rupture_2D.py

# -*- coding: utf-8 -*-
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# + {"code_folding": [0], "cell_type": "markdown"}
# <div style='background-image: url("../../share/images/header.svg") ; padding: 0px ; background-size: cover ; border-radius: 5px ; height: 250px'>
#     <div style="float: right ; margin: 50px ; padding: 20px ; background: rgba(255 , 255 , 255 , 0.7) ; width: 50% ; height: 150px">
#         <div style="position: relative ; top: 50% ; transform: translatey(-50%)">
#             <div style="font-size: xx-large ; font-weight: 900 ; color: rgba(0 , 0 , 0 , 0.8) ; line-height: 100%">Earthquake physics</div>
#             <div style="font-size: large ; padding-top: 20px ; color: rgba(0 , 0 , 0 , 0.5)"> Dynamic Rupture in 2D </div>
#         </div>
#     </div>
# </div>
# -

# ##### Authors:
# * Mitchell Williams
# * Kenneth Duru
# ---

# ## Description
#
# ### Background 
#
# This notebook simulates  dynamic earthquake ruptures and elastic wave propagation in two space dimensions (2D). Spontaneously propagating shear ruptures on a frictional interface in an elastic solid is a useful idealization of natural earthquakes. The conditions relating slip-rate and fault strength are often expressed as nonlinear friction laws.
#
# During an earthquake, two sides of the fault, initially held in contact by a high level of frictional resistance, slip suddenly when that resistance catastrophically decreases, generating strong ground shaking which is carried by seismic waves to remote areas, far away from fault zones.
#
# This notebook describes and implements a high order accurate finite difference method for enforcing nonlinear friction laws in a consistent and provably stable manner, suitable for efficient explicit time integration and accurate propagation of seismic waves in heterogeneous media with free surface boundary conditions.
#
# <br/><br/>
#
# ### The elastic wave equation
# --------------------------------------------------------------------
# Consider two elastic solids separated by a fault at the interface $x = 0$. In each elastic solid, wave motion is described by the elastic wave equation, in velocity-stress formulation: 
# <br/><br/> 
# \begin{align}
# {\rho} {\partial_t v_i} = {\frac{ \partial \sigma_{ij}}{\partial x_j},  \quad}  {\quad}   S_{klij}{\partial_t \sigma_{ij}} = \frac{1}{2}  \cdot (\frac{\partial v_k}{\partial x_l} + \frac{\partial v_l}{\partial x_k}), \quad i, j = 1, 2,
# \end{align}
# <br/> <br/>
#  with particle velocities ${v_i}$, stress tensor ${\sigma_{ij} }$, compliance tensor $S_{klij}$, and density ${\rho}$.
# <br/><br/>
#
# -----------------------------------------------------------------
#
# On the fault there is the velocity vector $\left(v_{n}, v_{m}\right)=\left(v_{x}, v_{y}\right)$, and the traction vector $\left(T_{n}, T_{m}\right)=\left(\sigma_{xx}, \sigma_{xy}\right)$, where $v_{n}$, $T_{n}$ are the normal components and $v_{m}$, $T_{m}$ being the shear components on the  interface. Denoting the fields in the positive and the negative parts of the fault with the superscripts $\pm$, we introduce the jumps in particle velocities by $\lbrack{v_j}\rbrack = {v_j}^{+}-{v_j}^{-}$, with $j = m, n$, and  the total traction by $T_j = T_{0j} + \Delta{T_j}$, where $T_{0j}$ are the initial background tractions and $\Delta{T_j}$ are the traction changes on the fault. The conditions on the fault connecting the two elastic solids are
# <br/><br/> 
# \begin{align}
# \text{force balance}&: \quad T_{j}^{+} = T_{j}^{-} = T_{j}, \quad j = m, n, \\
# \text{no opening}&: \quad \lbrack{v_n}\rbrack = 0,\\
# \text{friction laws}&: \quad T_m = \sigma_n\frac{f(V, \theta)}{V}\lbrack{v_m}\rbrack, \quad V = \left|\lbrack{v_m}\rbrack\right|.
