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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 micro-particle
19 interactions. Gravitational fields also present an opportunity to demonstrate
20 the saving and visualisation of vector data for Mayavi, and the construction of
21 variable sized meshes.
23 The gravitational potential $U$ at a point $P$ due to a region with a mass
24 distribution of density $\rho(P)$, is given by Poisson's equation
25 \citep{Blakely1995}
26 \begin{equation} \label{eqn:poisson}
27 \nabla^2 U(P) = -4\pi\gamma\rho(P)
28 \end{equation}
29 where $\gamma$ is the gravitational constant.
30 Consider now the \esc general form,
31 \autoref{eqn:poisson} requires only two coefficients,
32 $A$ and $Y$, thus the relevant terms of the general form are reduced to;
33 \begin{equation}
34 -\left(A_{jl} u_{,l} \right)_{,j} = Y
35 \end{equation}
36 One recognises that the LHS side is equivalent to
37 \begin{equation} \label{eqn:ex10a}
38 -\nabla A \nabla u
39 \end{equation}
40 and when $A=\delta_{jl}$, \autoref{eqn:ex10a} is equivalent to
41 \begin{equation*}
42 -\nabla^2 u
43 \end{equation*}
44 Thus Poisson's \autoref{eqn:poisson} satisfies the general form when
45 \begin{equation}
46 A=\delta_{jl} \text{ and } Y= 4\pi\gamma\rho
47 \end{equation}
48 The boundary condition is the last parameter that requires consideration. The
49 potential $U$ is related to the mass of a sphere by
50 \begin{equation}
51 U(P)=-\gamma \frac{m}{r^2}
52 \end{equation} where $m$ is the mass of the sphere and $r$ is the distance from
53 the center of the mass to the observation point $P$. Plainly, the magnitude
54 of the potential is governed by an inverse-square distance law. Thus, in the
55 limit as $r$ increases;
56 \begin{equation}
57 \lim_{r\to\infty} U(P) = 0
58 \end{equation}
59 Provided that the domain being solved is large enough, and the source mass is
60 contained within a central region of the domain, the potential will decay to
61 zero. This is a dirichlet boundary condition where $U=0$.
63 This boundary condition can be satisfied when there is some mass suspended in a
64 free-space. For geophysical models where the mass of interest may be an anomaly
65 inside a host rock, the anomaly can be isolated by subtracting the density of the
66 host rock from the model. This creates a fictitious free space model that will
67 satisfy the analytic boundary conditions. The result is that
68 $Y=4\pi\gamma\Delta\rho$, where $\Delta\rho=\rho-\rho_0$ and $\rho_0$ is the
69 baseline or host density. This of course means that the true gravity response is
70 not being modelled, but the response due to an anomalous mass with a
71 density contrast $\Delta\rho$.
73 For this example we have set all of the boundaries to zero but only one boundary
74 point needs to be set for the problem to be solvable. The normal flux condition
75 is also zero by default. Note, that for a more realistic and complicated models
76 it may be necessary to give careful consideration to the boundary conditions of the model,
77 which can have an influence upon the solution.
79 Setting the boundary condition is relatively simple using the \verb!q! and
80 \verb!r! variables of the general form. First \verb!q! is defined as a masking
81 function on the boundary using
82 \begin{python}
83 q=whereZero(x[1]-my)+whereZero(x[1])+whereZero(x[0])+whereZero(x[0]-mx)
84 mypde.setValue(q=q,r=0)
85 \end{python}
86 This identifies the points on the boundary and \verb!r! is simply
87 ser to \verb!r=0.0!. This is a Dirichlet boundary condition.
89 \clearpage
90 \section{Gravity Pole}
91 \sslist{example10a.py}
92 A gravity pole is used in this example to demonstrate the vector characteristics
93 of gravity, and also to demonstrate how this information can be exported for
94 visualisation to Mayavi or an equivalent using the VTK data format.
96 The solution script for this section is very simple. First the domain is
97 constructed, then the parameters of the model are set, and finally the steady
98 state solution is found. There are quite a few values that can now be derived
99 from the solution and saved to file for visualisation.
101 The potential $U$ is related to the gravitational response $\vec{g}$ via
102 \begin{equation}
103 \vec{g} = \nabla U
104 \end{equation}
105 $\vec{g}$ is a vector and thus, has a a vertical component $g_{z}$ where
106 \begin{equation}
107 g_{z}=\vec{g}\cdot\hat{z}
108 \end{equation}
109 Finally, there is the magnitude of the vertical component $g$ of
110 $g_{z}$
111 \begin{equation}
112 g=|g_{z}|
113 \end{equation}
114 These values are derived from the \esc solution \verb!sol! to the potential $U$
115 using the following commands
116 \begin{python}
117 g_field=grad(sol) #The gravitational acceleration g.
