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15 \section{Newtonian Potential}
17 In this chapter the gravitational potential field is developed for \esc.
18 Gravitational fields are present in many modelling scenarios, including
19 geophysical investigations, planetary motion and attraction and micro-particle
20 interactions. Gravitational fields also present an opportunity to demonstrate
21 the saving and visualisation of vector data for Mayavi, and the construction of
22 variable sized meshes.
24 The gravitational potential $U$ at a point $P$ due to a region with a mass
25 distribution of density $\rho(P)$, is given by Poisson's equation
26 \citep{Blakely1995}
27 \begin{equation} \label{eqn:poisson}
28 \nabla^2 U(P) = -4\pi\gamma\rho(P)
29 \end{equation}
30 where $\gamma$ is the gravitational constant.
31 Consider now the \esc general form,
32 \autoref{eqn:poisson} requires only two coefficients,
33 $A$ and $Y$, thus the relevant terms of the general form are reduced to;
34 \begin{equation}
35 -\left(A_{jl} u_{,l} \right)_{,j} = Y
36 \end{equation}
37 One recognises that the LHS side is equivalent to
38 \begin{equation} \label{eqn:ex10a}
39 -\nabla A \nabla u
40 \end{equation}
41 and when $A=\delta_{jl}$, \autoref{eqn:ex10a} is equivalent to
42 \begin{equation*}
43 -\nabla^2 u
44 \end{equation*}
45 Thus Poisson's \autoref{eqn:poisson} satisfies the general form when
46 \begin{equation}
47 A=\delta_{jl} \text{ and } Y= 4\pi\gamma\rho
48 \end{equation}
49 The boundary condition is the last parameter that requires consideration. The
50 potential $U$ is related to the mass of a sphere by
51 \begin{equation}
52 U(P)=-\gamma \frac{m}{r^2}
53 \end{equation} where $m$ is the mass of the sphere and $r$ is the distance from
54 the center of the mass to the observation point $P$. Plainly, the magnitude
55 of the potential is governed by an inverse-square distance law. Thus, in the
56 limit as $r$ increases;
57 \begin{equation}
58 \lim_{r\to\infty} U(P) = 0
59 \end{equation}
60 Provided that the domain being solved is large enough, and the source mass is
61 contained within a central region of the domain, the potential will decay to
62 zero. This is a dirichlet boundary condition where $U=0$.
64 This boundary condition can be satisfied when there is some mass suspended in a
65 free-space. For geophysical models where the mass of interest may be an anomaly
66 inside a host rock, the anomaly can be isolated by subtracting the density of the
67 host rock from the model. This creates a fictitious free space model that will
68 satisfy the analytic boundary conditions. The result is that
69 $Y=4\pi\gamma\Delta\rho$, where $\Delta\rho=\rho-\rho_0$ and $\rho_0$ is the
70 baseline or host density. This of course means that the true gravity response is
71 not being modelled, but the response due to an anomalous mass with a
72 density contrast $\Delta\rho$.
74 For this example we have set all of the boundaries to zero but only one boundary
75 point needs to be set for the problem to be solvable. The normal flux condition
76 is also zero by default. Note, that for a more realistic and complicated models
77 it may be necessary to give careful consideration to the boundary conditions of the model,
78 which can have an influence upon the solution.
80 Setting the boundary condition is relatively simple using the \verb!q! and
81 \verb!r! variables of the general form. First \verb!q! is defined as a masking
82 function on the boundary using
83 \begin{python}
84 q=whereZero(x[1]-my)+whereZero(x[1])+whereZero(x[0])+whereZero(x[0]-mx)
85 mypde.setValue(q=q,r=0)
86 \end{python}
87 This identifies the points on the boundary and \verb!r! is simply
88 ser to \verb!r=0.0!. This is a Dirichlet boundary condition.
