The fundamental analytical solution to the three-dimensional Lamb’s problem, the problem of determining the elastic disturbance resulting from a point force in a homogeneous half space, is implemented in this Ipython notebook. This solution provides fundamental information used as benchmark for comparison with entirely numerical solutions. A setup of the fundamental problem is illustrated below. The figure on the right hand side is published in [1] (Figure 1. System of coordinates)
Simulations of 3D elastic wave propagation need to be validated by the use of analytical solutions. In order to evaluate how healthy a numerical solution is, one may recreate conditions for which analytical solutions exist with the aim of reproducing and compare the different results.
We which to find the displacement wavefield $\mathbf{u}(\mathbf{x},t)$ at some distance $\mathbf{x}$ from a seismic source with $ \mathbf{F} = f_1\mathbf{\hat{x}_1} + f_2\mathbf{\hat{x}_2} + f_3\mathbf{\hat{x}_3}$.
For a uniform elastic material and a Cartesian co-ordinate system the equation for the conservation of linear momentum can be written
\begin{align*} \rho(x) \frac{\partial^2}{\partial t^2} \mathbf{u(\mathbf{x},t)} = (\lambda + \mu)\nabla(\nabla\mathbf{u(\mathbf{x},t)}) + \mu\nabla^2 \mathbf{u(\mathbf{x},t)} + \mathbf{f(\mathbf{x},t)} \end{align*}We will consider the case where the source function is localized in both time and space
\begin{align*} \mathbf{f(\mathbf{x},t)} = (f_1\mathbf{\hat{x}_1} + f_2\mathbf{\hat{x}_2} + f_3\mathbf{\hat{x}_3})\delta(x_1 - x^{'}_{1})\delta(x_2 - x^{'}_{2})\delta(x_3 - x^{'}_{3})\delta(t - t^{'}) \end{align*}For such a source we will refer to the displacement solution as a Green’s function, and use the standard notation
\begin{align*} \mathbf{u(\mathbf{x},t)} = g_1(\mathbf{x},t;\mathbf{x^{'}},t^{'})\mathbf{\hat{x}_1} + g_2(\mathbf{x},t;\mathbf{x^{'}},t^{'})\mathbf{\hat{x}_2} + g_3(\mathbf{x},t;\mathbf{x^{'}},t^{'})\mathbf{\hat{x}_3} \end{align*}The complete solution is found after applying the Laplace transform to the elastic wave equation, implementing the stress-free boundary condition, defining some transformations, and performing some algebraic manoeuvres. Then, the Green's function at the free surface is given:
\begin{align*} \begin{split} \mathbf{G}(x_1,x_2,0,t;0,0,x^{'}_{3},0) & = \dfrac{1}{\pi^2\mu r} \dfrac{\partial}{\partial t}\int_{0}^{((t/r)^2 - \alpha^{-2})^{1/2}}\mathbf{H}(t-r/\alpha)\mathbb{R}[\eta_\alpha\sigma^{-1}((t/r)^2 - \alpha^{-2} - p^2)^{-1/2}\mathbf{M}(q,p,0,t,x^{'}_{3})\mathbf{F}] dp \\ & + \dfrac{1}{\pi^2\mu r} \dfrac{\partial}{\partial t}\int_{0}^{p_2}\mathbf{H}(t-t_2)\mathbb{R}[\eta_\beta\sigma^{-1}((t/r)^2 - \beta^{-2} - p^2)^{-1/2}\mathbf{N}(q,p,0,t,x^{'}_{3})\mathbf{F}] dp \end{split} \end{align*}Details on the involved terms are found in the original paper [2]. The Green's $\mathbf{G}$ function consist of three components of displacement evolving from the application of three components of force $\mathbf{F}$. If we assume that each component of $\mathbf{F}$ provokes three components of displacement, then $\mathbf{G}$ is composed by nine independent components that correspond one to one to the matrices $\mathbf{M}$ and $\mathbf{N}$. Without losing generality it is shown that among them four are equal zero, and we end up only with five possible components.
[1] Eduardo Kausel - Lamb's problem at its simplest, 2012
[2] Lane R. Johnson - Green’s Function for Lamb’s Problem, 1974
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# Import all necessary libraries, this is a configuration step for the exercise.
# Please run it before the simulation code!
import numpy as np
import matplotlib.pyplot as plt
import os
from ricker import ricker
# Show the plots in the Notebook.
plt.switch_backend("nbagg")
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# Compile the source code (needs gfortran!)
