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12  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
13    
14    
15    
16  \chapter{The Module \pycad} \label{PYCAD CHAP}  \chapter{The Module \pycad} \label{PYCAD CHAP}
17    
18    
19  \section{Introduction}  \section{Introduction}
20    
21  \pycad provides a simple way to build a mesh for your finite element  \pycad provides a simple way to build a mesh for your finite element
22  simulation.  You begin by building what we call a {\it Design} using  simulation.  You begin by building what we call a \class{Design} using
23  primitive geometric objects, and then to go on to build a mesh from  primitive geometric objects, and then to go on to build a mesh from
24  the {\it Design}.  The final step of generating the mesh from a {\it  this.  The final step of generating the mesh from a \class{Design}
25  Design} uses freely available mesh generation software, such as \gmshextern.  uses freely available mesh generation software, such as \gmshextern.
26    
27  A {\it Design} is built by defining points, which are used to specify  A \class{Design} is built by defining points, which are used to specify
28  the corners of geometric objects and the vertices of curves.  Using  the corners of geometric objects and the vertices of curves.  Using
29  points you construct more interesting objects such as lines,  points you construct more interesting objects such as lines,
30  rectangles, and arcs.  By adding many of these objects into what we  rectangles, and arcs.  By adding many of these objects into what we
31  call a {\it Design}, you can build meshes for arbitrarily complex 2-D  call a \class{Design}, you can build meshes for arbitrarily complex 2-D
32  and 3-D structures.  and 3-D structures.
33    
34  The example included below shows how to use {\it pycad} to create a 2-D mesh  \section{The Unit Square}
35  in the shape of a trapezoid with a cutout area.  So the simplest geometry is the unit square. First we generate the
36    corner points
37    \begin{python}
38    from esys.pycad import *
39    p0=Point(0.,0.,0.)
40    p1=Point(1.,0.,0.)
41    p2=Point(1.,1.,0.)
42    p3=Point(0.,1.,0.)
43    \end{python}
44    which are then linked to define the edges of the square
45  \begin{python}  \begin{python}
46      from esys.pycad import *  l01=Line(p0,p1)
47      from esys.pycad.gmsh import Design  l12=Line(p1,p2)
48      from esys.finley import MakeDomain  l23=Line(p2,p3)
49    l30=Line(p3,p0)
     # A trapezoid  
     p0=Point(0.0, 0.0, 0.0)  
     p1=Point(1.0, 0.0, 0.0)  
     p2=Point(1.0, 0.5, 0.0)  
     p3=Point(0.0, 1.0, 0.0)  
     l01=Line(p0, p1)  
     l12=Line(p1, p2)  
     l23=Line(p2, p3)  
     l30=Line(p3, p0)  
     c=CurveLoop(l01, l12, l23, l30)  
   
     # A small triangular cutout  
     x0=Point(0.1, 0.1, 0.0)  
     x1=Point(0.5, 0.1, 0.0)  
     x2=Point(0.5, 0.2, 0.0)  
     x01=Line(x0, x1)  
     x12=Line(x1, x2)  
     x20=Line(x2, x0)  
     cutout=CurveLoop(x01, x12, x20)  
   
     # Create the surface with cutout  
     s=PlaneSurface(c, holes=[cutout])  
   
     # Create a Design which can make the mesh  
     d=Design(dim=2, element_size=0.05)  
   
     # Add the trapezoid with cutout  
     d.addItems(s)  
   
     # Create the geometry, mesh and Escript domain  
     d.setScriptFileName("trapezoid.geo")  
     d.setMeshFileName("trapezoid.msh")  
     domain=MakeDomain(d, integrationOrder=-1, reducedIntegrationOrder=-1,  
     optimizeLabeling=True)  
   
     # Create a file that can be read back in to python with  
     # mesh=ReadMesh(fileName)  
     domain.write("trapezoid.fly")  
50  \end{python}  \end{python}
51    The lines are put together to form a loop
52    \begin{python}
53    c=CurveLoop(l01,l12,l23,l30)
54    \end{python}
55    The orientation of the line defining the \class{CurveLoop} is important. It is assumed that the surrounded
56    area is to the left when moving along the lines from their starting points towards the end points. Moreover,
57    the line need to form a closed loop.  