# \end{align}
# <br/> <br/>
#  Here $\sigma_n > 0$ is the compressive normal stress and $ f(V, \theta) \ge 0 $ is the nonlinear fricition coefficient relating to the fault's shear strength, $\tau = |T_m| > 0$, and will be described  in more detail below.
# <br/><br/>
#
# -------------------------------------------------------------
#
# ### Fracture modes 
#
# There are three modes of fracture as illustrated below. Mode I fracture is an in plane motion where the direction of movement is perpendicular to the fracture itself, this opening can be described by a single component tensile component. Mode II is an in plane fracture that moves parallel to the fracture, it can be described by a compressive component that is perpendicular to the fracture (non-opening) and another parallel component that is creating shearing forces. This is a widely used mode for rupture dynamics because, both P and S waves emanate from this mode. This can allow 'super shear' ruptures, shear ruptures propagating above the S wave velocity. Mode III is an out of plane shearing motion and can be described in one out of plane component. This mode of rupture has a strict upper bound of the S wave velocity. Since earthquakes don't open wide chasms into the earth, mode I is not used here. <br/>
#
# <img src=fracmodes.png>
#
# Figure source: https://www.doitpoms.ac.uk/tlplib/brittle_fracture/same.php
#
# ### Friction laws
#
# It is necessary to consider the role of friction in all of this because it modulates the stresses and slip speed as the separate sides of the fault slip past each other. The first thing to consider is that as the fault slips it experiences shear strength ${\tau}$ which is exerted by the normal stresses on the fault ${\sigma_n}$. The way these two variables are related is through the friction coefficient ${f}$: ${\quad} {\tau} = f \cdot {\sigma_n}$. 
#
#  For the purposes of this notebook  we consider two widely used models of friction, *slip-weakening* and *rate-and-state*. <br/>
#
#
# <p style="width:75%;float:right;padding-left:50px">
# <img src=RS_graph.png>
# <span style="font-size:smaller">
# </span>
# </p>
# #### Slip Weakening Friction
# Slip weakening friction describes a friction behavior in which friction coefficient weakens linearly with slip.  When the fault is at rest the friction coefficient is denoted by static friction coefficient ${f_s}$, when the fault is in motion the friction coefficient is denoted by the dynamic friction coefficient ${f_d}$. An earthquake begins when the load $\tau$ at a finite patch on the fault overcomes the peak frictional strength, $\tau \ge \tau_p$, ${\quad \tau_p = f_s \cdot \sigma_n}$.  As the earthquake continues the fault surfaces slide relative to each other  and the frictional coefficient evolves from linearly from static friction coefficient $f_s$  to the dynamic friction coefficient ${f_d}$. The fault is fully weakened when the slip reaches the critical slip ${D_c}$. This can be seen in the figure above which plots the friction coefficient as a function of slip.
#  
# #### Rate and State Friction
#
# Rate and state friction is a constitutive law that was empirically found and is considered more realistic than slip weakening. It takes the form:  
#
# \begin{align}
#  &\tau = \sigma_n \cdot f(V, \theta), \quad f(V, \theta) = [ f_0 + a \cdot ln(\frac {V}{V_0}) + b \cdot ln( \frac{V_0 \theta}{D_c})], \\
#   &\frac {d \theta}{dt} = G(V, \theta).
#  \end{align}
#
# In this equation ${V}$ is the current slip velocity and ${V_0}$ is a reference slip velocity while ${f_0}$ is the steady state friction coefficient at the reference velocity: ${f_{V= V_0}}$. There is the state variable ${\theta}$, and the state evolution law $G(V, \theta)$. We consider specifically the aging law: ${ G(V, \theta) = 1 - \frac {V\theta }{D_c}.