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 1 ahallam 3153 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 \section{Newtonian Potential} 15 16 ahallam 3232 In this chapter the gravitational potential field is developed for \esc. 17 Gravitational fields are present in many modelling scenarios, including 18 geophysical investigations, planetary motion and attraction and microparticle 19 ahallam 3373 interactions. Gravitational fields also present an opportunity to demonstrate 20 ahallam 3232 the saving and visualisation of vector data for Mayavi. 21 ahallam 3153 22 The gravitational potential $U$ at a point $P$ due to a region with a mass 23 ahallam 3232 distribution of density $\rho(P)$, is given by Poisson's equation 24 \citep{Blakely1995} 25 ahallam 3153 \begin{equation} \label{eqn:poisson} 26 \nabla^2 U(P) = -4\pi\gamma\rho(P) 27 \end{equation} 28 where $\gamma$ is the gravitational constant. 29 ahallam 3373 Consider now the \esc general form, which has a simple relationship to 30 \autoref{eqn:poisson}. There are only two non-zero coefficients, $A$ and $Y$, 31 thus the relevant terms of the general form are reduced to; 32 ahallam 3153 \begin{equation} 33 jfenwick 3308 -\left(A_{jl} u_{,l} \right)_{,j} = Y 34 ahallam 3153 \end{equation} 35 ahallam 3373 One recognises that the LHS side is equivalent to 36 ahallam 3153 \begin{equation} \label{eqn:ex10a} 37 -\nabla A \nabla u 38 \end{equation} 39 ahallam 3373 and when a $A=\delta_{jl}$, \autoref{eqn:ex10a} is equivalent to 40 ahallam 3153 \begin{equation*} 41 -\nabla^2 u 42 \end{equation*} 43 ahallam 3373 Thus Poisson's \autoref{eqn:poisson} satisfies the general form when 44 ahallam 3153 \begin{equation} 45 jfenwick 3308 A=\delta_{jl} \text{ and } Y= 4\pi\gamma\rho 46 ahallam 3153 \end{equation} 47 ahallam 3373 The boundary condition is the last parameter that requires consideration. The 48 potential $U$ is related to the mass of a sphere by 49 \begin{equation} 50 U(P)=-\gamma \frac{m}{r^2} 51 \end{equation} where $m$ is the mass of the sphere and $r$ is the distance from 52 the center of the mass to the observation point $P$. Plainly, the magnitude 53 of the potential is goverened by an inverse-square distance law. Thus, in the 54 limit as $r$ increases; 55 \begin{equation} 56 \lim_{r\to\infty} U(P) = 0 57 \end{equation} 58 Provided that the domain being solved is large enough, the potential will decay 59 to zero. This stipulates a dirichlet condition where $U=0$ on the boundary of a 60 domain. 61 62 This boundary condition can be satisfied when there is some mass suspended in a 63 free-space. For geophysical models where the mass of interest may be an anomaly 64 inside a host rock, the anomaly can be isolated by subtracting the density of the 65 host rock from the model. This creates a ficticious free space model that will 66 obey the boundary conditions. The result is that $Y=4\pi\gamma\Delta\rho$, where 67 $\Delta\rho=\rho-\rho_0$ and $\rho_0$ is the baseline or host density. This of 68 course means that the true gravity response is not being modelled, but the 69 reponse due to an anomalous mass $\Delta g$. 70 71 For this example we have set all of the boundaries to zero but only one bondary 72 point needs to be set for the problem to be solvable. The normal flux condition 73 is also zero by default. For a more realistic and complicated models it may be 74 ahallam 3232 necessary to give careful consideration to the boundary conditions of the model, 75 which can have an influence upon the solution. 76 ahallam 3153 77 Setting the boundary condition is relatively simple using the \verb!q! and 78 ahallam 3232 \verb!r! variables of the general form. First \verb!q! is defined as a masking 79 function on the boundary using 80 ahallam 3153 \begin{python} 81 q=whereZero(x[1]-my)+whereZero(x[1])+whereZero(x[0])+whereZero(x[0]-mx) 82 \end{python} 83 ahallam 3232 This identifies the points on the boundary and \verb!r! is simply 84 ser to \verb!r=0.0!. This is a dirichlet boundary condition. 85 ahallam 3153 86 ahallam 3232 \section{Gravity Pole} 87 ahallam 3153 \sslist{example10a.py} 88 ahallam 3232 A gravity pole is used in this example to demonstrate the radial directionality 89 of gravity, and also to show how this information can be exported for 90 visualisation to Mayavi or an equivalent using the VTK data format. 91 ahallam 3153 92 ahallam 3232 The solution script for this section is very simple. First the domain is 93 constructed, then the parameters of the model are set, and finally the steady 94 state solution is found. There are quite a few values that can now be derived 95 from the solution and saved to file for visualisation. 96 97 The potential $U$ is related to the gravitational response $\vec{g}$ via 98 \begin{equation} 99 \vec{g} = \nabla U 100 \end{equation} 101 ahallam 3373 $\vec{g}$ is a vector and thus, has a a vertical component $g_{z}$ where 102 ahallam 3232 \begin{equation} 103 jfenwick 3308 g_{z}=\vec{g}\cdot\hat{z} 104 ahallam 3232 \end{equation} 105 Finally, there is the magnitude of the vertical component $g$ of 106 jfenwick 3308 $g_{z}$ 107 ahallam 3232 \begin{equation} 108 jfenwick 3308 g=|g_{z}| 109 ahallam 3232 \end{equation} 110 These values are derived from the \esc solution \verb!sol! to the potential $U$ 111 using the following commands 112 \begin{python} 113 g_field=grad(sol) #The graviational accelleration g. 114 g_fieldz=g_field*[0,1] #The vertical component of the g field. 