58    
59    We use the \class{CurveLoop} to define a surface
60    \begin{python}
61    s=PlaneSurface(c)
62    \end{python}
63    Notice there is difference between the \class{CurveLoop} defining the boundary
64    of the surface and the actually surface \class{PlaneSurface}. This difference becomes clearer in the next example with a hole. The direction of the lines is important.
65    New we are ready to define the geometry which described by an instance of \class{Design} class:
66    \begin{python}
67    d=Design(dim=2,element_size=0.05)
68    \end{python}
69    Here we use the two dimensional domain with a local element size in the finite element mesh of $0.05$.
70    We then add the surface \code{s} to the geometry
71    \begin{python}
72    d.addItems(s)
73    \end{python}
74    This will automatically import all items used to construct \code{s} into the \class{Design} \code{d}.
75    Now we are ready to construct a \finley FEM mesh and then write it to the file \file{quad.fly}:
76    \begin{python}
77    from esys.finley import MakeDomain
78    dom=MakeDomain(d)
79    dom.write("quad.fly")
80    \end{python}
81    In some cases it is useful to access the script used to generate the geometry. You can specify a specific name
82    for the script file. In our case we use
83    \begin{python}
84    d.setScriptFileName("quad.geo")
85    \end{python}
86    If we put everything together we get the script
87    \begin{python}
88    from esys.pycad import *
89    from esys.pycad.gmsh import Design
90    from esys.finley import MakeDomain
91    p0=Point(0.,0.,0.)
92    p1=Point(1.,0.,0.)
93    p2=Point(1.,1.,0.)
94    p3=Point(0.,1.,0.)
95    l01=Line(p0,p1)
96    l12=Line(p1,p2)
97    l23=Line(p2,p3)
98    l30=Line(p3,p0)
99    c=CurveLoop(l01,l12,l23,l30)
100    s=PlaneSurface(c)
101    d=Design(dim=2,element_size=0.05)
102    d.setScriptFileName("quad.geo")
103    d.setMeshFileName("quad.msh")
104    d.addItems(s)
105    pl1=PropertySet("sides",l01,l23)
106    pl2=PropertySet("top_and_bottom",l12,l30)
107    d.addItems(pl1, pl2)
108    dom=MakeDomain(d)
109    dom.write("quad.fly")
110    \end{python}
111  This example is included with the software in  This example is included with the software in
112  \code{pycad/examples/trapezoid.py}.  If you have gmsh installed you can  \file{quad.py} in the \ExampleDirectory.
113  run the example and view the geometry and mesh with:  
114    There are three extra statements which we have not discussed yet: By default the mesh used to subdivide
115    the boundary are not written into the mesh file mainly to reduce the size of the data file. One need to explicitly add the lines to the  \Design which should be present in the mesh data. Here we additionally labeled the
116    lines on the top and the bottom with the name ``top_and_bottom`` and the lines on the left and right hand side
117    with the name ``sides`` using \class{PropertySet} objects. The labeling is convenient
118    when using tagging \index{tagging}, see Chapter~\ref{ESCRIPT CHAP}.
119    
120    \begin{figure}
121    \centerline{\includegraphics[width=\figwidth]{figures/quad.eps}}
122    \caption{Trapozid with triangle Hole.}
123    \label{fig:PYCAD 0}
124    \end{figure}
125    
126    If you have \gmshextern installed you can run the example and view the geometry and mesh with:
127    \begin{python}
128    escript quad.py
129    gmsh quad.geo
130    gmsh quad.msh
131    \end{python}
132    See Figure~\ref{fig:PYCAD 0} for a result.
133    
134    In most cases it is best practice to generate the mesh and to solve the mathematical
135    model in to different scripts. In our example you can read the \finley mesh into your simulation
136    code using
137  \begin{python}  \begin{python}
138      python trapezoid.py  from finley import ReadMesh
139      gmsh trapezoid.geo  mesh=ReadMesh("quad.fly")
     gmsh trapezoid.msh  
140  \end{python}  \end{python}
141    
142    Note that the underlying mesh generation software will not accept all
143    the geometries you can create with \pycad.  For example, \pycad
144    will happily allow you to create a 2-D \class{Design} that is a
145    closed loop with some additional points or lines lying outside of the
146    enclosed area, but \gmshextern will fail to create a mesh for it.
