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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 % http://www.uq.edu.au/esscc
7 %
8 % Primary Business: Queensland, Australia
9 % Licensed under the Open Software License version 3.0
10 % http://www.opensource.org/licenses/osl-3.0.php
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

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