\quad}$ At steady state the state variable is proportional to the critical distance divided by velocity ${ \theta_{ss} = \frac {D_c}{V}}$  <br/>
#
# Here ${a}$ is the *direct effect* and is used to model how the system responds to velocity changes. ${b}$ is the *evolution effect* and describes the magnitude of the steady state friction.  ${D_c}$ is the critical distance and corresponds to the slip length over which the evolution of ${a \to b}$ occurs. The relationship of ${a}$ and ${b}$ to the slip weakening friction variables is as follows: ${\quad b = \frac {df_s}{d(ln(t))} \quad, \quad (a-b) = \frac {df_d}{d(ln(V))} \quad}$ where ${t}$ is the amount of time of contact without an earthquake. It is interesting to notice that there is a relationship between slip velocity ${V}$ and dynamic friction ${f_d}$ that depends on the difference ${(a-b)}$. If ${(a-b)>0}$ it shows that there is a velocity strengthening relationship ${\frac {df_d}{d(ln(V))} > 0}$. Any rupture that enters this region will arrest and is called a *stable* region. If ${(a-b)<0}$ it shows that there is a velocity weakening relationship ${\frac {df_d}{d(ln(V))} < 0}$, this is where earthquakes can nucleate and grow and is called *unstable*. On the border between *stable* and *unstable* regions is a zone of *conditional stability* where ruptures can sustain propagation while not growing or terminating. This notebook only exists in the unstable zone.
#
#
#
# The equations are discretized using the SBP-SAT finite difference scheme. Time integration is performed using the classical fourth order accurate Runge-Kutta method. For more elaborate discussions, we refer the reader to the references and the  notebooks on the SBP-SAT method. A summary of the numerical implementations used in the this notebook is presentated in the flow-chart below.
#
# <img src=2Dnotebook_flowchart.jpg>
#
# ### Putting it together
#
# In the parameter window you can modulate the material properties, run time, domain size, and CFL criterion. In addition, you can choose rate and state friction or slip weakening friction, notice the differences that occur with this choice. In the Calculations and plotting cell the resulting on fault tractions and slip velocities are solved for in the RK4_2D.elastic_RK4_2D function, which also calls on the other associated functions. The bottom cell contains 5 output plots of the rupture behavior. 
#
#
#
# The plot that is observed immediately under this cell is the particle velocity plotted in the entire domain. The fault runs vertically through the center of the model and is marked by a vertical line. When this animation finishes the seismograms of the earthquake pop up under this. In the last cell there are three plots that come from the data generated in the previous cell. These three plots show the on fault slip, slip rate, and tractions as they evlove through time. Lastly there is a plot that shows the slip rate on the fault through time.
#  
#
# #### References 
# Byerlee, J. “Friction of Rocks.” Rock Friction and Earthquake Prediction, vol. 116, 1978, pp. 615–626., doi:10.1007/978-3-0348-7182-2_4. <br/>
# Duru, Kenneth, and Eric M. Dunham. “Dynamic Earthquake Rupture Simulations on Nonplanar Faults Embedded in 3D 
# Geometrically Complex, <br/>${\quad}$ Heterogeneous Elastic Solids.” Journal of Computational Physics, vol. 305, 2016, pp. 185–207., doi: 10.1016/j.jcp.2015.10.0215<br/>
# Gustafsson, Bertil. High Order Difference Methods for Time Dependent PDE. Springer, 2008.<br/>
# Kozdon, J. E., E. M. Dunham, and J. Nordström (2012), Interaction of waves with frictional interfaces using summation-by-parts difference operators:<br/>${\quad}$  Weak enforcement of nonlinear boundary conditions, Journal of Scientific Computing, 50(2), 341-367, doi:10.1007/s10915-011-9485-3<br/>
# Leeman, John. “Modeling Rate and State Friction with Python.” SciPy 2016. Austin, USA.<br/>
# Scholz, Christopher H. “Earthquakes and Friction Laws.” Nature, vol. 391, no. 6662, 1998, pp. 37–42., doi:10.1038/34097.

