/[escript]/trunk/doc/user/finley.tex
ViewVC logotype

Diff of /trunk/doc/user/finley.tex

Parent Directory Parent Directory | Revision Log Revision Log | View Patch Patch

revision 2748 by gross, Tue Nov 17 07:32:59 2009 UTC revision 2793 by gross, Tue Dec 1 06:10:10 2009 UTC
# 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.  \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.
# Line 262  $7$, $10$, $15$ 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}  \section{Macro Elements}
276  \label{SEC FINLEY MACRO}  \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    
# Line 453  $\checkmark$  & 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 \footnote{If the stiffness matrix is non-regular \MKL may return without  \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  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 \optional{, \optional{integrationOrder=-1}, optimize=True}}  \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.
# Line 484  degree \var{integrationOrder} \index{int Line 505  degree \var{integrationOrder} \index{int
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  By default the labeling of mesh nodes and element distribution is
507  optimized. Set \var{optimize=False} to switch off relabeling and redistribution.  optimized. Set \var{optimize=False} to switch off relabeling and redistribution.
508  If \var{useMacroElements} is set, second order elements are interpreated as macro elements~\index{macro elements}.  If \var{useMacroElements} is set, second order elements are interpreted as macro elements~\index{macro elements}.
509  Currently \function{ReadGmsh} does not support MPI.    Currently \function{ReadGmsh} does not support MPI.  
510  \end{funcdesc}  \end{funcdesc}
511    
512  \begin{funcdesc}{MakeDomain}{design\optional{, integrationOrder=-1\optional{, optimizeLabeling=True\optional{, useMacroElements=False}}}}  \begin{funcdesc}{MakeDomain}{design\optional{, integrationOrder=-1\optional{, optimizeLabeling=True\optional{, useMacroElements=False}}}}
513  Creates a Finley \Domain from a \class{Design} object using \gmshextern.  Creates a Finley \Domain from a \class{Design} object from \pycad using \gmshextern.
514  The \class{Design} \var{design} defines the geometry.  The \class{Design} \var{design} defines the geometry.
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
# Line 503  Currently \function{MakeDomain} does not Line 524  Currently \function{MakeDomain} does not
524    
525  \begin{funcdesc}{load}{fileName}  \begin{funcdesc}{load}{fileName}
526  recovers a \Domain object from a dump file created by the \  recovers a \Domain object from a dump file created by the \
527  eateseates a \Domain object form the FEM mesh defined in  \function{dump} method of a \Domain object defined in
528  file \var{fileName}. The file must be given the \finley file format.  file \var{fileName}.
 If \var{integrationOrder} is positive, a numerical integration scheme  
 chosen which is accurate on each element up to a polynomial of  
 degree \var{integrationOrder} \index{integration order}. Otherwise  
 an appropriate integration order is chosen independently.  
529  \end{funcdesc}  \end{funcdesc}
530    
531    

Legend:
Removed from v.2748  
changed lines
  Added in v.2793

  ViewVC Help
Powered by ViewVC 1.1.26