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13    
14    \chapter{The \finley Module}\label{CHAPTER ON FINLEY}
15    %\declaremodule{extension}{finley}
16    %\modulesynopsis{Solving linear, steady partial differential equations using finite elements}
17    
18    {\it finley} is a library of C functions solving linear, steady partial
19    differential equations\index{partial differential equations} (PDEs) or systems
20    of PDEs using isoparametrical finite elements\index{FEM!isoparametrical}.
21    It supports unstructured 1D, 2D and 3D meshes.
22    The module \finley provides access to the library through the \LinearPDE class
23    of \escript supporting its full functionality.
24    {\it finley} is parallelized using the OpenMP\index{OpenMP} paradigm.
25    
26    \section{Formulation}
27    For a single PDE that has a solution with a single component the linear PDE is
28    defined in the following form:
29    \begin{equation}\label{FINLEY.SINGLE.1}
30    \begin{array}{cl} &
31    \displaystyle{
32    \int_{\Omega}
33    A_{jl} \cdot v_{,j}u_{,l}+ B_{j} \cdot v_{,j} u+ C_{l} \cdot v u_{,l}+D \cdot vu \; d\Omega }  \\
34    + & \displaystyle{\int_{\Gamma} d \cdot vu \; d{\Gamma} }
35    +  \displaystyle{\int_{\Gamma^{contact}} d^{contact} \cdot [v][u] \; d{\Gamma} } \\
36    = & \displaystyle{\int_{\Omega}  X_{j} \cdot v_{,j}+ Y \cdot v \; d\Omega }\\
37    + & \displaystyle{\int_{\Gamma} y \cdot v \; d{\Gamma}}  +
38    \displaystyle{\int_{\Gamma^{contact}} y^{contact}\cdot [v] \; d{\Gamma}} \\
39    \end{array}
40    \end{equation}
41    
42  \chapter{ The module \finley}  \section{Meshes}
43   \label{CHAPTER ON FINLEY}  \label{FINLEY MESHES}
44    
45  \begin{figure}  \begin{figure}
46  \centerline{\includegraphics[width=\figwidth]{figures/FinleyMesh.eps}}  \centerline{\includegraphics{FinleyMesh}}
47  \caption{Subdivision of an Ellipse into triangles order 1 (\finleyelement{Tri3})}  \caption{Subdivision of an Ellipse into triangles order 1 (\finleyelement{Tri3})}
48  \label{FINLEY FIG 0}  \label{FINLEY FIG 0}
49  \end{figure}  \end{figure}
50    
51    To understand the usage of \finley one needs to have an understanding of how
52    the finite element meshes\index{FEM!mesh} are defined.
53    \fig{FINLEY FIG 0} shows an example of the subdivision of an ellipse into
54    so-called elements\index{FEM!elements}\index{element}.
55    In this case, triangles have been used but other forms of subdivisions can be
56    constructed, e.g. quadrilaterals or, in the three-dimensional case, into
57    tetrahedra and hexahedra. The idea of the finite element method is to
58    approximate the solution by a function which is a polynomial of a certain order
59    and is continuous across its boundary to neighbour elements.
60    In the example of \fig{FINLEY FIG 0} a linear polynomial is used on each
61    triangle. As one can see, the triangulation is quite a poor approximation of
62    the ellipse. It can be improved by introducing a midpoint on each element edge
63    then positioning those nodes located on an edge expected to describe the
64    boundary, onto the boundary.
65    In this case the triangle gets a curved edge which requires a parameterization
66    of the triangle using a quadratic polynomial.
67    For this case, the solution is also approximated by a piecewise quadratic
68    polynomial (which explains the name isoparametrical elements),
69    see \Ref{Zienc,NumHand} for more details.
70    \finley also supports macro elements\index{macro elements}.
71    For these elements a piecewise linear approximation is used on an element which
72    is further subdivided (in the case of \finley halved).
73    As such, these elements do not provide more than a further mesh refinement but
74    should be used in the case of incompressible flows, see \class{StokesProblemCartesian}.
75    For these problems a linear approximation of the pressure across the element is
76    used (use the \ReducedSolutionFS) while the refined element is used to
77    approximate velocity. So a macro element provides a continuous pressure
78    approximation together with a velocity approximation on a refined mesh.
79    This approach is necessary to make sure that the incompressible flow has a
80    unique solution.
81    
82    The union of all elements defines the domain of the PDE.
83    Each element is defined by the nodes used to describe its shape.
84    In \fig{FINLEY FIG 0} the element, which has type \finleyelement{Tri3}, with
85    element reference number $19$\index{element!reference number} is defined by the
86    nodes with reference numbers $9$, $11$ and $0$\index{node!reference number}.
87    Notice that the order is counterclockwise.
88    The coefficients of the PDE are evaluated at integration nodes with each
89    individual element.
90    For quadrilateral elements a Gauss quadrature scheme is used.
91    In the case of triangular elements a modified form is applied.
92    The boundary of the domain is also subdivided into elements\index{element!face}.
93    In \fig{FINLEY FIG 0} line elements with two nodes are used.
94    The elements are also defined by their describing nodes, e.g. the face element
95    with reference number $20$, which has type \finleyelement{Line2}, is defined by
96    the nodes with the reference numbers $11$ and $0$.
97    Again the order is crucial, if moving from the first to second node the domain
98    has to lie on the left hand side (in the case of a two-dimensional surface
99    element the domain has to lie on the left hand side when moving
100    counterclockwise). If the gradient on the surface of the domain is to be
101    calculated rich face elements need to be used. Rich elements on a face are
102    identical to interior elements but with a modified order of nodes such that the
103    'first' face of the element aligns with the surface of the domain.
104    In \fig{FINLEY FIG 0} elements of the type \finleyelement{Tri3Face} are used.
105    The face element reference number $20$ as a rich face element is defined by the
106    nodes with reference numbers $11$, $0$ and $9$.
107    Notice that the face element $20$ is identical to the interior element $19$
108    except that, in this case, the order of the node is different to align the first
109    edge of the triangle (which is the edge starting with the first node) with the
110    boundary of the domain.
