17th GAMM-Seminar Leipzig on
Construction of Grid Generation Algorithms

Max-Planck-Institute for Mathematics in the Sciences
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  17th GAMM-Seminar
February, 1st-3rd, 2001
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  Abstract Stefan M. Holzer, Fri, 15.00-15.25 Previous Contents Next  
  Finite Element Meshes and Suitable Data Structures
Stefan M. Holzer (Uni Stuttgart)

The representation that is best suited for finite element and/or boundary element meshes is topological representation in a B-Rep (boundary representation) data structure. In such a data structure, geometric data as well as boundary conditions can be handled as attributes to the topological entities vertex (v), edge (e), and face (f). Various data structures are suitable for representing a vef-graph. However, most finite element programs still stick to the simplest possible representation, i.e. to a representation that stores only the (f,v) relation. If the spatial decomposition is a simplicial one (i.e., triangles in 2D or tetrahedra in 3D), then such a mesh can be represented in a relational database by tables with a fixed number of columns, based on the (f,v) relation.

Considering adaptive mesh refinement rather than density-based remeshing, such a data structure is no longer satisfactory. Local hierarchical mesh refinement (which is also needed for geometrical multigrid) requires bisection of edges, insertion of new nodes on old edges, splitting of old faces, and so on. Intermediate topologies that arise during such a refinement procedure are outside the scope of a data structure that is based on fixed-size tables for the (f,v) relation only. Also, topological queries such as adjacencies are expensive on such a simple data structure. Topological queries play a major role in adaptive methods (e.g., for evaluating Babuska-Miller error estimates) and in parallelization (e.g., graph-theory based domain partioning algorithms).

A two-dimensional finite element mesh as well as a surface mesh for a three-dimensional body can both be considered as two-manifolds. Data structures for two-manifolds are available. For the purpose of designing versatile finite element codes, the most attractive B-Rep data structure for FE meshes is the Winged Edge data structure introduced by Baumgart in the mid-80ies. We discuss the application of such a data structure in the context of uniform p-extension Finite Element methods for plate bending problems on locally refined meshes, as well as application in a greedy algorithm for dynamic domain partitioning. We also discuss possible data structures for three-dimensional finite element meshes, ranging from combined B-Rep and sweep representation to fully 3D non-manifold representations.

Finally, an algorithm for 3D finite element meshing of typical Civil Engineering structures is presented.

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[O->]Jens Burmeister (Uni Kiel), Kai Helms (MPI Leipzig)
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