Numerical Solution of Seepage Problem Using Quad-Tree Based Triangular Finite Elements

Abstract

A triangular mesh based on the quad-tree grid is applied in the finite element solution of seepage flow under a sheet pile. After obtaining the quad-tree grid, cells are directly transformed into triangles by dividing a cell into four to eight triangles. Cells at the boundaries are turned into triangles using the Delaunay criterion for cell corner nodes and intersection nodes. Different mesh arrangements are considered in order to compare the flow characteristics with changing mesh size. Mesh patterns and results from finite element method are presented graphically for two test cases

Keywords

• Seepage, quad-tree grid, finite element method, triangulation

References

1. Basu, P. P., Peano, A., Adaptivity in p-version finite element analysis, J. Struct. Engng., 109, 2310-2324, 1983.
2. Zienkiewicz, O. C., Zhu, J. Z., Gong, N. G., Effective and practical h–p adaptive analysis procedure for the finite element method, Int. J. Numer. Meth. Engng., Cilt 28, 879-891, 1989.
3. Alyavuz, B., Dairesel delikli dikdörtgen levhanin h-tipi sonlu elemanlar ile uyarmali analizi, Gazi Üniversitesi Müh. Mim. Fak. Dergisi, 22 (1), 39-46, 2007.
4. Yeh, G. T., Chang, J. R., Cheng, H. P., Sung, C. H., An adaptive local grid refinement based on the exact peak capture and oscillation free scheme to solve transport equations, Comput. Fluids, 24 (3), 293-332, 1995.
5. Cruz, L. S., Numerical solution of shallow water equations on quad-tree grids, Ph.D. Thesis, University of Oxford, 1997.
6. Rogers, B., Fujihara, M., Borthwick, A. G. L., Adaptive q-tree Gudunov type scheme for shallow water equations, Int. J. Numer. Meth. Fluids, 35, 247-280, 2001.
7. Borthwick, A. G. L., Leon, S. C., Josca, J., Adaptive quad-tree model of shallow-flow hydrodynamics, J. Hydraul. Res., 39 (4), 413-424, 2001.
8. Koçyiğit, Ö., Modelling of water quality and sediment transport in aquatic basins using an unstructured grid system, Ph.D. Thesis, Cardiff University, U.K., 2003.

9. Liang, Q. Du, G., Hall, J. W., Borthwick, A. G. L., Flood inundation modeling with an adaptive quad-tree grid shallow water equation solver, J. Hydraul. Engng., 134 (11), 1603-1610, 2008.
10. Liang, Q., Borthwick, A. G. L., Adaptive quad-tree simulation of shallow flows with wet–dry fronts over complex topography, Comput. Fluids, 38, 221 – 234, 2009.
11. Finkel, R. A., Bentley, J. L., Quad-trees: A data structure for retrieval on composite keys Acta Inform., 4 (1), 1-9, 1974.
12. Samet, H., Applications of spatial data structures, Addison Wesley Publishing Company, 1990.
13. Yiu, K. F. C., Greaves, D. M., Saalehi, A., Borthwick, A. G. L., Quad-tree grid generation: Information handling, boundary fitting and cfd applications”, Comput. Fluids, 25 (8), 759-769, 1996.
14. Wang, Z.J., A quad-tree-based adaptive cartesian/quad grid flow solver for Navier- Stokes equations, Comput. Fluids, 27 (4), 529-549, 1998.
15. Greaves, D. M., Borthwick, A. G. L., Hierarchical tree - based finite element mesh generation, Int. J. Numer. Meth. Engng., 45, 447-471, 1999.
16. Bern, M., Eppstein, D., Teng, S.-H., Parallel construction of quad-trees and quality triangulations, Int. J. Comput. Geom. App., 9 (6), pp. 517-532, 1999.
17. Quad-tree-Based Triangular Mesh Generation for Finite Element Analysis of Heterogeneous Spatial Data, ASAE Annual International Meeting, Sacramento, California, USA, 2001.
18. Cedergren, H. R., Seepage, drainage, and flow nets, John Wiley & Sons; 3rd edition, 1989.
19. Harr, H.E., Ground Water and Seepage, McGraw-Hill, New York, 1962.
20. Wang, H. F., Anderson M. P., Introduction to groundwater modeling: Finite difference and finite element methods, Academic Press, 1995.
21. Lo, S. H., A new mesh generation scheme for arbitrary planar domains, Int. J. Numer. Meth. Engng., 21, 1403-1426, 1985.
22. Zhu, Z. Q., Wang, P., Tuo, S. F., Liu, Z., A structured/unstructured grid generation method and its application, Acta Mech., 167, 197-211, 2004.
23. Watson, D.F., Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes, Comput. J., 24, 167-172, 1981.
24. Bowyer, A., Computing Dirichlet tessellations, Comput. J., 24 (2), 162-166 1981.
25. Bern, M., Plassmann, P., Handbook of Computational Geometry, Eds: J.R. Sack and J. Urritia, Elsevier Science, 303-308, 2000.
26. Field, D. A., Laplacian smoothing and Delaunay triangulations, Commun. Appl. Numer. Meth., 4, 709-712, 1998.
27. Hyun, S., Lindgren, L. E., Smoothing and adaptive remeshing schemes for graded element, Commun. Numer. Meth. Engng., 17 (1), 1-17, 2001.