# ### Exercises   
#
# Explore the following questions by interacting with the variables in the parameter windows. 
#
#
# * **1)** Explore the differences between slip weakening and rate and state frictions. How does this alter the rupture behavior? 
#
#
#
# * **2)** How does altering the resolution change the model time? At what point does poor resolution negatively affect the model.
#
#
# * ** 3)** Compare the differences between mode II and mode III. How does altering the P wave velocity alter this? 
#
#
#
# * **4) ** Modulate the background stresses in the friction and fracture parameter window. If the stresses are too high what happens to the rupture? If they are too low? 
#
#
#
# * **5)** When altering all of these parameters what changes do you see in the evolution of slip-rate? Which combination of parameters would create the most dangerous earthquake? 
#
#

# + {"code_folding": [0]}
# Import necessary routines
# %matplotlib notebook
import numpy as np
import matplotlib.pyplot as plt
import RK4_2D

#plt.switch_backend("TkAgg")          # plots in external window
plt.switch_backend("nbagg")           # plots within this notebook
# -

# In order to show a less time consuming solution the spatial and temporal resolution has been decreased. In the case you would like to increase it, you can modify the grid points in x and y (mantaining their proportions), the snapshot frequency and the fracture mode. 

# + {"code_folding": [0]}
# Parameter cell
Lx = 10.0     # length of the domain (x-axis)
Ly = 20.0     # width of the domain (y-axis)

nx = 50       # grid points in x
ny = 100      # grid points in y
tend = 0.98*3.464      # final time
CFL = 0.5

isnap = 5     # snapshot frequency
order = 4     # spatial order of accuracy (2,4,6)

nf = 5        # number of fields

cs_l = 3.464    # shear wave speed
cp_l = 6.0      # compresional wave speed
rho_l = 2.6702  # density

cs_r = 3.464    # shear wave speed
cp_r = 6.0      # compresional wave speed
rho_r = 2.6702  # density

# Bi-material:
#cs_r = 2.0      # shear wave speed
#cp_r = 4.0      # compresional wave speed
#rho_r = 2.6     # density

dx = Lx/nx    # spatial step in x
dy = Ly/ny    # spatial step in y

print('your x-axis resolution is '+str(dx)+' km')
print('your y-axis resolution is '+str(dy)+' km')

# + {"code_folding": [0]}
# Friction and Fracture parameter cell 

# Friction type
fric_law = 'SW'   #RS OR SW

# Fracture mode
mode = 'III'   #  'II' or 'III'

cs = np.max([cs_l, cs_r])
# cp = np.max([cp_l, cp_r])

# Set mode info
# Mode II crack: In-plane shear
if mode == 'II':
    nf = 5        # number of fields [vx, vy, sxx, syy, sxy] 
    dt = CFL/np.sqrt(cp**2 + cs**2)*dx    # Time step

# Mode III crack: out-of-plane (anti-plane) shear    
if mode == 'III':
    nf = 3        # number of fields  [vz, sxz, sxy] 
    dt = dx*CFL/cs    # Time step

nt = np.int(np.round(tend/dt)) # number of time steps

# To test your own friction parameters change the varible 'user_params' from 'default' to 'edit'
# then reasign your own values to the following variables 
user_params = 'default'

Tau_0_user = np.ones((ny, 1))                           # shear stress 
alp_s_user = np.ones((ny, 1))                           # stastic friction
alp_d_user = np.ones((ny, 1))                           # dynamic friction
D_c_user = np.ones((ny, 1))                             # critical slip
sigma_n_user = -np.ones((ny, 1))                        # normal stress 
L0_user = np.ones((ny, 1))                              # state evolution distance
f0_user = np.ones((ny, 1))                              # referance friction coeff
a_user = np.ones((ny, 1))                               # direct effect 
b_user = np.ones((ny, 1))                               # evolution parameter 
V0_user = np.ones((ny, 1))                              # reference slip rate


