Computational & Technology Resources
an online resource for computational,
engineering & technology publications
Computational Science, Engineering & Technology Series
ISSN 1759-3158
Edited by: B.H.V. Topping, Z. Bittnar
Chapter 7

Projection Techniques embedded in the PCGM for handling Hanging Nodes and Boundary Restrictions

A. Meyer

Faculty of Mathematics, Technical University of Chemnitz, Germany

Full Bibliographic Reference for this chapter
A. Meyer, "Projection Techniques embedded in the PCGM for handling Hanging Nodes and Boundary Restrictions", in B.H.V. Topping, Z. Bittnar, (Editors), "Engineering Computational Technology", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 7, pp 147-165, 2002. doi:10.4203/csets.8.7
Keywords: finite elements, adaptive work, preconditioned conjugate gradient.

We look for efficient implementations of adaptive finite element simulations for problems in mechanics and physics. Here, efficient means both with respect to memory requirements and with respect to time.
The first wish "low memory" requires data structures which allow quick execution of the main tasks
  • mesh handling (refinement/coarsening)

  • forming (element) matrix / right hand side

  • solving the linear systems

  • post processing / error estimation [1,4] / error control for mesh refinement
by storing as less information as possible. The second wish "short running times" require such data structures with effort for all the main tasks sketched above, when is the number of nodes (actually in one mesh of the adaptive series).

In our experiments we have defined three data structures:

"nodes": containing coordinates and solution values of each node
"edges": containing 3 nodes and refinement information to son-edges
  or geometry/boundary conditions of finest edges, resp.
"elements": containing its edges and nodes, material information
  and the element matrix and right hand side

For completing a finite element calculation on these data structures, we have to incorporate special projection procedures for handling "hanging nodes", (a necessary tool for most simple refinement ideas) and for all kinds of restrictions from boundary conditions.

We give a special variant of the well-known Preconditioned Conjugate Gradient Method running in a subspace of , which is defined from a special projection . Then we describe the structure of the projector for solving these problems for linear and qudratic elements:

  • In case of hanging nodes, calculate a solution that belongs to the conformal subspace (continuous function).
  • Satisfy boundary restrictions for some boundary nodes.
We give an example of a Signorini contact simulation [3,2], where all these projections are included. Here, the solve up to nodes requires near the same running time as one linear elastic problem with prescribed displacements.

M. Ainsworth, J. Oden, "A Posteriori Error Estimation in Finite Element Analysis", Comp.Meth.Appl.Mech.Eng. 142(1997)1-2, 1-88. doi:10.1016/S0045-7825(96)01107-3
J.Schöberl, "Solving the Signorini problem on the basis of domain decomposition techniques", Computiong 60 (1993), 323-344. doi:10.1007/BF02684379
A. Signorini, "Sopra akune questioni di elastistatica", Attil della Societa Italiana per il Progresso delle Scienzie (1993).
R. Verfürth, "A Review of a posteriori Error Estimation and Adaptive Mesh Refinement Techniques", Wiley-Teubner, Chichester, Stuttgart (1996).

purchase the full-text of this chapter (price £20)

go to the previous chapter
go to the next chapter
return to the table of contents
return to the book description
purchase this book (price £74 +P&P)