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2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3 %
4 % Copyright (c) 2003-2010 by University of Queensland
5 % Earth Systems Science Computational Center (ESSCC)
6 % http://www.uq.edu.au/esscc
7 %
8 % Primary Business: Queensland, Australia
9 % Licensed under the Open Software License version 3.0
10 % http://www.opensource.org/licenses/osl-3.0.php
11 %
12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
13
14 The acoustic wave equation governs the propagation of pressure waves. Wave
15 types that obey this law tend to travel in liquids or gases where shear waves
16 or longitudinal style wave motion is not possible. An obvious example is sound
17 waves.
18
19 The acoustic wave equation is defined as;
20 \begin{equation}
21 \nabla ^2 p - \frac{1}{c^2} \frac{\partial ^2 p}{\partial t^2} = 0
22 \label{eqn:acswave}
23 \end{equation}
24 where $p$ is the pressure, $t$ is the time and $c$ is the wave velocity. In this
25 chapter the acoustic wave equation is demonstrated. Important steps include the
26 translation of the Laplacian $\nabla^2$ to the \esc general form, the stiff
27 equation stability criterion and solving for the displacement or acceleration solution.
28
29 \section{The Laplacian in \esc}
30 The Laplacian operator which can be written as $\Delta$ or $\nabla^2$, is
31 calculated via the divergence of the gradient of the object, which in this
32 example is the scalar $p$. Thus we can write;
33 \begin{equation}
34 \nabla^2 p = \nabla \cdot \nabla p =
35 \sum_{i}^n
36 \frac{\partial^2 p}{\partial x^2_{i}}
37 \label{eqn:laplacian}
38 \end{equation}
39 For the two dimensional case in Cartesian coordinates \autoref{eqn:laplacian}
40 becomes;
41 \begin{equation}
42 \nabla^2 p = \frac{\partial^2 p}{\partial x^2}
43 + \frac{\partial^2 p}{\partial y^2}
44 \end{equation}
45
46 In \esc the Laplacian is calculated using the divergence representation and the
47 intrinsic functions \textit{grad()} and \textit{trace()}. The function
48 \textit{grad{}} will return the spatial gradients of an object.
49 For a rank 0 solution, this is of the form;
50 \begin{equation}
51 \nabla p = \left[
52 \frac{\partial p}{\partial x _{0}},
53 \frac{\partial p}{\partial x _{1}}
54 \right]
55 \label{eqn:grad}
56 \end{equation}
57 Larger ranked solution objects will return gradient tensors. For example, a
58 pressure field which acts in the directions $p _{0}$ and $p
59 _{1}$ would return;
60 \begin{equation}
61 \nabla p = \begin{bmatrix}
62 \frac{\partial p _{0}}{\partial x _{0}} &
63 \frac{\partial p _{1}}{\partial x _{0}} \\
64 \frac{\partial p _{0}}{\partial x _{1}} &
65 \frac{\partial p _{1}}{\partial x _{1}}
66 \end{bmatrix}
67 \label{eqn:gradrank1}
68 \end{equation}
69
70 \autoref{eqn:grad} corresponds to the Linear PDE general form value
71 $X$. Notice however, that the general form contains the term $X
72 _{i,j}$\footnote{This is the first derivative in the $j^{th}$
73 direction for the $i^{th}$ component of the solution.},
74 hence for a rank 0 object there is no need to do more then calculate the
75 gradient and submit it to the solver. In the case of the rank 1 or greater
76 object, it is also necessary to calculate the trace. This is the sum of the
77 diagonal in \autoref{eqn:gradrank1}.
78
79 Thus when solving for equations containing the Laplacian one of two things must
80 be completed. If the object \verb!p! is less then rank 1 the gradient is
81 calculated via;
82 \begin{python}
83 gradient=grad(p)
84 \end{python}
85 and if the object is greater then or equal to a rank 1 tensor, the trace of
86 the gradient is calculated.
87 \begin{python}
88 gradient=trace(grad(p))
89 \end{python}
90 These values can then be submitted to the PDE solver via the general form term
91 $X$. The Laplacian is then computed in the solution process by taking the
92 divergence of $X$.
93
94 Note, if you are unsure about the rank of your tensor, the \textit{getRank}
95 command will return the rank of the PDE object.
