28 


29 
\subsection{Meshes} 
\subsection{Meshes} 
30 
To understand the usage of \finley one needs to have an understanding of how the finite element meshes 
To understand the usage of \finley one needs to have an understanding of how the finite element meshes 
31 
\index{FEM!mesh} are defined. \fig{FINLEY FIG 0} shoes an example of the 
\index{FEM!mesh} are defined. \fig{FINLEY FIG 0} shows an example of the 
32 
subdivision of an ellipse into so called elements \index{FEM!elements} \index{element}. 
subdivision of an ellipse into so called elements \index{FEM!elements} \index{element}. 
33 
In this case, triangles have been used but other forms of subdivisions 
In this case, triangles have been used but other forms of subdivisions 
34 
can be constructed, e.g. into quadrilaterals or, in the three dimensional case, into tetrahedrons 
can be constructed, e.g. into quadrilaterals or, in the three dimensional case, into tetrahedrons 
35 
and hexahedrons. The idea of the finite element method is to approximate the solution by a function 
and hexahedrons. The idea of the finite element method is to approximate the solution by a function 
36 
which is a polynomial of a certain order and is continuous across it boundary to neighbour elements. 
which is a polynomial of a certain order and is continuous across it boundary to neighbour elements. 
37 
In the example of \fig{FINLEY FIG 0} a linear polynomial is used on each triangle. As one can seen, the triangulation 
In the example of \fig{FINLEY FIG 0} a linear polynomial is used on each triangle. As one can see, the triangulation 
38 
is a quite poor approximation of the ellipse. It can be improved by introducing a midpoint on each element edge and 
is quite a poor approximation of the ellipse. It can be improved by introducing a midpoint on each element edge then 
39 
position those nodes, which are located on an edge expecting to describing the boundary, onto the boundary. 
positioning those nodes located on an edge expected to describe the boundary, onto the boundary. 
40 
In this case the triangle gets a curved edge which requires a parametrization of the triangle using a 
In this case the triangle gets a curved edge which requires a parametrization of the triangle using a 
41 
quadratic polynomial. For this case, the solution is also approximated by a piecewise quadratic polynomial 
quadratic polynomial. For this case, the solution is also approximated by a piecewise quadratic polynomial 
42 
(which explains the name isoparametrical elements), see \Ref{Zienc,NumHand} for more details. 
(which explains the name isoparametrical elements), see \Ref{Zienc,NumHand} for more details. 
43 