90 \clearpage
91 \section{Gravity Pole}
92 \sslist{example10a.py}
93 A gravity pole is used in this example to demonstrate the vector characteristics
94 of gravity, and also to demonstrate how this information can be exported for
95 visualisation to Mayavi or an equivalent using the VTK data format.
97 The solution script for this section is very simple. First the domain is
98 constructed, then the parameters of the model are set, and finally the steady
99 state solution is found. There are quite a few values that can now be derived
100 from the solution and saved to file for visualisation.
102 The potential $U$ is related to the gravitational response $\vec{g}$ via
103 \begin{equation}
104 \vec{g} = \nabla U
105 \end{equation}
106 $\vec{g}$ is a vector and thus, has a a vertical component $g_{z}$ where
107 \begin{equation}
108 g_{z}=\vec{g}\cdot\hat{z}
109 \end{equation}
110 Finally, there is the magnitude of the vertical component $g$ of
111 $g_{z}$
112 \begin{equation}
113 g=|g_{z}|
114 \end{equation}
115 These values are derived from the \esc solution \verb!sol! to the potential $U$
116 using the following commands
117 \begin{python}
118 g_field=grad(sol) #The gravitational acceleration g.
119 g_fieldz=g_field*[0,1] #The vertical component of the g field.
120 gz=length(g_fieldz) #The magnitude of the vertical component.
121 \end{python}
122 This data can now be simply exported to a VTK file via
123 \begin{python}
124 # Save the output to file.
125 saveVTK(os.path.join(save_path,"ex10a.vtu"),\
126 grav_pot=sol,g_field=g_field,g_fieldz=g_fieldz,gz=gz)
127 \end{python}
129 It is quite simple to visualise the data from the gravity solution in Mayavi2.
130 With Mayavi2 open go to File, Load data, Open file \ldots as in
131 \autoref{fig:mayavi2openfile} and select the saved data file. The data will
132 have then been loaded and is ready for visualisation. Notice that under the data
133 object in the Mayavi2 navigation tree the 4 values saved to the VTK file are
134 available (\autoref{fig:mayavi2data}). There are two vector values,
135 \verb|gfield| and \verb|gfieldz|. Note that to plot both of these on the same
136 chart requires that the data object be imported twice.
138 The point scalar data \verb|grav_pot| is the gravitational potential and it is
139 easily plotted using a surface module. To visualise the cell data a filter is
140 required that converts to point data. This is done by right clicking on the data
141 object in the explorer tree and selecting the cell to point filter as in
142 \autoref{fig:mayavi2cell2point}.
144 The settings can then be modified to suit personal taste. An example of the
145 potential and gravitational field vectors is illustrated in
146 \autoref{fig:ex10pot}.
148 \begin{figure}[ht]
149 \centering
150 \includegraphics[width=0.75\textwidth]{figures/mayavi2_openfile.png}
151 \caption{Open a file in Mayavi2}
152 \label{fig:mayavi2openfile}
153 \end{figure}
155 \begin{figure}[ht]
156 \centering
157 \includegraphics[width=0.75\textwidth]{figures/mayavi2_data.png}
158 \caption{The 4 types of data in the imported VTK file.}
159 \label{fig:mayavi2data}
160 \end{figure}
162 \begin{figure}[ht]
163 \centering
164 \includegraphics[width=0.75\textwidth]{figures/mayavi2_cell2point.png}
165 \caption{Converting cell data to point data.}
166 \label{fig:mayavi2cell2point}
167 \end{figure}
169 \begin{figure}[ht]
170 \centering
171 \includegraphics[width=0.75\textwidth]{figures/ex10apot.png}
172 \caption{Newtonian potential with $\vec{g}$ field directionality. The magnitude
173 of the field is reflected in the size of the vector arrows.}
174 \label{fig:ex10pot}
175 \end{figure}
176 \clearpage
178 \section{Gravity Well}
179 \sslist{example10b.py}
180 Let us now investigate the effect of gravity in three dimensions. Consider a
181 volume which contains a spherical mass anomaly and a gravitational potential
182 which decays to zero at the base of the model.
