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revision 3231 by ahallam, Wed Sep 8 23:01:25 2010 UTC revision 3232 by ahallam, Fri Oct 1 02:08:38 2010 UTC
# Line 35  waves, $\mu$ is the propagation material Line 35  waves, $\mu$ is the propagation material
35  In a similar process to the previous chapter, we will use the acceleration  In a similar process to the previous chapter, we will use the acceleration
36  solution to solve this PDE. By substituting $a$ directly for  solution to solve this PDE. By substituting $a$ directly for
37  $\frac{\partial^{2}u\hackscore{i}}{\partial t^2}$ we can derive the  $\frac{\partial^{2}u\hackscore{i}}{\partial t^2}$ we can derive the
38  displacement solution. Using $a$ we can see that \refEq{eqn:wav} becomes  displacement solution. Using $a$ we can see that \autoref{eqn:wav} becomes
39  \begin{equation} \label{eqn:wava}  \begin{equation} \label{eqn:wava}
40  \rho a\hackscore{i} - \frac{\partial  \rho a\hackscore{i} - \frac{\partial
41  \sigma\hackscore{ij}}{\partial x\hackscore{j}} = 0  \sigma\hackscore{ij}}{\partial x\hackscore{j}} = 0
# Line 53  distinct terms: Line 53  distinct terms:
53  \mu (u\hackscore{i,j} + u\hackscore{j,i})  \mu (u\hackscore{i,j} + u\hackscore{j,i})
54  \end{equation}  \end{equation}
55  \end{subequations}  \end{subequations}
56  One simply recognizes in \ref{eqn:sigtrace} that $u\hackscore{k,k}$ is the  One simply recognizes in \autoref{eqn:sigtrace} that $u\hackscore{k,k}$ is the
57  trace of the displacement solution and that $\delta\hackscore{ij}$ is the  trace of the displacement solution and that $\delta\hackscore{ij}$ is the
58  kronecker delta function with dimensions equivalent to $u$. The second term  kronecker delta function with dimensions equivalent to $u$. The second term
59  \ref{eqn:sigtrans} is the sum of $u$ with its own transpose. Putting these facts  \autoref{eqn:sigtrans} is the sum of $u$ with its own transpose. Putting these
60  together we see that the spatial differential of the stress is given by the  facts together we see that the spatial differential of the stress is given by the
61  gradient of $u$ and the aforementioned opperations. This value is then submitted  gradient of $u$ and the aforementioned opperations. This value is then submitted
62  to the \esc PDE as $X$.  to the \esc PDE as $X$.
63  \begin{python}  \begin{python}
# Line 109  $escript -t 4 example08a.py Line 109  $escript -t 4 example08a.py
109  \end{verbatim}  \end{verbatim}
110  would call the script in this section and solve it using 4 threads.  would call the script in this section and solve it using 4 threads.
111    
112  The computation times on an increasing number of cores is outlines in Table  The computation times on an increasing number of cores is outlines in
113  \ref{tab:wpcores}  \autoref{tab:wpcores}.
114    
115  \begin{table}[ht]  \begin{table}[ht]
116  \begin{center}  \begin{center}
# Line 222  It is quite Line 222  It is quite
222  simple to implement a source which is smooth in time. In addition to the  simple to implement a source which is smooth in time. In addition to the
223  original source function the only extra requirement is a time function. For  original source function the only extra requirement is a time function. For
224  this example the time variant source will be the derivative of a gausian curve  this example the time variant source will be the derivative of a gausian curve
225  defined by the required dominant frequency (Figure \ref{fig:tvsource}).  defined by the required dominant frequency (\autoref{fig:tvsource}).
226  \begin{python}  \begin{python}
227  #Creating the time function of the source.  #Creating the time function of the source.
228  dfeq=50 #Dominant Frequency  dfeq=50 #Dominant Frequency
# Line 285  It is inpractical to calculate the solut Line 285  It is inpractical to calculate the solut
285  allow us the create an approximate solution with small to zero boundary effects  allow us the create an approximate solution with small to zero boundary effects
286  on a model with a solvable size.  on a model with a solvable size.
287    
288  To dampen the waves, the method of Cerjan(1985)  To dampen the waves, the method of \citet{Cerjan1985}
 \footnote{\textit{A nonreflecting boundary condition for discrete acoustic and  
 elastic wave equations}, 1985, Cerjan C, Geophysics 50, doi:10.1190/1.1441945}  
289  where the solution and the stress are multiplied by a damping function defined  where the solution and the stress are multiplied by a damping function defined
290  on $n$ nodes of the domain adjacent to the boundary, given by;  on $n$ nodes of the domain adjacent to the boundary, given by;
291  \begin{equation}  \begin{equation}
# Line 340  abc=abcleft*abcright*abcbottom Line 338  abc=abcleft*abcright*abcbottom
338  Note that the boundary conditions are not applied to the surface, as this is  Note that the boundary conditions are not applied to the surface, as this is
339  effectively a free surface where normal reflections would be experienced.  effectively a free surface where normal reflections would be experienced.
340  Special conditions can be introduced at this surface if they are known. The  Special conditions can be introduced at this surface if they are known. The
341  resulting boundary damping function can be viewed in Figure  resulting boundary damping function can be viewed in
342  \ref{fig:abconds}.  \autoref{fig:abconds}.
343    
344  \section{Second order Meshing}  \section{Second order Meshing}
345  For stiff problems like the wave equation it is often prudent to implement  For stiff problems like the wave equation it is often prudent to implement
# Line 392  the mesh by half. Line 390  the mesh by half.
390  \sslist{example08c.py}  \sslist{example08c.py}
391  To make the problem more interesting we will now introduce an interface to the  To make the problem more interesting we will now introduce an interface to the
392  middle of the domain. Infact we will use the same domain as we did for heat flux  middle of the domain. Infact we will use the same domain as we did for heat flux
393  in Chapter \ref{CHAP HEAT 2}. The domain contains a syncline with two set of  in \autoref{CHAP HEAT 2}. The domain contains a syncline with two set of
394  material properties on either side of the interface.  material properties on either side of the interface.
395    
396  \begin{figure}[ht]  \begin{figure}[ht]

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