96 \begin{python}
97 rank = p.getRank()
98 \end{python}
99
100
101 \section{Numerical Solution Stability} \label{sec:nsstab}
102 Unfortunately, the wave equation belongs to a class of equations called
103 \textbf{stiff} PDEs. These types of equations can be difficult to solve
104 numerically as they tend to oscillate about the exact solution, which can
105 eventually lead to a catastrophic failure. To counter this problem, explicitly
106 stable schemes like the backwards Euler method, and correct parameterisation of
107 the problem are required.
108
109 There are two variables which must be considered for
110 stability when numerically trying to solve the wave equation. For linear media,
111 the two variables are related via;
112 \begin{equation} \label{eqn:freqvel}
113 f=\frac{v}{\lambda}
114 \end{equation}
115 The velocity $v$ that a wave travels in a medium is an important variable. For
116 stability the analytical wave must not propagate faster then the numerical wave
117 is able to, and in general, needs to be much slower then the numerical wave.
118 For example, a line 100m long is discretised into 1m intervals or 101 nodes. If
119 a wave enters with a propagation velocity of 100m/s then the travel time for
120 the wave between each node will be 0.01 seconds. The time step, must therefore
121 be significantly less then this. Of the order $10E-4$ would be appropriate.
122 This stability criterion is known as the Courant–Friedrichs–Lewy
123 condition given by
124 \begin{equation}
125 dt=f\cdot \frac{dx}{v}
126 \end{equation}
127 where $dx$ is the mesh size and $f$ is a safety factor. To obtain a time step of
128 $10E-4$, a safety factor of $f=0.1$ was used.
129
130 The wave frequency content also plays a part in numerical stability. The
131 nyquist-sampling theorem states that a signals bandwidth content will be
132 accurately represented when an equispaced sampling rate $f _{n}$ is
133 equal to or greater then twice the maximum frequency of the signal
134 $f_{s}$, or;
135 \begin{equation} \label{eqn:samptheorem}
136 f_{n} \geqslant f_{s}
137 \end{equation}
138 For example, a 50Hz signal will require a sampling rate greater then 100Hz or
139 one sample every 0.01 seconds. The wave equation relies on a spatial frequency,
140 thus the sampling theorem in this case applies to the solution mesh spacing.
141 This relationship confirms that the frequency content of the input signal
142 directly affects the time discretisation of the problem.
143
144 To accurately model the wave equation with high resolutions and velocities
145 means that very fine spatial and time discretisation is necessary for most
146 problems. This requirement makes the wave equation arduous to
147 solve numerically due to the large number of time iterations required in each
148 solution. Models with very high velocities and frequencies will be the worst
149 affected by this problem.
150
151 \section{Displacement Solution}
152 \sslist{example07a.py}
153
154 We begin the solution to this PDE with the centred difference formula for the
155 second derivative;
156 \begin{equation}
157 f''(x) \approx \frac{f(x+h - 2f(x) + f(x-h)}{h^2}
158 \label{eqn:centdiff}
159 \end{equation}
160 substituting \autoref{eqn:centdiff} for $\frac{\partial ^2 p }{\partial t ^2}$
161 in \autoref{eqn:acswave};
162 \begin{equation}
163 \nabla ^2 p - \frac{1}{c^2h^2} \left[p_{(t+1)} - 2p_{(t)} +
164 p_{(t-1)} \right]
165 = 0
166 \label{eqn:waveu}
167 \end{equation}
168 Rearranging for $p_{(t+1)}$;
169 \begin{equation}
170 p_{(t+1)} = c^2 h^2 \nabla ^2 p_{(t)} +2p_{(t)} -
171 p_{(t-1)}
172 \end{equation}
173 this can be compared with the general form of the \modLPDE module and it
174 becomes clear that $D=1$, $X_{i,j}=-c^2 h^2 \nabla ^2 p_{(t)}$ and
175 $Y=2p_{(t)} - p_{(t-1)}$.
176
177 The solution script is similar to others that we have created in previous
178 chapters. The general steps are;
179 \begin{enumerate}
180 \item The necessary libraries must be imported.
181 \item The domain needs to be defined.
182 \item The time iteration and control parameters need to be defined.
183 \item The PDE is initialised with source and boundary conditions.
184 \item The time loop is started and the PDE is solved at consecutive time steps.
185 \item All or select solutions are saved to file for visualisation later on.
186 \end{enumerate}
187
188 Parts of the script which warrant more attention are the definition of the
189 source, visualising the source, the solution time loop and the VTK data export.
