"Mesh Generation and Grid Generation on the Web"

Ian MacPhedran and Roger Young's pages on finite element resources:

The Geometry Center

MGNet has codes, preprints, virtual proceedings, a large bibliography, and more dealing with multigrid and/or domain decomposition methods for solving PDE's. There has also been a monthly electronic newsletter for the past 6 years.

The CFD (computational fluid dynamics) codes list

Robert Schneiders site on mesh generation is a large site with info on people, literature, software, and numerous research efforts worldwide.

PDEtools is a Maple package for working with PDEs, solving systems of ODEs and making general changes of variables.

OpenFEM is an opensource finite element toolbox for Matlab and Scilab.

Newsgroups for PDE and FEM

There are also 2 mailing lists:

FEA-L: To get on this list, send email to listserv@sailor.itis.com. The first line of the body of the message should contain the statement SUBSCRIBE FEA-L.

feusers: To get on this list, send email to mailbase@mailbase.ac.uk. The first 2 lines of the body of the message should contain
JOIN FEUSERS
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Books and References for PDE and FEM

Axelsson, O.; Barker, V.A. 1984, Finite Element Solution of Boundary Value Problems: Theory and Computation, Academic Press

Bornemann, F., An Adaptive Multilevel Approach to Parabolic Equations in Two Space Dimensions, Dissertation, Freie Universitaet Berlin, 1991.

Bornemann, F.; Erdmann, B.; Kornhuber, R., Adaptive Multilevel Methods in Three Space Dimensions, Int. J. Numer. Meths. in Eng., 36, 1993.

Braess, Dietrich. 1992, Finite Elemente, Springer, (In German)

Braess, Dietrich. 1992, Finite Elements: Theory, fast solvers, and applications in solid mechanics translated from the 2nd German Edition by Larry Schumaker. The book is directed to graduate students in mathematics and to young researchers. We take into account that most of the research is now concentrated on finite elements for hard problems and on the fast solution of the resulting equations. Therefore the nonconforming elements and saddle point problems get as much space as the conforming elements. Moreover we treat the method of conjugate gradients and multigrid methods. Finally we present an introduction to the application of finite elements in solid mechanics which up to now does not exist in textbooks in a similar way.

Brenner; Scott. 1994, The Mathematical Theory of Finite Elements, Springer-Verlag

Celia; Grey., Numerical Methods for Differential Equations, [G. Scott Lett] "very good".

Ciarlet, Philippe G. 1978, The finite element method for elliptic problems, Amsterdam: North-Holland

Deuflhard, P.; Leinen, P.; Yserentant, H., Concepts of an Adaptive Hierarchical Finite Element Code., IMPACT, 1, 1989.

Grossmann; Roos., Numerik Partieller Differentialgleichungen, Teubner., (In German).

Johnson, Claes. Numerical Solution of Partial Differential Equations by the Finite Element Method. [Hans Sandholt] I found great pleasure to read it.

Hackbusch, Wolfgang, Elliptic Differential Equations. Theory and Numerical Treatment, Springer Series in Computational Mathematics 18, Springer-Verlag. See Hackbush's page.

Hackbusch, Wolfgang, Iterative Solution of Large Sparse Systems of Equations, Applied Mathematical Sciences, Vol. 95., Springer-Verlag. See Hackbush's page. [G. Scott Lett] A good book to start; helping to bridge the gap between recipes and understanding.

Hughes, T.J.R. 1987. The Finite Element Method

Kornhuber, R. Monotone Multigrid Methods for Nonlinear Variational Problems. Habilitationsschrift, Freie Universitaet Berlin, 1995.

Oden, J.T.; Reddy, J. N. 1976. An introduction to the mathematical theory of finite elements. New York: Wiley-Interscience

Schwarz, H.R. 1984. Methode der Finiten Elemente. Teubner. (In German)

Strang; Fix. 1973. An Analysis of the Finite Element Method. Prentice Hall

Wait, R.; Mitchell, A.R. 1985. Finite Element Analysis and Applications. John Wiley & Sons.