# + {"code_folding": [0, 11, 16, 24, 41, 69, 108]}
# Initializaion cell 
# Do not alter

# Friction condition
Y_fault =np.zeros((ny, 1)) 
Y0 = 10
slip = np.zeros((ny, 1))
psi = np.zeros((ny, 1))
FaultOutput = np.zeros((ny, nt, 6)) # c
FaultOutput0 = np.zeros((ny, 6))

for j in range(0, ny):
        #Y_fault[j, 0] = j*dy
        
        if np.abs(j*dy-Y0) <= 1.5:
            Y_fault[j, 0] = 1.0
if fric_law not in ('SW', 'RS'):
    # Choose friction law: fric_law
    # Slip-weakening (SW)
    # Rate-and-state friction law (RS)
    
    print('friction law not implemented. choose fric_law = SW or fric_law = RS')
    exit(-1)
    
if user_params not in ('edit', 'default'):

    print('friction law not implemented. choose user_params = edit or fric_law = default')
    exit(-1)
    
if user_params == 'edit':
    Tau_0 = Tau_0_user                          # shear stress 
    alp_s = alp_s_user                           # stastic friction
    alp_d = alp_d_user                         # dynamic friction
    D_c = D_c_user                             # critical slip
    sigma_n = sigma_n_user                        # normal stress 
    L0 = L0_user                               # state evolution distance
    f0 = f0_user                             # referance friction coeff
    a = a_user                              # direct effect 
    b = b_user                              # evolution parameter 
    V0 = V0_user                             # reference slip rate
    
if fric_law  in ('SW'):
     alpha = np.ones((ny, 1))*1e1000000                      # initial friction coefficient
     slip = np.zeros((ny, 1))                        # initial slip (in m)
     slip_new = np.zeros((ny, 1))
     #Tau_0 = np.ones((ny, 1))*(70+11.6*(np.exp(-(Y_fault-7.0)**2/(2*9))))               # initial load (81.24 in MPa), slight increase will unlock the fault
     
     if user_params == 'default':
        Tau_0 = np.ones((ny, 1))*(70+11.6*Y_fault)   
        alp_s = np.ones((ny, 1))*0.677                          # stastic friction
        alp_d = np.ones((ny, 1))*0.525                          # dynamic friction
        D_c = np.ones((ny, 1))*0.4                              # critical slip
        sigma_n = -np.ones((ny, 1))*120.0                        # normal stress 
     
        
     # These are not needed for the slip weakening case   
     psi = np.ones((ny, 1))*0.0                         # initial condition for the state variable in friction law
     psi_new = np.ones((ny, 1))*0.0
     L0 = np.ones((ny, 1))*1.0                               # state evolution distance
     f0 = np.ones((ny, 1))*1.0                               # referance friction coeff
     a = np.ones((ny, 1))*1.0                                # direct effect 
     b = np.ones((ny, 1))*1.0                                # evolution parameter 
     V0 = np.ones((ny, 1))*1.0                               # reference slip rate
        
     FaultOutput[:, 0, 2] = sigma_n[:,0]   
     FaultOutput[:, 0, 3] = Tau_0[:,0] 
     FaultOutput[:, 0, 4] = slip[:,0] 
     FaultOutput[:, 0, 5] = psi[:,0]   
    
if fric_law  in ('RS'):
    alpha = np.ones((ny, 1))*1e1000000                      # initial friction coefficient                                                                                   
    slip = np.ones((ny, 1))*0.0                         # initial slip (in m) 
    slip_new = np.zeros((ny, 1))
    #Tau_0 = np.ones((ny, 1))*81.24+0.1*0.36                 # initial load (81.24 in MPa), slight increase will unlock the fault   
    psi = np.ones((ny, 1))*0.4367                      # initial condition for the state variable in friction law
    psi_new = np.ones((ny, 1))*0.0
    
    if user_params == 'default':
        sigma_n = -np.ones((ny, 1))*120.0                        # background normal stress 
        Tau_0 = np.ones((ny, 1))*75 #-2*0.2429*sigma_n*Y_fault
        L0 = np.ones((ny, 1))*0.02                              # state evolution distance
        f0 = np.ones((ny, 1))*0.6                               # referance friction coeff
        a = np.ones((ny, 1))*0.008                              # direct effect 
        b = np.ones((ny, 1))*0.012                              # evolution parameter 
        V0 = np.ones((ny, 1))*1.0e-6                            # reference slip rate
    
    
    Vin = np.ones((ny, 1))*2.0e-12 
    theta = L0/V0*np.exp(((a*np.log(2.0*np.sinh(75/(a*120)))-f0-a*np.log(Vin/V0))/b))
    psi[:,0] = f0[:,0] + b[:,0]*np.log(V0[:,0]/L0[:,0]*theta[:,0])
                          