115 gz=length(g_fieldz) #The magnitude of the vertical component. 116 \end{python} 117 This data can now be simply exported to a VTK file via 118 \begin{python} 119 # Save the output to file. 120 saveVTK(os.path.join(save_path,"ex10a.vtu"),\ 121 grav_pot=sol,g_field=g_field,g_fieldz=g_fieldz,gz=gz) 122 \end{python} 123 124 ahallam 3373 It is quite simple to visualise the data from the gravity solution in Mayavi2. 125 With Mayavi2 open go to File, Load data, Open file \ldots as in 126 \autoref{fig:mayavi2openfile} and select the saved data file. The data will 127 have then been loaded and is ready for visualisation. Notice that under the data 128 object in the Mayavi2 navigation tree the 4 values saved to the VTK file are 129 available (\autoref{fig:mayavi2data}). There are two vector values, 130 \verb|gfield| and \verb|gfieldz|. Note that to plot both of these on the same 131 chart requires that the data object be imported twice. 132 133 The point scalar data \verb|grav_pot| is the gravitational potential and it is 134 easily plotted using a surface module. To visualise the cell data a filter is 135 required that converts to point data. This is done by right clicking on the data 136 object in the explorer tree and sellecting the cell to point filter as in 137 \autoref{fig:mayavi2cell2point}. 138 139 The settings can then be modified to suit personal taste. An example of the 140 potential and gravitational field vectors is illustrated in 141 \autoref{fig:ex10pot}. 142 143 ahallam 3153 \begin{figure}[ht] 144 \centering 145 ahallam 3373 \includegraphics[width=0.75\textwidth]{figures/mayavi2_openfile.png} 146 \caption{Open a file in Mayavi2} 147 \label{fig:mayavi2openfile} 148 \end{figure} 149 150 \begin{figure}[ht] 151 \centering 152 \includegraphics[width=0.75\textwidth]{figures/mayavi2_data.png} 153 \caption{The 4 types of data in the imported VTK file.} 154 \label{fig:mayavi2data} 155 \end{figure} 156 157 \begin{figure}[ht] 158 \centering 159 \includegraphics[width=0.75\textwidth]{figures/mayavi2_cell2point.png} 160 \caption{Converting cell data to point data.} 161 \label{fig:mayavi2cell2point} 162 \end{figure} 163 164 \begin{figure}[ht] 165 \centering 166 ahallam 3153 \includegraphics[width=0.75\textwidth]{figures/ex10apot.png} 167 \caption{Newtonian potential with g field directionality.} 168 \label{fig:ex10pot} 169 \end{figure} 170 ahallam 3373 \clearpage 171 ahallam 3153 172 \section{Gravity Well} 173 \sslist{example10b.py} 174 ahallam 3232 Let us now investigate the effect of gravity in three dimensions. Consider a 175 volume which contains a sphericle mass anomaly and a gravitational potential 176 ahallam 3373 which decays to zero at the base of the model. 177 ahallam 3153 178 ahallam 3232 The script used to solve this model is very similar to that used for the gravity 179 pole in the previous section, but with an extra spatial dimension. As for all 180 the 3D problems examined in this cookbook, the extra dimension is easily 181 integrated into the \esc solultion script. 182 183 The domain is redefined from a rectangle to a Brick; 184 \begin{python} 185 domain = Brick(l0=mx,l1=my,n0=ndx, n1=ndy,l2=mz,n2=ndz) 186 \end{python} 187 the source is relocated along $z$; 188 \begin{python} 189 x=x-[mx/2,my/2,mz/2] 190 \end{python} 191 and, the boundary conditions are updated. 192 \begin{python} 193 q=whereZero(x[2]-inf(x[2])) 194 \end{python} 195 No modifications to the PDE solution section are required. This make the 196 migration of 2D to a 3D problem almost trivial. 197 198 \autoref{fig:ex10bpot} illustrates the strength of a PDE solution. Three 199 different visualisation types help define and illuminate properties of the data. 200 The cut surfaces of the potential are similar to a 2D section for a given x or y 201 and z. The iso-surfaces illuminate the 3D shape of the gravity field, as well as 202 its strength which is give by the colour. Finally, the streamlines highlight the 203 directional flow of the gravity field in this example. 204 205 ahallam 3153 \begin{figure}[htp] 206 \centering 207 \includegraphics[width=0.75\textwidth]{figures/ex10bpot.png} 208 \caption{Gravity well with iso surfaces and streamlines of the vector 209 gravitational potential.} 210 \label{fig:ex10bpot} 211 \end{figure} 212 213 \section{Gravity Surface over a fault model.} 214 \sslist{example10c.py,example10m.py} 215 ahallam 3232 This model demonstrates the gravity result for a more complicated domain which 216 contains a fault. Additional information will be added when geophysical boundary 217 conditions for a gravity scenario have been established. 218 219 ahallam 3153 \begin{figure}[htp] 220 \centering 221 \subfigure[The geometry of the fault model in example10c.py.] 222 {\label{fig:ex10cgeo} 223 \includegraphics[width=0.8\textwidth]{figures/ex10potfaultgeo.png}} \\ 224 \subfigure[The fault of interest from the fault model in 225 example10c.py.] 226 {\label{fig:ex10cmsh} 227 \includegraphics[width=0.8\textwidth]{figures/ex10potfaultmsh.png}} 228 \end{figure} 229 230 \begin{figure}[htp] 231 \centering 232 \includegraphics[width=0.8\textwidth]{figures/ex10cpot.png} 233 \caption{Gravitational potential of the fault model with primary layers and 234 faults identified as isosurfaces.} 235 \label{fig:ex10cpot} 236 \end{figure} 237