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 1 ahallam 3003 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 % 4 % Copyright (c) 2003-2010 by University of Queensland 5 % Earth Systems Science Computational Center (ESSCC) 6 7 % 8 % Primary Business: Queensland, Australia 9 % Licensed under the Open Software License version 3.0 10 11 % 12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 13 14 15 16 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 18 ahallam 3004 or longitudinal style wave motion is not possible. An obvious example is sound 19 ahallam 3003 waves. 20 21 ahallam 3004 The acoustic wave equation is defined as; 22 ahallam 3003 \begin{equation} 23 \nabla ^2 p - \frac{1}{c^2} \frac{\partial ^2 p}{\partial t^2} = 0 24 \label{eqn:acswave} 25 \end{equation} 26 ahallam 3004 where $p$ is the pressure, $t$ is the time and $c$ is the wave velocity. 27 ahallam 3003 28 ahallam 3004 \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 ahallam 3029 \nabla^2 p = \nabla \cdot \nabla p = 34 \sum\hackscore{i}^n 35 \frac{\partial^2 p}{\partial x^2\hackscore{i}} 36 ahallam 3004 \label{eqn:laplacian} 37 \end{equation} 38 ahallam 3232 For the two dimensional case in Cartesian coordinates \autoref{eqn:laplacian} 39 ahallam 3004 becomes; 40 \begin{equation} 41 \nabla^2 p = \frac{\partial^2 p}{\partial x^2} 42 + \frac{\partial^2 p}{\partial y^2} 43 \end{equation} 44 ahallam 3003 45 ahallam 3004 In \esc the Laplacian is calculated using the divergence representation and the 46 intrinsic functions \textit{grad()} and \textit{trace()}. The fucntion 47 \textit{grad{}} will return the spatial gradients of an object. 48 For a rank 0 solution, this is of the form; 49 \begin{equation} 50 \nabla p = \left[ 51 \frac{\partial p}{\partial x \hackscore{0}}, 52 \frac{\partial p}{\partial x \hackscore{1}} 53 \right] 54 \label{eqn:grad} 55 \end{equation} 56 Larger ranked solution objects will return gradient tensors. For example, a 57 pressure field which acts in the directions $p \hackscore{0}$ and $p 58 \hackscore{1}$ would return; 59 \begin{equation} 60 \nabla p = \begin{bmatrix} 61 \frac{\partial p \hackscore{0}}{\partial x \hackscore{0}} & 62 \frac{\partial p \hackscore{1}}{\partial x \hackscore{0}} \\ 63 \frac{\partial p \hackscore{0}}{\partial x \hackscore{1}} & 64 \frac{\partial p \hackscore{1}}{\partial x \hackscore{1}} 65 \end{bmatrix} 66 \label{eqn:gradrank1} 67 \end{equation} 68 ahallam 3003 69 ahallam 3232 \autoref{eqn:grad} corresponds to the Linear PDE general form value 70 ahallam 3029 $X$. Notice however that the general form contains the term $X 71 \hackscore{i,j}$\footnote{This is the first derivative in the $j^{th}$ 72 direction for the $i^{th}$ component of the solution.}, 73 ahallam 3004 hence for a rank 0 object there is no need to do more than calculate the 74 gradient and submit it to the solver. In the case of the rank 1 or greater 75 object, it is nesscary to calculate the trace also. This is the sum of the 76 ahallam 3232 diagonal in \autoref{eqn:gradrank1}. 77 ahallam 3004 78 Thus when solving for equations containing the Laplacian one of two things must 79 ahallam 3063 be completed. If the object \verb!p! is less than rank 1 the gradient is 80 ahallam 3004 calculated via; 81 ahallam 3025 \begin{python} 82 ahallam 3063 gradient=grad(p) 83 ahallam 3025 \end{python} 84 ahallam 3004 and if the object is greater thank or equal to a rank 1 tensor, the trace of 85 the gradient is calculated. 86 ahallam 3025 \begin{python} 87 ahallam 3004 gradient=trace(grad(p)) 88 ahallam 3025 \end{python} 89 ahallam 3004 These valuse can then be submitted to the PDE solver via the general form term 90 $X$. The Laplacian is then computed in the solution process by taking the 91 divergence of $X$. 92 93 ahallam 3025 Note, if you are unsure about the rank of your tensor, the \textit{getRank} 94 command will return the rank of the PDE object. 95 \begin{python} 96 rank = p.getRank() 97 \end{python} 98 99 100 \section{Numerical Solution Stability} \label{sec:nsstab} 101 ahallam 3004 Unfortunately, the wave equation belongs to a class of equations called 102 \textbf{stiff} PDEs. These types of equations can be difficult to solve 103 ahallam 3025 numerically as they tend to oscilate about the exact solution which can 104 eventually lead to a catastrophic failure in the solution. To counter this 105 problem, explicitly stable schemes like 106 ahallam 3004 the backwards Euler method are required. There are two variables which must be 107 considered for stability when numerically trying to solve the wave equation. 