118 g_fieldz=g_field*[0,1] #The vertical component of the g field.
119 gz=length(g_fieldz) #The magnitude of the vertical component.
120 \end{python}
121 This data can now be simply exported to a VTK file via
122 \begin{python}
123 # Save the output to file.
124 saveVTK(os.path.join(save_path,"ex10a.vtu"),\
125 grav_pot=sol,g_field=g_field,g_fieldz=g_fieldz,gz=gz)
126 \end{python}
128 It is quite simple to visualise the data from the gravity solution in Mayavi2.
129 With Mayavi2 open go to File, Load data, Open file \ldots as in
130 \autoref{fig:mayavi2openfile} and select the saved data file. The data will
131 have then been loaded and is ready for visualisation. Notice that under the data
132 object in the Mayavi2 navigation tree the 4 values saved to the VTK file are
133 available (\autoref{fig:mayavi2data}). There are two vector values,
134 \verb|gfield| and \verb|gfieldz|. Note that to plot both of these on the same
135 chart requires that the data object be imported twice.
137 The point scalar data \verb|grav_pot| is the gravitational potential and it is
138 easily plotted using a surface module. To visualise the cell data a filter is
139 required that converts to point data. This is done by right clicking on the data
140 object in the explorer tree and selecting the cell to point filter as in
141 \autoref{fig:mayavi2cell2point}.
143 The settings can then be modified to suit personal taste. An example of the
144 potential and gravitational field vectors is illustrated in
145 \autoref{fig:ex10pot}.
147 \begin{figure}[ht]
148 \centering
149 \includegraphics[width=0.75\textwidth]{figures/mayavi2_openfile.png}
150 \caption{Open a file in Mayavi2}
151 \label{fig:mayavi2openfile}
152 \end{figure}
154 \begin{figure}[ht]
155 \centering
156 \includegraphics[width=0.75\textwidth]{figures/mayavi2_data.png}
157 \caption{The 4 types of data in the imported VTK file.}
158 \label{fig:mayavi2data}
159 \end{figure}
161 \begin{figure}[ht]
162 \centering
163 \includegraphics[width=0.75\textwidth]{figures/mayavi2_cell2point.png}
164 \caption{Converting cell data to point data.}
165 \label{fig:mayavi2cell2point}
166 \end{figure}
168 \begin{figure}[ht]
169 \centering
170 \includegraphics[width=0.75\textwidth]{figures/ex10apot.png}
171 \caption{Newtonian potential with $\vec{g}$ field directionality. The magnitude
172 of the field is reflected in the size of the vector arrows.}
173 \label{fig:ex10pot}
174 \end{figure}
175 \clearpage
177 \section{Gravity Well}
178 \sslist{example10b.py}
179 Let us now investigate the effect of gravity in three dimensions. Consider a
180 volume which contains a spherical mass anomaly and a gravitational potential
181 which decays to zero at the base of the model.
183 The script used to solve this model is very similar to that used for the gravity
184 pole in the previous section, but with an extra spatial dimension. As for all
185 the 3D problems examined in this cookbook, the extra dimension is easily
186 integrated into the \esc solution script.
188 The domain is redefined from a rectangle to a Brick;
189 \begin{python}
190 domain = Brick(l0=mx,l1=my,n0=ndx, n1=ndy,l2=mz,n2=ndz)
191 \end{python}
192 the source is relocated along $z$;
193 \begin{python}
194 x=x-[mx/2,my/2,mz/2]
195 \end{python}
196 and, the boundary conditions are updated.
197 \begin{python}
198 q=whereZero(x[2]-inf(x[2]))
199 \end{python}
200 No modifications to the PDE solution section are required. Thus the migration
201 from a 2D to a 3D problem is almost trivial.
203 \autoref{fig:ex10bpot} illustrates the strength of a PDE solution. Three
204 different visualisation types help define and illuminate properties of the data.
205 The cut surfaces of the potential are similar to a 2D section for a given x or y
206 and z. The iso-surfaces illuminate the 3D shape of the gravity field, as well as
207 its strength which is illustrated by the colour. Finally, the streamlines
208 highlight the directional flow of the gravity field in this example.
210 The boundary conditions were discussed previously, but not thoroughly
211 investigated. It was stated, that in the limit as the boundary becomes more
212 remote from the source, the potential will reduce to zero.
213 \autoref{fig:ex10bpot2} is the solution to the same gravity problem
214 but with a slightly larger domain. It is obvious in this case that
215 the previous domain size was too small to accurately represent the
216 solution. The profiles in \autoref{fig:ex10p} further demonstrates the affect
217 the domain size has upon the solution. As the domain size increases, the
218 profiles begin to converge. This convergence is a good indicator of an
219 appropriately dimensioned model for the problem, and and sampling location.
