development of an explicit numerical …...development of the numerical manifold method for dynamic...
TRANSCRIPT
QU XIAOLEI
SCHOOL OF CIVIL AND RESOURCE ENGINEERING
THE UNIVERSITY OF WESTERN AUSTRALIA
SEPTEMBER 2013
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD
FOR DYNAMIC STABILITYANALYSIS OF ROCK
SLOPE
DEVELOPMENT OF THE NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY ANALYSIS OF ROCK SLOPE FAILURE
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DEVELOPMENT OF AN EXPLICIT
NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITYANALYSIS
OF ROCK SLOPE
QU XIAOLEI
School of Civil and Resource Engineering
A thesis submitted to
The University of Western Australia
in fulfillment of the requirement for the degree of
Doctor of Philosophy
SEPTEMBER 2013
DEVELOPMENT OF THE NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY ANALYSIS OF ROCK SLOPE FAILURE
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TABLE OF CONTENTS
Table of Contents ............................................................................................................ ii ACKNOWLEDGEMENT ............................................................................................. vi SUMMARY ................................................................................................................... vii LIST OF FIGURES ....................................................................................................... ix LIST OF TABLES ....................................................................................................... xiv LIST OF SYMBOLS .................................................................................................... xv LIST OF ABBREVIATION ...................................................................................... xviii Chapter 1. Introduction ............................................................................................... 1.1
1.1 Background ..................................................................................................... 1.1
1.2 Objectives of research ..................................................................................... 1.5
1.3 Origanization of the thesis .............................................................................. 1.6
Chapter 2. Literature review ...................................................................................... 2.9 2.1 Origin of the manifold .................................................................................... 2.9
2.2 Basic concepts of the NMM ......................................................................... 2.10
2.3 Current developments of the NMM .............................................................. 2.11
2.3.1 Improvement of the accuracy of the NMM ................................. 2.11
2.3.2 Extension of the NMM for discontinuity problems ..................... 2.12
2.3.3 Development of 3-D NMM ......................................................... 2.12
2.3.4 Other developments and applications of the NMM ..................... 2.13
2.4 Comparison with other numerical methods .................................................. 2.14
2.4.1 Comparison with FEM ................................................................. 2.14
2.4.2 Comparison with DDA ................................................................ 2.15
2.4.3 Comparison with DEM ................................................................ 2.16
2.5 Time integration algorithms for numerical methods .................................... 2.19
2.5.1 Numerical properties for time integration .................................... 2.21
2.5.2 Numerical examples for time integration .................................... 2.29
2.6 Methods for dynamic stability analysis of rock slope .................................. 2.33
2.6.1 LEM ............................................................................................. 2.36
2.6.2 Newmark method ......................................................................... 2.39
2.6.3 Numerical methods ...................................................................... 2.44
Chapter 3. Theory of the numerical manifold method and its integration schemes ........................................................................................................................ 3.46
3.1 Introduction ................................................................................................... 3.46
3.2 Fundamentals of the NMM ........................................................................... 3.46
3.2.1 Finite cover system in the NMM ................................................. 3.47
3.2.2 Contact algorithm ......................................................................... 3.57
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3.3 Integration schemes in the NMM .................................................................. 3.62
3.3.1 Simplex integration ...................................................................... 3.63
3.3.2 Time integration ........................................................................... 3.64
3.4 Summary ....................................................................................................... 3.65
Chapter 4. An explicit time integration scheme for the numerical manifold method ......................................................................................................................... 4.66
4.1 Introduction ................................................................................................... 4.66
4.2 Brief descrptions of the NMM ...................................................................... 4.67
4.3 Explicit time integration for the NMM ......................................................... 4.68
4.3.1 Mass matrix .................................................................................. 4.69
4.3.2 Internal force ................................................................................ 4.71
4.3.3 Damping algorithm ...................................................................... 4.72
4.4 Contact force in the ENMM .......................................................................... 4.74
4.4.1 Contact force calculation approach .............................................. 4.74
4.4.2 Calculation of contact force ......................................................... 4.75
4.5 Open-close algorithm in the ENMM ............................................................ 4.78
4.6 Numerical simulations .................................................................................. 4.80
4.6.1 Simply-supported beam subjected a concentrated load ............... 4.81
4.6.2 Numerical simulation of plan stress field problem ...................... 4.82
4.6.3 Single block sliding along the inclined surface ........................... 4.84
4.6.4 Highly fractured rock slope stability analysis .............................. 4.85
4.6.5 Rock tunnel stability analysis ...................................................... 4.91
4.7 Summary ....................................................................................................... 4.94
Chapter 5. Verification of computational efficiency and accuracy of the explicit numerical manifold method with wave propagation problems ............................. 5.96