111    
112    Be aware that face elements and elements in the interior of the domain must
113    match, i.e. a face element must be the face of an interior element or, in case
114    of a rich face element, it must be identical to an interior element.
115    If no face elements are specified \finley implicitly assumes homogeneous
116    natural boundary conditions\index{natural boundary conditions!homogeneous},
117    i.e. \var{d}=$0$ and \var{y}=$0$, on the entire boundary of the domain.
118    For inhomogeneous natural boundary conditions\index{natural boundary conditions!inhomogeneous},
119    the boundary must be described by face elements.
120    
121  \begin{figure}  \begin{figure}
122  \centerline{\includegraphics[width=\figwidth]{figures/FinleyContact.eps}}  \centerline{\includegraphics{FinleyContact}}
123  \caption{Mesh around a contact region (\finleyelement{Rec4})}  \caption{Mesh around a contact region (\finleyelement{Rec4})}
124  \label{FINLEY FIG 01}  \label{FINLEY FIG 01}
125  \end{figure}  \end{figure}
126    
127  \declaremodule{extension}{finley} \modulesynopsis{Solving linear, steady partial differential equations using  If discontinuities of the PDE solution are considered, contact
128  finite elements}  elements\index{element!contact}\index{contact conditions} are introduced to
129    describe the contact region $\Gamma^{contact}$ even if $d^{contact}$ and
130  {\it finley} is a library of C functions solving linear, steady partial differential equations  $y^{contact}$ are zero.
131  \index{partial differential equations} (PDEs) or systems of PDEs using isoparametrical finite  \fig{FINLEY FIG 01} shows a simple example of a mesh of rectangular elements
132  elements \index{FEM!isoparametrical}.  around a contact region $\Gamma^{contact}$\index{element!contact}.
133  It supports unstructured, 1D, 2D and 3D meshes. The module \finley provides an access to the  The contact region is described by the elements $4$, $3$ and $6$.
134  library through the \LinearPDE class of \escript supporting its full functionality. {\it finley}  Their element type is \finleyelement{Line2_Contact}.
135  is parallelized using the OpenMP \index{OpenMP} paradigm.  The nodes $9$, $12$, $6$ and $5$ define contact element $4$, where the
136    coordinates of nodes $12$ and $5$ and nodes $4$ and $6$ are identical, with the
137  \section{Formulation}  idea that nodes $12$ and $9$ are located above and nodes $5$ and $6$ below the
138    contact region.
139  For a single PDE with a solution with a single component the linear PDE is defined in the  Again, the order of the nodes within an element is crucial.
140  following form:  There is also the option of using rich elements if the gradient is to be
141  \begin{equation}\label{FINLEY.SINGLE.1}  calculated on the contact region. Similarly to the rich face elements these
142  \begin{array}{cl} &  are constructed from two interior elements by reordering the nodes such that
143  \displaystyle{  the 'first' face of the element above and the 'first' face of the element below
144  \int\hackscore{\Omega}  the contact regions line up. The rich version of element $4$ is of type
145  A\hackscore{jl} \cdot v\hackscore{,j}u\hackscore{,l}+ B\hackscore{j} \cdot v\hackscore{,j} u+ C\hackscore{l} \cdot v u\hackscore{,l}+D \cdot vu \; d\Omega }  \\  \finleyelement{Rec4Face_Contact} and is defined by the nodes $9$, $12$, $16$,
146  + & \displaystyle{\int\hackscore{\Gamma} d \cdot vu \; d{\Gamma} }  $18$, $6$, $5$, $0$ and $2$.
147  +  \displaystyle{\int\hackscore{\Gamma^{contact}} d^{contact} \cdot [v][u] \; d{\Gamma} } \\  \tab{FINLEY TAB 1} shows the interior element types and the corresponding
148  = & \displaystyle{\int\hackscore{\Omega}  X\hackscore{j} \cdot v\hackscore{,j}+ Y \cdot v \; d\Omega }\\  element types to be used on the face and contacts.
149  + & \displaystyle{\int\hackscore{\Gamma} y \cdot v \; d{\Gamma}}  +  \fig{FINLEY.FIG:1}, \fig{FINLEY.FIG:2} and \fig{FINLEY.FIG:4} show the ordering
150  \displaystyle{\int\hackscore{\Gamma^{contact}} y^{contact}\cdot [v] \; d{\Gamma}} \\  of the nodes within an element.
 \end{array}  
 \end{equation}  
   
 \section{Meshes}  
 To understand the usage of \finley one needs to have an understanding of how the finite element meshes  
 \index{FEM!mesh} are defined. \fig{FINLEY FIG 0} shows an example of the  
 subdivision of an ellipse into so called elements \index{FEM!elements} \index{element}.  
 In this case, triangles have been used but other forms of subdivisions  
 can be constructed, e.g. into quadrilaterals or, in the three dimensional case, into tetrahedrons  
 and hexahedrons. The idea of the finite element method is to approximate the solution by a function  
 which is a polynomial of a certain order and is continuous across it boundary to neighbour elements.  
 In the example of \fig{FINLEY FIG 0} a linear polynomial is used on each triangle. As one can see, the triangulation  
 is quite a poor approximation of the ellipse. It can be improved by introducing a midpoint on each element edge then  
 positioning those nodes located on an edge expected to describe the boundary, onto the boundary.  
 In this case the triangle gets a curved edge which requires a parametrization of the triangle using a  
 quadratic polynomial. For this case, the solution is also approximated by a piecewise quadratic polynomial  
 (which explains the name isoparametrical elements), see \Ref{Zienc,NumHand} for more details.    
   
 The union of all elements defines the domain of the PDE.  
 Each element is defined by the nodes used to describe its shape. In \fig{FINLEY FIG 0} the element,  
 which has type \finleyelement{Tri3},  
 with element reference number $19$ \index{element!reference number} is defined by the nodes  
 with reference numbers $9$, $11$ and $0$ \index{node!reference number}. Notice that the order is counterclockwise.  
 The coefficients of the PDE are evaluated at integration nodes with each individual element.  
