.exp(-(np.log(x)-m_log)**2/(2*s_log**2))/(x*s_log*np.sqrt(2*np.pi))
    return p

#-------------------------------------------------------------------------------

def cementation():
    '''
    Generate cementation between grains.
    '''
    # generate the list of cohesive surface area and its list of weight
    x_min = 1e2 # µm2
    x_max = 4e4 # µm2
    n_x = 20000
    x_L = np.linspace(x_min, x_max, n_x)
    p_x_L = []
    for x in x_L :
        p_x_L.append(pdf_lognormal(x,m_log,s_log))
    # counter
    global counter_bond0
    counter_bond0 = 0
    # iterate on interactions
    for i in O.interactions:
        # only grain-grain contact can be cemented
        if isinstance(O.bodies[i.id1].shape, Sphere) and isinstance(O.bodies[i.id2].shape, Sphere) :
            # only a fraction of the contact is cemented
            if random.uniform(0,1) < f_cemented :
                counter_bond0 = counter_bond0 + 1
                # creation of cohesion
                i.phys.cohesionBroken = False
                # determine the cohesive surface
                cohesiveSurface = (random.choices(x_L,p_x_L)[0]-x_max*mass_removal)*1e-12 # m2
                if cohesiveSurface < 0:
                    cohesiveSurface = 0
                    counter_bond0 = counter_bond0 - 1 
                    i.phys.cohesionBroken = True
                else :
                    # set normal and shear adhesions
                    i.phys.normalAdhesion = tensileCohesion*cohesiveSurface
                    i.phys.shearAdhesion = shearCohesion*cohesiveSurface
                # set normal and shear adhesions
                i.phys.normalAdhesion = tensileCohesion*cohesiveSurface
                i.phys.shearAdhesion = shearCohesion*cohesiveSurface
                # local law E(Ab)
                localYoungModulus = (YoungModulus-80e6)*cohesiveSurface/Ab_mean + 80e6
                i.phys.kn = localYoungModulus*(O.bodies[i.id1].shape.radius*2*O.bodies[i.id2].shape.radius*2)/(O.bodies[i.id1].shape.radius*2+O.bodies[i.id2].shape.radius*2)
                i.phys.ks = 0.25*localYoungModulus*(O.bodies[i.id1].shape.radius*2*O.bodies[i.id2].shape.radius*2)/(O.bodies[i.id1].shape.radius*2+O.bodies[i.id2].shape.radius*2) # 0.25 is the Poisson ratio
                i.phys.kr = i.phys.ks*alphaKrReal*O.bodies[i.id1].shape.radius*O.bodies[i.id2].shape.radius
                i.phys.ktw = i.phys.ks*alphaKtwReal*O.bodies[i.id1].shape.radius*O.bodies[i.id2].shape.radius
    # write in the report
    simulation_report = open(simulation_report_name, 'a')
    simulation_report.write(str(counter_bond0)+" contacts cemented initially\n\n")
    simulation_report.close()
    print('\n'+str(counter_bond0)+" contacts cemented\n")

    # time step
    O.dt = factor_dt_crit_2 * PWaveTimeStep()

    # next time, do not call this function anymore, but the next one instead
    checker.command = 'checkUnbalanced_load_confinement_ic()'
    checker.iterPeriod = 200
    # change the vertical pressure applied
    O.engines = O.engines[:-1] + [PyRunner(command='controlWalls()', iterPeriod = 1)]

#-------------------------------------------------------------------------------

def checkUnbalanced_load_confinement_ic():
    '''
    Wait to reach the vertical/lateral pressure targetted for confinement.
    '''
    global i_load, n_try_i_load, P_load, L_unbalanced_ite, L_confinement_z_ite, L_count_bond

    addPlotData_confinement_ic()
    saveData_ic()
    # check the force applied
    if abs(O.forces.f(plate_z.id)[2] - P_load*plate_x.state.pos[0]*plate_y.state.pos[1])/(P_load*plate_x.state.pos[0]*plate_y.state.pos[1]) > 0.005 :
        return
    if unbalancedForce() > unbalancedForce_criteria :
        return
    # characterize the ic algorithm
    global tic, iter_0
    tac = time.perf_counter()
    hours = (tac-tic)//(60*60)
    minutes = (tac-tic -hours*60*60)//(60)
    seconds = int(tac-tic -hours*60*60 -minutes*60)
    tic = tac