221 \begin{figure}[htp]
222 \centering
223 \includegraphics[width=0.75\textwidth]{figures/ex10bpot.png}
224 \caption{Gravity well with iso surfaces and streamlines of the vector
225 gravitational potential \textemdash small model.}
226 \label{fig:ex10bpot}
227 \end{figure}
229 \begin{figure}[htp]
230 \centering
231 \includegraphics[width=0.75\textwidth]{figures/ex10bpot2.png}
232 \caption{Gravity well with iso surfaces and streamlines of the vector
233 gravitational potential \textemdash large model.}
234 \label{fig:ex10bpot2}
235 \end{figure}
237 \begin{figure}[htp]
238 \centering
239 \includegraphics[width=0.85\textwidth]{figures/ex10p_boundeff.pdf}
240 \caption{Profile of the graviational provile along x where $y=0,z=250$ for
241 various sized domains.}
242 \label{fig:ex10p}
243 \end{figure}
244 \clearpage
246 \section{Gravity Surface over a fault model.}
247 \sslist{example10c.py,example10m.py}
248 This model demonstrates the gravity result for a more complicated domain which
249 contains a fault. Additional information will be added when geophysical boundary
250 conditions for a gravity scenario have been established.
252 \begin{figure}[htp]
253 \centering
254 \subfigure[The geometry of the fault model in example10c.py.]
255 {\label{fig:ex10cgeo}
256 \includegraphics[width=0.8\textwidth]{figures/ex10potfaultgeo.png}} \\
257 \subfigure[The fault of interest from the fault model in
258 example10c.py.]
259 {\label{fig:ex10cmsh}
260 \includegraphics[width=0.8\textwidth]{figures/ex10potfaultmsh.png}}
261 \end{figure}
263 \begin{figure}[htp]
264 \centering
265 \includegraphics[width=0.8\textwidth]{figures/ex10cpot.png}
266 \caption{Gravitational potential of the fault model with primary layers and
267 faults identified as isosurfaces.}
268 \label{fig:ex10cpot}
269 \end{figure}
270 \clearpage
272 \section{Variable mesh-element sizes}
273 \sslist{example10m.py}
274 We saw in a previous section that the domain needed to be sufficiently
275 large for the boundary conditions to be satisfied. This can be troublesome when
276 trying to solve problems that require a dense mesh, either for solution
277 resolution of stability reasons. The computational cost of solving a large
278 number of elements can be prohibitive.S
280 With the help of Gmsh, it is possible to create a mesh for \esc, which has a
281 variable element size. Such an approach can significantly reduce the number of
282 elements that need to be solved, and a large domain can be created that has
283 sufficient resolution close to the source and extends to distances large enough
284 that the infinity clause is satisfied.
286 To create a variable size mesh, multiple meshing domains are required. The
287 domains must share points, boundaries and surfaces so that they are joined; and
288 no sub-domains are allowed to overlap. Whilst this initially seems complicated,
289 it is quite simple to implement.
291 This example creates a mesh which contains a high resolution sub-domain at its
292 center. We begin by defining two curve loops which describe the large or
293 big sub-domain and the smaller sub-domain which is to contain the high
294 resolution portion of the mesh.
295 \begin{python}
296 ################################################BIG DOMAIN
298 width=10000. #width of model
299 depth=10000. #depth of model
300 bele_size=500. #big element size
302 p0=Point(0.0, 0.0)
303 p1=Point(width, 0.0)
304 p2=Point(width, depth)
305 p3=Point(0.0, depth)
306 # Join corners in anti-clockwise manner.
307 l01=Line(p0, p1)
308 l12=Line(p1, p2)
309 l23=Line(p2, p3)
310 l30=Line(p3, p0)
312 cbig=CurveLoop(l01,l12,l23,l30)
314 ################################################SMALL DOMAIN
316 xwidth=2000.0 #x width of model
317 zdepth=2000.0 #y width of model
318 sele_size=10. #small element size
320 xshift=width/2-xwidth/2
321 zshift=depth/2-zdepth/2
323 p4=Point(xshift, zshift)
324 p5=Point(xwidth+xshift, zshift)
325 p6=Point(xwidth+xshift, zdepth+zshift)
326 p7=Point(xshift, zdepth+zshift)
327 # Join corners in anti-clockwise manner.
328 l45=Line(p4, p5)
329 l56=Line(p5, p6)
330 l67=Line(p6, p7)
331 l74=Line(p7, p4)
333 csmall=CurveLoop(l45,l56,l67,l74)
334 \end{python}
335 The small sub-domain curve can then be used to create a surface.
336 \begin{python}
337 ssmall=PlaneSurface(csmall)
338 \end{python}
339 However, so that the two domains do not overlap, when the big sub-domain
340 curveloop is used to create a surface it must contain a hole. The hole is
341 defined by the small sub-domain curveloop.