184 The script used to solve this model is very similar to that used for the gravity
185 pole in the previous section, but with an extra spatial dimension. As for all
186 the 3D problems examined in this cookbook, the extra dimension is easily
187 integrated into the \esc solution script.
189 The domain is redefined from a rectangle to a Brick;
190 \begin{python}
191 domain = Brick(l0=mx,l1=my,n0=ndx, n1=ndy,l2=mz,n2=ndz)
192 \end{python}
193 the source is relocated along $z$;
194 \begin{python}
195 x=x-[mx/2,my/2,mz/2]
196 \end{python}
197 and, the boundary conditions are updated.
198 \begin{python}
199 q=whereZero(x[2]-inf(x[2]))
200 \end{python}
201 No modifications to the PDE solution section are required. Thus the migration
202 from a 2D to a 3D problem is almost trivial.
204 \autoref{fig:ex10bpot} illustrates the strength of a PDE solution. Three
205 different visualisation types help define and illuminate properties of the data.
206 The cut surfaces of the potential are similar to a 2D section for a given x or y
207 and z. The iso-surfaces illuminate the 3D shape of the gravity field, as well as
208 its strength which is illustrated by the colour. Finally, the streamlines
209 highlight the directional flow of the gravity field in this example.
211 The boundary conditions were discussed previously, but not thoroughly
212 investigated. It was stated, that in the limit as the boundary becomes more
213 remote from the source, the potential will reduce to zero.
214 \autoref{fig:ex10bpot2} is the solution to the same gravity problem
215 but with a slightly larger domain. It is obvious in this case that
216 the previous domain size was too small to accurately represent the
217 solution. The profiles in \autoref{fig:ex10p} further demonstrates the affect
218 the domain size has upon the solution. As the domain size increases, the
219 profiles begin to converge. This convergence is a good indicator of an
220 appropriately dimensioned model for the problem, and and sampling location.
222 \begin{figure}[htp]
223 \centering
224 \includegraphics[width=0.75\textwidth]{figures/ex10bpot.png}
225 \caption{Gravity well with iso surfaces and streamlines of the vector
226 gravitational potential \textemdash small model.}
227 \label{fig:ex10bpot}
228 \end{figure}
230 \begin{figure}[htp]
231 \centering
232 \includegraphics[width=0.75\textwidth]{figures/ex10bpot2.png}
233 \caption{Gravity well with iso surfaces and streamlines of the vector
234 gravitational potential \textemdash large model.}
235 \label{fig:ex10bpot2}
236 \end{figure}
238 \begin{figure}[htp]
239 \centering
240 \includegraphics[width=0.85\textwidth]{figures/ex10p_boundeff.pdf}
241 \caption{Profile of the graviational provile along x where $y=0,z=250$ for
242 various sized domains.}
243 \label{fig:ex10p}
244 \end{figure}
245 \clearpage
247 % \section{Gravity Surface over a fault model.}
248 % \sslist{example10c.py,example10m.py}
249 % This model demonstrates the gravity result for a more complicated domain which
250 % contains a fault. Additional information will be added when geophysical boundary
251 % conditions for a gravity scenario have been established.
252 %
253 % \begin{figure}[htp]
254 % \centering
255 % \subfigure[The geometry of the fault model in example10c.py.]
256 % {\label{fig:ex10cgeo}
257 % \includegraphics[width=0.8\textwidth]{figures/ex10potfaultgeo.png}} \\
258 % \subfigure[The fault of interest from the fault model in
259 % example10c.py.]