    # compute mean overlap/diameter
    L_over_diam = []
    for contact in O.interactions:
        if isinstance(O.bodies[contact.id1].shape, Sphere) and isinstance(O.bodies[contact.id2].shape, Sphere):
            b1_x = O.bodies[contact.id1].state.pos[0]
            b1_y = O.bodies[contact.id1].state.pos[1]
            b1_z = O.bodies[contact.id1].state.pos[2]
            b2_x = O.bodies[contact.id2].state.pos[0]
            b2_y = O.bodies[contact.id2].state.pos[1]
            b2_z = O.bodies[contact.id2].state.pos[2]
            dist = math.sqrt((b1_x-b2_x)**2+(b1_y-b2_y)**2+(b1_z-b2_z)**2)
            over = O.bodies[contact.id1].shape.radius + O.bodies[contact.id2].shape.radius - dist
            diam = 1/(1/(O.bodies[contact.id1].shape.radius*2)+1/(O.bodies[contact.id2].shape.radius*2))
            L_over_diam.append(over/diam)
    m_over_diam = np.mean(L_over_diam)
    print('Mean Overlap/Diameter', m_over_diam)

    # report
    simulation_report = open(simulation_report_name, 'a')
    simulation_report.write("Pressure (Confinement) applied : "+str(hours)+" hours "+str(minutes)+" minutes "+str(seconds)+" seconds\n")
    simulation_report.write(str(O.iter-iter_0)+' Iterations\n')
    simulation_report.write(str(n_grains)+' grains\n')
    simulation_report.write('Mean Overlap/Diameter ' + str(m_over_diam) + '\n')
    simulation_report.write('IC generation ends\n\n')
    simulation_report.close()
    print("\nPressure (Confinement) applied : "+str(hours)+" hours "+str(minutes)+" minutes "+str(seconds)+" seconds\n")

    # reset plot (IC done, simulation starts)
    plot.reset()
    # save new reference position for walls
    plate_x.state.refPos = plate_x.state.pos
    plate_y.state.refPos = plate_y.state.pos
    plate_z.state.refPos = plate_z.state.pos

    # next time, do not call this function anymore, but the next one instead
    iter_0 = O.iter
    checker.command = 'checkUnbalanced()'
    checker.iterPeriod = 500

    # trackers    
    i_load = 1
    n_try_i_load = 0
    P_load = P_load + 1/n_load * d_P
    L_unbalanced_ite = []
    L_confinement_z_ite = []
    L_count_bond = []

    # user print
    print('Loading step :', i_load, '/', n_load)

#-------------------------------------------------------------------------------

def addPlotData_cementation_ic():
    """
    Save data in plot.
    """
    # add forces applied on walls
    sx = O.forces.f(plate_x.id)[0]/(plate_y.state.pos[1]*plate_z.state.pos[2])
    sy = O.forces.f(plate_y.id)[1]/(plate_x.state.pos[0]*plate_z.state.pos[2])
    sz = O.forces.f(plate_z.id)[2]/(plate_x.state.pos[0]*plate_y.state.pos[1])
    # add data
    plot.addData(i=O.iter-iter_0, porosity=porosity(), coordination=avgNumInteractions(), unbalanced=unbalancedForce(), counter_bond=0,\
                 Sx=sx, Sy=sy, Sz=sz,\
                 conf_verified=sz/P_cementation*100,\
                 strain_z=100*(plate_z.state.pos[2]-plate_z.state.refPos[2])/plate_z.state.refPos[2])

