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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 3370 where $p$ is the pressure, $t$ is the time and $c$ is the wave velocity. In this 27 chapter the acoustic wave equation is demonstrated. Important steps include the 28 translation of the Laplacian $\nabla^2$ to the \esc general form, the stiff 29 ahallam 3373 equation stability criterion and solving for the displacement or acceleration solution. 30 ahallam 3003 31 ahallam 3004 \section{The Laplacian in \esc} 32 ahallam 3373 The Laplacian operator which can be written as $\Delta$ or $\nabla^2$, is 33 ahallam 3370 calculated via the divergence of the gradient of the object, which in this 34 example is the scalar $p$. Thus we can write; 35 ahallam 3004 \begin{equation} 36 ahallam 3029 \nabla^2 p = \nabla \cdot \nabla p = 37 jfenwick 3308 \sum_{i}^n 38 \frac{\partial^2 p}{\partial x^2_{i}} 39 ahallam 3004 \label{eqn:laplacian} 40 \end{equation} 41 ahallam 3232 For the two dimensional case in Cartesian coordinates \autoref{eqn:laplacian} 42 ahallam 3004 becomes; 43 \begin{equation} 44 \nabla^2 p = \frac{\partial^2 p}{\partial x^2} 45 + \frac{\partial^2 p}{\partial y^2} 46 \end{equation} 47 ahallam 3003 48 ahallam 3004 In \esc the Laplacian is calculated using the divergence representation and the 49 ahallam 3370 intrinsic functions \textit{grad()} and \textit{trace()}. The function 50 ahallam 3004 \textit{grad{}} will return the spatial gradients of an object. 51 For a rank 0 solution, this is of the form; 52 \begin{equation} 53 \nabla p = \left[ 54 jfenwick 3308 \frac{\partial p}{\partial x _{0}}, 55 \frac{\partial p}{\partial x _{1}} 56 ahallam 3004 \right] 57 \label{eqn:grad} 58 \end{equation} 59 Larger ranked solution objects will return gradient tensors. For example, a 60 jfenwick 3308 pressure field which acts in the directions $p _{0}$ and $p 61 _{1}$ would return; 62 ahallam 3004 \begin{equation} 63 \nabla p = \begin{bmatrix} 64 jfenwick 3308 \frac{\partial p _{0}}{\partial x _{0}} & 65 \frac{\partial p _{1}}{\partial x _{0}} \\ 66 \frac{\partial p _{0}}{\partial x _{1}} & 67 \frac{\partial p _{1}}{\partial x _{1}} 68 ahallam 3004 \end{bmatrix} 69 \label{eqn:gradrank1} 70 \end{equation} 71 ahallam 3003 72 ahallam 3232 \autoref{eqn:grad} corresponds to the Linear PDE general form value 73 ahallam 3373 $X$. Notice however, that the general form contains the term $X 74 jfenwick 3308 _{i,j}$\footnote{This is the first derivative in the $j^{th}$ 75 ahallam 3029 direction for the $i^{th}$ component of the solution.}, 76 ahallam 3370 hence for a rank 0 object there is no need to do more then calculate the 77 ahallam 3004 gradient and submit it to the solver. In the case of the rank 1 or greater 78 ahallam 3373 object, it is also necessary to calculate the trace. This is the sum of the 79 ahallam 3232 diagonal in \autoref{eqn:gradrank1}. 80 ahallam 3004 81 Thus when solving for equations containing the Laplacian one of two things must 82 ahallam 3370 be completed. If the object \verb!p! is less then rank 1 the gradient is 83 ahallam 3004 calculated via; 84 ahallam 3025 \begin{python} 85 ahallam 3063 gradient=grad(p) 86 ahallam 3025 \end{python} 87 ahallam 3370 and if the object is greater then or equal to a rank 1 tensor, the trace of 88 ahallam 3004 the gradient is calculated. 89 ahallam 3025 \begin{python} 90 ahallam 3004 gradient=trace(grad(p)) 91 ahallam 3025 \end{python} 92 ahallam 3370 These values can then be submitted to the PDE solver via the general form term 93 ahallam 3004 $X$. The Laplacian is then computed in the solution process by taking the 94 divergence of $X$. 