# Contents of /trunk/doc/cookbook/example10.tex

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Sun Sep 5 01:31:49 2010 UTC (9 years ago) by ahallam
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Updates to cookbook. Fixes to SWP and new Gravitational Potential

 1 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 In this chapter we examine the gravitational potential field due to source 17 bodies and geological structure. The use of vector data and visualisation in 18 Mayavi is investigated. 19 20 The gravitational potential $U$ at a point $P$ due to a region with a mass 21 distribution $\rho(P)$ is given by Poisson's equation 22 \begin{equation} \label{eqn:poisson} 23 \nabla^2 U(P) = -4\pi\gamma\rho(P) 24 \end{equation} 25 where $\gamma$ is the gravitational constant. 26 If this is now related to the simplified general form 27 \begin{equation} 28 -\left(A\hackscore{jl} u\hackscore{,l} \right)\hackscore{,j} = Y 29 \end{equation} 30 where the LHS side is equivalent to 31 \begin{equation} \label{eqn:ex10a} 32 -\nabla A \nabla u 33 \end{equation} 34 If $A=\delta\hackscore{jl}$ then equation \ref{eqn:ex10a} is equivalent to 35 \begin{equation*} 36 -\nabla^2 u 37 \end{equation*} 38 and thus Poisson's Equation \ref{eqn:poisson} satisfies the general form when 39 \begin{equation} 40 A=\delta\hackscore{jl} \text{ and } Y= 4\pi\gamma\rho 41 \end{equation} 42 At least one boundary point must be set for the problem to be solvable. For this 43 example we have set all of the boundaries to zero. The normal flux condition is 44 also zero by default. 45 46 Setting the boundary condition is relatively simple using the \verb!q! and 47 \verb!r! variables of the general form, where 48 \begin{python} 49 q=whereZero(x[1]-my)+whereZero(x[1])+whereZero(x[0])+whereZero(x[0]-mx) 50 \end{python} 51 identifies the points on the boundary and \verb!r=0.0! sets the dirichlet 52 condition. 53 54 \section{Gravity Profile} 55 \sslist{example10a.py} 56 57 \begin{figure}[ht] 58 \centering 59 \includegraphics[width=0.75\textwidth]{figures/ex10apot.png} 60 \caption{Newtonian potential with g field directionality.} 61 \label{fig:ex10pot} 62 \end{figure} 63 64 \section{Gravity Well} 65 \sslist{example10b.py} 66 67 \begin{figure}[htp] 68 \centering 69 \includegraphics[width=0.75\textwidth]{figures/ex10bpot.png} 70 \caption{Gravity well with iso surfaces and streamlines of the vector 71 gravitational potential.} 72 \label{fig:ex10bpot} 73 \end{figure} 74 75 \section{Gravity Surface over a fault model.} 76 \sslist{example10c.py,example10m.py} 77 \begin{figure}[htp] 78 \centering 79 \subfigure[The geometry of the fault model in example10c.py.] 80 {\label{fig:ex10cgeo} 81 \includegraphics[width=0.8\textwidth]{figures/ex10potfaultgeo.png}} \\ 82 \subfigure[The fault of interest from the fault model in 83 example10c.py.] 84 {\label{fig:ex10cmsh} 85 \includegraphics[width=0.8\textwidth]{figures/ex10potfaultmsh.png}} 86 \end{figure} 87 88 \begin{figure}[htp] 89 \centering 90 \includegraphics[width=0.8\textwidth]{figures/ex10cpot.png} 91 \caption{Gravitational potential of the fault model with primary layers and 92 faults identified as isosurfaces.} 93 \label{fig:ex10cpot} 94 \end{figure} 95