Zienkiewicz, O.C.; Morgan, K. 1983. Finite Elements and Approximation. J. Wiley & Sons

Sandia's Comparison of Meshing Software Packages

Sandia survey of meshing software (1998)

Albert Hines' Comparison of Meshing Software Packages

[Albert M. Hines, Feb 1996]:
This table reflects the best information that I can gather
to date on some of the more popular meshers.  Many others exist
and may be better or worse than those listed here. This study was
conducted by Howmet, a manufacturing corporation with no incentive
to support or reject any package over another. Portions of the
study were also funded by your generous tax donations through an
ICCA program designed to enhance the capability of US investment
casting through (among many other things) automeshing software.


The meshes were used for heat transfer and fluid flow predominantly.
A limited amount of stress analysis and electromagnetics were also
investigated.  The desired modeling analysis is a crucial factor
in determining whether a meshing package will support the
mesh quality and refinement necessary for a simulation.
(e.g. mesh quality is almost a non-issue for heat transfer,
but not for stress!)   The geometries that I have considered
are aerospace turbine engine components: complex 3-D thin walled
structures. Bulky or spindly geometries may behave differently
(e.g. engine blocks, oil rigs, etc.).


The scale used is 0 to 10. A zero means that it meets none of my
expectations. A 10 means that it meets all of my expectations.
A cumulative score should not be obtained by adding columns.
Many entries are more important than others. Weighting for
the various tasks are different for each analyst/application.
The scores are often subjective and should be considered at best
approximate. They do not necessarily reflect the opinion of Howmet
Corporation or, for that matter, of anyone other than myself.
They are, however, the result of careful and expensive investigation
into the market.

tty
                            SOFTWARE(5): OCTREE  PATRAN   MESHCAST  HYPERMESH   ALG

      ----------------------------------------------------------------------------

      3-D SURFACE MESH (*)

              automesh robustness (a)         9       8       7       4       3

              automesh flexibilty (b)         4       6       8       7       5

              number of elements (c)         10       6       7       6       8

              meshing speed (d)               3       1       9       5       3

              typical element quality (e)     1       8       7       4       5

              semi-auto effectiveness        N/A      7       3      10       6

              mesh editing                   N/A      8       1      10       2

ttyb
      3-D SOLID MESH

              automesh robustness (a)         8       7 (4)   7       N/A     4

              automesh options (b)            6       7       7       N/A     6

              number of elements (c)         10       4 (4)   5       N/A     3

              meshing speed (d)               4       1       9       N/A     3

              typical element quality (e)     2       6 (4)   7       N/A     4

              semi-auto effectiveness        N/A      4      N/A      9       6

              mesh editing                   N/A      8       3      10       6

              software vendor meshing (f)     8       2       9       5       8

              auto-hex meshing               N/A     N/A     N/A     N/A      3

ttyb
      GENERAL

              Ease of use (g)                 3       4       9       8       4

              I/O options                     4       8       7       9       7

              Error reporting                 3       4       3       3       3

              Barrier-free tech. support      8       1       10      6       3

              Competent/helpful tech. support 6       6       8       7       8

              Ease of install/maintenance     8       4       8       7       8

              "Bug-free-ness"                 6       7       3       4       4

              COST(1)                    $12,000 $6800(2) $10,000(3) $4500  $5,525



NOTES


*    OCTREE surface mesh obtained by extracting faces of its solid elements.


(a)  Should mesh the first time with given parameters. This refers to
tetrahedra only.


(b)  Includes things like a finely meshed boundary with coarsely meshed
interior, ignoring features smaller than a given size, multiple element
layers through thin sections, multiple meshing algorithms, etc. Refers
to tetrahedral elements.