95 96 ahallam 3025 Note, if you are unsure about the rank of your tensor, the \textit{getRank} 97 command will return the rank of the PDE object. 98 \begin{python} 99 rank = p.getRank() 100 \end{python} 101 102 103 \section{Numerical Solution Stability} \label{sec:nsstab} 104 ahallam 3004 Unfortunately, the wave equation belongs to a class of equations called 105 \textbf{stiff} PDEs. These types of equations can be difficult to solve 106 ahallam 3373 numerically as they tend to oscillate about the exact solution, which can 107 ahallam 3370 eventually lead to a catastrophic failure. To counter this problem, explicitly 108 ahallam 3373 stable schemes like the backwards Euler method, and correct parameterisation of 109 ahallam 3370 the problem are required. 110 111 There are two variables which must be considered for 112 stability when numerically trying to solve the wave equation. For linear media, 113 the two variables are related via; 114 ahallam 3004 \begin{equation} \label{eqn:freqvel} 115 f=\frac{v}{\lambda} 116 \end{equation} 117 ahallam 3025 The velocity $v$ that a wave travels in a medium is an important variable. For 118 ahallam 3370 stability the analytical wave must not propagate faster then the numerical wave 119 is able to, and in general, needs to be much slower then the numerical wave. 120 ahallam 3003 For example, a line 100m long is discretised into 1m intervals or 101 nodes. If 121 a wave enters with a propagation velocity of 100m/s then the travel time for 122 the wave between each node will be 0.01 seconds. The time step, must therefore 123 ahallam 3370 be significantly less then this. Of the order $10E-4$ would be appropriate. 124 ahallam 3003 125 ahallam 3004 The wave frequency content also plays a part in numerical stability. The 126 nyquist-sampling theorem states that a signals bandwidth content will be 127 jfenwick 3308 accurately represented when an equispaced sampling rate $f _{n}$ is 128 ahallam 3370 equal to or greater then twice the maximum frequency of the signal 129 jfenwick 3308 $f_{s}$, or; 130 ahallam 3004 \begin{equation} \label{eqn:samptheorem} 131 jfenwick 3308 f_{n} \geqslant f_{s} 132 ahallam 3004 \end{equation} 133 ahallam 3370 For example a 50Hz signal will require a sampling rate greater then 100Hz or 134 ahallam 3004 one sample every 0.01 seconds. The wave equation relies on a spatial frequency, 135 ahallam 3373 thus the sampling theorem in this case applies to the solution mesh spacing. 136 This relationship confirms that the frequency content of the input signal 137 directly affects the time discretisation of the problem. 138 ahallam 3004 139 To accurately model the wave equation with high resolutions and velocities 140 means that very fine spatial and time discretisation is necessary for most 141 problems. 142 This requirement makes the wave equation arduous to 143 ahallam 3003 solve numerically due to the large number of time iterations required in each 144 ahallam 3004 solution. Models with very high velocities and frequencies will be the worst 145 ahallam 3029 affected by this problem. 