Dealing with corners and changing geometry - The Boundary Element Method with Programming - page 346

accurately computed and this error accumulates load case after load case. Obviously
some improvements are possible by adjusting the stress distribution so the resultant
excavation force is closer to the actual one.

12.5 CONCLUSIONS

The correct treatment of corners and edges is of great importance for some applications,
in particular for applications where the boundary conditions as well as the geometry are
changing during the calculation process. It was found out, that from all possibilities to
improve the results at corner nodes discontinuous elements give the best results. Of
course additional degrees of freedom are introduced by this method. For simplicity all
elements have been treated as discontinuous here. This increases the size of the equation
system drastically, especially in 3D. It is much more efficient to use discontinuous nodes
only where they are needed, i.e. only at corner and edge nodes where the traction is
discontinuous. The manner in which the interpolation functions are presented in chapter
3 makes possible a mixture of discontinuous and continuous functions in one element.
When dealing with changing geometries as in sequential excavation problems the multi-
region analysis with discontinuous elements gives good results. However, the effort can
be quite considerable especially for 3-D applications because with each excavation stage
modelled the number of regions and hence the interface degrees of freedom increase. An
alternative method that involves only one region seems attractive but the accuracy still
has to be improved.

12.6

REFERENCES

1. Beer G. and Watson J.O. (1995) Introduction to Finite and Boundary Element
Methods for Engineers. J. Wiley.
2. Gao X.W. and Davies T. (2001) Boundary element programming in mechanics.
Cambridge University Press, London.
3. Sladek V. and Sladek J. (1991) Why use double nodes in BEM? Engineering
Analysis with Boundary Elements 8 : 109-112.
4 . Aliabadi M. H. (2002) The Boundary Element Method (Volume 2). J. Wiley.
5. Stroud, A.H. and Secrest, D. (1966) Gaussian Quadrature Formulas. Prentice-Hall,
Englewood Cliffs, New Jersey.
6. Duenser C. (2007) Simulation of sequential tunnel excavation with the Boundary
Element Method. Monographic Series TU Graz,Austria.
7. Duenser C., Beer G. (2001) Boundary element analysis of sequential tunnel advance.
Proceedings of the ISRM regional symposium, Eurock: 475-480.