High-Order FEM Formulation for 3-D Slope Instability

Affiliation(s)

Geo-Disaster Laboratory, Graduate School of Science and Engineering, Ehime University, Matsuyama, Japan.

Geo-Disaster Laboratory, Graduate School of Science and Engineering, Ehime University, Matsuyama, Japan.

ABSTRACT

High-order finite element method (FEM) formulation also referred to as spectral element method (SEM) formulation is currently implemented in this paper for 3-dimensional (3-D) elasto-plastic problems in stability assessment of large- scale slopes (vegetated and barren slopes) in different instability conditions such as seismic and saturation. We have reviewed the SEM formulation, and have sought its applicability for vegetated slopes. Utilizing p (high-order polynomial degree or spectral degrees) and h (mesh operation for quality meshing in required elemental budgets) refining techniques in the existing FEM, the complexity of problem domain can be well addressed in greater numerical stability. Unlike the existing FEM formulation, this high-order FEM employs the same integration and interpolation points to achieve a progressive response of the instability, which drastically reduces the computational costs (formation of diagonalized mass matrix) and offers significant benefits to slope instability computations for serial and parallel implementations. With this formulation, we have achieved the following three qualities in slope instability modeling: 1) geometric flexibility of the finite elements, 2) high computational efficiency, and 3) reliable spectral accuracy. A sample problem has also been presented in this paper, which has accommodated all aforesaid numerical qualities.

Cite this paper

T. Chandra, B. Prakash and Y. Ryuichi, "High-Order FEM Formulation for 3-D Slope Instability,"*Applied Mathematics*, Vol. 4 No. 5, 2013, pp. 8-17. doi: 10.4236/am.2013.45A002.

T. Chandra, B. Prakash and Y. Ryuichi, "High-Order FEM Formulation for 3-D Slope Instability,"

References

[1] P. Cupillard, E. Delavaud, G. Burgos, G. Festa, J. P. Vi lotte, Y. Capdeville and J. P. Montagner, “RegSEM: A Versatile Code Based on the Spectral Element Method to Compute Seismic Wave Propagation at the Regional Scale,” Geophysical Journal International, Vol. 188, No. 3, 2012, pp. 1203-1220. doi:10.1111/j.1365-246X.2011.05311.x

[2] D. Jonas, D. Basabe and M. K. Sen, “Stability of the High-Order Finite Elements for Acoustic or Elastic Wave Propagation with High-Order Time Stepping,” Geophysical Journal International, Vol. 181, No. 1, 2010, pp. 577-590. doi:10.1111/j.1365-246X.2010.04536.x

[3] I. Khan and B. H. V. Topping, “Parallel Finite Element Analysis Using Jacobi-35 Conditioned Conjugate Gradi ent Algorithm,” Advances in Engineering Software, Vol. 25, No. 2-3, 1996, pp. 309-319. doi:10.1016/0965-9978(95)00111-5

[4] J. Y. Kim and S. R. Lee, “An Improved Search Strategy for the Critical Slip Surface Using Finite Stress Fields,” Computers and Geotechnics, Vol. 21, No. 4, 1997, pp. 295-313. doi:10.1016/S0266-352X(97)00027-X

[5] D. Komatitsch and J. Tromp, “Introduction to the Spectral Element Method for Three-Dimensional Seismic Wave Propagation,” Geophysical Journal International, Vol. 139, No. 3, 1999, pp. 806-822. doi:10.1046/j.1365-246x.1999.00967.x

[6] D. Komatitsch and J. Tromp, “Spectral-Element Simulations of Global Seismic Wave Propagation I. Validation,” Geophysical Journal International, Vol. 149, No. 2, 2002, pp. 390-412. doi:10.1046/j.1365-246X.2002.01653.x

[7] D. Komatitsch, S. Tsuboi and J. Tromp, “The Spectral Element Method in Seismology. Seismic Earth: Array Analysis of Broadband Seismograms,” Geophysical Mo nograph Series, Vol. 157, 2005, pp. 205-227. doi:10.1029/156GM13

[8] D. Komatitsch and J. P. Vilotte, “The Spectral Element Method: An Efficient Tool to Simulate the Seismic Response of 2D and 3D Geological Structures,” Bulletin of the Seismological Society of America, Vol. 88, No. 2, 1998, pp. 368-392.