5.1 Introduction ................................................................................................... 5.96
5.2 The brief overview of the NMM ................................................................... 5.98
5.2.1 The NMM and its cover system ................................................... 5.98
5.2.2 The explicit scheme of the NMM ................................................ 5.98
5.3 Stress wave propagation in a continous bar .................................................. 5.99
5.3.1 Effect of mesh size ..................................................................... 5.100
5.3.2 Effect of time step ...................................................................... 5.103
5.3.3 Computational efficiency ........................................................... 5.107
5.4 Stress wave propagation through fractured rock mass ................................ 5.108
5.4.1 P-wave propagation through homogeneous medium ................. 5.108
5.4.2 P-wave propagation through joint between different mediums . 5.109
5.4.3 Stress wave propagation through the multiple parallel joints .... 5.110
5.5 Seismic wave effect for a fractured rock slope ........................................... 5.112
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5.6 Summary ..................................................................................................... 5.115
Chapter 6. The temporal coupled explicit-implicit algorithm for dynamic problems using the numerical manifold method ................................................... 6.117
6.1 Introduction ................................................................................................. 6.117
6.2 The NMM and its cover system .................................................................. 6.119
6.2.1 Dual cover system in the NMM ................................................. 6.119
6.2.2 Contact problems in the NMM .................................................. 6.120
6.3 Temporal coupled explicit-implicit algorithm in the NMM ....................... 6.121
6.3.1 Summary of equations of motion and time integration.............. 6.121
6.3.2 The coupled explicit-implicit algorithm .................................... 6.123
6.3.3 Transfer algorithm for the E-I algorithm ................................... 6.124
6.4 Contact algorithm of the coupled algorithm ............................................... 6.126
6.4.1 Contact force calculation ........................................................... 6.126
6.4.2 Damping algorithm .................................................................... 6.127
6.5 Numerical examples .................................................................................... 6.128
6.5.1 Calibration of the temporal coupled E-I algorithm .................... 6.128
6.5.2 Open-pit mining stability analysis ............................................. 6.129
6.6 Summary ..................................................................................................... 6.141
Chapter 7. The spatial coupled explicit-implicit algorithm for dynamic problems using the numerical manifold method ................................................... 7.142
7.1 Introduction ................................................................................................. 7.142
7.2 Coupled algorithm ...................................................................................... 7.145
7.2.1 Summary of equations of motion and time integration.............. 7.145
7.2.2 Coupled explicit-implicit algorithm in the NMM ...................... 7.147
7.3 An alternative approach for the coupled E-I ALGORITHM ...................... 7.152
7.3.1 Onefold cover system ................................................................ 7.152
7.3.2 Contact algorithm on the onefold cover system ........................ 7.154
7.3.3 Contact matrices of the coupled E-I algorithm .......................... 7.157
7.3.4 Spring stiffness problems ........................................................... 7.162
7.4 Numerical examples .................................................................................... 7.164
7.4.1 Calibration of the spatial coupled E-I algorithm ........................ 7.164
7.4.2 Simulation of discrete blocks sliding on an inclined surface ..... 7.168
7.5 Summary ..................................................................................................... 7.169
Chapter 8. Dynamic stability analysis of rock slope failure using the explicit numerical manifold method .................................................................................... 8.171
8.1 Introduction ................................................................................................. 8.171
8.2 Numerical methods for rock slope dynamic stability ................................. 8.174
8.2.1 Continuum methods ................................................................... 8.175
8.2.2 Discontinuum methods .............................................................. 8.177
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8.2.3 Hybrid methods .......................................................................... 8.180
8.3 The new development of the NMM for dynamic stability analysis of rock slope ............................................................................................................ 8.181
8.3.1 Explicit NMM ............................................................................ 8.183
8.3.2 Coupled E-I NMM ..................................................................... 8.184
8.4 The parallel computation of the NMM ....................................................... 8.187
8.4.1 Parallelization with openMP ...................................................... 8.188
8.4.2 Speedup ...................................................................................... 8.191
8.5 Numerical examples .................................................................................... 8.194
8.5.1 A dynamic case study of rock slope stability analysis ............... 8.195
8.5.2 Dynamic stability analysis of Jinping I hydropower station ...... 8.198
8.6 Summary ..................................................................................................... 8.200
Chapter 9. Conclusions and recommendations ..................................................... 9.202 9.1 Summaries ................................................................................................... 9.202
9.2 Conclusions ................................................................................................. 9.203
9.3 Recommendations ....................................................................................... 9.206
References ................................................................................................................. 9.208
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ACKNOWLEDGEMENT
First and foremost, I would like to express my sincere gratitude to my supervisor,
Professor Ma Guowei for his warm encouragement, patient guidance and constant
persistence throughout this research. His unwavering enthusiasm and interest in
scientific research especially on the development of the numerical manifold method are
much appreciated and unforgettable. It is honoured and proud to work with him.
