Three-Dimensional Numerical Manifold Method Based on Hexahedron Element with Full First-Order Cover Function and its Application in Underground Excavation


Article Preview

In this paper, three-dimensional(3D) Numerical Manifold Method (NNM) based on hexahedron element cover with full first-order cover function is proposed and the shape function of C8 isoparametric element in FEM is used as the cover weight function. All sub-matrices in equilibrium equations, including stiffness matrix, initial stress matrix, point force matrix, surface force matrix, body force matrix, inertia force matrix, contact matrix and friction matrix, are derived. Different with 2D contact, the direction of shear stiffness and friction force can not be easily defined in 3D contact. A new iterative method based on vector theory to detect the contact direction is developed. The application of 3D NMM in underground excavation is also presented and show good agreement with real engineering.



Key Engineering Materials (Volumes 340-341)

Edited by:

N. Ohno and T. Uehara




Y.H. Zhang et al., "Three-Dimensional Numerical Manifold Method Based on Hexahedron Element with Full First-Order Cover Function and its Application in Underground Excavation", Key Engineering Materials, Vols. 340-341, pp. 365-370, 2007

Online since:

June 2007




[1] G.H. Shi: Discontinuous Deformation Analysis-A New Numerical Model for The Static and Dynamics of Block Systems, Ph. D Dissertation, Department of Civil Engineering, University of California, Berkeley, USA. (1988).

[2] G.H. Shi: Manifold method. Proc. of 1 st Int. Forum on DDA and Simulations of Discontinuous Media, Berkeley, California, USA (1996), pp.52-204.

[3] G.H. Shi: Numerical Manifold Method, Proc. of 2 nd International Conference on Analysis of Discontinuous Deformation, Japan (1997), pp.1-35.

[4] Y.M. Cheng, Y.H. Zhang and W.S. Cheng: Wilson non-conforming Element in Numerical Manifold Method. Communications in Numerical Methods in Engineering, Vol. 18 (2002), pp.877-884.


[5] P.A. Cundall & R.D. Hart: Development of Generalized 2-D and 3-D Distinct Element Programs for Modeling Jointed Rock, U.S. Army Corps of Engineers, Washington DC (1985).

[6] Y.Y. Jiao: Three Dimensional DEM and Its Application, Ph. D Dissertation. Wuhan Institute of Rock & Soil Mechanics (1998).

[7] Y.M. Cheng, W.S. Chen and X.R. Ger: Procedure to the detect of three-dimensional blocks using penetration edges method. Geotechnical Special Publication no. 117, ASCE (2002), pp.79-85.

[8] Q.H. Jiang: Research on Three Dimensional Discontinuous Deformation Analysis Method, Ph. D Dissertation, Wuhan Institute of Rock & Soil Mechanics (2000).

[9] S.M. Luo, X.W. Zhang, W.G. Lv and D.R. Jiang: Theoretical Study of Three-dimensional Numerical Manifold Method, Applied Math. And Mech., Vol. 26 (2005), pp.1126-1131.