147    
148    \begin{figure}
149    \centerline{\includegraphics[width=\figwidth]{figures/trap.eps}}
150    \caption{Trapozid with triangle Hole.}
151    \label{fig:PYCAD 1}
152    \end{figure}
153    
154    
155    \section{Holes}
156    The example included below shows how to use \pycad to create a 2-D mesh
157    in the shape of a trapezoid with a cut-out area, see  Figure~\ref{fig:PYCAD 1}:
158    \begin{python}
159    from esys.pycad import *
160    from esys.pycad.gmsh import Design
161    from esys.finley import MakeDomain
162    
163    # A trapezoid
164    p0=Point(0.0, 0.0, 0.0)
165    p1=Point(1.0, 0.0, 0.0)
166    p2=Point(1.0, 0.5, 0.0)
167    p3=Point(0.0, 1.0, 0.0)
168    l01=Line(p0, p1)
169    l12=Line(p1, p2)
170    l23=Line(p2, p3)
171    l30=Line(p3, p0)
172    c=CurveLoop(l01, l12, l23, l30)
173    
174    # A small triangular cutout
175    x0=Point(0.1, 0.1, 0.0)
176    x1=Point(0.5, 0.1, 0.0)
177    x2=Point(0.5, 0.2, 0.0)
178    x01=Line(x0, x1)
179    x12=Line(x1, x2)
180    x20=Line(x2, x0)
181    cutout=CurveLoop(x01, x12, x20)
182    
183    # Create the surface with cutout
184    s=PlaneSurface(c, holes=[cutout])
185    
186    # Create a Design which can make the mesh
187    d=Design(dim=2, element_size=0.05)
188    
189    # Add the trapezoid with cutout
190    d.addItems(s)
191    
192    # Create the geometry, mesh and Escript domain
193    d.setScriptFileName("trapezoid.geo")
194    d.setMeshFileName("trapezoid.msh")
195    domain=MakeDomain(d)
196    # write mesh to a finley file:
197    domain.write("trapezoid.fly")
198    \end{python}
199    This example is included with the software in
200    \file{trapezoid.py} in the \ExampleDirectory.
201    
202  A \code{CurveLoop} is used to connect several lines into a single curve.  A \code{CurveLoop} is used to connect several lines into a single curve.
203  It is used in the example above to create the trapezoidal outline for the grid  It is used in the example above to create the trapezoidal outline for the grid
204  and also for the triangular cutout area.  and also for the triangular cutout area.
205  You can use any number of lines when creating a \code{CurveLoop}, but  You can use any number of lines when creating a \class{CurveLoop}, but
206  the end of one line must be identical to the start of the next.  the end of one line must be identical to the start of the next.
207    
 Sometimes you might see us write \code{-c} where \code{c} is a  
 \code{CurveLoop}.  This is the reverse curve of the curve \code{c}.  
 It is identical to the original except that its points are traversed  
 in the opposite order.  This may make it easier to connect two curves  
 in a \code{CurveLoop}.  
   
 The example python script above calls both  
 \code{d.setScriptFileName()} and \code{d.setMeshFileName()}.  You need  
 only call these if you wish to save the gmsh geometry and mesh files.  
208    
209  Note that the underlying mesh generation software will not accept all  \begin{figure}
210  the geometries you can create with {\it pycad}.  For example, {\it  \centerline{\includegraphics[width=\figwidth]{figures/brick.eps}}
211  pycad} will happily allow you to create a 2-D {\it Design} that is a  \caption{Three dimensional Block.}
212  closed loop with some additional points or lines lying outside of the  \label{fig:PYCAD 2}
213  enclosed area, but gmsh will fail to create a mesh for it.  \end{figure}
214    
215    \section{A 3D example}
216    In this section we discuss the definition of 3D geometries. The example is the unit cube, see Figure~\ref{fig:PYCAD 2}. First we generate the vertices of the cube:
217    \begin{python}
218    from esys.pycad import *
219    p0=Point(0.,0.,0.)
220    p1=Point(1.,0.,0.)
221    p2=Point(0.,1.,0.)
222    p3=Point(1.,1.,0.)
223    p4=Point(0.,0.,1.)
224    p5=Point(1.,0.,1.)
225    p6=Point(0.,1.,1.)
226    p7=Point(1.,1.,1.)