44 
The union of all elements defines the domain of the PDE. 
The union of all elements defines the domain of the PDE. 
45 
Each element is defined by the nodes used to describe is shape. In \fig{FINLEY FIG 0} the element, 
Each element is defined by the nodes used to describe its shape. In \fig{FINLEY FIG 0} the element, 
46 
which have type \finleyelement{Tri3}, 
which has type \finleyelement{Tri3}, 
47 
with the element reference number $19$ \index{element!reference number} is defined by the nodes 
with element reference number $19$ \index{element!reference number} is defined by the nodes 
48 
with the reference numbers $9$, $11$ and $0$ \index{node!reference number}. Notice that the order is counterclockwise. 
with reference numbers $9$, $11$ and $0$ \index{node!reference number}. Notice that the order is counterclockwise. 
49 
The coefficients of the PDE are evaluated at integration nodes with each individual element. 
The coefficients of the PDE are evaluated at integration nodes with each individual element. 
50 
For quadrilateral elements a Gauss quadrature scheme is used. In the case of triangular elements a 
For quadrilateral elements a Gauss quadrature scheme is used. In the case of triangular elements a 
51 
modified from is applied. The boundary of the domain is also subdivided into elements \index{element!face}. In \fig{FINLEY FIG 0} 
modified form is applied. The boundary of the domain is also subdivided into elements. \index{element!face} In \fig{FINLEY FIG 0} 
52 
line elements with two nodes are used. The elements are also defined by their describing nodes, e.g. 
line elements with two nodes are used. The elements are also defined by their describing nodes, e.g. 
53 
the face element reference number $20$ which has type \finleyelement{Line2} is defined by the nodes 
the face element reference number $20$ which has type \finleyelement{Line2} is defined by the nodes 
54 
with the reference numbers $11$ and $0$. Again the order is crucial, if moving from the first 
with the reference numbers $11$ and $0$. Again the order is crucial, if moving from the first 
55 
to second node the domain has to lay on the left hand side (in case of a two dimension surface element 
to second node the domain has to lie on the left hand side (in the case of a two dimension surface element 
56 
the domain has to lay on left hand side when moving counterclockwise). If the gradient on the 
the domain has to lie on the left hand side when moving counterclockwise). If the gradient on the 
57 
surface of the domain wants to be calculated rich face elements face to be used. Rich elements on a face 
surface of the domain is to be calculated rich face elements face to be used. Rich elements on a face 
58 
is identical to interior elements but with a modified order of nodes such that the 'first' face of the element aligns 
are identical to interior elements but with a modified order of nodes such that the 'first' face of the element aligns 
59 
with the surface of the domian. In \fig{FINLEY FIG 0} 
with the surface of the domian. In \fig{FINLEY FIG 0} 
60 
elements of the type \finleyelement{Tri3Face} are used. 
elements of the type \finleyelement{Tri3Face} are used. 
61 
The face element reference number $20$ as a rich face element is defined by the nodes 
The face element reference number $20$ as a rich face element is defined by the nodes 
62 
with the reference numbers $11$, $0$ and $9$. Notice that the face element $20$ is identical to the 
with reference numbers $11$, $0$ and $9$. Notice that the face element $20$ is identical to the 
63 
interior element $19$ however, in this case, the order of the node is different to align the first 
interior element $19$ except that, in this case, the order of the node is different to align the first 
64 
edge of the triangle (which is the edge starting with the first node) with the boundary of the domain. 
edge of the triangle (which is the edge starting with the first node) with the boundary of the domain. 
65 


66 
Be aware that face elements and elements in the interior of the domain must match, i.e. a face element must be the face 
Be aware that face elements and elements in the interior of the domain must match, i.e. a face element must be the face 
67 
of an interior element or, in case of a rich face element, must be identical to an interior element. 
of an interior element or, in case of a rich face element, it must be identical to an interior element. 
68 
If no face elements are specified 
If no face elements are specified 
69 
\finley implicitly assumes homogeneous natural boundary conditions \index{natural boundary conditions!homogeneous}, 
\finley implicitly assumes homogeneous natural boundary conditions \index{natural boundary conditions!homogeneous}, 
70 
i.e. \var{d}=$0$ and \var{y}=$0$, on the entire boundary of the domain. For 
i.e. \var{d}=$0$ and \var{y}=$0$, on the entire boundary of the domain. For 
77 
of rectangular elements around a contact region $\Gamma^{contact}$ \index{element!contact}. 
of rectangular elements around a contact region $\Gamma^{contact}$ \index{element!contact}. 
78 
The contact region is described by the 
The contact region is described by the 
79 
elements $4$, $3$ and $6$. Their element type is \finleyelement{Line2_Contact}. 
elements $4$, $3$ and $6$. Their element type is \finleyelement{Line2_Contact}. 
80 
The nodes $9$, $12$, $6$, $5$ are defining contact element $4$, where the coordinates of nodes $12$ and $5$ and 
The nodes $9$, $12$, $6$, $5$ define contact element $4$, where the coordinates of nodes $12$ and $5$ and 
81 
nodes $4$ and $6$ are identical with the idea that nodes $12$ and $9$ are located above and 
nodes $4$ and $6$ are identical with the idea that nodes $12$ and $9$ are located above and 
82 
nodes $5$ and $6$ are below the contact region. 
nodes $5$ and $6$ below the contact region. 
83 
Again, the order of the nodes within an element is crucial. There is also the option of using rich elements 
Again, the order of the nodes within an element is crucial. There is also the option of using rich elements 
84 
if the gradient wants to be calculated on the contact region. Similar to the rich face elements 
if the gradient is to be calculated on the contact region. Similarly to the rich face elements 
85 
they are constructed from two interior elements with reordering the nodes such that 
these are constructed from two interior elements by reordering the nodes such that 
86 
the 'first' face of the element above and the 'first' face of the element below the 
the 'first' face of the element above and the 'first' face of the element below the 
87 
contact regions are lining up. The rich version of element 
contact regions line up. The rich version of element 
88 
$4$ is of type \finleyelement{Rec4Face_Contact} and is defined by the nodes $9$, $12$, $16$, $18$, $6$, $5$, $0$ and 
$4$ is of type \finleyelement{Rec4Face_Contact} and is defined by the nodes $9$, $12$, $16$, $18$, $6$, $5$, $0$ and 
89 
$2$. 
$2$. 
90 