 For quadrilateral elements a Gauss quadrature scheme is used. In the case of triangular elements a  
 modified form is applied. The boundary of the domain is also subdivided into elements. \index{element!face} In \fig{FINLEY FIG 0}  
 line elements with two nodes are used. The elements are also defined by their describing nodes, e.g.  
 the face element reference number $20$ which has type \finleyelement{Line2} is defined by the nodes  
 with the reference numbers $11$ and $0$. Again the order is crucial, if moving from the first  
 to second node the domain has to lie on the left hand side (in the case of a two dimension surface element  
 the domain has to lie on the left hand side when moving counterclockwise). If the gradient on the  
 surface of the domain is to be calculated rich face elements face to be used. Rich elements on a face  
 are identical to interior elements but with a modified order of nodes such that the 'first' face of the element aligns  
 with the surface of the domian. In \fig{FINLEY FIG 0}  
 elements of the type \finleyelement{Tri3Face} are used.  
 The face element reference number $20$ as a rich face element is defined by the nodes  
 with reference numbers $11$, $0$ and $9$. Notice that the face element $20$ is identical to the  
 interior element $19$ except that, in this case, the order of the node is different to align the first  
 edge of the triangle (which is the edge starting with the first node) with the boundary of the domain.  
   
 Be aware that face elements and elements in the interior of the domain must match, i.e. a face element must be the face  
 of an interior element or, in case of a rich face element, it must be identical to an interior element.  
 If no face elements are specified  
 \finley implicitly assumes homogeneous natural boundary conditions \index{natural boundary conditions!homogeneous},  
 i.e. \var{d}=$0$ and \var{y}=$0$, on the entire boundary of the domain. For    
 inhomogeneous natural boundary conditions \index{natural boundary conditions!inhomogeneous},  
 the boundary must be described by face elements.  
   
 If discontinuities of the PDE solution are considered contact elements  
 \index{element!contact}\index{contact conditions} are introduced to describe the contact region $\Gamma^{contact}$  
 even if $d^{contact}$ and $y^{contact}$ are zero. \fig{FINLEY FIG 01} shows a simple example of a mesh  
 of rectangular elements around a contact region $\Gamma^{contact}$ \index{element!contact}.  
 The contact region is described by the  
 elements $4$, $3$ and $6$. Their element type is \finleyelement{Line2_Contact}.  
 The nodes $9$, $12$, $6$, $5$ define contact element $4$, where the coordinates of nodes $12$ and $5$ and  
 nodes $4$ and $6$ are identical with the idea that nodes $12$ and $9$ are located above and  
 nodes $5$ and $6$ below the contact region.    
 Again, the order of the nodes within an element is crucial. There is also the option of using rich elements  
 if the gradient is to be calculated on the contact region. Similarly to the rich face elements  
 these are constructed from two interior elements by reordering the nodes such that  
 the 'first' face of the element above and the 'first' face of the element below the  
 contact regions line up.  The rich version of element  
 $4$ is of type \finleyelement{Rec4Face_Contact} and is defined by the nodes $9$, $12$, $16$, $18$, $6$, $5$, $0$ and  
 $2$.  
   
 \tab{FINLEY TAB 1} shows the interior element types and the corresponding element types to be used  
 on the face and contacts. \fig{FINLEY.FIG:1}, \fig{FINLEY.FIG:2} and \fig{FINLEY.FIG:4} show the ordering of  
 the nodes within an element.  
151    
152  \begin{table}  \begin{table}
153  \begin{tablev}{l|llll}{textrm}{interior}{face}{rich face}{contact}{rich contact}  \centering
154  \linev{\finleyelement{Line2}}{\finleyelement{Point1}}{\finleyelement{Line2Face}}{\finleyelement{Point1_Contact}}{\finleyelement{Line2Face_Contact}}  \begin{tabular}{l|llll}
155  \linev{\finleyelement{Line3}}{\finleyelement{Point1}}{\finleyelement{Line3Face}}{\finleyelement{Point1_Contact}}{\finleyelement{Line3Face_Contact}}  \textbf{interior}&\textbf{face}&\textbf{rich face}&\textbf{contact}&\textbf{rich contact}\\
156  \linev{\finleyelement{Tri3}}{\finleyelement{Line2}}{\finleyelement{Tri3Face}}{\finleyelement{Line2_Contact}}{\finleyelement{Tri3Face_Contact}}  \hline
157  \linev{\finleyelement{Tri6}}{\finleyelement{Line3}}{\finleyelement{Tri6Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Tri6Face_Contact}}  \finleyelement{Line2} & \finleyelement{Point1} & \finleyelement{Line2Face} & \finleyelement{Point1_Contact} & \finleyelement{Line2Face_Contact}\\
158  \linev{\finleyelement{Rec4}}{\finleyelement{Line2}}{\finleyelement{Rec4Face}}{\finleyelement{Line2_Contact}}{\finleyelement{Rec4Face_Contact}}  \finleyelement{Line3} & \finleyelement{Point1} & \finleyelement{Line3Face} & \finleyelement{Point1_Contact} & \finleyelement{Line3Face_Contact}\\
159  \linev{\finleyelement{Rec8}}{\finleyelement{Line3}}{\finleyelement{Rec8Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Rec8Face_Contact}}  \finleyelement{Tri3} & \finleyelement{Line2} & \finleyelement{Tri3Face} & \finleyelement{Line2_Contact} & \finleyelement{Tri3Face_Contact}\\