227    \end{python}
228    We connect the points to form the bottom and top surfaces of the cube:
229    \begin{python}
230    l01=Line(p0,p1)
231    l13=Line(p1,p3)
232    l32=Line(p3,p2)
233    l20=Line(p2,p0)
234    bottom=PlaneSurface(CurveLoop(l01,l13,l32,l20))
235    \end{python}
236    and
237    \begin{python}
238    l45=Line(p4,p5)
239    l57=Line(p5,p7)
240    l76=Line(p7,p6)
241    l64=Line(p6,p4)
242    top=PlaneSurface(CurveLoop(l45,l57,l76,l64))
243    \end{python}
244    To form the front face we introduce the two additional lines connecting the left and right front
245    points of the the \code{top} and \code{bottom} face:
246    \begin{python}
247    l15=Line(p1,p5)
248    l40=Line(p4,p0)
249    \end{python}
250    To form the front face we encounter the problem as the line \code{l45} used to define the
251    \code{top} face is pointing the wrong direction.  In \pycad you can reversing direction of an
252    object by changing its sign. So we write \code{-l45} to indicate that the direction is to be reversed. With this notation we can write
253    \begin{python}
254    front=PlaneSurface(CurveLoop(l01,l15,-l45,l40))
255    \end{python}
256    Keep in mind that if you use \code{Line(p4,p5)} instead \code{-l45} both objects are treated as different although the connecting the same points with a straight line in the same direction. The resulting geometry would include an opening along the \code{p4}--\code{p5} connection. This will lead to an inconsistent mesh and may result in a failure of the volumetric mesh generator. Similarly we can define the other sides of the cube:
257    \begin{python}
258    l37=Line(p3,p7)
259    l62=Line(p6,p2)
260    back=PlaneSurface(CurveLoop(l32,-l62,-l76,-l37))
261    left=PlaneSurface(CurveLoop(-l40,-l64,l62,l20))
262    right=PlaneSurface(CurveLoop(-l15,l13,l37,-l57))
263    \end{python}
264    We can now put the six surfaces together to form a \class{SurfaceLoop} defining the
265    boundary of the volume of the cube:
266    \begin{python}
267    sl=SurfaceLoop(top,-bottom,front,back,left,right)
268    v=Volume(sl)
269    \end{python}
270    Similar to the definition of a \code{CurvedLoop} the orientation of the surfaces \code{SurfaceLoop} is relevant. In fact the surface normal direction defined by the the right hand rule needs to point outwards as indicated by the surface normals in
271    Figure~\ref{fig:PYCAD 2}. As the \code{bottom} face is directed upwards it is inserted with the minus sign
272    into the \code{SurfaceLoop} in order to adjust the orientation of the surface.
273    
274    As in the 2D case, the \class{Design} class is used to define the geometry:
275    \begin{python}
276    from esys.pycad.gmsh import Design
277    from esys.finley import MakeDomain
278    
279    des=Design(dim=3, element_size = 0.1, keep_files=True)
280    des.setScriptFileName("brick.geo")
281    des.addItems(v, top, bottom, back, front, left , right)
282    
283    dom=MakeDomain(des)
284    dom.write("brick.fly")
285    \end{python}
286    Note that the \finley mesh file \file{brick.fly} will contain the
287    triangles used to define the surfaces as they are added to the \class{Design}.
288    The example script of the cube is included with the software in
289    \file{brick.py} in the \ExampleDirectory.
290    
291    \begin{figure}
292    \centerline{\includegraphics[width=\figwidth]{figures/refine1.eps}}
293    \caption{Local refinement at the origin by \code{local_scale=0.01} with \code{element_size=0.3} and number of elements on the top set to 10.}
294    \label{fig:PYCAD 5}
295    \end{figure}
296    
297    \section{Element Sizes}
298    The element size used globally is defined by the
299    \code{element_size} argument of the \class{Design}. The mesh generator
300    will try to use this mesh size everywhere in the geometry. In some cases it can be
301    desirable to use locally a finer mesh. A local refinement can be defined at each
302    \class{Point}:
303    \begin{python}
304    p0=Point(0.,0.,0.,local_scale=0.01)
305    \end{python}
306    Here the mesh generator will create a mesh with an element size which is by the factor \code{0.01}
307    times smaller than the global mesh size \code{element_size=0.3}, see Figure~\ref{fig:PYCAD 5}. The point where a refinement is defined must be a point of curve used to define the geometry.
