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revision 2558 by gross, Mon Jul 27 05:03:32 2009 UTC revision 2881 by jfenwick, Thu Jan 28 02:03:15 2010 UTC
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1    
2  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3  %  %
4  % Copyright (c) 2003-2009 by University of Queensland  % Copyright (c) 2003-2010 by University of Queensland
5  % Earth Systems Science Computational Center (ESSCC)  % Earth Systems Science Computational Center (ESSCC)
6  % http://www.uq.edu.au/esscc  % http://www.uq.edu.au/esscc
7  %  %
# Line 55  A\hackscore{jl} \cdot v\hackscore{,j}u\h Line 55  A\hackscore{jl} \cdot v\hackscore{,j}u\h
55  \end{equation}  \end{equation}
56    
57  \section{Meshes}  \section{Meshes}
58    \label{FINLEY MESHES}
59  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
60  \index{FEM!mesh} are defined. \fig{FINLEY FIG 0} shows an example of the  \index{FEM!mesh} are defined. \fig{FINLEY FIG 0} shows an example of the
61  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}.
# Line 65  which is a polynomial of a certain order Line 66  which is a polynomial of a certain order
66  In the example of \fig{FINLEY FIG 0} a linear polynomial is used on each triangle. As one can see, the triangulation  In the example of \fig{FINLEY FIG 0} a linear polynomial is used on each triangle. As one can see, the triangulation
67  is quite a poor approximation of the ellipse. It can be improved by introducing a midpoint on each element edge then  is quite a poor approximation of the ellipse. It can be improved by introducing a midpoint on each element edge then
68  positioning those nodes located on an edge expected to describe the boundary, onto the boundary.  positioning those nodes located on an edge expected to describe the boundary, onto the boundary.
69  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 parameterization of the triangle using a
70  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
71  (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.  
72    \finley supports macro elements\index{macro elements}. For these elements a piecewise linear approximation is used on an element which is further subdivided (in the case \finley halved). As such these elements do not provide more than a further mesh refinement but should be used in the case of incompressible flows, see \class{StokesProblemCartesian}. For these problems a linear approximation of the pressure across the element is used (use the \ReducedSolutionFS \FunctionSpace) while the refined element is used to approximate velocity. So a macro element provides a continuous pressure approximation together with a velocity approximation on a refined mesh. This approach is necessary to make sure that the  incompressible flow has a unique solution.
73    
74  The union of all elements defines the domain of the PDE.  The union of all elements defines the domain of the PDE.
75  Each element is defined by the nodes used to describe its 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,
# Line 116  contact regions line up.  The rich versi Line 118  contact regions line up.  The rich versi
118  $4$ 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
119  $2$.  $2$.
120    
121    
122    
123  \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
124  on the 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
125  the nodes within an element.  the nodes within an element.
# Line 133  the nodes within an element. Line 137  the nodes within an element.
137  \linev{\finleyelement{Tet10}}{\finleyelement{Tri9}}{\finleyelement{Tet10Face}}{\finleyelement{Tri9_Contact}}{\finleyelement{Tet10Face_Contact}}  \linev{\finleyelement{Tet10}}{\finleyelement{Tri9}}{\finleyelement{Tet10Face}}{\finleyelement{Tri9_Contact}}{\finleyelement{Tet10Face_Contact}}
138  \linev{\finleyelement{Hex8}}{\finleyelement{Rec4}}{\finleyelement{Hex8Face}}{\finleyelement{Rec4_Contact}}{\finleyelement{Hex8Face_Contact}}  \linev{\finleyelement{Hex8}}{\finleyelement{Rec4}}{\finleyelement{Hex8Face}}{\finleyelement{Rec4_Contact}}{\finleyelement{Hex8Face_Contact}}
139  \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}}
140    \linev{\finleyelement{Hex27}}{\finleyelement{Rec9}}{N\textbackslash A}{N\textbackslash A}{N\textbackslash A}
141    \linev{\finleyelement{Hex27Macro}}{\finleyelement{Rec9Macro}}{N\textbackslash A}{N\textbackslash A}{N\textbackslash A}
142    \linev{\finleyelement{Tet10Macro}}{\finleyelement{Tri6Macro}}{N\textbackslash A}{N\textbackslash A}{N\textbackslash A}
143    \linev{\finleyelement{Rec9Macro}}{\finleyelement{Line3Macro}}{N\textbackslash A}{N\textbackslash A}{N\textbackslash A}
144    \linev{\finleyelement{Tri6Macro}}{\finleyelement{Line3Macro}}{N\textbackslash A}{N\textbackslash A}{N\textbackslash A}
145  \end{tablev}  \end{tablev}
146  \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.