[9] R. Pasquetti and F. Rapetti, “Spectral Element Methods on Unstructured Meshes: Comparisions and Recent Advances,” Journal of Scientific Computing, Vol. 27, No. 1-3, 2006, pp. 377-387. doi:10.1007/s10915-005-9048-6

[10] J. Tromp, D. Komatitsch and Q. Liu, “Spectral-Element and Adjoint Methods in Seismology,” Communications in Computational Physics, Vol. 3, No. 1, 2008, pp. 1-32.

[11] H. N. Gharti, D. Komatitsch, V. Oye, R. Martin and J. Tromp, “Specfem 3D Geotech User Manual, 1.1 Beta,” 2012. http://www.geodynamic.org/cig/software/specfem3d-geotech

[12] R. C. Tiwari, N. P. Bhandary, R. Yatabe and D. R. Bhat, “New Numerical Scheme in the Finite Element Method for Evaluating Root-Reinforcement Effect on Soil Slope Stability,” Geotechnique, Vol. 63, No. 2, 2013, pp. 129 139. doi:10.1680/geot.11.P.039

[13] M. Genet, A. Stokes, F. Salin, S. B. Mickovski, T. Four caud, J. Dumail and L. P. H. Van Beek, “The Influence of Cellulose Content on Tensile Strength in Tree Roots,” Plant and Soil, Vol. 278, No. 1-2, 2005, pp. 1-9.

[14] D. H. Gray and R. D. Sotir, “Biotechnical Soil Bioengineering Slope Stabilization: A Practical Guide for Erosion Control,” John Wiley & Sons Ltd., New York, 1996.

[15] L. J. Waldron, “The Shear Resistance of Root-Permeated Homogenous and Stratified Soil,” Soil Science Society of America Journal, Vol. 41, No. 5, 1977, pp. 843-849. doi:10.2136/sssaj1977.03615995004100050005x

[16] T. H. Wu, M. C. Kinnell, W. P. McKinnell III and D. N. Swanston, “Strength of Tree Roots and Landslides on Prince of Wales Island,” Canadian Geotechnical Journal, Vol. 16, No. 7, 1979, pp. 19-33. doi:10.1139/t79-003

[17] G. B. Bischetti, E. A. Chaiaradia, T. Epis and E. Morlotti, “Root Cohesion of Forest Species in the Italian Alps,” Plant and Soil, Vol. 324, No. 1, 2009, pp. 71-89. doi:10.1007/s11104-009-9941-0

[18] H. N. Gharti, D. Komatitsch, V. Oye, R. Martin and J. Tromp, “Application of an Elastoplastic Spectral-Element Method to 3D Slope Stability Analysis,” International Journal for Numerical Methods in Engineering, Vol. 91, No. 1, 2012, pp. 1-26. doi:10.1002/nme.3374

[19] O. C. Zienkiewicz, C. Humpheson and R. W. Lewis, “Associated and Non-Associated Visco-Plasticity and Plasticity in Soil Mechanics,” Geotechnique, Vol. 25, No. 4, 1975, pp. 671-689. doi:10.1680/geot.1975.25.4.671

[20] M. M. Berilgen, “Investigation of Stability of Slopes under Drawdown Conditions,” Computers and Geotechnics, Vol. 34, No. 2, 2007, pp. 81-91. doi:10.1016/j.compgeo.2006.10.004

[21] D. V. Griffiths, J. Huang and G. F. Dewolfe, “Numerical and Analytical Observations on Long and Infinite Slopes,” International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 35, No. 5, 2011, pp. 569-585. doi:10.1002/nag.909

[22] D. V. Griffiths and P. A. Lane, “Slope Stability Analysis by Finite Elements,” Geotechnique, Vol. 49, No. 3, 1999, pp. 387-403. doi:10.1680/geot.1999.49.3.387

[23] M. S. Huang and C. Q. Jia, “Strength Reduction FEM in Stability Analysis of Soil Slope Subjected to Transient Unsaturated Seepage,” Computer and Geotechnics, Vol. 36, No. 1-2, 2009, pp. 93-101. doi:10.1016/j.compgeo.2008.03.006

[24] P. A. Lane and D. V. Griffiths, “Assessment of Stability of Slopes under Draindown Conditions,” Journal of Geo technical and Geoenvironmental Engineering (ASCE), Vol. 126, No. 5, 2000, pp. 443-450.