146 ahallam 3003 147 \section{Displacement Solution} 148 \sslist{example07a.py} 149 150 We begin the solution to this PDE with the centred difference formula for the 151 second derivative; 152 \begin{equation} 153 f''(x) \approx \frac{f(x+h - 2f(x) + f(x-h)}{h^2} 154 \label{eqn:centdiff} 155 \end{equation} 156 ahallam 3232 substituting \autoref{eqn:centdiff} for $\frac{\partial ^2 p }{\partial t ^2}$ 157 in \autoref{eqn:acswave}; 158 ahallam 3003 \begin{equation} 159 jfenwick 3308 \nabla ^2 p - \frac{1}{c^2h^2} \left[p_{(t+1)} - 2p_{(t)} + 160 p_{(t-1)} \right] 161 ahallam 3003 = 0 162 \label{eqn:waveu} 163 \end{equation} 164 Rearranging for $p_{(t+1)}$; 165 \begin{equation} 166 jfenwick 3308 p_{(t+1)} = c^2 h^2 \nabla ^2 p_{(t)} +2p_{(t)} - 167 p_{(t-1)} 168 ahallam 3003 \end{equation} 169 this can be compared with the general form of the \modLPDE module and it 170 jfenwick 3308 becomes clear that $D=1$, $X_{i,j}=-c^2 h^2 \nabla ^2 p_{(t)}$ and 171 ahallam 3004 $Y=2p_{(t)} - p_{(t-1)}$. 172 ahallam 3003 173 ahallam 3025 The solution script is similar to others that we have created in previous 174 ahallam 3004 chapters. The general steps are; 175 \begin{enumerate} 176 \item The necessary libraries must be imported. 177 \item The domain needs to be defined. 178 \item The time iteration and control parameters need to be defined. 179 \item The PDE is initialised with source and boundary conditions. 180 \item The time loop is started and the PDE is solved at consecutive time steps. 181 ahallam 3370 \item All or select solutions are saved to file for visualisation later on. 182 ahallam 3004 \end{enumerate} 183 184 Parts of the script which warrant more attention are the definition of the 185 source, visualising the source, the solution time loop and the VTK data export. 186 187 \subsection{Pressure Sources} 188 As the pressure is a scalar, one need only define the pressure for two 189 time steps prior to the start of the solution loop. Two known solutions are 190 required because the wave equation contains a double partial derivative with 191 respect to time. This is often a good opportunity to introduce a source to the 192 solution. This model has the source located at it's centre. The source should 193 be smooth and cover a number of samples to satisfy the frequency stability 194 ahallam 3370 criterion. Small sources will generate high frequency signals. Here, when using 195 a rectangular domain, the source is defined by a cosine function. 196 ahallam 3025 \begin{python} 197 ahallam 3004 U0=0.01 # amplitude of point source 198 xc=[500,500] #location of point source 199 # define small radius around point xc 200 src_radius = 30 201 # for first two time steps 202 ahallam 3025 u=U0*(cos(length(x-xc)*3.1415/src_radius)+1)*\ 203 whereNegative(length(x-xc)-src_radius) 204 ahallam 3004 u_m1=u 205 ahallam 3025 \end{python} 206 ahallam 3004 207 ahallam 3025 \subsection{Visualising the Source} 208 There are two options for visualising the source. The first is to export the 209 initial conditions of the model to VTK, which can be interpreted as a scalar 210 ahallam 3370 surface in Mayavi2. The second is to take a cross section of the model which 211 will require the \textit{Locator} function. 212 ahallam 3063 First \verb!Locator! must be imported; 213 ahallam 3025 \begin{python} 214 from esys.escript.pdetools import Locator 215 \end{python} 216 The function can then be used on the domain to locate the nearest domain node 217 to the point or points of interest. 218 219 It is now necessary to build a list of $(x,y)$ locations that specify where are 220 ahallam 3370 model slice will go. This is easily implemented with a loop; 221 ahallam 3025 \begin{python} 222 cut_loc=[] 223 src_cut=[] 224 for i in range(ndx/2-ndx/10,ndx/2+ndx/10): 225 cut_loc.append(xstep*i) 226 src_cut.append([xstep*i,xc[1]]) 227 \end{python} 228 ahallam 3063 We then submit the output to \verb!Locator! and finally return the appropriate 229 values using the \verb!getValue! function. 