147  The rich types have to be used if the gradient of function is to be calculated on faces and contacts, respectively.}  The rich types have to be used if the gradient of function is to be calculated on faces and contacts, respectively.}
# Line 254  $7$, $10$, $15$ and $20$, respectively. Line 263  $7$, $10$, $15$ and $20$, respectively.
263  \input{finleyelements}  \input{finleyelements}
264  \clearpage  \clearpage
265    
266    \begin{figure}[th]
267    \begin{center}
268    \subfigure[Triangle]{\label{FINLEY MACRO TRI}\includegraphics[scale=0.25]{figures/FinleyMacroTri}}
269    \subfigure[Quadrilateral]{\label{FINLEY MACRO REC}\includegraphics[scale=0.25]{figures/FinleyMacroRec}}
270    \includegraphics[scale=0.2]{figures/FinleyMacroLeg}
271    \end{center}
272    Macro elements in \finley.
273    \end{figure}
274    
275    \section{Macro Elements}
276    \label{SEC FINLEY MACRO}
277    \finley supports the usage of macro elements~\index{macro elements} which can be used to
278    achieve LBB compliance when solving incompressible fluid flow problems. LBB compliance is required to
279    get a problem which has a unique solution for pressure and velocity. For macro elements the
280    pressure and velocity are approximated by a polynomial of order 1 but the velocity approximation bases on a refinement of the element. The nodes of a triangle and quadrilateral element is shown in Figures~\ref{FINLEY MACRO TRI} and~\ref{FINLEY MACRO REC}, respectively. In essence, the velocity uses the same nodes like a quadratic polynomial approximation but replaces the quadratic polynomial by piecewise linear polynomials. In fact, this is the
281    way \finley is defining the macro elements. In particular \finley uses the same local ordering of the nodes for the macro element as for the corresponding quadratic element. Another interpretation is that
282    one uses a linear approximation of the velocity together with a linear approximation of the pressure but on elements
283    created by combining elements to macro elements. Notice that the macro elements still use quadratic interpolation to represent the element and domain boundary. However, if elements have linear boundary
284    a macro element approximation for the velocity is equivalent to using a linear approximation on a mesh which is created through a one step, global refinement.
285    Typically macro elements are only required to use when an incompressible fluid flow problem
286    is solved, e.g the Stokes problem in Section \ref{STOKES PROBLEM}. Please see Section~\ref{FINLEY MESHES} for
287    more details on the supported macro elements.
288    
289    
290    
291  \begin{table}  \begin{table}
292  {\small  {\scriptsize
293  \begin{tabular}{l||c|c|c|c|c|c|c|c}  \begin{tabular}{l||c|c|c|c|c|c|c|c}
294  \member{setSolverMethod} & \member{DIRECT}& \member{PCG} & \member{GMRES} & \member{TFQMR} & \member{MINRES} & \member{PRES20} & \member{BICGSTAB} & \member{LUMPING} \\  \member{setSolverMethod} & \member{DIRECT}& \member{PCG} & \member{GMRES} & \member{TFQMR} & \member{MINRES} & \member{PRES20} & \member{BICGSTAB} & \member{LUMPING} \\
295  \hline  \hline
# Line 267  $7$, $10$, $15$ and $20$, respectively. Line 300  $7$, $10$, $15$ and $20$, respectively.
300     \hline\member{setIterMax} &  & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\     \hline\member{setIterMax} &  & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
301   \hline\member{setTolerance} &  & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\   \hline\member{setTolerance} &  & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
302   \hline\member{setAbsoluteTolerance} &  & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\   \hline\member{setAbsoluteTolerance} &  & $\checkmark$& $\checkmark$ & $\checkmark$& $\checkmark$& $\checkmark$ & $\checkmark$ \\
303    \hline\member{setReordering} & $\checkmark$ & & & & & & & \\
304  \end{tabular}  \end{tabular}
305  }  }
306  \caption{Solvers available for \finley and the \PASO package and the relevant options in \class{SolverOptions}  \caption{Solvers available for
307   \label{TAB FINLEY SOLVER OPTIONS 1}}  \finley
308    and the \PASO package and the relevant options in \class{SolverOptions}.
309    \MKL supports
310    \MINIMUMFILLIN
311    and
312    \NESTEDDESCTION
313    reordering.