308    
309    Alternatively, one can define a mesh size along a curve by defining the number of elements to be used to subdivide the curve. For instance, to use $20$ element on line \code{l23} on uses:
310    \begin{python}
311    l23=Line(p2, p3)
312    l23.setElementDistribution(20)
313    \end{python}
314    Setting the number of elements on a curve overwrites the global mesh size \code{element_size}. The result is shown in Figure~\ref{fig:PYCAD 5}.
315    
316    
317    
# Line 121  enclosed area, but gmsh will fail to cre Line 321  enclosed area, but gmsh will fail to cre
321    
322  \subsection{Primitives}  \subsection{Primitives}
323    
324  Some of the most commonly-used objects in {\it pycad} are listed here. For a more complete  Some of the most commonly-used objects in \pycad are listed here. For a more complete
325  list see the full API documentation.  list see the full API documentation.
326    
327  \begin{classdesc}{Point}{x=0.,y=0.,z=0.\optional{,local_scale=1.}}  \begin{classdesc}{Point}{x=0.,y=0.,z=0.\optional{,local_scale=1.}}
# Line 132  Create a point with from coordinates wit Line 332  Create a point with from coordinates wit
332  Create a line with between starting and ending points.  Create a line with between starting and ending points.
333  \end{classdesc}  \end{classdesc}
334  \begin{methoddesc}[Line]{setElementDistribution}{n\optional{,progression=1\optional{,createBump=\False}}}  \begin{methoddesc}[Line]{setElementDistribution}{n\optional{,progression=1\optional{,createBump=\False}}}
335  Defines the number of elements on the line. If set it overwrites the local length setting which would be applied. The progression factor \var{progression} defines the change of element size between naighboured elements. If \var{createBump} is set  Defines the number of elements on the line. If set it overwrites the local length setting which would be applied. The progression factor \var{progression} defines the change of element size between neighboured elements. If \var{createBump} is set
336  progression is applied towards the center of the line.  progression is applied towards the centre of the line.
337  \end{methoddesc}  \end{methoddesc}
338  \begin{methoddesc}[Line]{resetElementDistribution}{}  \begin{methoddesc}[Line]{resetElementDistribution}{}
339  removes a previously set element distribution from the line.  removes a previously set element distribution from the line.
# Line 149  no element distribution is set None is r Line 349  no element distribution is set None is r
349  A spline curve defined by a list of points \var{point0}, \var{point1},....  A spline curve defined by a list of points \var{point0}, \var{point1},....
350  \end{classdesc}  \end{classdesc}
351  \begin{methoddesc}[Spline]{setElementDistribution}{n\optional{,progression=1\optional{,createBump=\False}}}  \begin{methoddesc}[Spline]{setElementDistribution}{n\optional{,progression=1\optional{,createBump=\False}}}
352  Defines the number of elements on the line. If set it overwrites the local length setting which would be applied. The progression factor \var{progression} defines the change of element size between naighboured elements. If \var{createBump} is set  Defines the number of elements on the line. If set it overwrites the local length setting which would be applied. The progression factor \var{progression} defines the change of element size between neighboured elements. If \var{createBump} is set
353  progression is applied towards the center of the line.  progression is applied towards the centre of the line.
354  \end{methoddesc}  \end{methoddesc}
355  \begin{methoddesc}[Spline]{resetElementDistribution}{}  \begin{methoddesc}[Spline]{resetElementDistribution}{}
356  removes a previously set element distribution from the line.  removes a previously set element distribution from the line.
# Line 165  no element distribution is set None is r Line 365  no element distribution is set None is r
365  A B-spline curve defined by a list of points \var{point0}, \var{point1},....  A B-spline curve defined by a list of points \var{point0}, \var{point1},....
366  \end{classdesc}  \end{classdesc}
367  \begin{methoddesc}[BSpline]{setElementDistribution}{n\optional{,progression=1\optional{,createBump=\False}}}  \begin{methoddesc}[BSpline]{setElementDistribution}{n\optional{,progression=1\optional{,createBump=\False}}}
368  Defines the number of elements on the line. If set it overwrites the local length setting which would be applied. The progression factor \var{progression} defines the change of element size between naighboured elements. If \var{createBump} is set  Defines the number of elements on the line. If set it overwrites the local length setting which would be applied. The progression factor \var{progression} defines the change of element size between neighboured elements. If \var{createBump} is set
369  progression is applied towards the center of the line.  progression is applied towards the centre of the line.
