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4 % Copyright (c) 2003-2010 by University of Queensland
5 % Earth Systems Science Computational Center (ESSCC)
6 % http://www.uq.edu.au/esscc
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8 % Primary Business: Queensland, Australia
9 % Licensed under the Open Software License version 3.0
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12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
14 \section{Newtonian Potential}
16 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 interactions. Gravitational fields also present an opportunity to demonstrate
20 the saving and visualisation of vector data for Mayavi.
22 The gravitational potential $U$ at a point $P$ due to a region with a mass
23 distribution of density $\rho(P)$, is given by Poisson's equation
24 \citep{Blakely1995}
25 \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 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 \begin{equation}
33 -\left(A_{jl} u_{,l} \right)_{,j} = Y
34 \end{equation}
35 One recognises that the LHS side is equivalent to
36 \begin{equation} \label{eqn:ex10a}
37 -\nabla A \nabla u
38 \end{equation}
39 and when a $A=\delta_{jl}$, \autoref{eqn:ex10a} is equivalent to
40 \begin{equation*}
41 -\nabla^2 u
42 \end{equation*}
43 Thus Poisson's \autoref{eqn:poisson} satisfies the general form when
44 \begin{equation}
45 A=\delta_{jl} \text{ and } Y= 4\pi\gamma\rho
46 \end{equation}
47 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.
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 $\Delta g$.
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 necessary to give careful consideration to the boundary conditions of the model,
75 which can have an influence upon the solution.
77 Setting the boundary condition is relatively simple using the \verb!q! and
78 \verb!r! variables of the general form. First \verb!q! is defined as a masking
79 function on the boundary using
80 \begin{python}
81 q=whereZero(x[1]-my)+whereZero(x[1])+whereZero(x[0])+whereZero(x[0]-mx)
82 \end{python}
83 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.
86 \section{Gravity Pole}
87 \sslist{example10a.py}
88 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.
92 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.
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 $\vec{g}$ is a vector and thus, has a a vertical component $g_{z}$ where
102 \begin{equation}
103 g_{z}=\vec{g}\cdot\hat{z}
104 \end{equation}
105 Finally, there is the magnitude of the vertical component $g$ of
106 $g_{z}$
107 \begin{equation}
108 g=|g_{z}|
109 \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}
124 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.
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}.
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}.
143 \begin{figure}[ht]
144 \centering
145 \includegraphics[width=0.75\textwidth]{figures/mayavi2_openfile.png}
146 \caption{Open a file in Mayavi2}
147 \label{fig:mayavi2openfile}
148 \end{figure}
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}
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}
164 \begin{figure}[ht]
165 \centering
166 \includegraphics[width=0.75\textwidth]{figures/ex10apot.png}
167 \caption{Newtonian potential with g field directionality.}
168 \label{fig:ex10pot}
169 \end{figure}
170 \clearpage
172 \section{Gravity Well}
173 \sslist{example10b.py}
174 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 which decays to zero at the base of the model.
178 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.
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.
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.
205 \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}
213 \section{Gravity Surface over a fault model.}
214 \sslist{example10c.py,example10m.py}
215 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.
219 \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}
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}

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