160  \linev{\finleyelement{Rec9}}{\finleyelement{Line3}}{\finleyelement{Rec9Face}}{\finleyelement{Line3_Contact}}{\finleyelement{Rec9Face_Contact}}  \finleyelement{Tri6} & \finleyelement{Line3} & \finleyelement{Tri6Face} & \finleyelement{Line3_Contact} & \finleyelement{Tri6Face_Contact}\\
161  \linev{\finleyelement{Tet4}}{\finleyelement{Tri6}}{\finleyelement{Tet4Face}}{\finleyelement{Tri6_Contact}}{\finleyelement{Tet4Face_Contact}}  \finleyelement{Rec4} & \finleyelement{Line2} & \finleyelement{Rec4Face} & \finleyelement{Line2_Contact} & \finleyelement{Rec4Face_Contact}\\
162  \linev{\finleyelement{Tet10}}{\finleyelement{Tri9}}{\finleyelement{Tet10Face}}{\finleyelement{Tri9_Contact}}{\finleyelement{Tet10Face_Contact}}  \finleyelement{Rec8} & \finleyelement{Line3} & \finleyelement{Rec8Face} & \finleyelement{Line3_Contact} & \finleyelement{Rec8Face_Contact}\\
163  \linev{\finleyelement{Hex8}}{\finleyelement{Rec4}}{\finleyelement{Hex8Face}}{\finleyelement{Rec4_Contact}}{\finleyelement{Hex8Face_Contact}}  \finleyelement{Rec9} & \finleyelement{Line3} & \finleyelement{Rec9Face} & \finleyelement{Line3_Contact} & \finleyelement{Rec9Face_Contact}\\
164  \linev{\finleyelement{Hex20}}{\finleyelement{Rec8}}{\finleyelement{Hex20Face}}{\finleyelement{Rec8_Contact}}{\finleyelement{Hex20Face_Contact}}  \finleyelement{Tet4} & \finleyelement{Tri6} & \finleyelement{Tet4Face} & \finleyelement{Tri6_Contact} & \finleyelement{Tet4Face_Contact}\\
165  \end{tablev}  \finleyelement{Tet10} & \finleyelement{Tri9} & \finleyelement{Tet10Face} & \finleyelement{Tri9_Contact} & \finleyelement{Tet10Face_Contact}\\
166  \caption{Finley elements and corresponding elements to be used on domain faces and contacts.  \finleyelement{Hex8} & \finleyelement{Rec4} & \finleyelement{Hex8Face} & \finleyelement{Rec4_Contact} & \finleyelement{Hex8Face_Contact}\\
167  The rich types have to be used if the gradient of function is to be calculated on faces and contacts, resepctively.}  \finleyelement{Hex20} & \finleyelement{Rec8} & \finleyelement{Hex20Face} & \finleyelement{Rec8_Contact} & \finleyelement{Hex20Face_Contact}\\
168    \finleyelement{Hex27} & \finleyelement{Rec9} & N/A & N/A & N/A\\
169    \finleyelement{Hex27Macro} & \finleyelement{Rec9Macro} & N/A & N/A & N/A\\
170    \finleyelement{Tet10Macro} & \finleyelement{Tri6Macro} & N/A & N/A & N/A\\
171    \finleyelement{Rec9Macro} & \finleyelement{Line3Macro} & N/A & N/A & N/A\\
172    \finleyelement{Tri6Macro} & \finleyelement{Line3Macro} & N/A & N/A & N/A\\
173    \end{tabular}
174    \caption{Finley elements and corresponding elements to be used on domain faces
175    and contacts.
176    The rich types have to be used if the gradient of the function is to be
177    calculated on faces and contacts, respectively.}
178  \label{FINLEY TAB 1}  \label{FINLEY TAB 1}
179  \end{table}  \end{table}
180    
181  The native \finley file format is defined as follows.  The native \finley file format is defined as follows.
182  Each node \var{i} has \var{dim} spatial coordinates \var{Node[i]}, a reference number  Each node \var{i} has \var{dim} spatial coordinates \var{Node[i]}, a reference
183  \var{Node_ref[i]}, a degree of freedom \var{Node_DOF[i]} and tag \var{Node_tag[i]}.  number \var{Node_ref[i]}, a degree of freedom \var{Node_DOF[i]} and a tag
184  In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however, for periodic boundary conditions,  \var{Node_tag[i]}.
185  \var{Node_DOF[i]} is chosen differently, see example below. The tag can be used to mark nodes sharing  In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however, for periodic
186  the same properties. Element \var{i} is defined by the \var{Element_numNodes} nodes \var{Element_Nodes[i]}  boundary conditions, \var{Node_DOF[i]} is chosen differently, see example below.
187  which is a list of node reference numbers. The order is crucial.  The tag can be used to mark nodes sharing the same properties.
188  It has a reference number \var{Element_ref[i]} and a tag \var{Element_tag[i]}. The tag  Element \var{i} is defined by the \var{Element_numNodes} nodes
189  can be used to mark elements  sharing the same properties. For instance elements above  \var{Element_Nodes[i]} which is a list of node reference numbers.
190  a contact region are marked with $2$ and elements below a contact region are marked with $1$.  The order of these is crucial. Each element has a reference number
191  \var{Element_Type} and \var{Element_Num} give the element type and the number of elements in the mesh.  \var{Element_ref[i]} and a tag \var{Element_tag[i]}.
192  Analogue notations are used for face and contact elements. The following Python script  The tag can be used to mark elements sharing the same properties.
193  prints the mesh definition in the \finley file format:  For instance elements above a contact region are marked with tag $2$ and
194    elements below a contact region are marked with tag $1$.
195    \var{Element_Type} and \var{Element_Num} give the element type and the number
196    of elements in the mesh.