108 ahallam 3025 For linear media, the two variables are related via; 109 ahallam 3004 \begin{equation} \label{eqn:freqvel} 110 f=\frac{v}{\lambda} 111 \end{equation} 112 ahallam 3025 The velocity $v$ that a wave travels in a medium is an important variable. For 113 stability the analytical wave must not propagate faster than the numerical wave 114 is able to, and in general, needs to be much slower than the numerical wave. 115 ahallam 3003 For example, a line 100m long is discretised into 1m intervals or 101 nodes. If 116 a wave enters with a propagation velocity of 100m/s then the travel time for 117 the wave between each node will be 0.01 seconds. The time step, must therefore 118 be significantly less than this. Of the order $10E-4$ would be appropriate. 119 120 ahallam 3004 The wave frequency content also plays a part in numerical stability. The 121 nyquist-sampling theorem states that a signals bandwidth content will be 122 accurately represented when an equispaced sampling rate $f \hackscore{n}$ is 123 equal to or greater than twice the maximum frequency of the signal 124 $f\hackscore{s}$, or; 125 \begin{equation} \label{eqn:samptheorem} 126 f\hackscore{n} \geqslant f\hackscore{s} 127 \end{equation} 128 For example a 50Hz signal will require a sampling rate greater than 100Hz or 129 one sample every 0.01 seconds. The wave equation relies on a spatial frequency, 130 thus the sampling theorem in this case applies to the solution mesh spacing. In 131 this way, the frequency content of the input signal directly affects the time 132 discretisation of the problem. 133 134 To accurately model the wave equation with high resolutions and velocities 135 means that very fine spatial and time discretisation is necessary for most 136 problems. 137 This requirement makes the wave equation arduous to 138 ahallam 3003 solve numerically due to the large number of time iterations required in each 139 ahallam 3004 solution. Models with very high velocities and frequencies will be the worst 140 ahallam 3029 affected by this problem. 141 ahallam 3003 142 \section{Displacement Solution} 143 \sslist{example07a.py} 144 145 We begin the solution to this PDE with the centred difference formula for the 146 second derivative; 147 \begin{equation} 148 f''(x) \approx \frac{f(x+h - 2f(x) + f(x-h)}{h^2} 149 \label{eqn:centdiff} 150 \end{equation} 151 ahallam 3232 substituting \autoref{eqn:centdiff} for $\frac{\partial ^2 p }{\partial t ^2}$ 152 in \autoref{eqn:acswave}; 153 ahallam 3003 \begin{equation} 154 ahallam 3004 \nabla ^2 p - \frac{1}{c^2h^2} \left[p\hackscore{(t+1)} - 2p\hackscore{(t)} + 155 p\hackscore{(t-1)} \right] 156 ahallam 3003 = 0 157 \label{eqn:waveu} 158 \end{equation} 159 Rearranging for $p_{(t+1)}$; 160 \begin{equation} 161 ahallam 3004 p\hackscore{(t+1)} = c^2 h^2 \nabla ^2 p\hackscore{(t)} +2p\hackscore{(t)} - 162 p\hackscore{(t-1)} 163 ahallam 3003 \end{equation} 164 this can be compared with the general form of the \modLPDE module and it 165 ahallam 3004 becomes clear that $D=1$, $X\hackscore{i,j}=-c^2 h^2 \nabla ^2 p_{(t)}$ and 166 $Y=2p_{(t)} - p_{(t-1)}$. 167 ahallam 3003 168 ahallam 3025 The solution script is similar to others that we have created in previous 169 ahallam 3004 chapters. The general steps are; 170 \begin{enumerate} 171 \item The necessary libraries must be imported. 172 \item The domain needs to be defined. 173 \item The time iteration and control parameters need to be defined. 174 \item The PDE is initialised with source and boundary conditions. 175 \item The time loop is started and the PDE is solved at consecutive time steps. 176 \item All or select solutions are saved to file for visualisation lated on. 177 \end{enumerate} 178 179 Parts of the script which warrant more attention are the definition of the 180 source, visualising the source, the solution time loop and the VTK data export. 