230 ahallam 3025 \begin{python} 231 ahallam 3373 src=Locator(mydomain,src_cut) 232 ahallam 3025 src_cut=src.getValue(u) 233 \end{python} 234 ahallam 3370 It is then a trivial task to plot and save the output using \mpl 235 (\autoref{fig:cxsource}). 236 ahallam 3025 \begin{python} 237 ahallam 3373 pl.plot(cut_loc,src_cut) 238 ahallam 3025 pl.axis([xc[0]-src_radius*3,xc[0]+src_radius*3,0.,2*U0]) 239 pl.savefig(os.path.join(savepath,"source_line.png")) 240 \end{python} 241 \begin{figure}[h] 242 ahallam 3029 \centering 243 ahallam 3063 \includegraphics[width=6in]{figures/sourceline.png} 244 ahallam 3025 \caption{Cross section of the source function.} 245 \label{fig:cxsource} 246 \end{figure} 247 248 249 \subsection{Point Monitoring} 250 In the more general case where the solution mesh is irregular or specific 251 locations need to be monitored, it is simple enough to use the \textit{Locator} 252 function. 253 \begin{python} 254 rec=Locator(mydomain,[250.,250.]) 255 \end{python} 256 ahallam 3029 When the solution \verb u is updated we can extract the value at that point 257 ahallam 3025 via; 258 \begin{python} 259 u_rec=rec.getValue(u) 260 \end{python} 261 ahallam 3063 For consecutive time steps one can record the values from \verb!u_rec! in an 262 array initialised as \verb!u_rec0=[]! with; 263 ahallam 3025 \begin{python} 264 u_rec0.append(rec.getValue(u)) 265 \end{python} 266 267 It can be useful to monitor the value at a single or multiple individual points 268 in the model during the modelling process. This is done using 269 ahallam 3063 the \verb!Locator! function. 270 ahallam 3025 271 272 ahallam 3003 \section{Acceleration Solution} 273 \sslist{example07b.py} 274 275 ahallam 3373 An alternative method to the displacement solution, is to solve for the 276 acceleration $\frac{\partial ^2 p}{\partial t^2}$ directly. The displacement can 277 then be derived from the acceleration after a solution has been calculated 278 The acceleration is given by a modified form of \autoref{eqn:waveu}; 279 ahallam 3003 \begin{equation} 280 \nabla ^2 p - \frac{1}{c^2} a = 0 281 \label{eqn:wavea} 282 \end{equation} 283 jfenwick 3308 and can be solved directly with $Y=0$ and $X=-c^2 \nabla ^2 p_{(t)}$. 284 ahallam 3003 After each iteration the displacement is re-evaluated via; 285 \begin{equation} 286 jfenwick 3308 p_{(t+1)}=2p_{(t)} - p_{(t-1)} + h^2a 287 ahallam 3003 \end{equation} 288 289 ahallam 3029 \subsection{Lumping} 290 ahallam 3004 For \esc, the acceleration solution is prefered as it allows the use of matrix 291 lumping. Lumping or mass lumping as it is sometimes known, is the process of 292 aggressively approximating the density elements of a mass matrix into the main 293 diagonal. The use of Lumping is motivaed by the simplicity of diagonal matrix 294 inversion. As a result, Lumping can significantly reduce the computational 295 ahallam 3029 requirements of a problem. Care should be taken however, as this 296 function can only be used when the $A$, $B$ and $C$ coefficients of the 297 general form are zero. 298 ahallam 3003 299 ahallam 3004 To turn lumping on in \esc one can use the command; 300 ahallam 3025 \begin{python} 301 ahallam 3004 mypde.getSolverOptions().setSolverMethod(mypde.getSolverOptions().LUMPING) 302 ahallam 3025 \end{python} 303 ahallam 3004 It is also possible to check if lumping is set using; 304 ahallam 3025 \begin{python} 305 ahallam 3004 print mypde.isUsingLumping() 306 ahallam 3025 \end{python} 307 ahallam 3004 308 \section{Stability Investigation} 309 It is now prudent to investigate the stability limitations of this problem. 