197    Analogue notations are used for face and contact elements.
198    The following \PYTHON script prints the mesh definition in the \finley file
199    format:
200  \begin{python}  \begin{python}
201  print "%s\n"%mesh_name    print("%s\n"%mesh_name)
202  # node coordinates:    # node coordinates:
203  print "%dD-nodes %d\n"%(dim,numNodes)    print("%dD-nodes %d\n"%(dim, numNodes))
204  for i in range(numNodes):    for i in range(numNodes):
205     print "%d %d %d"%(Node_ref[i],Node_DOF[i],Node_tag[i])       print("%d %d %d"%(Node_ref[i], Node_DOF[i], Node_tag[i]))
206     for j in range(dim): print " %e"%Node[i][j]       for j in range(dim): print(" %e"%Node[i][j])
207     print "\n"       print("\n")
208  # interior elements    # interior elements
209  print "%s %d\n"%(Element_Type,Element_Num)    print("%s %d\n"%(Element_Type, Element_Num))
210  for i in range(Element_Num):    for i in range(Element_Num):
211     print "%d %d"%(Element_ref[i],Element_tag[i])       print("%d %d"%(Element_ref[i], Element_tag[i]))
212     for j in range(Element_numNodes): print " %d"%Element_Nodes[i][j]       for j in range(Element_numNodes): print(" %d"%Element_Nodes[i][j])
213     print "\n"       print("\n")
214  # face elements    # face elements
215  print "%s %d\n"%(FaceElement_Type,FaceElement_Num)    print("%s %d\n"%(FaceElement_Type, FaceElement_Num))
216  for i in range(FaceElement_Num):    for i in range(FaceElement_Num):
217     print "%d %d"%(FaceElement_ref[i],FaceElement_tag[i])       print("%d %d"%(FaceElement_ref[i], FaceElement_tag[i]))
218     for j in range(FaceElement_numNodes): print " %d"%FaceElement_Nodes[i][j]       for j in range(FaceElement_numNodes): print(" %d"%FaceElement_Nodes[i][j])
219     print "\n"       print("\n")
220  # contact elements    # contact elements
221  print "%s %d\n"%(ContactElement_Type,ContactElement_Num)    print("%s %d\n"%(ContactElement_Type, ContactElement_Num))
222  for i in range(ContactElement_Num):    for i in range(ContactElement_Num):
223     print "%d %d"%(ContactElement_ref[i],ContactElement_tag[i])       print("%d %d"%(ContactElement_ref[i], ContactElement_tag[i]))
224     for j in range(ContactElement_numNodes): print " %d"%ContactElement_Nodes[i][j]       for j in range(ContactElement_numNodes): print(" %d"%ContactElement_Nodes[i][j])
225     print "\n"       print("\n")
226  # point sources (not supported yet)    # point sources (not supported yet)
227  write("Point1 0",face_element_typ,numFaceElements)    print("Point1 0")
228  \end{python}  \end{python}
229    
230  The following example of a mesh file defines the mesh shown in \fig{FINLEY FIG 01}:  The following example of a mesh file defines the mesh shown in \fig{FINLEY FIG 01}:
# Line 214  Line2_Contact 3 Line 262  Line2_Contact 3
262  Point1 0  Point1 0
263  \end{verbatim}  \end{verbatim}
264  Notice that the order in which the nodes and elements are given is arbitrary.  Notice that the order in which the nodes and elements are given is arbitrary.
265  In the case that rich contact elements are used the contact element section gets  In the case that rich contact elements are used the contact element section
266   the form  gets the form
267  \begin{verbatim}  \begin{verbatim}
268  Rec4Face_Contact 3  Rec4Face_Contact 3
269   4 0  9 12 16 18  6  5  0  2   4 0  9 12 16 18  6  5  0  2
270   3 0 13  9 18 19  8  6  2  3   3 0 13  9 18 19  8  6  2  3
271   6 0 15 13 19 20 10  8  3  7   6 0 15 13 19 20 10  8  3  7
272  \end{verbatim}  \end{verbatim}
273  Periodic boundary condition \index{boundary conditions!periodic} can be introduced by altering \var{Node_DOF}.  Periodic boundary conditions\index{boundary conditions!periodic} can be
274  It allows identification of nodes even if they have different physical locations. For instance, to  introduced by altering \var{Node_DOF}.
275  enforce periodic boundary conditions at the face $x_0=0$ and $x_0=1$ one identifies  It allows identification of nodes even if they have different physical locations.
276  the degrees of freedom for nodes $0$, $5$, $12$ and $16$ with the degrees of freedom for  For instance, to enforce periodic boundary conditions at the face $x_0=0$ and
277  $7$, $10$, $15$ and $20$, respectively. The node section of the \finley mesh gets now the form:    $x_0=1$ one identifies the degrees of freedom for nodes $0$, $5$, $12$ and $16$
278    with the degrees of freedom for $7$, $10$, $15$ and $20$, respectively.
279    The node section of the \finley mesh now reads:
280  \begin{verbatim}  \begin{verbatim}
281  2D Nodes 16  2D Nodes 16
282  0   0 0 0.   0.  0   0 0 0.   0.
# Line 247  $7$, $10$, $15$ and $20$, respectively. Line 297  $7$, $10$, $15$ and $20$, respectively.
297  20 16 0 1.0  1.0  20 16 0 1.0  1.0
298  \end{verbatim}  \end{verbatim}
299    
300    \clearpage
301    \input{finleyelements}
302    \clearpage
303    
304    \section{Macro Elements}
305    \label{SEC FINLEY MACRO}
306    
307    \begin{figure}[th]
308    \begin{center}
309    \includegraphics{FinleyMacroLeg}\\
310    \subfigure[Triangle]{\label{FINLEY MACRO TRI}\includegraphics{FinleyMacroTri}}\quad
311    \subfigure[Quadrilateral]{\label{FINLEY MACRO REC}\includegraphics{FinleyMacroRec}}
312    \end{center}
313    \caption{Macro elements in \finley}
314    \end{figure}
315    
316  \include{finleyelements}  \finley supports the usage of macro elements\index{macro elements} which can be
317    used to achieve LBB compliance when solving incompressible fluid flow problems.
318    LBB compliance is required to get a problem which has a unique solution for
319    pressure and velocity. For macro elements the pressure and velocity are
320    approximated by a polynomial of order 1 but the velocity approximation bases on
321    a refinement of the elements. The nodes of a triangle and quadrilateral element
322    are shown in Figures~\ref{FINLEY MACRO TRI} and~\ref{FINLEY MACRO REC},
323    respectively. In essence, the velocity uses the same nodes like a quadratic
324    polynomial approximation but replaces the quadratic polynomial by piecewise
325    linear polynomials. In fact, this is the way \finley defines the macro elements.
326    In particular \finley uses the same local ordering of the nodes for the macro
327    element as for the corresponding quadratic element. Another interpretation is
328    that one uses a linear approximation of the velocity together with a linear
329    approximation of the pressure but on elements created by combining elements to
330    macro elements. Notice that the macro elements still use quadratic
331    interpolation to represent the element and domain boundary.
332    However, if elements have linear boundaries a macro element approximation for
333    the velocity is equivalent to using a linear approximation on a mesh which is
334    created through a one-step global refinement.
335    Typically macro elements are only required to use when an incompressible fluid
336    flow problem is solved, e.g. the Stokes problem in \Sec{STOKES PROBLEM}.