181 182 \subsection{Pressure Sources} 183 As the pressure is a scalar, one need only define the pressure for two 184 time steps prior to the start of the solution loop. Two known solutions are 185 required because the wave equation contains a double partial derivative with 186 respect to time. This is often a good opportunity to introduce a source to the 187 solution. This model has the source located at it's centre. The source should 188 be smooth and cover a number of samples to satisfy the frequency stability 189 criterion. Small sources will generate high frequency signals. Here, the source 190 is defined by a cosine function. 191 ahallam 3025 \begin{python} 192 ahallam 3004 U0=0.01 # amplitude of point source 193 xc=[500,500] #location of point source 194 # define small radius around point xc 195 src_radius = 30 196 # for first two time steps 197 ahallam 3025 u=U0*(cos(length(x-xc)*3.1415/src_radius)+1)*\ 198 whereNegative(length(x-xc)-src_radius) 199 ahallam 3004 u_m1=u 200 ahallam 3025 \end{python} 201 ahallam 3004 When using a rectangular domain 202 203 ahallam 3025 \subsection{Visualising the Source} 204 There are two options for visualising the source. The first is to export the 205 initial conditions of the model to VTK, which can be interpreted as a scalar 206 suface in mayavi. The second is to take a cross section of the model. 207 208 For the later, we will require the \textit{Locator} function. 209 ahallam 3063 First \verb!Locator! must be imported; 210 ahallam 3025 \begin{python} 211 from esys.escript.pdetools import Locator 212 \end{python} 213 The function can then be used on the domain to locate the nearest domain node 214 to the point or points of interest. 215 216 It is now necessary to build a list of $(x,y)$ locations that specify where are 217 model slice will go. This is easily implemeted with a loop; 218 \begin{python} 219 cut_loc=[] 220 src_cut=[] 221 for i in range(ndx/2-ndx/10,ndx/2+ndx/10): 222 cut_loc.append(xstep*i) 223 src_cut.append([xstep*i,xc]) 224 \end{python} 225 ahallam 3063 We then submit the output to \verb!Locator! and finally return the appropriate 226 values using the \verb!getValue! function. 227 ahallam 3025 \begin{python} 228 src=Locator(mydomain,src_cut) 229 src_cut=src.getValue(u) 230 \end{python} 231 It is then a trivial task to plot and save the output using \mpl. 232 \begin{python} 233 pl.plot(cut_loc,src_cut) 234 pl.axis([xc-src_radius*3,xc+src_radius*3,0.,2*U0]) 235 pl.savefig(os.path.join(savepath,"source_line.png")) 236 \end{python} 237 \begin{figure}[h] 238 ahallam 3029 \centering 239 ahallam 3063 \includegraphics[width=6in]{figures/sourceline.png} 240 ahallam 3025 \caption{Cross section of the source function.} 241 \label{fig:cxsource} 242 \end{figure} 243 244 245 \subsection{Point Monitoring} 246 In the more general case where the solution mesh is irregular or specific 247 locations need to be monitored, it is simple enough to use the \textit{Locator} 248 function. 249 \begin{python} 250 rec=Locator(mydomain,[250.,250.]) 251 \end{python} 252 ahallam 3029 When the solution \verb u is updated we can extract the value at that point 253 ahallam 3025 via; 254 \begin{python} 255 u_rec=rec.getValue(u) 256 \end{python} 257 ahallam 3063 For consecutive time steps one can record the values from \verb!u_rec! in an 258 array initialised as \verb!u_rec0=[]! with; 259 ahallam 3025 \begin{python} 260 u_rec0.append(rec.getValue(u)) 261 \end{python} 262 263 It can be useful to monitor the value at a single or multiple individual points 264 in the model during the modelling process. This is done using 265 ahallam 3063 the \verb!Locator! function. 266 ahallam 3025 267 268 ahallam 3003 \section{Acceleration Solution} 269 \sslist{example07b.py} 270 271 An alternative method is to solve for the acceleration $\frac{\partial ^2 272 ahallam 3029 p}{\partial t^2}$ directly, and derive the displacement solution from the 273 ahallam 3232 PDE solution. \autoref{eqn:waveu} is thus modified; 274 ahallam 3003 \begin{equation} 275 \nabla ^2 p - \frac{1}{c^2} a = 0 276 \label{eqn:wavea} 277 \end{equation} 278 ahallam 3004 and can be solved directly with $Y=0$ and $X=-c^2 \nabla ^2 p\hackscore{(t)}$. 