314    Currently the \UMFPACK interface does not support any reordering.
315    \label{TAB FINLEY SOLVER OPTIONS 1} }
316  \end{table}  \end{table}
317    
318  \begin{table}  \begin{table}
# Line 432  $\checkmark$  & Line 474  $\checkmark$  &
474  \caption{Preconditioners available for \finley and the \PASO package and the relevant options in \class{SolverOptions}. \label{TAB FINLEY SOLVER OPTIONS 2}}  \caption{Preconditioners available for \finley and the \PASO package and the relevant options in \class{SolverOptions}. \label{TAB FINLEY SOLVER OPTIONS 2}}
475  \end{table}  \end{table}
476    
477  \subsection{Linear Solvers in \SolverOptions}  \section{Linear Solvers in \SolverOptions}
478  Table~\ref{TAB FINLEY SOLVER OPTIONS 1} and  Table~\ref{TAB FINLEY SOLVER OPTIONS 1} and
479  Table~\ref{TAB FINLEY SOLVER OPTIONS 2} show the solvers and preconditioners supported by  Table~\ref{TAB FINLEY SOLVER OPTIONS 2} show the solvers and preconditioners supported by
480  \finley through the \PASO library. Currently direct solvers are not supported under MPI.  \finley through the \PASO library. Currently direct solvers are not supported under MPI.
481  By default, \finley is using the iterative solvers \PCG for symmetric and \BiCGStab for non-symmetric problems.  By default, \finley is using the iterative solvers \PCG for symmetric and \BiCGStab for non-symmetric problems.
482  If the direct solver is selected which can be useful when solving very ill-posedequations  If the direct solver is selected which can be useful when solving very ill-posed equations
483  \finley uses the \MKL solver package. If \MKL is not available \UMFPACK is used. If \UMFPACK is not available  \finley uses the \MKL \footnote{If the stiffness matrix is non-regular \MKL may return without
484    returning a proper error code. If you observe suspicious solutions when using MKL, this may cause by a non-invertible operator. } solver package. If \MKL is not available \UMFPACK is used. If \UMFPACK is not available
485  a suitable iterative solver from the \PASO is used.  a suitable iterative solver from the \PASO is used.
486    
487  \subsection{Functions}  \section{Functions}
488  \begin{funcdesc}{ReadMesh}{fileName,integrationOrder=-1}  \begin{funcdesc}{ReadMesh}{fileName \optional{, \optional{integrationOrder=-1}, optimize=True}}
489  creates a \Domain object form the FEM mesh defined in  creates a \Domain object form the FEM mesh defined in
490  file \var{fileName}. The file must be given the \finley file format.  file \var{fileName}. The file must be given the \finley file format.
491  If \var{integrationOrder} is positive, a numerical integration scheme  If \var{integrationOrder} is positive, a numerical integration scheme
492  chosen which is accurate on each element up to a polynomial of  chosen which is accurate on each element up to a polynomial of
493  degree \var{integrationOrder} \index{integration order}. Otherwise  degree \var{integrationOrder} \index{integration order}. Otherwise
494    an appropriate integration order is chosen independently.
495    By default the labeling of mesh nodes and element distribution is
496    optimized. Set \var{optimize=False} to switch off relabeling and redistribution.
497    \end{funcdesc}
498    
499    \begin{funcdesc}{ReadGmsh}{fileName \optional{, \optional{integrationOrder=-1}, optimize=True\optional{, useMacroElements=False}}}
500    creates a \Domain object form the FEM mesh defined in
501    file \var{fileName}. The file must be given the \gmshextern file format.
502    If \var{integrationOrder} is positive, a numerical integration scheme
503    chosen which is accurate on each element up to a polynomial of
504    degree \var{integrationOrder} \index{integration order}. Otherwise
505  an appropriate integration order is chosen independently.  an appropriate integration order is chosen independently.
506    By default the labeling of mesh nodes and element distribution is
507    optimized. Set \var{optimize=False} to switch off relabeling and redistribution.
508    If \var{useMacroElements} is set, second order elements are interpreted as macro elements~\index{macro elements}.
509    Currently \function{ReadGmsh} does not support MPI.  
510  \end{funcdesc}  \end{funcdesc}
511    
512  \begin{funcdesc}{load}{fileName}  \begin{funcdesc}{MakeDomain}{design\optional{, integrationOrder=-1\optional{, optimizeLabeling=True\optional{, useMacroElements=False}}}}
513  recovers a \Domain object from a dump file created by the \  Creates a Finley \Domain from a \class{Design} object from \pycad using \gmshextern.