342 \begin{python}
343 sbig=PlaneSurface(cbig,holes=[csmall])
344 \end{python}
345 The two sub-domains now have a common geometry and no over-laping features as
346 per \autoref{fig:ex10mgeo}. Notice, that both domains have a normal in the
347 same direction.
349 The next step, is exporting each sub-domain individually, with an appropriate
350 element size. This is carried out using the \pycad Design command.
351 \begin{python}
352 # Design the geometry for the big mesh.
353 d1=Design(dim=2, element_size=bele_size, order=1)
354 d1.addItems(sbig)
355 d1.addItems(PropertySet(l01,l12,l23,l30))
356 d1.setScriptFileName(os.path.join(save_path,"example10m_big.geo"))
357 MakeDomain(d1)
359 # Design the geometry for the small mesh.
360 d2=Design(dim=2, element_size=sele_size, order=1)
361 d2.addItems(ssmall)
362 d2.setScriptFileName(os.path.join(save_path,"example10m_small.geo"))
363 MakeDomain(d2)
364 \end{python}
365 Finally, a system call to Gmsh is required to merge and then appropriately
366 mesh the two sub-domains together.
367 \begin{python}
368 # Join the two meshes using Gmsh and then apply a 2D meshing algorithm.
369 # The small mesh must come before the big mesh in the merging call!!@!!@!
370 sp.call("gmsh -2 "+
371 os.path.join(save_path,"example10m_small.geo")+" "+
372 os.path.join(save_path,"example10m_big.geo")+" -o "+
373 os.path.join(save_path,"example10m.msh"),shell=True)
374 \end{python}
375 The ``-2'' option is responsible for the 2D meshing, and the ``-o'' option
376 provides the output path. The resulting mesh is depicted in
377 \autoref{fig:ex10mmsh}
379 To use the Gmsh ``*.msh'' file in the solution script, the mesh reading function
380 ``ReadGmsh'' is required. It can be imported via;
381 \begin{python}
382 from esys.finley import ReadGmsh
383 \end{python}
384 To read in the file the function is called
385 \begin{python}
386 domain=ReadGmsh(os.path.join(mesh_path,'example10m.msh'),2) # create the domain
387 \end{python}
388 where the integer argument is the number of domain dimensions.
389 %
390 \begin{figure}[ht]
391 \centering
392 \includegraphics[width=0.8\textwidth]{figures/ex10m_geo.png}
393 \caption{Geometry of two surfaces for a single domain.}
394 \label{fig:ex10mgeo}
395 \end{figure}
397 \begin{figure}[ht]
398 \centering
399 \includegraphics[width=0.8\textwidth]{figures/ex10m_msh.png}
400 \caption{Mesh of merged surfaces, showing variable element size. Elements
401 range from 10m in the centroid to 500m at the boundary.}
402 \label{fig:ex10mmsh}
403 \end{figure}
404 \clearpage
406 \section{Unbounded problems}
407 With a variable element-size, it is now possible to solve the potential problem
408 over a very large mesh. To test the accuracy of the solution, we will compare
409 the \esc result with the analytic solution for the vertical gravitational
410 acceleration $g_z$ of an infinite horizontal cylinder.
412 For a horizontal cylinder with a circular cross-section with infinite strike,
413 the analytic solution is give by
414 \begin{equation}
415 g_z = 2\gamma\pi R^2 \Delta\rho \frac{z}{(x^2+z^2)}
416 \end{equation}
417 where $\gamma$ is the gravitational constant (as defined previously), $R$ is the
418 radius of the cylinder, $\Delta\rho$ is the density contrast and $x$ and $z$ are
419 geometric factors, relating the observation point to the center of the source
420 via the horizontal and vertical displacements respectively.
422 The accuracy of the solution was tested using a square domain. For each test the
423 dimensions of the domain were modified, being set to 5, 10, 20 and 40 Km. The
424 results are compared with the analytic solution and are depicted in
425 \autoref{fig:ex10q boundeff} and \autoref{fig:ex10q boundeff zoom}. Clearly,
426 as the domain size increases, the results are valid at greater
427 distances from the source. The same is true at the anomaly peak, where the
428 variation around the source diminishes with an increasing domain size.
430 \begin{figure}[ht]
431 \centering
432 \includegraphics[width=0.8\textwidth]{figures/ex10q_boundeff.pdf}
433 \caption{Solution profile 1000.0 meters from the source as the domain size
434 increases.}
435 \label{fig:ex10q boundeff}
436 \end{figure}
438 \begin{figure}[ht]
439 \centering
440 \includegraphics[width=0.8\textwidth]{figures/ex10q_boundeff_zoom.pdf}
441 \caption{Magnification of \autoref{fig:ex10q boundeff}.}
442 \label{fig:ex10q boundeff zoom}
443 \end{figure}
444 \clearpage
446 \subsection{Inversion using scipy}

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