#-------------------------------------------------------------------------------

def addPlotData_confinement_ic():
    """
    Save data in plot.
    """
    # add forces applied on walls
    sx = O.forces.f(plate_x.id)[0]/(plate_y.state.pos[1]*plate_z.state.pos[2])
    sy = O.forces.f(plate_y.id)[1]/(plate_x.state.pos[0]*plate_z.state.pos[2])
    sz = O.forces.f(plate_z.id)[2]/(plate_x.state.pos[0]*plate_y.state.pos[1])
    # add data
    plot.addData(i=O.iter-iter_0, porosity=porosity(), coordination=avgNumInteractions(), unbalanced=unbalancedForce(), counter_bond=count_bond(),\
                 Sx=sx, Sy=sy, Sz=sz,\
                 conf_verified=sz/P_load*100, \
                 strain_z=100*(plate_z.state.pos[2]-plate_z.state.refPos[2])/plate_z.state.refPos[2])

#-------------------------------------------------------------------------------

def saveData_ic():
    """
    Save data in .txt file during the ic.
    """
    plot.saveDataTxt('data/IC_'+O.tags['d.id']+'.txt')
    # post-proccess
    L_sigma_x = []
    L_sigma_y = []
    L_sigma_z = []
    L_confinement = []
    L_coordination = []
    L_unbalanced = []
    L_ite  = []
    L_strain_z = []
    L_n_bond = []
    file = 'data/IC_'+O.tags['d.id']+'.txt'
    data = np.genfromtxt(file, skip_header=1)
    file_read = open(file, 'r')
    lines = file_read.readlines()
    file_read.close()
    if len(lines) >= 3:
        for i in range(len(data)):
            L_sigma_x.append(abs(data[i][0]))
            L_sigma_y.append(abs(data[i][1]))
            L_sigma_z.append(abs(data[i][2]))
            L_confinement.append(data[i][3])
            L_coordination.append(data[i][4])
            L_n_bond.append(data[i][5])
            L_ite.append(data[i][6])
            L_strain_z.append(data[i][8])
            L_unbalanced.append(data[i][9])

        # plot
        fig, ((ax1, ax2, ax3), (ax4, ax5, ax6)) = plt.subplots(2,3, figsize=(20,10),num=1)

        ax1.plot(L_ite, L_sigma_x, label = r'$\sigma_x$')
        ax1.plot(L_ite, L_sigma_y, label = r'$\sigma_y$')
        ax1.plot(L_ite, L_sigma_z, label = r'$\sigma_z$')
        ax1.legend()
        ax1.set_title('Stresses (Pa)')

        ax2.plot(L_ite, L_unbalanced, 'b')
        ax2.set_ylabel('Unbalanced (-)', color='b')
        ax2.set_ylim(ymin=0, ymax=2*unbalancedForce_criteria)
        ax2b = ax2.twinx()
        ax2b.plot(L_ite, L_confinement, 'r')
        ax2b.set_ylabel('Confinement (%)', color='r')
        ax2b.set_ylim(ymin=0, ymax=150)
        ax2b.set_title('Steady-state indices')

        ax3.plot(L_ite, L_n_bond)
        ax3.set_title('Number of bond (-)')

        ax4.plot(L_ite, L_strain_z, label=r'$\epsilon_z$ (%)')
        ax4.legend()
        ax4.set_title('Strains (%)')

        ax5.plot(L_ite, L_coordination)
        ax5.set_title('Coordination number (-)')

        plt.savefig('plot/IC_'+O.tags['d.id']+'.png')

        plt.close()

#-------------------------------------------------------------------------------
#Load
#-------------------------------------------------------------------------------

def controlWalls():
    '''
    Control the upper wall to applied a defined confinement force.

    The displacement of the wall depends on the force difference. A maximum value is defined.
    '''
    Fz = O.forces.f(plate_z.id)[2]
    if Fz == 0:
        plate_z.state.pos =  (plate_x.state.pos[0]/2, plate_y.state.pos[1]/2, max([b.state.pos[2]+0.99*b.shape.radius for b in O.bodies if isinstance(b.shape, Sphere)]))
    else :
        dF = Fz - P_load*plate_x.state.pos[0]*plate_y.state.pos[1]
        v_plate_max = rMean*k_v_max/O.dt
        v_try_abs = abs(kp*dF)/O.dt
        # maximal speed is applied to top wall
        if v_try_abs < v_plate_max :
            plate_z.state.vel = (0, 0, np.sign(dF)*v_try_abs)
        else :
            plate_z.state.vel = (0, 0, np.sign(dF)*v_plate_max)