514  eateseates a \Domain object form the FEM mesh defined in  The \class{Design} \var{design} defines the geometry.
 file \var{fileName}. The file must be given the \finley file format.  
515  If \var{integrationOrder} is positive, a numerical integration scheme  If \var{integrationOrder} is positive, a numerical integration scheme
516  chosen which is accurate on each element up to a polynomial of  chosen which is accurate on each element up to a polynomial of
517  degree \var{integrationOrder} \index{integration order}. Otherwise  degree \var{integrationOrder} \index{integration order}. Otherwise
518  an appropriate integration order is chosen independently.  an appropriate integration order is chosen independently.
519    Set \var{optimizeLabeling=False} to switch off relabeling and redistribution (not recommended).
520    If \var{useMacroElements} is set, macro elements~\index{macro elements} are used.
521    Currently \function{MakeDomain} does not support MPI.  
522  \end{funcdesc}  \end{funcdesc}
523    
524    
525    \begin{funcdesc}{load}{fileName}
526    recovers a \Domain object from a dump file created by the \
527    \function{dump} method of a \Domain object defined in
528    file \var{fileName}.
529    \end{funcdesc}
530    
531    
532  \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, \\
533    periodic0=\False,periodic1=\False,useElementsOnFace=\False,optimize=\False}    periodic0=\False, periodic1=\False, useElementsOnFace=\False, useMacroElements=\False, optimize=\False}
534  Generates a \Domain object representing a two dimensional rectangle between  Generates a \Domain object representing a two dimensional rectangle between
535  $(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with  $(0,0)$ and $(l0,l1)$ with orthogonal edges. The rectangle is filled with
536  \var{n0} elements along the $x_0$-axis and  \var{n0} elements along the $x_0$-axis and
# Line 473  For \var{order}=1 and \var{order}=2 Line 541  For \var{order}=1 and \var{order}=2
541  In the case of \var{useElementsOnFace}=\False,  In the case of \var{useElementsOnFace}=\False,
542  \finleyelement{Line2} and    \finleyelement{Line2} and  
543  \finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively.  \finleyelement{Line3} are used to subdivide the edges of the rectangle, respectively.
544    If \var{order}=-1, \finleyelement{Rec8Macro} and \finleyelement{Line3Macro}~\index{macro elements}. This option should be used when solving incompressible fluid flow problem, e.g. \class{StokesProblemCartesian}.
545  In the case of \var{useElementsOnFace}=\True (this option should be used if gradients  In the case of \var{useElementsOnFace}=\True (this option should be used if gradients
546  are calculated on domain faces),  are calculated on domain faces),
547  \finleyelement{Rec4Face} and    \finleyelement{Rec4Face} and  
# Line 491  If \var{optimize}=\True mesh node relabe Line 560  If \var{optimize}=\True mesh node relabe
560  \end{funcdesc}  \end{funcdesc}
561    
562  \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, \\
563    periodic0=\False,periodic1=\False,periodic2=\False,useElementsOnFace=\False,optimize=\False}    periodic0=\False,periodic1=\False,periodic2=\False,useElementsOnFace=\False, useMacroElements=\False, optimize=\False}
564  Generates a \Domain object representing a three dimensional brick between  Generates a \Domain object representing a three dimensional brick between
565  $(0,0,0)$ 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
566  \var{n0} elements along the $x_0$-axis,  \var{n0} elements along the $x_0$-axis,
# Line 507  In the case of \var{useElementsOnFace}=\ Line 576  In the case of \var{useElementsOnFace}=\
576  are calculated on domain faces),  are calculated on domain faces),
577  \finleyelement{Hex8Face} and    \finleyelement{Hex8Face} and  
578  \finleyelement{Hex20Face} are used on the brick faces, respectively.    \finleyelement{Hex20Face} are used on the brick faces, respectively.  
579    If \var{order}=-1, \finleyelement{Hex20Macro} and \finleyelement{Rec8Macro}~\index{macro elements}. This option should be used when solving incompressible fluid flow problem, e.g. \class{StokesProblemCartesian}.
580  If \var{integrationOrder} is positive, a numerical integration scheme  If \var{integrationOrder} is positive, a numerical integration scheme
581  chosen which is accurate on each element up to a polynomial of  chosen which is accurate on each element up to a polynomial of
582  degree \var{integrationOrder} \index{integration order}. Otherwise  degree \var{integrationOrder} \index{integration order}. Otherwise

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