(c)  Being in industry, practicality is a must. Meshes > ~100,000 nodes
do not run in a reasonable amount of time in our lab.  If a
program cannot mesh a part within this limitation, I consider
it unable to mesh the part, even if it might successfully
generate the mesh. Automeshers should allow the specification
of a target number of nodes. Any mesh above this size should
send a flag to remesh, ignoring small features.


(d)  Again, from a practical point of view, there are limits beyond which I
consider the program to fail. The most glaring comparison is
between PATRAN and MeshCAST. A typical part that meshes less
than 10 minutes with MeshCAST normally takes several days to
mesh in PATRAN. It is likely that given infinite time, PATRAN
would succeed (after all, it contains the RPI finite octree
mesher as one of its algorithms!), however, I can't justify
tying a machine up for a week to find out if it will mesh (and often it
crashes at the end of the meshing cycle!).


(e)  There are many if's and's & but's here. Basically I use aspect ratio and
dihedral angle for tetrahedra, jacobian at the gauss points for hex'es,
wedge's, and quad's, and interior angle for triangles. These are the only
element types that I use.


(f)  For many problems, I could not successfully mesh the part, so I sent the
geometry to the software vendor. In other cases, I sent it specifically
to evaluate the software run under the developer's care. Score reflects
willingness, speed, size, and quality of the mesh received.


(g)  I am partial to a Mac-like interface, which I consider to be the computer
world's 'reference standard'. Program should require minimal training and
manuals. All options should be available on the menus (no hidden or
cryptic commands). This is not just personal preference, but time and
dollars in training.


(1)  Cost is annual lease, US dollars, except as noted. Platform is HP-735
network license.  Cost is what we pay, not necessarily list price.


(2)  Unigraphics common database module $3,000 extra (i.e. $9,800). Base
license includes one translator, we chose IGES.


(3)  This price includes the fluids, thermal, radiation, solver modules. I am
not sure of the price of just MeshCAST by itself.


(4)  This is very approximated assuming infinite meshing time.


(5)  Versions tested: OCTREE: March 1,1995, P3: v1.4, Meshcast: v3.0.3,
Hypermesh: v2.00b, Algor: July, 1995.

Furthermore, the toolbox includes extensions for handling systems of equations and example algorithms for nonlinear methods used in obstacle, porous media or Stefan problems.

Core of the program is a variety og multilevel/multigrid preconditioners for the arising linear systems.

This object-oriented code is written in C++ and can be compiled with Gnu g++, version 2.7.2, and some other compilers. If you want to compile with Gnu g++, version 2.6.3, you have to change the definition of the complex-type in the file general.h.

Tri

Tripoint takes input from text files as the polygon to triangulate. Output is also as text files, using the same data format. Points are represented by their coordinates, and edges by the indecies of their endpoints. There is a matlab code front end, called drawmesh.m. This allows the graphical input of the input polygon using the mouse, and the display with zooming of the output triangulation. There is also matlab code called dispmesh.m, which allows the displaying of output only. This allows the user freedom over how to generate the input.

The algorithm is similar to one in "Provably Good Mesh Generation"[1]. The main differences include: Tripoint uses connected component information while generating the quadtree, as in [2]. Tripoint does not cut dangling acute triangles from the input as a first step as in [1]. Ideally, as in [2], tripoint would center input verticies in boxes, and use a very general and easy algorithm for triangulating individual boxes. Instead, vertices are not centered, and the ideal algorithm is used with added special case analysis for boxes near an input vertex.

QMG

The main new feature of QMG2.0 (compared to QMG1.1) is its ability to handle true curved geometry. QMG2.0 permits boundaries defined by Bezier curves, triangular Bezier patches and quadrilateral tensor-product Bezier patches. Representations of certain simple curved geometries (cylinders, spheres, tori) using Bezier patches are shipped with QMG.

Sinda/Fluint

deal.II can be downloaded for free and is distributed under an Open Source license.

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