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

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