    
        
        

     # These are not needed for the rate and state case   
    alp_s = np.ones((ny, 1))*1.0                             # stastic friction
    alp_d = np.ones((ny, 1))*1.0                             # dynamic friction
    D_c = np.ones((ny, 1))*1.0                               # critical slip
        
    FaultOutput[:, 0, 2] = sigma_n[:,0]  
    FaultOutput[:, 0, 3] = Tau_0[:,0]
    FaultOutput[:, 0, 4] = slip[:,0]
    FaultOutput[:, 0, 5] = psi[:,0] 
       
friction_parameters = np.zeros((12, ny))    
#friction_parameters = [alpha, alpha, Tau_0, L0, f0, a, b, V0, sigma_n, alp_s, alp_d, D_c]    
#                        0         1      2     3   4  5  6   7    8       9      10  11
for j in range(0, ny):
        friction_parameters[0, j] = alpha[j, 0]
        friction_parameters[1, j] = alpha[j, 0]
        friction_parameters[2, j] = Tau_0[j, 0]
        friction_parameters[3, j] = L0[j, 0]
        friction_parameters[4, j] = f0[j, 0]
        friction_parameters[5, j] = a[j, 0]
        
        
        friction_parameters[6, j] = b[j, 0]
        friction_parameters[7, j] = V0[j, 0]
        friction_parameters[8, j] = sigma_n[j, 0]
        friction_parameters[9, j] = alp_s[j, 0]
        friction_parameters[10, j] = alp_d[j, 0]
        friction_parameters[11, j] = D_c[j, 0]
# end friction condition

# source parameters
x0 = -15.0        # [km]
y0 = 7.5         # [km]
t0 = 0.0         # [s]
T =  0.1         # [s]
M0 = 00.0      # [MPa]
M = [0, 0, 1., 1., 0]
source_type = 'Gaussian' # 'Gaussian', 'Brune'
source_parameter = [x0, y0, t0, T, M0, source_type, M]



# extract Lame parameters
mu_l = rho_l*cs_l**2
Lambda_l = rho_l*cp_l**2-2.0*mu_l

mu_r = rho_r*cs_r**2
Lambda_r = rho_r*cp_r**2-2.0*mu_r

# Model type, available are "homogeneous", "random":
model_type = "homogeneous"

# Initialize velocity model
Mat_l = np.zeros((nx, ny, 3))

Mat_r = np.zeros((nx, ny, 3))

if model_type == "homogeneous":
    Mat_l[:,:,0] += rho_l
    Mat_l[:,:,1] += Lambda_l
    Mat_l[:,:,2] += mu_l
    
    Mat_r[:,:,0] += rho_r
    Mat_r[:,:,1] += Lambda_r
    Mat_r[:,:,2] += mu_r

elif model_type == "random":
    pert = 0.4
    
    rho_pert = np.zeros((nx+nx-1, ny))
    mu_pert = np.zeros((nx+nx-1, ny))
    lambda_pert = np.zeros((nx+nx-1, ny))
    
    
    rho_pert = 2.0 * (np.random.rand(nx+nx-1, ny) - 0.5) * pert
    mu_pert = 2.0 * (np.random.rand(nx+nx-1, ny) - 0.5) * pert
    lambda_pert = 2.0 * (np.random.rand(nx+nx-1, ny) - 0.5) * pert 
    
    
    r_rho_l = rho_pert[0:nx, :]
    r_mu_l =  mu_pert[0:nx, :]
    r_lambda_l =  lambda_pert[0:nx, :]
    
    
    
    Mat_l[:,:,0] += rho_l*(1.0 + r_rho_l)
    Mat_l[:,:,1] += Lambda_l*(1.0 + r_lambda_l)
    Mat_l[:,:,2] += mu_l*(1.0 + r_mu_l)
    
    
    r_rho_r = rho_pert[nx-1:2*(nx-1)+1, :]
    r_mu_r =  mu_pert[nx-1:2*(nx-1)+1, :]
    r_lambda_r =  lambda_pert[nx-1:2*(nx-1)+1, :]
    