#-------------------------------------------------------------------------------

def count_bond():
    '''
    Count the number of bond.
    '''
    counter_bond = 0
    for i in O.interactions:
        if isinstance(O.bodies[i.id1].shape, Sphere) and isinstance(O.bodies[i.id2].shape, Sphere):
            if not i.phys.cohesionBroken :
                counter_bond = counter_bond + 1
    return counter_bond

#-------------------------------------------------------------------------------

def checkUnbalanced():
    """
    Look for the equilibrium during the loading phase.
    """
    global n_try_i_load, i_load, P_load, L_unbalanced_ite, L_confinement_z_ite, L_count_bond
    # count the number of load
    n_try_i_load = n_try_i_load + 1
    # track and plot unbalanced
    L_unbalanced_ite.append(unbalancedForce())
    # track and plot confinement
    L_confinement_z_ite.append(O.forces.f(plate_z.id)[2]/(P_load*plate_x.state.pos[0]*plate_y.state.pos[1])*100)
    # track and plot bonds number
    L_count_bond.append(count_bond())

    # plot
    if len(L_unbalanced_ite)>2:
        fig, ((ax1, ax2, ax3)) = plt.subplots(1,3, figsize=(16,9),num=1)
        # unbalanced
        ax1.plot(L_unbalanced_ite)
        ax1.set_title('unbalanced force (-)')
        ax1.set_ylim(ymin=0, ymax=2*unbalancedForce_criteria)
        # confinement
        ax2.plot(L_confinement_z_ite)
        ax2.set_ylim(ymin=0, ymax=150)
        ax2.set_title('confinements (%)')
        # number of bond
        ax3.plot(L_count_bond)
        ax3.set_title('Number of bond (-)')
        # close
        fig.savefig('plot/tracking_ite.png')
        plt.close()

    # minimum of time 
    if n_try_i_load < 10:
        return
    # verify confinement pressure applied
    if abs(O.forces.f(plate_z.id)[2] - P_load*plate_x.state.pos[0]*plate_y.state.pos[1])/(P_load*plate_x.state.pos[0]*plate_y.state.pos[1]) > 0.005: 
        return
    # verify unbalanced force criteria
    if unbalancedForce() < unbalancedForce_criteria:
        # reset trackers
        n_try_i_load = 0
        i_load = i_load + 1
        P_load = P_load + 1/n_load * d_P
        L_unbalanced_ite = []
        L_confinement_z_ite = []
        L_count_bond = []

        # save data
        saveData()
        
        # check simulation stop conditions
        if i_load > n_load:
            stopLoad()
        else :
            # user print
            print('Loading step :', i_load, '/', n_load)


#-------------------------------------------------------------------------------

def stopLoad():
    """
    Close simulation.
    """
    # close yade
    O.pause()
    # characterize the dem step
    tac = time.perf_counter()
    hours = (tac-tic)//(60*60)
    minutes = (tac-tic -hours*60*60)//(60)
    seconds = int(tac-tic -hours*60*60 -minutes*60)
    # report
    simulation_report = open(simulation_report_name, 'a')
    simulation_report.write("Oedometric loading test : "+str(hours)+" hours "+str(minutes)+" minutes "+str(seconds)+" seconds\n")
    simulation_report.write(str(count_bond())+" contacts cemented finally\n\n")
    simulation_report.close()
    print("Oedometric loading test : "+str(hours)+" hours "+str(minutes)+" minutes "+str(seconds)+" seconds")
    print('\n'+str(count_bond())+" contacts cemented\n")
    # characterize the last DEM step and the simulation
    hours = (tac-tic_0)//(60*60)
    minutes = (tac-tic_0 -hours*60*60)//(60)
    seconds = int(tac-tic_0 -hours*60*60 -minutes*60)
    # report
    simulation_report = open(simulation_report_name, 'a')
    simulation_report.write("Simulation time : "+str(hours)+" hours "+str(minutes)+" minutes "+str(seconds)+" seconds\n\n")
    simulation_report.close()
    print("\nSimulation time : "+str(hours)+" hours "+str(minutes)+" minutes "+str(seconds)+" seconds\n")
    