260 % {\label{fig:ex10cmsh}
261 % \includegraphics[width=0.8\textwidth]{figures/ex10potfaultmsh.png}}
262 % \end{figure}
263 %
264 % \begin{figure}[htp]
265 % \centering
266 % \includegraphics[width=0.8\textwidth]{figures/ex10cpot.png}
267 % \caption{Gravitational potential of the fault model with primary layers and
268 % faults identified as isosurfaces.}
269 % \label{fig:ex10cpot}
270 % \end{figure}
271 % \clearpage
273 \section{Variable mesh-element sizes}
274 \sslist{example10m.py}
275 We saw in a previous section that the domain needed to be sufficiently
276 large for the boundary conditions to be satisfied. This can be troublesome when
277 trying to solve problems that require a dense mesh, either for solution
278 resolution of stability reasons. The computational cost of solving a large
279 number of elements can be prohibitive.S
281 With the help of Gmsh, it is possible to create a mesh for \esc, which has a
282 variable element size. Such an approach can significantly reduce the number of
283 elements that need to be solved, and a large domain can be created that has
284 sufficient resolution close to the source and extends to distances large enough
285 that the infinity clause is satisfied.
287 To create a variable size mesh, multiple meshing domains are required. The
288 domains must share points, boundaries and surfaces so that they are joined; and
289 no sub-domains are allowed to overlap. Whilst this initially seems complicated,
290 it is quite simple to implement.
292 This example creates a mesh which contains a high resolution sub-domain at its
293 center. We begin by defining two curve loops which describe the large or
294 big sub-domain and the smaller sub-domain which is to contain the high
295 resolution portion of the mesh.
296 \begin{python}
297 ################################################BIG DOMAIN
299 width=10000. #width of model
300 depth=10000. #depth of model
301 bele_size=500. #big element size
303 p0=Point(0.0, 0.0)
304 p1=Point(width, 0.0)
305 p2=Point(width, depth)
306 p3=Point(0.0, depth)
307 # Join corners in anti-clockwise manner.
308 l01=Line(p0, p1)
309 l12=Line(p1, p2)
310 l23=Line(p2, p3)
311 l30=Line(p3, p0)
313 cbig=CurveLoop(l01,l12,l23,l30)
315 ################################################SMALL DOMAIN
317 xwidth=2000.0 #x width of model
318 zdepth=2000.0 #y width of model
319 sele_size=10. #small element size
321 xshift=width/2-xwidth/2
322 zshift=depth/2-zdepth/2
324 p4=Point(xshift, zshift)
325 p5=Point(xwidth+xshift, zshift)
326 p6=Point(xwidth+xshift, zdepth+zshift)
327 p7=Point(xshift, zdepth+zshift)
328 # Join corners in anti-clockwise manner.
329 l45=Line(p4, p5)
330 l56=Line(p5, p6)
331 l67=Line(p6, p7)
332 l74=Line(p7, p4)
334 csmall=CurveLoop(l45,l56,l67,l74)
335 \end{python}
336 The small sub-domain curve can then be used to create a surface.
337 \begin{python}
338 ssmall=PlaneSurface(csmall)
339 \end{python}
340 However, so that the two domains do not overlap, when the big sub-domain
341 curveloop is used to create a surface it must contain a hole. The hole is
342 defined by the small sub-domain curveloop.
343 \begin{python}
344 sbig=PlaneSurface(cbig,holes=[csmall])
345 \end{python}
346 The two sub-domains now have a common geometry and no over-laping features as
347 per \autoref{fig:ex10mgeo}. Notice, that both domains have a normal in the
348 same direction.
350 The next step, is exporting each sub-domain individually, with an appropriate
351 element size. This is carried out using the \pycad Design command.
352 \begin{python}
353 # Design the geometry for the big mesh.
354 d1=Design(dim=2, element_size=bele_size, order=1)
355 d1.addItems(sbig)
356 d1.addItems(PropertySet(l01,l12,l23,l30))
357 d1.setScriptFileName(os.path.join(save_path,"example10m_big.geo"))
358 MakeDomain(d1)
360 # Design the geometry for the small mesh.
361 d2=Design(dim=2, element_size=sele_size, order=1)
362 d2.addItems(ssmall)
363 d2.setScriptFileName(os.path.join(save_path,"example10m_small.geo"))
364 MakeDomain(d2)
365 \end{python}
366 Finally, a system call to Gmsh is required to merge and then appropriately
367 mesh the two sub-domains together.