91 
\tab{FINLEY TAB 1} shows the interior element types and the corresponding element types to be used 
\tab{FINLEY TAB 1} shows the interior element types and the corresponding element types to be used 
92 
on face and contacts. \fig{FINLEY.FIG:1}, \fig{FINLEY.FIG:2} and \fig{FINLEY.FIG:4} show the ordering of 
on the face and contacts. \fig{FINLEY.FIG:1}, \fig{FINLEY.FIG:2} and \fig{FINLEY.FIG:4} show the ordering of 
93 
the nodes within an element. 
the nodes within an element. 
94 


95 
\begin{table} 
\begin{table} 
107 
\linev{\finleyelement{Hex20}}{\finleyelement{Rec8}}{\finleyelement{Hex20Face}}{\finleyelement{Rec8_Contact}}{\finleyelement{Hex20Face_Contact}} 
\linev{\finleyelement{Hex20}}{\finleyelement{Rec8}}{\finleyelement{Hex20Face}}{\finleyelement{Rec8_Contact}}{\finleyelement{Hex20Face_Contact}} 
108 
\end{tablev} 
\end{tablev} 
109 
\caption{Finley elements and corresponding elements to be used on domain faces and contacts. 
\caption{Finley elements and corresponding elements to be used on domain faces and contacts. 
110 
The rich types have to be used if the gradient of function wants to calculated on faces and contacts, resepctively.} 
The rich types have to be used if the gradient of function is to be calculated on faces and contacts, resepctively.} 
111 
\label{FINLEY TAB 1} 
\label{FINLEY TAB 1} 
112 
\end{table} 
\end{table} 
113 


114 
The native \finley file format is defined as follows. 
The native \finley file format is defined as follows. 
115 
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 number 
116 
\var{Node_ref[i]}, a degree of freedom \var{Node_DOF[i]} and tag \var{Node_tag[i]}. 
\var{Node_ref[i]}, a degree of freedom \var{Node_DOF[i]} and tag \var{Node_tag[i]}. 
117 
In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however for periodic boundary conditions 
In most cases \var{Node_DOF[i]}=\var{Node_ref[i]} however, for periodic boundary conditions, 
118 
\var{Node_DOF[i]} is chosen differently, see example below. The tag can be used to mark nodes sharing 
\var{Node_DOF[i]} is chosen differently, see example below. The tag can be used to mark nodes sharing 
119 
the same properties. Element \var{i} is defined by the \var{Element_numNodes} nodes \var{Element_Nodes[i]} 
the same properties. Element \var{i} is defined by the \var{Element_numNodes} nodes \var{Element_Nodes[i]} 
120 
which is a list of node reference numbers. The order is crucial. 
which is a list of node reference numbers. The order is crucial. 
121 
It has a reference number \var{Element_ref[i]} and a tag \var{Element_tag[i]}. The tag 
It has a reference number \var{Element_ref[i]} and a tag \var{Element_tag[i]}. The tag 
122 
can be used to mark elements sharing the same properties. For instance elements above 
can be used to mark elements sharing the same properties. For instance elements above 
123 
a contact region are marked with $2$ and element below a contact region are marked with $1$. 
a contact region are marked with $2$ and elements below a contact region are marked with $1$. 
124 
\var{Element_Type} and \var{Element_Num} give the element type and the number of elements in the mesh. 
\var{Element_Type} and \var{Element_Num} give the element type and the number of elements in the mesh. 
125 
Analogue notations are used for face and contact elements. The following Python script 
Analogue notations are used for face and contact elements. The following Python script 
126 
prints the mesh definition in the \finley file format: 
prints the mesh definition in the \finley file format: 
189 
Point1 0 
Point1 0 
190 
\end{verbatim} 
\end{verbatim} 
191 
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. 
192 
In that case rich contact element are used the contact element section get the form 
In the case that rich contact elements are used the contact element section gets 
193 