370  \end{methoddesc}  \end{methoddesc}
371  \begin{methoddesc}[BSpline]{resetElementDistribution}{}  \begin{methoddesc}[BSpline]{resetElementDistribution}{}
372  removes a previously set element distribution from the line.  removes a previously set element distribution from the line.
# Line 181  no element distribution is set None is r Line 381  no element distribution is set None is r
381  A Brezier spline curve defined by a list of points \var{point0}, \var{point1},....  A Brezier spline curve defined by a list of points \var{point0}, \var{point1},....
382  \end{classdesc}  \end{classdesc}
383  \begin{methoddesc}[BezierCurve]{setElementDistribution}{n\optional{,progression=1\optional{,createBump=\False}}}  \begin{methoddesc}[BezierCurve]{setElementDistribution}{n\optional{,progression=1\optional{,createBump=\False}}}
384  Defines the number of elements on the line. If set it overwrites the local length setting which would be applied. The progression factor \var{progression} defines the change of element size between naighboured elements. If \var{createBump} is set  Defines the number of elements on the line. If set it overwrites the local length setting which would be applied. The progression factor \var{progression} defines the change of element size between neighboured elements. If \var{createBump} is set
385  progression is applied towards the center of the line.  progression is applied towards the centre of the line.
386  \end{methoddesc}  \end{methoddesc}
387  \begin{methoddesc}[BezierCurve]{resetElementDistribution}{}  \begin{methoddesc}[BezierCurve]{resetElementDistribution}{}
388  removes a previously set element distribution from the line.  removes a previously set element distribution from the line.
# Line 193  number of elements, progression factor a Line 393  number of elements, progression factor a
393  no element distribution is set None is returned.  no element distribution is set None is returned.
394  \end{methoddesc}  \end{methoddesc}
395    
396  \begin{classdesc}{Arc}{center_point, start_point, end_point}  \begin{classdesc}{Arc}{centre_point, start_point, end_point}
397  Create an arc by specifying a center for a circle and start and end points. An arc may subtend an angle of at most $\pi$ radians.  Create an arc by specifying a centre for a circle and start and end points. An arc may subtend an angle of at most $\pi$ radians.
398  \end{classdesc}  \end{classdesc}
399  \begin{methoddesc}[Arc]{setElementDistribution}{n\optional{,progression=1\optional{,createBump=\False}}}  \begin{methoddesc}[Arc]{setElementDistribution}{n\optional{,progression=1\optional{,createBump=\False}}}
400  Defines the number of elements on the line. If set it overwrites the local length setting which would be applied. The progression factor \var{progression} defines the change of element size between naighboured elements. If \var{createBump} is set  Defines the number of elements on the line. If set it overwrites the local length setting which would be applied. The progression factor \var{progression} defines the change of element size between neighboured elements. If \var{createBump} is set
401  progression is applied towards the center of the line.  progression is applied towards the centre of the line.
402  \end{methoddesc}  \end{methoddesc}
403  \begin{methoddesc}[Arc]{resetElementDistribution}{}  \begin{methoddesc}[Arc]{resetElementDistribution}{}
404  removes a previously set element distribution from the line.  removes a previously set element distribution from the line.
# Line 229  Set \var{max_deviation}=\var{None} to re Line 429  Set \var{max_deviation}=\var{None} to re
429  applies 2D transfinite meshing to the surface.  applies 2D transfinite meshing to the surface.
430  \var{orientation} defines the orientation of triangles. Allowed values  \var{orientation} defines the orientation of triangles. Allowed values
431  are \var{``Left''}, \var{``Right''} or \var{``Alternate''}. The  are \var{``Left''}, \var{``Right''} or \var{``Alternate''}. The
432  boundary of the surface muist be defined by three or four lines where an  boundary of the surface must be defined by three or four lines where an
433  element distribution must be defined on all faces where opposite  element distribution must be defined on all faces where opposite
434  faces uses the same element distribution. No holes must be present.  faces uses the same element distribution. No holes must be present.