337    Please see \Sec{FINLEY MESHES} for more details on the supported macro elements.
338    
339    \section{Linear Solvers in \SolverOptions}
340    
341    Table~\ref{TAB FINLEY SOLVER OPTIONS 1} and
342    Table~\ref{TAB FINLEY SOLVER OPTIONS 2} show the solvers and preconditioners
343    supported by \finley through the \PASO library.
344    Currently direct solvers are not supported under \MPI.
345    By default, \finley uses the iterative solvers \PCG for symmetric and \BiCGStab
346    for non-symmetric problems.
347    If the direct solver is selected, which can be useful when solving very
348    ill-posed equations, \finley uses the \MKL\footnote{If the stiffness matrix is
349    non-regular \MKL may return without a proper error code. If you observe
350    suspicious solutions when using \MKL, this may be caused by a non-invertible
351    operator.} solver package. If \MKL is not available \UMFPACK is used.
352    If \UMFPACK is not available a suitable iterative solver from \PASO is used.
353    
354  \subsection{Linear Solvers in \LinearPDE}  \begin{table}
355  Currently \finley supports the linear solvers \PCG, \GMRES, \PRESTWENTY and \BiCGStab.  \centering
356  For \GMRES the options \var{trancation} and \var{restart} of the \method{getSolution} can be  {\scriptsize
357  used to control the trunction and restart during iteration. Default values are  \begin{tabular}{l||c|c|c|c|c|c|c|c}
358  \var{truncation}=5 and \var{restart}=20.  \member{setSolverMethod} & \member{DIRECT}& \member{PCG} & \member{GMRES} & \member{TFQMR} & \member{MINRES} & \member{PRES20} & \member{BICGSTAB} & \member{LUMPING} \\
359  The default solver is \BiCGStab  but if the symmetry flag is set \PCG is the default solver.  \hline
360  \finley supports the solver options \var{iter_max} which specifies the maximum number of iterations steps,   \hline
361  \var{verbose}=\True or \False and \var{preconditioner}=\constant{JACOBI} or \constant {ILU0}.   \member{setReordering} & $\checkmark$ & & & & & &\\
362  In some installations \finley supports the \Direct solver and the   \hline  \member{setRestart} &  & & $\checkmark$ & & & $20$ & \\
363  solver options \var{reordering}=\constant{util.NO_REORDERING},   \hline\member{setTruncation} &  & & $\checkmark$ & & & $5$ & \\
364  \constant{util.MINIMUM_FILL_IN} or \constant{util.NESTED_DISSECTION} (default is \constant{util.NO_REORDERING}),     \hline\member{setIterMax} &  & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
365  \var{drop_tolerance} specifying the threshold for values to be dropped in the   \hline\member{setTolerance} &  & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
366  incomplete elimation process (default is 0.01) and \var{drop_storage} specifying the maximum increase   \hline\member{setAbsoluteTolerance} &  & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
367  in storage allowed in the  \hline\member{setReordering} & $\checkmark$ & & & & & & & \\
368  incomplete elimation process (default is 1.20).  \end{tabular}
369    }
370  \subsection{Functions}  \caption{Solvers available for \finley and the \PASO package and the relevant
371  \begin{funcdesc}{Mesh}{fileName,integrationOrder=-1}  options in \class{SolverOptions}.
372  creates a \Domain object form the FEM mesh defined in  \MKL supports \member{MINIMUM_FILL_IN}\index{linear solver!minimum fill-in ordering}\index{minimum fill-in ordering}
373  file \var{fileName}. The file must be given the \finley file format.  and \member{NESTED_DISSECTION}\index{linear solver!nested dissection ordering}\index{nested dissection}
374  If \var{integrationOrder} is positive, a numerical integration scheme  reordering.
375  chosen which is accurate on each element up to a polynomial of  Currently the \UMFPACK interface does not support any reordering.
376  degree \var{integrationOrder} \index{integration order}. Otherwise  \label{TAB FINLEY SOLVER OPTIONS 1}}
377  an appropriate integration order is chosen independently.  \end{table}
378    
379    \begin{table}
380    {\scriptsize
381    \begin{tabular}{l||c|c|c|c|c|c|c|c}
382    \member{setPreconditioner}&
383    \member{NO_PRECONDITIONER}&
384    \member{AMG}&
385    \member{JACOBI}&
386    \member{GAUSS_SEIDEL}&
387    \member{REC_ILU}&
388    \member{RILU}&
389    \member{ILU0}&
390    \member{DIRECT}\\
391    \hline
392    status:&later&later&$\checkmark$&$\checkmark$&$\checkmark$&later&$\checkmark$&later\\
393    \hline
394    \hline
395    \member{setCoarsening}& &$\checkmark$& & & & & &\\
396    \hline
397    \member{setLevelMax}& &$\checkmark$& & & & & &\\
398    \hline
399    \member{setCoarseningThreshold}& &$\checkmark$& & & & & &\\
400    \hline
401    \member{setMinCoarseMatrixSize}& &$\checkmark$& & & & & &\\
402    \hline
403    \member{setNumSweeps}& & &$\checkmark$&$\checkmark$& & & &\\
404    \hline
405    \member{setNumPreSweeps}& &$\checkmark$& & & & & &\\
406    \hline
407    \member{setNumPostSweeps}& &$\checkmark$& & & & & &\\
408    \hline
409    \member{setInnerTolerance}& & & & & & & &\\
410    \hline
411    \member{setDropTolerance}& & & & & & & &\\
412    \hline
413    \member{setDropStorage}& & & & & & & &\\
414    \hline
415    \member{setRelaxationFactor}& & & & & &$\checkmark$& &\\
416    \hline
417    \member{adaptInnerTolerance}& & & & & & & &\\
418    \hline
419    \member{setInnerIterMax}& & & & & & & &\\
420    \end{tabular}
421    }
422    \caption{Preconditioners available for \finley and the \PASO package and the
423    relevant options in \class{SolverOptions}.
424    \label{TAB FINLEY SOLVER OPTIONS 2}}
425    \end{table}
426    
427    \section{Functions}
428    \begin{funcdesc}{ReadMesh}{fileName \optional{, \optional{integrationOrder=-1}, optimize=True}}
429    creates a \Domain object from the FEM mesh defined in file \var{fileName}.