190
191 \subsection{Pressure Sources}
192 As the pressure is a scalar, one need only define the pressure for two
193 time steps prior to the start of the solution loop. Two known solutions are
194 required because the wave equation contains a double partial derivative with
195 respect to time. This is often a good opportunity to introduce a source to the
196 solution. This model has the source located at it's centre. The source should
197 be smooth and cover a number of samples to satisfy the frequency stability
198 criterion. Small sources will generate high frequency signals. Here, when using
199 a rectangular domain, the source is defined by a cosine function.
200 \begin{python}
201 U0=0.01 # amplitude of point source
202 xc=[500,500] #location of point source
203 # define small radius around point xc
204 src_radius = 30
205 # for first two time steps
206 u=U0*(cos(length(x-xc)*3.1415/src_radius)+1)*\
207 whereNegative(length(x-xc)-src_radius)
208 u_m1=u
209 \end{python}
210
211 \subsection{Visualising the Source}
212 There are two options for visualising the source. The first is to export the
213 initial conditions of the model to VTK, which can be interpreted as a scalar
214 surface in Mayavi2. The second is to take a cross section of the model which
215 will require the \textit{Locator} function.
216 First \verb!Locator! must be imported;
217 \begin{python}
218 from esys.escript.pdetools import Locator
219 \end{python}
220 The function can then be used on the domain to locate the nearest domain node
221 to the point or points of interest.
222
223 It is now necessary to build a list of $(x,y)$ locations that specify where are
224 model slice will go. This is easily implemented with a loop;
225 \begin{python}
226 cut_loc=[]
227 src_cut=[]
228 for i in range(ndx/2-ndx/10,ndx/2+ndx/10):
229 cut_loc.append(xstep*i)
230 src_cut.append([xstep*i,xc[1]])
231 \end{python}
232 We then submit the output to \verb!Locator! and finally return the appropriate
233 values using the \verb!getValue! function.
234 \begin{python}
235 src=Locator(mydomain,src_cut)
236 src_cut=src.getValue(u)
237 \end{python}
238 It is then a trivial task to plot and save the output using \mpl
239 (\autoref{fig:cxsource}).
240 \begin{python}
241 pl.plot(cut_loc,src_cut)
242 pl.axis([xc[0]-src_radius*3,xc[0]+src_radius*3,0.,2*U0])
243 pl.savefig(os.path.join(savepath,"source_line.png"))
244 \end{python}
245 \begin{figure}[h]
246 \centering
247 \includegraphics[width=6in]{figures/sourceline.png}
248 \caption{Cross section of the source function.}
249 \label{fig:cxsource}
250 \end{figure}
251
252
253 \subsection{Point Monitoring}
254 In the more general case where the solution mesh is irregular or specific
255 locations need to be monitored, it is simple enough to use the \textit{Locator}
256 function.
257 \begin{python}
258 rec=Locator(mydomain,[250.,250.])
259 \end{python}
260 When the solution \verb u is updated we can extract the value at that point
261 via;
262 \begin{python}
263 u_rec=rec.getValue(u)
264 \end{python}
265 For consecutive time steps one can record the values from \verb!u_rec! in an
266 array initialised as \verb!u_rec0=[]! with;
267 \begin{python}
268 u_rec0.append(rec.getValue(u))
269 \end{python}
270
271 It can be useful to monitor the value at a single or multiple individual points
272 in the model during the modelling process. This is done using
273 the \verb!Locator! function.
274
275
276 \section{Acceleration Solution}
277 \sslist{example07b.py}
278
279 An alternative method to the displacement solution, is to solve for the
280 acceleration $\frac{\partial ^2 p}{\partial t^2}$ directly. The displacement can
281 then be derived from the acceleration after a solution has been calculated
282 The acceleration is given by a modified form of \autoref{eqn:waveu};
283 \begin{equation}
284 \nabla ^2 p - \frac{1}{c^2} a = 0
285 \label{eqn:wavea}
286 \end{equation}
287 and can be solved directly with $Y=0$ and $X=-c^2 \nabla ^2 p_{(t)}$.