435  \end{methoddesc}  \end{methoddesc}
# Line 251  Set \var{max_deviation}=\var{None} to re Line 451  Set \var{max_deviation}=\var{None} to re
451  applies 2D transfinite meshing to the surface.  applies 2D transfinite meshing to the surface.
452  \var{orientation} defines the orientation of triangles. Allowed values  \var{orientation} defines the orientation of triangles. Allowed values
453  are \var{``Left''}, \var{``Right''} or \var{``Alternate''}. The  are \var{``Left''}, \var{``Right''} or \var{``Alternate''}. The
454  boundary of the surface muist be defined by three or four lines where an  boundary of the surface must be defined by three or four lines where an
455  element distribution must be defined on all faces where opposite  element distribution must be defined on all faces where opposite
456  faces uses the same element distribution. No holes must be present.  faces uses the same element distribution. No holes must be present.
457  \end{methoddesc}  \end{methoddesc}
# Line 287  into a \numpy object of shape $(3,)$. Line 487  into a \numpy object of shape $(3,)$.
487  defines a rotation by \var{angle} around axis through point \var{point} and direction \var{axis}.  defines a rotation by \var{angle} around axis through point \var{point} and direction \var{axis}.
488  \var{axis} and \var{point} can be any object that can be converted  \var{axis} and \var{point} can be any object that can be converted
489  into a \numpy object of shape $(3,)$.  into a \numpy object of shape $(3,)$.
490  \var{axis} does not have to be normalized but must have positive length. The right hand rule~\cite{RIGHTHANDRULE}  \var{axis} does not have to be normalised but must have positive length. The right hand rule~\cite{RIGHTHANDRULE}
491  applies.  applies.
492  \end{classdesc}  \end{classdesc}
493    
494    
495  \begin{classdesc}{Dilation}{\optional{factor=1., \optional{center=[0,0,0]}}}  \begin{classdesc}{Dilation}{\optional{factor=1., \optional{centre=[0,0,0]}}}
496  defines a dilation by the expansion/contraction \var{factor} with  defines a dilation by the expansion/contraction \var{factor} with
497  \var{center} as the dilation center.  \var{centre} as the dilation centre.
498  \var{center} can be any object that can be converted  \var{centre} can be any object that can be converted
499  into a \numpy object of shape $(3,)$.  into a \numpy object of shape $(3,)$.
500  \end{classdesc}  \end{classdesc}
501    
# Line 304  defines a reflection on a plane defined Line 504  defines a reflection on a plane defined
504  where $n$ is the surface normal \var{normal} and $d$ is the plane \var{offset}.  where $n$ is the surface normal \var{normal} and $d$ is the plane \var{offset}.
505  \var{normal} can be any object that can be converted  \var{normal} can be any object that can be converted
506  into a \numpy object of shape $(3,)$.  into a \numpy object of shape $(3,)$.
507  \var{normal} does not have to be normalized but must have positive length.  \var{normal} does not have to be normalised but must have positive length.
508  \end{classdesc}  \end{classdesc}
509    
510  \begin{datadesc}{DEG}  \begin{datadesc}{DEG}
# Line 374  returns the tag used for this property s Line 574  returns the tag used for this property s
574  \modulesynopsis{Python geometry description and meshing interface}  \modulesynopsis{Python geometry description and meshing interface}
575    
576  The class and methods described here provide an interface to the mesh  The class and methods described here provide an interface to the mesh
577  generation software, which is currently gmsh.  This interface could be  generation software, which is currently \gmshextern.  This interface could be
578  adopted to triangle or another mesh generation package if this is  adopted to triangle or another mesh generation package if this is
579  deemed to be desirable in the future.  deemed to be desirable in the future.
580    
# Line 440  returns \True if work files are kept. Ot Line 640  returns \True if work files are kept. Ot
640  \end{methoddesc}  \end{methoddesc}
641    
642  \begin{methoddesc}[Design]{setScriptFileName}{\optional{name=None}}  \begin{methoddesc}[Design]{setScriptFileName}{\optional{name=None}}
643  set the filename for the gmsh input script. if no name is given a name with extension "geo" is generated.  set the file name for the gmsh input script. if no name is given a name with extension "geo" is generated.
644  \end{methoddesc}  \end{methoddesc}
645    
646  \begin{methoddesc}[Design]{getScriptFileName}{}  \begin{methoddesc}[Design]{getScriptFileName}{}

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