430    The file must be in the \finley file format.
431    If \var{integrationOrder} is positive, a numerical integration scheme is chosen
432    which is accurate on each element up to a polynomial of degree
433    \var{integrationOrder}\index{integration order}.
434    Otherwise an appropriate integration order is chosen independently.
435    By default the labeling of mesh nodes and element distribution is optimized.
436    Set \var{optimize=False} to switch off relabeling and redistribution.
437  \end{funcdesc}  \end{funcdesc}
438    
439  \begin{funcdesc}{Interval}{n0,order=1,l0=1.,integrationOrder=-1, \\  \begin{funcdesc}{ReadGmsh}{fileName \optional{, \optional{integrationOrder=-1}, optimize=True\optional{, useMacroElements=False}}}
440    periodic0=\False,useElementsOnFace=\False}  creates a \Domain object from the FEM mesh defined in file \var{fileName}.
441  Generates a \Domain object representing a interval $[0,l0]$. The interval is filled with  The file must be in the \gmshextern file format.
442  \var{n0} elements.  If \var{integrationOrder} is positive, a numerical integration scheme is chosen
443  For \var{order}=1 and \var{order}=2  which is accurate on each element up to a polynomial of degree
444  \finleyelement{Line2} and    \var{integrationOrder}\index{integration order}.
445  \finleyelement{Line3} are used, respectively.  Otherwise an appropriate integration order is chosen independently.
446  In the case of \var{useElementsOnFace}=\False,  By default the labeling of mesh nodes and element distribution is optimized.
447  \finleyelement{Point1} are used to describe the boundary points.  Set \var{optimize=False} to switch off relabeling and redistribution.
448  In the case of \var{useElementsOnFace}=\True (this option should be used if gradients  If \var{useMacroElements} is set, second order elements are interpreted as
449  are calculated on domain faces),  macro elements\index{macro elements}.
450  \finleyelement{Line2} and    Currently \function{ReadGmsh} does not support \MPI.
451  \finleyelement{Line3} are used on both ends of the interval.    \end{funcdesc}
452  If \var{integrationOrder} is positive, a numerical integration scheme  
453  chosen which is accurate on each element up to a polynomial of  \begin{funcdesc}{MakeDomain}{design\optional{, integrationOrder=-1\optional{, optimizeLabeling=True\optional{, useMacroElements=False}}}}
454  degree \var{integrationOrder} \index{integration order}. Otherwise  creates a \finley \Domain from a \pycad \class{Design} object using \gmshextern.
455  an appropriate integration order is chosen independently. If  The \class{Design} \var{design} defines the geometry.
456  \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions}  If \var{integrationOrder} is positive, a numerical integration scheme is chosen
457  along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley  which is accurate on each element up to a polynomial of degree
458  the value at $x_0=0$ will be identical to the values at $x_0=\var{l0}$.  \var{integrationOrder}\index{integration order}.
459    Otherwise an appropriate integration order is chosen independently.
460    Set \var{optimizeLabeling=False} to switch off relabeling and redistribution
461    (not recommended).
462    If \var{useMacroElements} is set, macro elements\index{macro elements} are used.
463    Currently \function{MakeDomain} does not support \MPI.
464    \end{funcdesc}
465    
466    \begin{funcdesc}{load}{fileName}
467    recovers a \Domain object from a dump file \var{fileName} created by the
468    \function{dump} method of a \Domain object.
469  \end{funcdesc}  \end{funcdesc}
470    
471  \begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\  \begin{funcdesc}{Rectangle}{n0,n1,order=1,l0=1.,l1=1., integrationOrder=-1, \\
472    periodic0=\False,periodic1=\False,useElementsOnFace=\False}    periodic0=\False, periodic1=\False, useElementsOnFace=\False, useMacroElements=\False,\\ optimize=\False}
473  Generates a \Domain object representing a two dimensional rectangle between  generates a \Domain object representing a two-dimensional rectangle between
474  $(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with  $(0,0)$ and $(l0,l1)$ with orthogonal edges.
475  \var{n0} elements along the $x_0$-axis and  The rectangle is filled with \var{n0} elements along the $x_0$-axis and
476  \var{n1} elements along the $x_1$-axis.  \var{n1} elements along the $x_1$-axis.
477  For \var{order}=1 and \var{order}=2  For \var{order}=1 and \var{order}=2, elements of type \finleyelement{Rec4} and
478  \finleyelement{Rec4} and    \finleyelement{Rec8} are used, respectively.
479  \finleyelement{Rec8} are used, respectively.  In the case of \var{useElementsOnFace}=\False, \finleyelement{Line2} and
480  In the case of \var{useElementsOnFace}=\False,  \finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively.
481  \finleyelement{Line2} and    If \var{order}=-1, \finleyelement{Rec8Macro} and \finleyelement{Line3Macro}\index{macro elements}
482  \finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively.  are used. This option should be used when solving incompressible fluid flow
483  In the case of \var{useElementsOnFace}=\True (this option should be used if gradients  problems, e.g. \class{StokesProblemCartesian}.
484  are calculated on domain faces),  In the case of \var{useElementsOnFace}=\True (this option should be used if
485  \finleyelement{Rec4Face} and    gradients are calculated on domain faces), \finleyelement{Rec4Face} and
486  \finleyelement{Rec8Face} are used on the edges, respectively.    \finleyelement{Rec8Face} are used on the edges, respectively.
487  If \var{integrationOrder} is positive, a numerical integration scheme  If \var{integrationOrder} is positive, a numerical integration scheme is chosen
488  chosen which is accurate on each element up to a polynomial of  which is accurate on each element up to a polynomial of degree
489  degree \var{integrationOrder} \index{integration order}. Otherwise  \var{integrationOrder}\index{integration order}.
490  an appropriate integration order is chosen independently. If  Otherwise an appropriate integration order is chosen independently.
491  \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions}  If \var{periodic0}=\True, periodic boundary conditions\index{periodic boundary conditions}
492  along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley  along the $x_0$-direction are enforced.