    
    Mat_r[:,:,0] += rho_r*(1.0 + r_rho_r)
    Mat_r[:,:,1] += Lambda_r*(1.0 + r_lambda_r)
    Mat_r[:,:,2] += mu_r*(1.0 + r_mu_r)
    

# Initialize pressure at different time steps and the second
# derivatives in each direction
F_l = np.zeros((nx, ny, nf))
Fnew_l = np.zeros((nx, ny, nf))
X_l = np.zeros((nx, ny))
Y_l = np.zeros((nx, ny))
p_l = np.zeros((nx, ny))
v = 2.0
F_r = np.zeros((nx, ny, nf))
Fnew_r = np.zeros((nx, ny, nf))
X_r = np.zeros((nx, ny))
Y_r = np.zeros((nx, ny))
p_r = np.zeros((nx, ny))


for i in range(0, nx):
    for j in range(0, ny):
        X_l[i,j] = -Lx + i*dx
        Y_l[i,j] = j*dy
        
        X_r[i,j] = i*dx
        Y_r[i,j] = j*dy

               
# Set up seismograms     
# Receiver locations left
rx_l = np.array([0, 0])
ry_l = np.array([0, Y0])

irx_l = np.array([1, 1])
iry_l = np.array([0, 0])

for i in range(len(rx_l)):
    irx_l[i] = (np.round(rx_l[i]/dx))+(nx-1)
    iry_l[i] = (np.round(ry_l[i]/dy))

seisvx_l = np.zeros((len(irx_l), nt))
seisvy_l = np.zeros((len(irx_l), nt))

# Receiver locations right

rx_r = np.array([0, 0])
ry_r = np.array([0, Y0])

irx_r = np.array([1, 1])
iry_r = np.array([0, 0])

for i in range(len(rx_r)):
    irx_r[i] = (np.round(rx_r[i]/dx))
    iry_r[i] = (np.round(ry_r[i]/dy))

seisvx_r = np.zeros((len(irx_r), nt))
seisvy_r = np.zeros((len(irx_r), nt))

# Boundary reflection coefficients: 0<= r[j] <= 1
r_l = np.array([0.,0.,1.,0.])
r_r = np.array([0.,0.,1.,0.])

# required for seismograms
ir_l = np.arange(len(irx_l))
ir_r = np.arange(len(irx_r))

print('The final model time will be '+str(nt*dt)+' seconds')

# + {"code_folding": [0]}
# Plotting cell

# create plot
image_r = plt.imshow(np.squeeze(np.append([p_l.transpose()],[p_r.transpose()],axis=2)), aspect='auto',extent=[-Lx,Lx,Ly,0]
          ,cmap = 'seismic', vmin=-v, vmax=+v, animated=True, 
          interpolation='none')

# Plot the receivers
for x, y in zip(rx_l, ry_l):
    plt.text(x, y, '+')
    
for x, y in zip(rx_r, ry_r):
    plt.text(x, y, '+')

plt.text(x0, y0, 'o')
plt.colorbar()
plt.xlabel('x')
plt.ylabel('y')
plt.show()


# Time-stepping 
for it in range(nt):
    
    t = it*dt
    
    #4th order Runge-Kutta 
    RK4_2D.elastic_RK4_2D(Fnew_l, F_l, Mat_l, X_l, Y_l, t, nf, nx, ny, dx, dy, dt, order, r_l, source_parameter, Fnew_r, F_r, Mat_r, X_r, Y_r, r_r, friction_parameters, slip,  psi, slip_new, psi_new, fric_law, FaultOutput0, Y0, mode)
    