368 \begin{python}
369 # Join the two meshes using Gmsh and then apply a 2D meshing algorithm.
370 # The small mesh must come before the big mesh in the merging call!!@!!@!
371 sp.call("gmsh -2 "+
372 os.path.join(save_path,"example10m_small.geo")+" "+
373 os.path.join(save_path,"example10m_big.geo")+" -o "+
374 os.path.join(save_path,"example10m.msh"),shell=True)
375 \end{python}
376 The ``-2'' option is responsible for the 2D meshing, and the ``-o'' option
377 provides the output path. The resulting mesh is depicted in
378 \autoref{fig:ex10mmsh}
380 To use the Gmsh ``*.msh'' file in the solution script, the mesh reading function
381 ``ReadGmsh'' is required. It can be imported via;
382 \begin{python}
383 from esys.finley import ReadGmsh
384 \end{python}
385 To read in the file the function is called
386 \begin{python}
387 domain=ReadGmsh(os.path.join(mesh_path,'example10m.msh'),2) # create the domain
388 \end{python}
389 where the integer argument is the number of domain dimensions.
390 %
391 \begin{figure}[ht]
392 \centering
393 \includegraphics[width=0.8\textwidth]{figures/ex10m_geo.png}
394 \caption{Geometry of two surfaces for a single domain.}
395 \label{fig:ex10mgeo}
396 \end{figure}
398 \begin{figure}[ht]
399 \centering
400 \includegraphics[width=0.8\textwidth]{figures/ex10m_msh.png}
401 \caption{Mesh of merged surfaces, showing variable element size. Elements
402 range from 10m in the centroid to 500m at the boundary.}
403 \label{fig:ex10mmsh}
404 \end{figure}
405 \clearpage
407 \section{Unbounded problems}
408 With a variable element-size, it is now possible to solve the potential problem
409 over a very large mesh. To test the accuracy of the solution, we will compare
410 the \esc result with the analytic solution for the vertical gravitational
411 acceleration $g_z$ of an infinite horizontal cylinder.
413 For a horizontal cylinder with a circular cross-section with infinite strike,
414 the analytic solution is give by
415 \begin{equation}
416 g_z = 2\gamma\pi R^2 \Delta\rho \frac{z}{(x^2+z^2)}
417 \end{equation}
418 where $\gamma$ is the gravitational constant (as defined previously), $R$ is the
419 radius of the cylinder, $\Delta\rho$ is the density contrast and $x$ and $z$ are
420 geometric factors, relating the observation point to the center of the source
421 via the horizontal and vertical displacements respectively.
423 The accuracy of the solution was tested using a square domain. For each test the
424 dimensions of the domain were modified, being set to 5, 10, 20 and 40 Km. The
425 results are compared with the analytic solution and are depicted in
426 \autoref{fig:ex10q boundeff} and \autoref{fig:ex10q boundeff zoom}. Clearly, as
427 the domain size increases, the \esc approximation becomes more accurate at
428 greater distances from the source. The same is true at the anomaly peak, where
429 the variation around the source diminishes with an increasing domain size.
431 \begin{figure}[ht]
432 \centering
433 \includegraphics[width=0.8\textwidth]{figures/ex10q_boundeff.pdf}
434 \caption{Solution profile 1000.0 meters from the source as the domain size
435 increases.}
436 \label{fig:ex10q boundeff}
437 \end{figure}
439 \begin{figure}[ht]
440 \centering
441 \includegraphics[width=0.8\textwidth]{figures/ex10q_boundeff_zoom.pdf}
442 \caption{Magnification of \autoref{fig:ex10q boundeff}.}
443 \label{fig:ex10q boundeff zoom}
444 \end{figure}
446 There is a methodology which can help establish an appropriate zero mass region
447 to a domain.
448 \clearpage

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