493  the value on the line $x_0=0$ will be identical to the values on $x_0=\var{l0}$.  That means for any solution of a PDE solved by \finley the values on the line
494  Correspondingly,  $x_0=0$ will be identical to the values on $x_0=\var{l0}$.
495  \var{periodic1}=\False sets periodic boundary conditions  Correspondingly, \var{periodic1}=\True sets periodic boundary conditions in the
496  in $x_1$-direction.  $x_1$-direction.
497    If \var{optimize}=\True mesh node relabeling will be attempted to reduce the
498    computation and also ParMETIS will be used to improve the mesh partition if
499    running on multiple CPUs with \MPI.
500  \end{funcdesc}  \end{funcdesc}
501    
502  \begin{funcdesc}{Brick}{n0,n1,n2,order=1,l0=1.,l1=1.,l2=1., integrationOrder=-1, \\  \begin{funcdesc}{Brick}{n0,n1,n2,order=1,l0=1.,l1=1.,l2=1., integrationOrder=-1,
503    periodic0=\False,periodic1=\False,periodic2=\False,useElementsOnFace=\False}    periodic0=\False, periodic1=\False, \\ periodic2=\False, useElementsOnFace=\False, useMacroElements=\False, optimize=\False}
504  Generates a \Domain object representing a three dimensional brick between  generates a \Domain object representing a three-dimensional brick between
505  $(0,0,0)$ and $(l0,l1,l2)$ with orthogonal faces. The brick is filled with  $(0,0,0)$ and $(l0,l1,l2)$ with orthogonal faces. The brick is filled with
506  \var{n0} elements along the $x_0$-axis,  \var{n0} elements along the $x_0$-axis,
507  \var{n1} elements along the $x_1$-axis and  \var{n1} elements along the $x_1$-axis and
508  \var{n2} elements along the $x_2$-axis.  \var{n2} elements along the $x_2$-axis.
509  For \var{order}=1 and \var{order}=2  For \var{order}=1 and \var{order}=2, elements of type \finleyelement{Hex8} and
510  \finleyelement{Hex8} and    \finleyelement{Hex20} are used, respectively.
511  \finleyelement{Hex20} are used, respectively.  In the case of \var{useElementsOnFace}=\False, \finleyelement{Rec4} and
512  In the case of \var{useElementsOnFace}=\False,  \finleyelement{Rec8} are used to subdivide the faces of the brick, respectively.
513  \finleyelement{Rec4} and    In the case of \var{useElementsOnFace}=\True (this option should be used if
514  \finleyelement{Rec8} are used to subdivide the faces of the brick, respectively.  gradients are calculated on domain faces), \finleyelement{Hex8Face} and
515  In the case of \var{useElementsOnFace}=\True (this option should be used if gradients  \finleyelement{Hex20Face} are used on the brick faces, respectively.
516  are calculated on domain faces),  If \var{order}=-1, \finleyelement{Hex20Macro} and \finleyelement{Rec8Macro}\index{macro elements}
517  \finleyelement{Hex8Face} and    are used. This option should be used when solving incompressible fluid flow
518  \finleyelement{Hex20Face} are used on the brick faces, respectively.    problems, e.g. \class{StokesProblemCartesian}.
519  If \var{integrationOrder} is positive, a numerical integration scheme  If \var{integrationOrder} is positive, a numerical integration scheme is chosen
520  chosen which is accurate on each element up to a polynomial of  which is accurate on each element up to a polynomial of degree
521  degree \var{integrationOrder} \index{integration order}. Otherwise  \var{integrationOrder}\index{integration order}.
522  an appropriate integration order is chosen independently. If  Otherwise an appropriate integration order is chosen independently.
523  \var{periodic0}=\True, periodic boundary conditions \index{periodic boundary conditions}  If \var{periodic0}=\True, periodic boundary conditions\index{periodic boundary conditions}
524  along the $x_0$-directions are enforced. That means when for any solution of a PDE solved by \finley  along the $x_0$-direction are enforced.
525  the value on the plane $x_0=0$ will be identical to the values on $x_0=\var{l0}$. Correspondingly,  That means for any solution of a PDE solved by \finley the values on the plane
526  \var{periodic1}=\False and \var{periodic2}=\False sets periodic boundary conditions  $x_0=0$ will be identical to the values on $x_0=\var{l0}$.
527  in $x_1$-direction and $x_2$-direction, respectively.  Correspondingly, \var{periodic1}=\True and \var{periodic2}=\True sets periodic
528    boundary conditions in the $x_1$-direction and $x_2$-direction, respectively.
529    If \var{optimize}=\True mesh node relabeling will be attempted to reduce the
530    computation and also ParMETIS will be used to improve the mesh partition if
531    running on multiple CPUs with \MPI.
532  \end{funcdesc}  \end{funcdesc}
533    
534  \begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}  \begin{funcdesc}{GlueFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
535  Generates a new \Domain object from the list \var{mehList} of \finley meshes.  generates a new \Domain object from the list \var{meshList} of \finley meshes.
536  Nodes in face elements whose difference of coordinates is less then \var{tolerance} times the  Nodes in face elements whose difference of coordinates is less than
537  diameter of the domain are merged. The corresponding face elements are removed from the mesh.    \var{tolerance} times the diameter of the domain are merged.
538    The corresponding face elements are removed from the mesh.
539  TODO: explain \var{safetyFactor} and show an example.  %TODO: explain \var{safetyFactor} and show an example.
540  \end{funcdesc}  \end{funcdesc}
541    
542  \begin{funcdesc}{JoinFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}  \begin{funcdesc}{JoinFaces}{meshList,safetyFactor=0.2,tolerance=1.e-13}
543  Generates a new \Domain object from the list \var{mehList} of \finley meshes.  generates a new \Domain object from the list \var{meshList} of \finley meshes.
544  Face elements whose nodes coordinates have difference is less then \var{tolerance} times the  Face elements whose node coordinates differ by less than \var{tolerance} times
545  diameter of the domain are combined to form a contact element \index{element!contact}  the diameter of the domain are combined to form a contact element\index{element!contact}.
546  The corresponding face elements are removed from the mesh.    The corresponding face elements are removed from the mesh.
547    %TODO: explain \var{safetyFactor} and show an example.
 TODO: explain \var{safetyFactor} and show an example.  
548  \end{funcdesc}  \end{funcdesc}
549    

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