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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 172

Different Approaches to the Corner Problem in the Boundary Element Method with Application to Tunnel Excavation

U. Eberwien, C. Duenser and G. Beer

Institute for Structural Analysis, Graz University of Technology, Austria

Full Bibliographic Reference for this paper
U. Eberwien, C. Duenser, G. Beer, "Different Approaches to the Corner Problem in the Boundary Element Method with Application to Tunnel Excavation", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 172, 2006. doi:10.4203/ccp.83.172
Keywords: boundary elements, multi region, corner problem, discontinuous elements, tunnelling, sequential excavation.

Summary
The boundary element method (BEM) is well suited for modelling problems in infinite domains. And this situation is present in the case of sequential excavation in tunnelling.

Within an infinite domain, which is the rock mass, there exist numerous finite subdomains, whose outer boundary forms the geometry of the tunnel. Each of the finite subdomains represents the rock volume to be excavated during one excavation step. Hence, between two subsequent analysis steps, the problem geometry changes such, that one or more of the finite subdomains become inactive, i.e., are excavated. While the first analysis step always applies to the primary stress field of the intact rock mass, the last step refers to the completely excavated tunnel shape. Thus, using a BEM approach for this kind of problem involves the analysis of multiple subdomains [1,2] in multiple analysis steps with subsequently changing geometry, boundary conditions and loading. A thorough treatment of each single analysis step is essential since in an excavation analysis the result of one analysis step serves as loading for the next. Introducing errors would interfere subsequent results accumulatively.

The particular geometry conditions of tunnel excavation problems always show corner and edge nodes with discontinuous normal vectors. In the case of Dirichlet boundary conditions, traction discontinuities appear at these locations. This is well known as the corner problem. Evaluating the underlying boundary integral equation (BIE) by means of a collocation scheme with continuous, isoparametric elements results in a system of equations which is not suited for the solution of all unknowns at the corner points. In these cases, additional equations are required to allow for the evaluation of the remaining unknowns.

The full paper deals with three different approaches. Two of them introduce additional equations. In [3,4] the auxiliary equation approach is derived from the symmetry condition of the stress tensor. Sladek and Sladek [5] make up additional equations from the known displacement field at the corner node. In the third approach, discontinuous elements are used. The aim is, to determine the one best suited for sequential excavation analyses.

The application of the auxiliary equation approach is shown in two examples. By means of a benchmark example, the accuracy of the named approach is proved. The comparision with results obtained without treatment of the corner problem makes the need for a treatment obvious.

The second example concerns a realistic excavation problem. Some selected results are presented. At first, a finite element analysis serves as a reference solution to demonstrate the validity of the auxiliary equation approach. The consistency of the results is quite convincing. Second, it is shown, how the wrong, or not at all, treatment of the corner problem affects the traction results. And wrong tractions make up a incorrect loading for the next excavation step. Using the auxiliary equation approach gives the expected smooth traction distributions.

The ongoing work in this field will allow the authors to present results for the other approaches discussed to the corner problem.

References
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G. Beer, "Programming the Boundary Element Method", Wiley, 2001.
2
C. Duenser, G. Beer, "Boundary element analysis of sequential tunnel advance", Proceedings of the ISRM regional symposium Eurock, 475-480, 2001, A.A. Balkema, Netherlands.
3
C. Duenser, "Simulation of sequential tunnel excavation with the Boundary Element Method", PhD Thesis, 2001, Graz University of Technology.
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X.-W. Gao, T. G. Davies, "Boundary Element Programming in Mechanics", Cambridge University Press, 2002.
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V. Sladek, J. Sladek, "Why use double nodes in BEM?", EABE, V8, No 2, 109-112, 1991. doi:10.1016/0955-7997(91)90060-7
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L. Gaul, M. Koegl, M. Wagner, "Boundary Element Methods for Engineers and Scientists", Springer, 2003.
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C.A. Brebbia, J.C.F. Telles, L.C. Wrobel, "Boundary element techniques: theory and application in engineering", Springer, New York, 1984.
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X. Zhang, X. Zhang, "Exact integrations of two-dimensional high-order discontinuous boundary elements of elastostatics problems", EABE, V28,725-732, 2004. doi:10.1016/j.enganabound.2004.02.003

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