    # update fields and extract parameters
    F_l = Fnew_l
    F_r = Fnew_r
    slip = slip_new 
    psi = psi_new
    FaultOutput[:, it, :] = FaultOutput0
    if mode == 'II': 
        p_l = F_l[:,:,1]
        p_r = F_r[:,:,1]
    if mode == 'III': 
        p_l = F_l[:,:,0]
        p_r = F_r[:,:,0] 
    
    t = it*dt
    
    # Plot every isnap-th iteration
    if it % isnap == 0:                    # you can change the speed of the plot by increasing the plotting interval
        p_b = np.squeeze(np.append([p_l.transpose()],[p_r.transpose()],axis=2)) # p for both, transposed to plot
        plt.title("time: %.2f" % t)
        plt.axvline(x=0.0)
        image_r.set_data(p_b)
        plt.gcf().canvas.draw()

    
    # Save seismograms
    if mode == 'II':
        seisvy_l[ir_l, it] = F_l[irx_l[ir_l], iry_l[ir_l], 1]
        seisvy_r[ir_r, it] = F_r[irx_r[ir_r], iry_r[ir_r], 1]
        
    if mode == 'III':
        seisvy_l[ir_l, it] = F_l[irx_l[ir_l], iry_l[ir_l], 0]
        seisvy_r[ir_r, it] = F_r[irx_r[ir_r], iry_r[ir_r], 0]    

# Plot seismogram    
plt.ioff()
plt.figure(figsize=(10, 10))

plt.subplot(2,1,1)
ymax = seisvx_l.ravel().max()
time = np.arange(nt) * dt
for ir_l in range(len(seisvx_l)):
    plt.plot(time, seisvy_l[ir_l, :] + ymax * ir_l)
    plt.title('Left block')
    plt.xlabel('Time (s)')
    plt.ylabel('vy (m/s)')

plt.subplot(2,1,2)
ymax = seisvy_l.ravel().max()
for ir_l in range(len(seisvy_r)):
    plt.plot(time, seisvy_r[ir_l, :] + ymax * ir_l)
    plt.title('Right block')
    plt.xlabel('Time (s)')
    plt.ylabel('vy (m/s)')
    

plt.show()

# + {"code_folding": [0]}
# View slip, slip rate and traction. 

fig1 = plt.figure(figsize=(10,10))
ax3 = fig1.add_subplot(3,1,1)
line3 = ax3.plot(0,0,'g')
plt.title('Slip')
plt.xlabel('Depth [km]')
plt.ylabel('Slip [m]')

ax4 = fig1.add_subplot(3,1,2)
line4 = ax4.plot(0,0,'g')
plt.title('Slip rate')
plt.xlabel('Depth [km]')
plt.ylabel('Slip rate[m/s]]')

ax5 = fig1.add_subplot(3,1,3)
line5 = ax5.plot(0,0,'g')
plt.title('Traction')
plt.xlabel('Depth [km]')
plt.ylabel('stress [MPa]')

plt.tight_layout()
plt.ion()
plt.show()

y_fault = Y_l[-1,:]

for it in range(nt):
    slip_ = FaultOutput[:, it, 4]
    sliprate_ = np.sqrt(FaultOutput[:, it, 0]**2 + FaultOutput[:, it, 1]**2)
    traction_ = FaultOutput[:, it, 3]
    if it % 1 == 0: 
        
        for l in line3:
            l.remove()
            del l               
        for l in line4:
            l.remove()
            del l
        for l in line5:
            l.remove()
            del l

        # Display lines
        line3 = ax3.plot(y_fault,slip_,'g')
        line4 = ax4.plot(y_fault,sliprate_,'g')
        line5 = ax5.plot(y_fault,traction_,'g')
        plt.gcf().canvas.draw()
                   
    
plt.ioff()
plt.show()
    

# + {"code_folding": [0]}
#Plot time history of the slip-rate on the fault 

VT = np.zeros((nt, ny))

for it in range(nt):
    for j in range(ny):
        
        VT[it, j] = np.sqrt(FaultOutput[j, it, 0]**2 + FaultOutput[j, it, 1]**2)
        
        

v = 2.5
image_f = plt.imshow(VT, aspect='auto',extent=[0,Ly, nt*dt, 0]
          ,cmap = 'viridis', vmin=0, vmax=+v, animated=True, interpolation='none')
        
plt.title('slip-rate')
plt.colorbar()
plt.xlabel('fault [km]')
plt.ylabel('t [s]')
plt.show()