310 ahallam 3025 First, we let the frequency content of the source be very small. If we define 311 ahallam 3370 the source as a cosine input, then the wavlength of the input is equal to the 312 ahallam 3025 radius of the source. Let this value be 5 meters. Now, if the maximum velocity 313 ahallam 3373 of the model is $c=380.0ms^{-1}$, then the source 314 jfenwick 3308 frequency is $f_{r} = \frac{380.0}{5} = 76.0 Hz$. This is a worst case 315 ahallam 3025 scenario with a small source and the models maximum velocity. 316 317 ahallam 3232 Furthermore, we know from \autoref{sec:nsstab}, that the spatial sampling 318 ahallam 3025 frequency must be at least twice this value to ensure stability. If we assume 319 the model mesh is a square equispaced grid, 320 then the sampling interval is the side length divided by the number of samples, 321 given by $\Delta x = \frac{1000.0m}{400} = 2.5m$ and the maximum sampling 322 frequency capable at this interval is 323 jfenwick 3308 $f_{s}=\frac{380.0ms^{-1}}{2.5m}=152Hz$ this is just equal to the 324 ahallam 3232 required rate satisfying \autoref{eqn:samptheorem}. 325 ahallam 3004 326 ahallam 3232 \autoref{fig:ex07sampth} depicts three examples where the grid has been 327 ahallam 3025 undersampled, sampled correctly, and over sampled. The grids used had 328 200, 400 and 800 nodes per side respectively. Obviously, the oversampled grid 329 retains the best resolution of the modelled wave. 330 ahallam 3004 331 ahallam 3025 The time step required for each of these examples is simply calculated from 332 the propagation requirement. For a maximum velocity of $380.0ms^{-1}$, 333 \begin{subequations} 334 \begin{equation} 335 \Delta t \leq \frac{1000.0m}{200} \frac{1}{380.0} = 0.013s 336 \end{equation} 337 \begin{equation} 338 \Delta t \leq \frac{1000.0m}{400} \frac{1}{380.0} = 0.0065s 339 \end{equation} 340 \begin{equation} 341 \Delta t \leq \frac{1000.0m}{800} \frac{1}{380.0} = 0.0032s 342 \end{equation} 343 \end{subequations} 344 ahallam 3373 Observe that for each doubling of the number of nodes in the mesh, we halve 345 the time step. To illustrate the impact this has, consider our model. If the 346 ahallam 3025 source is placed at the center, it is $500m$ from the nearest boundary. With a 347 velocity of $380.0ms^{-1}$ it will take $\approx1.3s$ for the wavefront to 348 reach that boundary. In each case, this equates to $100$, $200$ and $400$ time 349 steps. This is again, only a best case scenario, for true stability these time 350 ahallam 3373 values may need to be halved and possibly halved again. 351 ahallam 3004 352 ahallam 3025 \begin{figure}[ht] 353 \centering 354 \subfigure[Undersampled Example]{ 355 ahallam 3373 \includegraphics[width=0.45\textwidth,trim=0cm 6cm 5cm 6cm 356 ,clip]{figures/ex07usamp.png} 357 ahallam 3025 \label{fig:ex07usamp} 358 } 359 \subfigure[Just sampled Example]{ 360 ahallam 3373 \includegraphics[width=0.45\textwidth,trim=0cm 6cm 5cm 6cm 361 ,clip]{figures/ex07jsamp.png} 362 ahallam 3025 \label{fig:ex07jsamp} 363 } 364 \subfigure[Over sampled Example]{ 365 ahallam 3373 \includegraphics[width=0.45\textwidth,trim=0cm 6cm 5cm 6cm 366 ,clip]{figures/ex07nsamp.png} 367 ahallam 3025 \label{fig:ex07nsamp} 368 } 369 \caption{Sampling Theorem example for stability 370 investigation.} 371 ahallam 3373 \label{fig:ex07sampth} 372 ahallam 3025 \end{figure} 373 ahallam 3004 374 ahallam 3025 375 376 377