279 ahallam 3003 After each iteration the displacement is re-evaluated via; 280 \begin{equation} 281 ahallam 3004 p\hackscore{(t+1)}=2p\hackscore{(t)} - p\hackscore{(t-1)} + h^2a 282 ahallam 3003 \end{equation} 283 284 ahallam 3029 \subsection{Lumping} 285 ahallam 3004 For \esc, the acceleration solution is prefered as it allows the use of matrix 286 lumping. Lumping or mass lumping as it is sometimes known, is the process of 287 aggressively approximating the density elements of a mass matrix into the main 288 diagonal. The use of Lumping is motivaed by the simplicity of diagonal matrix 289 inversion. As a result, Lumping can significantly reduce the computational 290 ahallam 3029 requirements of a problem. Care should be taken however, as this 291 function can only be used when the $A$, $B$ and $C$ coefficients of the 292 general form are zero. 293 ahallam 3003 294 ahallam 3004 To turn lumping on in \esc one can use the command; 295 ahallam 3025 \begin{python} 296 ahallam 3004 mypde.getSolverOptions().setSolverMethod(mypde.getSolverOptions().LUMPING) 297 ahallam 3025 \end{python} 298 ahallam 3004 It is also possible to check if lumping is set using; 299 ahallam 3025 \begin{python} 300 ahallam 3004 print mypde.isUsingLumping() 301 ahallam 3025 \end{python} 302 ahallam 3004 303 \section{Stability Investigation} 304 It is now prudent to investigate the stability limitations of this problem. 305 ahallam 3025 First, we let the frequency content of the source be very small. If we define 306 the source as a cosine input, than the wavlength of the input is equal to the 307 radius of the source. Let this value be 5 meters. Now, if the maximum velocity 308 of the model is $c=380.0ms^{-1}$ then the source 309 frequency is $f\hackscore{r} = \frac{380.0}{5} = 76.0 Hz$. This is a worst case 310 scenario with a small source and the models maximum velocity. 311 312 ahallam 3232 Furthermore, we know from \autoref{sec:nsstab}, that the spatial sampling 313 ahallam 3025 frequency must be at least twice this value to ensure stability. If we assume 314 the model mesh is a square equispaced grid, 315 then the sampling interval is the side length divided by the number of samples, 316 given by $\Delta x = \frac{1000.0m}{400} = 2.5m$ and the maximum sampling 317 frequency capable at this interval is 318 $f\hackscore{s}=\frac{380.0ms^{-1}}{2.5m}=152Hz$ this is just equal to the 319 ahallam 3232 required rate satisfying \autoref{eqn:samptheorem}. 320 ahallam 3004 321 ahallam 3232 \autoref{fig:ex07sampth} depicts three examples where the grid has been 322 ahallam 3025 undersampled, sampled correctly, and over sampled. The grids used had 323 200, 400 and 800 nodes per side respectively. Obviously, the oversampled grid 324 retains the best resolution of the modelled wave. 325 ahallam 3004 326 ahallam 3025 The time step required for each of these examples is simply calculated from 327 the propagation requirement. For a maximum velocity of $380.0ms^{-1}$, 328 \begin{subequations} 329 \begin{equation} 330 \Delta t \leq \frac{1000.0m}{200} \frac{1}{380.0} = 0.013s 331 \end{equation} 332 \begin{equation} 333 \Delta t \leq \frac{1000.0m}{400} \frac{1}{380.0} = 0.0065s 334 \end{equation} 335 \begin{equation} 336 \Delta t \leq \frac{1000.0m}{800} \frac{1}{380.0} = 0.0032s 337 \end{equation} 338 \end{subequations} 339 We can see, that for each doubling of the number of nodes in the mesh, we halve 340 the timestep. To illustrate the impact this has, consider our model. If the 341 source is placed at the center, it is $500m$ from the nearest boundary. With a 342 velocity of $380.0ms^{-1}$ it will take $\approx1.3s$ for the wavefront to 343 reach that boundary. In each case, this equates to $100$, $200$ and $400$ time 344 steps. This is again, only a best case scenario, for true stability these time 345 values may need to be halved and possibly havled again. 346 ahallam 3004 347 ahallam 3025 \begin{figure}[ht] 348 \centering 349 \subfigure[Undersampled Example]{ 350 ahallam 3054 \includegraphics[width=3in]{figures/ex07usamp.png} 351 ahallam 3025 \label{fig:ex07usamp} 352 } 353 \subfigure[Just sampled Example]{ 354 ahallam 3054 \includegraphics[width=3in]{figures/ex07jsamp.png} 355 ahallam 3025 \label{fig:ex07jsamp} 356 } 357 \subfigure[Over sampled Example]{ 358 ahallam 3054 \includegraphics[width=3in]{figures/ex07nsamp.png} 359 ahallam 3025 \label{fig:ex07nsamp} 360 } 361 \label{fig:ex07sampth} 362 \caption{Sampling Theorem example for stability 363 investigation.} 364 \end{figure} 365 ahallam 3004 366 ahallam 3025 367 368 369