288 After each iteration the displacement is re-evaluated via;
289 \begin{equation}
290 p_{(t+1)}=2p_{(t)} - p_{(t-1)} + h^2a
291 \end{equation}
292
293 \subsection{Lumping}
294 For \esc, the acceleration solution is prefered as it allows the use of matrix
295 lumping. Lumping or mass lumping as it is sometimes known, is the process of
296 aggressively approximating the density elements of a mass matrix into the main
297 diagonal. The use of Lumping is motivaed by the simplicity of diagonal matrix
298 inversion. As a result, Lumping can significantly reduce the computational
299 requirements of a problem. Care should be taken however, as this
300 function can only be used when the $A$, $B$ and $C$ coefficients of the
301 general form are zero.
302
303 More information about the lumping implementation used in \esc and its accuracy
304 can be found in the user guide.
305
306 To turn lumping on in \esc one can use the command;
307 \begin{python}
308 mypde.getSolverOptions().setSolverMethod(mypde.getSolverOptions().HRZ_LUMPING)
309 \end{python}
310 It is also possible to check if lumping is set using;
311 \begin{python}
312 print mypde.isUsingLumping()
313 \end{python}
314
315 \section{Stability Investigation}
316 It is now prudent to investigate the stability limitations of this problem.
317 First, we let the frequency content of the source be very small. If we define
318 the source as a cosine input, then the wavlength of the input is equal to the
319 radius of the source. Let this value be 5 meters. Now, if the maximum velocity
320 of the model is $c=380.0ms^{-1}$, then the source
321 frequency is $f_{r} = \frac{380.0}{5} = 76.0 Hz$. This is a worst case
322 scenario with a small source and the models maximum velocity.
323
324 Furthermore, we know from \autoref{sec:nsstab}, that the spatial sampling
325 frequency must be at least twice this value to ensure stability. If we assume
326 the model mesh is a square equispaced grid,
327 then the sampling interval is the side length divided by the number of samples,
328 given by $\Delta x = \frac{1000.0m}{400} = 2.5m$ and the maximum sampling
329 frequency capable at this interval is
330 $f_{s}=\frac{380.0ms^{-1}}{2.5m}=152Hz$ this is just equal to the
331 required rate satisfying \autoref{eqn:samptheorem}.
332
333 \autoref{fig:ex07sampth} depicts three examples where the grid has been
334 undersampled, sampled correctly, and over sampled. The grids used had
335 200, 400 and 800 nodes per side respectively. Obviously, the oversampled grid
336 retains the best resolution of the modelled wave.
337
338 The time step required for each of these examples is simply calculated from
339 the propagation requirement. For a maximum velocity of $380.0ms^{-1}$,
340 \begin{subequations}
341 \begin{equation}
342 \Delta t \leq \frac{1000.0m}{200} \frac{1}{380.0} = 0.013s
343 \end{equation}
344 \begin{equation}
345 \Delta t \leq \frac{1000.0m}{400} \frac{1}{380.0} = 0.0065s
346 \end{equation}
347 \begin{equation}
348 \Delta t \leq \frac{1000.0m}{800} \frac{1}{380.0} = 0.0032s
349 \end{equation}
350 \end{subequations}
351 Observe that for each doubling of the number of nodes in the mesh, we halve
352 the time step. To illustrate the impact this has, consider our model. If the
353 source is placed at the center, it is $500m$ from the nearest boundary. With a
354 velocity of $380.0ms^{-1}$ it will take $\approx1.3s$ for the wavefront to
355 reach that boundary. In each case, this equates to $100$, $200$ and $400$ time
356 steps. This is again, only a best case scenario, for true stability these time
357 values may need to be halved and possibly halved again.
358
359 \begin{figure}[ht]
360 \centering
361 \subfigure[Undersampled Example]{
362 \includegraphics[width=0.45\textwidth,trim=0cm 6cm 5cm 6cm,clip]{figures/ex07usamp.png}
363 \label{fig:ex07usamp}
364 }
365 \subfigure[Just sampled Example]{
366 \includegraphics[width=0.45\textwidth,trim=0cm 6cm 5cm 6cm,clip]{figures/ex07jsamp.png}
367 \label{fig:ex07jsamp}
368 }
369 \subfigure[Over sampled Example]{
370 \includegraphics[width=0.45\textwidth,trim=0cm 6cm 5cm 6cm,clip]{figures/ex07nsamp.png}
371 \label{fig:ex07nsamp}
372 }
373 \caption{Sampling Theorem example for stability investigation}
374 \label{fig:ex07sampth}
375 \end{figure}
376

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