    # save simulation
    os.mkdir('../Data_Rock_Oedometer_MassRemoval/'+O.tags['d.id'])
    shutil.copytree('data','../Data_Rock_Oedometer_MassRemoval/'+O.tags['d.id']+'/data')
    shutil.copytree('plot','../Data_Rock_Oedometer_MassRemoval/'+O.tags['d.id']+'/plot')
    shutil.copy('Rock_Oedometer_MassRemoval.py','../Data_Rock_Oedometer_MassRemoval/'+O.tags['d.id']+'/Rock_Oedometer_MassRemoval.py')
    shutil.copy(O.tags['d.id']+'_report.txt','../Data_Rock_Oedometer_MassRemoval/'+O.tags['d.id']+'/'+O.tags['d.id']+'_report.txt')

#-------------------------------------------------------------------------------

def addPlotData():
    """
    Save data in plot.
    """
    # add forces applied on wall x and z
    sx = O.forces.f(plate_x.id)[0]/(plate_y.state.pos[1]*plate_z.state.pos[2])
    sy = O.forces.f(plate_y.id)[1]/(plate_x.state.pos[0]*plate_z.state.pos[2])
    sz = O.forces.f(plate_z.id)[2]/(plate_x.state.pos[0]*plate_y.state.pos[1])
    # add data
    plot.addData(i=O.iter-iter_0, porosity=porosity(), coordination=avgNumInteractions(), unbalanced=unbalancedForce(), counter_bond=count_bond(), \
                Sx=sx, Sy=sy, Sz=sz, \
                X_plate=plate_x.state.pos[0], Y_plate=plate_y.state.pos[1], Z_plate=plate_z.state.pos[2],\
                conf_verified=sz/P_load*100, \
                strain_z=100*(plate_z.state.pos[2]-plate_z.state.refPos[2])/plate_z.state.refPos[2])

#-------------------------------------------------------------------------------

def saveData():
    """
    Save data in .txt file during the steps.
    """
    addPlotData()
    plot.saveDataTxt('data/'+O.tags['d.id']+'.txt')
    # post-proccess
    L_coordination = []
    L_n_bond = []
    L_sigma_x = []
    L_sigma_y = []
    L_sigma_z = []
    L_strain_z = []
    file = 'data/'+O.tags['d.id']+'.txt'
    data = np.genfromtxt(file, skip_header=1)
    file_read = open(file, 'r')
    lines = file_read.readlines()
    file_read.close()
    if len(lines) >= 3:
        for i in range(len(data)):
            L_sigma_x.append(data[i][0])
            L_sigma_y.append(data[i][1])
            L_sigma_z.append(data[i][2])
            L_coordination.append(data[i][7])
            L_n_bond.append(data[i][8])
            L_strain_z.append(abs(data[i][11]))

        # plot
        fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2,2, figsize=(16,9),num=1)

        ax1.plot(L_coordination)
        ax1.set_title('Coordination (-)')

        ax2.plot(L_n_bond)
        ax2.set_title('Number of bond (-)')

        ax3.plot(L_strain_z, L_sigma_z)
        ax3.set_xlabel(r'$\epsilon_z$ (%)')
        ax3.set_ylabel(r'vertical stresses (Pa)')

        ax4.plot(L_strain_z, L_sigma_x, label=r'$\sigma_x$')
        ax4.plot(L_strain_z, L_sigma_y, label=r'$\sigma_y$')
        ax4.plot(L_strain_z, L_sigma_z, label=r'$\sigma_z$')
        ax4.set_xlabel(r'$\epsilon_z$ (%)')
        ax4.set_ylabel(r'stresses (Pa)')
        ax4.legend()

        plt.suptitle(r'Trackers - loading step (-)')
        plt.savefig('plot/'+O.tags['d.id']+'.png')
        plt.close()

#-------------------------------------------------------------------------------
# start simulation
#-------------------------------------------------------------------------------

O.run()
waitIfBatch()