[25] T. Matsui and K. C. San, “Finite Element Slope Stability Analysis by Strength Reduction Technique,” Soils and Foundation, Vol. 32, No. 1, 1992, pp. 59-70. doi:10.3208/sandf1972.32.59

[26] D. Peter, D. Komatitsch, Y. Luo, R. Martin, N. Le Goff, E. Casarotti, P. Le Loher, F. Magnoni, Q. Liu and Peeligrini, “SCOTCH User’s Guide Version 5.1,” 2010. http://www.labri.fr/perso/pelegrin/scotch/

[27] M. A. Price and C. G. Armstrong, “Hexahedral Mesh Generation by Medial Surface Subdivision: Part II. Solids with Flat and Concave Edges,” International Journal for Numerical Methods in Engineering, Vol. 40, No. 1, 1997, pp. 111-136. doi:10.1002/(SICI)1097-0207(19970115)40:1<111::AID-NME56>3.0.CO;2-K

[28] J. Shepherd and C. Johnson, “Hexahedral Mesh Generation Constraints,” Engineering with Computers, Vol. 24, No. 3, 2008, pp. 195-213. doi:10.1007/s00366-008-0091-4

[29] T. J. Tautges, “The Generation of Hexahedral Meshes for Assembly Geometry: Survey and Progress,” International Journal for Numerical Methods in Engineering, Vol. 50, No. 12, 2001, pp. 2617-2642. doi:10.1002/nme.139

[30] CUBIT Development Team, “CUBIT Mesh Generation Environment Volume 1 User’s Manual,” Sandia National Lab, 1999. http://cubit.sandia.gov/cubitprogram.html

[31] Paraview Develops Team, “Paraview,” Kitware Inc., Clifton Park, 2010. http://paraview.org/wiki/paraview

[32] Open MPI Team, “The Open MPI Project, Indiana University,” Bloogminton, 2011. http://www.open-mpi.org

[33] Peeligrini, “SCOTCH User’s Guide Version 5.1,” 2010. http://www.labri.fr/perso/pelegrin/scotch/

[34] K. Terzaghi, “Mechanisms of Landslides,” Geotechnical Society of America, Berkeley, 1950, pp. 83-125.

[35] J. Krahenbuhl and A. Wagner, “Survey, Design, and Construction of Trail Suspension Bridges for Remote Areas, SKAT,” Swiss Center for Appropriate Technology, St. Gallen, 1983.

[1] P. Cupillard, E. Delavaud, G. Burgos, G. Festa, J. P. Vi lotte, Y. Capdeville and J. P. Montagner, “RegSEM: A Versatile Code Based on the Spectral Element Method to Compute Seismic Wave Propagation at the Regional Scale,” Geophysical Journal International, Vol. 188, No. 3, 2012, pp. 1203-1220. doi:10.1111/j.1365-246X.2011.05311.x

[2] D. Jonas, D. Basabe and M. K. Sen, “Stability of the High-Order Finite Elements for Acoustic or Elastic Wave Propagation with High-Order Time Stepping,” Geophysical Journal International, Vol. 181, No. 1, 2010, pp. 577-590. doi:10.1111/j.1365-246X.2010.04536.x

[3] I. Khan and B. H. V. Topping, “Parallel Finite Element Analysis Using Jacobi-35 Conditioned Conjugate Gradi ent Algorithm,” Advances in Engineering Software, Vol. 25, No. 2-3, 1996, pp. 309-319. doi:10.1016/0965-9978(95)00111-5