!rm -rf lamb.exe output.txt input.txt
!gfortran canhfs.for -o lamb.exe
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# Initialization of setup:
# Figure 4 in Lane R. Johnson - Green’s Function for Lamb’s Problem, 1974
# is reproduced when the following parameters are given
# -----------------------------------------------------------------------------
r = 10.0 # km
vp = 8.0 # P-wave velocity km/s
vs = 4.62 # s-wave velocity km/s
rho = 3.3 # Density kg/m^3
nt = 512 # Number of time steps
dt = 0.01 # Time step s
h = 0.2 # Source position km (0.01 to reproduce Fig 2.16 of the book)
ti = 0.0 # Initial time s
var = [vp, vs, rho, nt, dt, h, r, ti]
# -----------------------------------------------------------------------------
# Execute fortran code
# -----------------------------------------------------------------------------
with open('input.txt', 'w') as f:
for i in var:
print(i, file=f, end=' ') # Write input for fortran code
f.close()
os.system("./lamb.exe") # Code execution
# -----------------------------------------------------------------------------
# Load the solution
# -----------------------------------------------------------------------------
G = np.genfromtxt('output.txt')
u_rx = G[:,0] # Radial displacement owing to horizontal load
u_tx = G[:,1] # Tangential displacement due to horizontal load
u_zx = G[:,2] # Vertical displacement owing to horizontal load
u_rz = G[:,3] # Radial displacement owing to a vertical load
u_zz = G[:,4] # Vertical displacement owing to vertical load
t = np.linspace(dt, nt*dt, nt) # Time axis
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# Plotting
# -----------------------------------------------------------------------------
seis = [u_rx, u_tx, u_zx, u_rz, u_zz] # Collection of seismograms
labels = ['$u_{rx}(t) [cm]$','$u_{tx}(t)[cm]$','$u_{zx}(t)[cm]$','$u_{rz}(t)[cm]$','$u_{zz}(t)[cm]$']
cols = ['b','r','k','g','c']
# Initialize animated plot
fig = plt.figure(figsize=(12,8), dpi=80)
fig.suptitle("Green's Function for Lamb's problem", fontsize=16)
plt.ion() # set interective mode
plt.show()
for i in range(5):
st = seis[i]
ax = fig.add_subplot(2, 3, i+1)
ax.plot(t, st, lw = 1.5, color=cols[i])
ax.set_xlabel('Time(s)')
ax.text(0.8*nt*dt, 0.8*max(st), labels[i], fontsize=16)
plt.ticklabel_format(style='sci', axis='y', scilimits=(0,0))
ax.spines['left'].set_position('zero')
ax.spines['right'].set_color('none')
ax.spines['bottom'].set_position('zero')
ax.spines['top'].set_color('none')
ax.spines['left'].set_smart_bounds(True)
ax.spines['bottom'].set_smart_bounds(True)
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
plt.show()
Let $S(t)$ be a general source time function, then the displacent seismogram is given in terms of the Green's function $G$ via
\begin{equation} u(\mathbf{x},t) = G(\mathbf{x},t; \mathbf{x}',t') \ast S(t) \end{equation}Compute the convolution of the source time function 'ricker' with the Green's function of a Vertical displacement due to vertical loads. Plot the resulting displacement.
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# call the source time function
T = 1/5 # Period
src = ricker(dt,T)
# Normalize source time function
src = src/max(src)
# Initialize source time function
f = np.zeros(nt)
f[0:int(2 * T/dt)] = src
# Compute convolution
u = np.convolve(u_zz, f)
u = u[0:nt]
# ---------------------------------------------------------------
# Plot Seismogram
# ---------------------------------------------------------------
fig = plt.figure(figsize=(12,4), dpi=80)
plt.subplot(1,3,1)
plt.plot(t, u_zz, color='r', lw=2)
plt.title('Green\'s function')
plt.xlabel('time [s]', size=16)
plt.ylabel('Displacement [cm]', size=14)
plt.xlim([0,nt*dt])
plt.ticklabel_format(style='sci', axis='y', scilimits=(0,0))
plt.subplot(1,3,2)
plt.plot(t, f, color='k', lw=2)
plt.title('Source time function')
plt.xlabel('time [s]', size=16)
plt.xlim([0,nt*dt])
plt.ticklabel_format(style='sci', axis='y', scilimits=(0,0))
plt.subplot(1,3,3)
plt.plot(t, u, color='b', lw=2)
plt.title('Displacement')
plt.xlabel('time [s]', size=16)
plt.xlim([0,nt*dt])
plt.ticklabel_format(style='sci', axis='y', scilimits=(0,0))
plt.grid(True)
plt.show()
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