the form 
194 
\begin{verbatim} 
\begin{verbatim} 
195 
Rec4Face_Contact 3 
Rec4Face_Contact 3 
196 
4 0 9 12 16 18 6 5 0 2 
4 0 9 12 16 18 6 5 0 2 
198 
6 0 15 13 19 20 10 8 3 7 
6 0 15 13 19 20 10 8 3 7 
199 
\end{verbatim} 
\end{verbatim} 
200 
Periodic boundary condition \index{boundary conditions!periodic} can be introduced by altering \var{Node_DOF}. 
Periodic boundary condition \index{boundary conditions!periodic} can be introduced by altering \var{Node_DOF}. 
201 
It allows to identify nodes even if they have different physical locations. For instance, to 
It allows identification of nodes even if they have different physical locations. For instance, to 
202 
enforce periodic boundary conditions at the face $x_0=0$ and $x_0=1$ one identifies 
enforce periodic boundary conditions at the face $x_0=0$ and $x_0=1$ one identifies 
203 
the degrees of freedom for nodes $0$, $5$, $12$ and $16$ with the degrees of freedom for 
the degrees of freedom for nodes $0$, $5$, $12$ and $16$ with the degrees of freedom for 
204 
$7$, $10$, $15$ and $20$, respectively. The node section of the \finley mesh gets now the form: 
$7$, $10$, $15$ and $20$, respectively. The node section of the \finley mesh gets now the form: 
226 
\include{finleyelements} 
\include{finleyelements} 
227 


228 
\subsection{Linear Solvers in \LinearPDE} 
\subsection{Linear Solvers in \LinearPDE} 
229 
Currently \finley support the linear solvers \PCG, \GMRES, \PRESTWENTY and \BiCGStab. 
Currently \finley supports the linear solvers \PCG, \GMRES, \PRESTWENTY and \BiCGStab. 
230 
For \GMRES the options \var{trancation} and \var{restart} of the \method{getSolution} can be 
For \GMRES the options \var{trancation} and \var{restart} of the \method{getSolution} can be 
231 
used to control the trunction and restart during iteration. Default values are 
used to control the trunction and restart during iteration. Default values are 
232 
\var{truncation}=5 and \var{restart}=20. 
\var{truncation}=5 and \var{restart}=20. 
233 
The default solver is \BiCGStab but if the symmetry flag is set \PCG is the default solver. 
The default solver is \BiCGStab but if the symmetry flag is set \PCG is the default solver. 
234 
\finley supports the solver options \var{iter_max} which specifies the maximum number of iterations steps, 
\finley supports the solver options \var{iter_max} which specifies the maximum number of iterations steps, 
235 
\var{verbose}=\True or \False and \var{preconditioner}=\constant{JACOBI} or \constant {ILU0}. 
\var{verbose}=\True or \False and \var{preconditioner}=\constant{JACOBI} or \constant {ILU0}. 
236 
In some installation \finley supports \Direct solver and the 
In some installations \finley supports the \Direct solver and the 
237 
solver options \var{reordering}=\constant{util.NO_REORDERING}, 
solver options \var{reordering}=\constant{util.NO_REORDERING}, 
238 
\constant{util.MINIMUM_FILL_IN} or \constant{util.NESTED_DISSECTION} (default is \constant{util.NO_REORDERING}), 
\constant{util.MINIMUM_FILL_IN} or \constant{util.NESTED_DISSECTION} (default is \constant{util.NO_REORDERING}), 
239 
\var{drop_tolerance} specifying the threshold for values to be dropped in the 
\var{drop_tolerance} specifying the threshold for values to be dropped in the 