[4] J. Y. Kim and S. R. Lee, “An Improved Search Strategy for the Critical Slip Surface Using Finite Stress Fields,” Computers and Geotechnics, Vol. 21, No. 4, 1997, pp. 295-313. doi:10.1016/S0266-352X(97)00027-X

[5] D. Komatitsch and J. Tromp, “Introduction to the Spectral Element Method for Three-Dimensional Seismic Wave Propagation,” Geophysical Journal International, Vol. 139, No. 3, 1999, pp. 806-822. doi:10.1046/j.1365-246x.1999.00967.x

[6] D. Komatitsch and J. Tromp, “Spectral-Element Simulations of Global Seismic Wave Propagation I. Validation,” Geophysical Journal International, Vol. 149, No. 2, 2002, pp. 390-412. doi:10.1046/j.1365-246X.2002.01653.x

[7] D. Komatitsch, S. Tsuboi and J. Tromp, “The Spectral Element Method in Seismology. Seismic Earth: Array Analysis of Broadband Seismograms,” Geophysical Mo nograph Series, Vol. 157, 2005, pp. 205-227. doi:10.1029/156GM13

[8] D. Komatitsch and J. P. Vilotte, “The Spectral Element Method: An Efficient Tool to Simulate the Seismic Response of 2D and 3D Geological Structures,” Bulletin of the Seismological Society of America, Vol. 88, No. 2, 1998, pp. 368-392.

[9] R. Pasquetti and F. Rapetti, “Spectral Element Methods on Unstructured Meshes: Comparisions and Recent Advances,” Journal of Scientific Computing, Vol. 27, No. 1-3, 2006, pp. 377-387. doi:10.1007/s10915-005-9048-6

[10] J. Tromp, D. Komatitsch and Q. Liu, “Spectral-Element and Adjoint Methods in Seismology,” Communications in Computational Physics, Vol. 3, No. 1, 2008, pp. 1-32.

[11] H. N. Gharti, D. Komatitsch, V. Oye, R. Martin and J. Tromp, “Specfem 3D Geotech User Manual, 1.1 Beta,” 2012. http://www.geodynamic.org/cig/software/specfem3d-geotech

[12] R. C. Tiwari, N. P. Bhandary, R. Yatabe and D. R. Bhat, “New Numerical Scheme in the Finite Element Method for Evaluating Root-Reinforcement Effect on Soil Slope Stability,” Geotechnique, Vol. 63, No. 2, 2013, pp. 129 139. doi:10.1680/geot.11.P.039

[13] M. Genet, A. Stokes, F. Salin, S. B. Mickovski, T. Four caud, J. Dumail and L. P. H. Van Beek, “The Influence of Cellulose Content on Tensile Strength in Tree Roots,” Plant and Soil, Vol. 278, No. 1-2, 2005, pp. 1-9.

[14] D. H. Gray and R. D. Sotir, “Biotechnical Soil Bioengineering Slope Stabilization: A Practical Guide for Erosion Control,” John Wiley & Sons Ltd., New York, 1996.

[15] L. J. Waldron, “The Shear Resistance of Root-Permeated Homogenous and Stratified Soil,” Soil Science Society of America Journal, Vol. 41, No. 5, 1977, pp. 843-849. doi:10.2136/sssaj1977.03615995004100050005x

[16] T. H. Wu, M. C. Kinnell, W. P. McKinnell III and D. N. Swanston, “Strength of Tree Roots and Landslides on Prince of Wales Island,” Canadian Geotechnical Journal, Vol. 16, No. 7, 1979, pp. 19-33. doi:10.1139/t79-003

[17] G. B. Bischetti, E. A. Chaiaradia, T. Epis and E. Morlotti, “Root Cohesion of Forest Species in the Italian Alps,” Plant and Soil, Vol. 324, No. 1, 2009, pp. 71-89. doi:10.1007/s11104-009-9941-0

[18] H. N. Gharti, D. Komatitsch, V. Oye, R. Martin and J. Tromp, “Application of an Elastoplastic Spectral-Element Method to 3D Slope Stability Analysis,” International Journal for Numerical Methods in Engineering, Vol. 91, No. 1, 2012, pp. 1-26. doi:10.1002/nme.3374

[19] O. C. Zienkiewicz, C. Humpheson and R. W. Lewis, “Associated and Non-Associated Visco-Plasticity and Plasticity in Soil Mechanics,” Geotechnique, Vol. 25, No. 4, 1975, pp. 671-689. doi:10.1680/geot.1975.25.4.671

[20] M. M. Berilgen, “Investigation of Stability of Slopes under Drawdown Conditions,” Computers and Geotechnics, Vol. 34, No. 2, 2007, pp. 81-91. doi:10.1016/j.compgeo.2006.10.004

[21] D. V. Griffiths, J. Huang and G. F. Dewolfe, “Numerical and Analytical Observations on Long and Infinite Slopes,” International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 35, No. 5, 2011, pp. 569-585. doi:10.1002/nag.909

[22] D. V. Griffiths and P. A. Lane, “Slope Stability Analysis by Finite Elements,” Geotechnique, Vol. 49, No. 3, 1999, pp. 387-403. doi:10.1680/geot.1999.49.3.387

[23] M. S. Huang and C. Q. Jia, “Strength Reduction FEM in Stability Analysis of Soil Slope Subjected to Transient Unsaturated Seepage,” Computer and Geotechnics, Vol. 36, No. 1-2, 2009, pp. 93-101. doi:10.1016/j.compgeo.2008.03.006

[24] P. A. Lane and D. V. Griffiths, “Assessment of Stability of Slopes under Draindown Conditions,” Journal of Geo technical and Geoenvironmental Engineering (ASCE), Vol. 126, No. 5, 2000, pp. 443-450.

[25] T. Matsui and K. C. San, “Finite Element Slope Stability Analysis by Strength Reduction Technique,” Soils and Foundation, Vol. 32, No. 1, 1992, pp. 59-70. doi:10.3208/sandf1972.32.59

[26] D. Peter, D. Komatitsch, Y. Luo, R. Martin, N. Le Goff, E. Casarotti, P. Le Loher, F. Magnoni, Q. Liu and Peeligrini, “SCOTCH User’s Guide Version 5.1,” 2010. http://www.labri.fr/perso/pelegrin/scotch/

[27] M. A. Price and C. G. Armstrong, “Hexahedral Mesh Generation by Medial Surface Subdivision: Part II. Solids with Flat and Concave Edges,” International Journal for Numerical Methods in Engineering, Vol. 40, No. 1, 1997, pp. 111-136. doi:10.1002/(SICI)1097-0207(19970115)40:1<111::AID-NME56>3.0.CO;2-K

[28] J. Shepherd and C. Johnson, “Hexahedral Mesh Generation Constraints,” Engineering with Computers, Vol. 24, No. 3, 2008, pp. 195-213. doi:10.1007/s00366-008-0091-4

[29] T. J. Tautges, “The Generation of Hexahedral Meshes for Assembly Geometry: Survey and Progress,” International Journal for Numerical Methods in Engineering, Vol. 50, No. 12, 2001, pp. 2617-2642. doi:10.1002/nme.139

[30] CUBIT Development Team, “CUBIT Mesh Generation Environment Volume 1 User’s Manual,” Sandia National Lab, 1999. http://cubit.sandia.gov/cubitprogram.html

[31] Paraview Develops Team, “Paraview,” Kitware Inc., Clifton Park, 2010. http://paraview.org/wiki/paraview

[32] Open MPI Team, “The Open MPI Project, Indiana University,” Bloogminton, 2011. http://www.open-mpi.org

[33] Peeligrini, “SCOTCH User’s Guide Version 5.1,” 2010. http://www.labri.fr/perso/pelegrin/scotch/

[34] K. Terzaghi, “Mechanisms of Landslides,” Geotechnical Society of America, Berkeley, 1950, pp. 83-125.

[35] J. Krahenbuhl and A. Wagner, “Survey, Design, and Construction of Trail Suspension Bridges for Remote Areas, SKAT,” Swiss Center for Appropriate Technology, St. Gallen, 1983.