I would also like to thank Professor Shi Genhua for his advices and suggestions to
my research work. He has enhanced me a lot in my knowledge in the theoretical part.
I extend my gratitude to Professor Jing Lanru from Royal Institute of Technology
(KTH) for his suggestions, Sweden and Dr. Li Xu from Bejing Jiaotong University
(BJU), Beijing for his great helps in my research.
I am thankful to our whole NMM group, including Prof Ma Guowei, Dr. Hu
Jianhua, Dr Zhang Huihua, Mr Fu Guoyang, Mr Ren Fen, Mr Li Jinde, Mr Xu Zhenhao,
Mr Yi Xiawei, Mr Yang Shikou, Mr Wu Wei. We had a regular meeting each week. We
discussed the problems we met and tried to figure out ways to solve the problems
together. I benefited a lot from the discussions with them.
I take this opportunity to thank Professor Andrew Deeks, Professor Cheng Liang
and Dr. James Doherty for their helps and recommendations for my research work. I
would like to thank Professor Hao Hong to be as my vice advisor in my research.
The scholarship provided by China Scholarship Council (CSC) joints the
University of Western Australia (UWA) is gratefully acknowledged.
Last but not the least, I would like to dedicate this work to my beloved wife, Zhang
Li, who gives me infinite support in the shade and brings me a warm happiness family,
and to my parents for their love and support throughout the past years.
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SUMMARY
In this thesis, an explicit version of the numerical manifold method (NMM) has
been developed for dynamic stability analysis of rock slope.
Firstly, Newmark integration scheme used in the NMM is brief introduced and
derived in detail to further deepen understanding of the NMM and its implementations.
The numerical results present the explicit scheme is more efficient in solving the
nonlinear dynamic systems and such problems compared to implicit scheme. Then, an
explicit time integration scheme for the NMM is proposed to improve the computational
efficiency. The developed explicit NMM (ENMM) is validated by several examples.
The calibration study of the ENMM on P-wave propagation across a rock bar has been
conducted. Various considerations in the numerical simulations are discussed and
parametric studies have been carried out to obtain an insight into the influencing factors
in wave propagation simulation. The numerical results from the ENMM and NMM
modelling are accordant well with the theoretical solutions, but the ENMM is more
efficient than the original NMM.
The temporal and spatial coupled explicit-implicit (E-I) algorithms for the
numerical manifold method (NMM) are proposed. The time integration schemes,
transfer algorithm, contact algorithm and damping algorithm are studied in the temporal
coupled E-I algorithm to merge both merits of the explicit and implicit algorithms in
terms of efficiency and accuracy. In particular, onefold cover system is drawn into the
coupled spatial E-I algorithm, in which the contact algorithm based on the onefold cover
system is discussed and derived in detail. The simulated results are well agreement with
the implicit and explicit algorithms simulations, but the efficiency is improved
evidently.
The dynamic stability analysis of rock slope failure using the NMM is studied.
Conservational pseudo-static methods (PSMs), Newmark method and numerical
methods applying into the seismic stability analyses are investigated, the advantages and
limitations of which are studied by contrast of the NMM. An alternative ENMM and
coupled E-I algorithms are applied to study the seismic stability of rock slope.
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Furthermore, parallel computing with OpenMP is evaluated to improve efficiency of the
NMM. To reveal the validity and applicability of the developed ENMM, some
numerical examples of rock slope stability analysis are investigated, in which one
example of rock slope is taken into account to present the coupled ENMM with
discontinuous deformation analysis (DDA) in terms of efficiency. Simulated results of
the NMM will compare with the field measurements to illustrate the applicability of the
NMM.
The present study showed the developed ENMM is more efficient while without
losing the accuracy, comparing to the original implicit version of the NMM. Therefore,
it can be predicted that the proposed ENMM is promising and can be extend applied to
larger scale project of rock slope with dynamic stability analysis.
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LIST OF FIGURES
Figure 2.1 The scales of fish and fish body. ................................................................ 2.10
Figure 2.2 Size separation process with the vibrating screen modeled by the NMM (Ma et al. 2010). ...................................................................................... 2.14
Figure 2.3 Corner-to-corner contact in the NMM and DEM: (a) The shortest path method in the NMM; (b) The corner rounding technique in the DEM. .. 2.19
Figure 2.4 Efficiency versus DOFs between explicit and implicit algorithms. ........... 2.20
Figure 2.5 Spectral radius versus sampling frequency for Newmark integration. ....... 2.24
Figure 2.6 Period errors for the undamped case of Newmark integration methods. ... 2.25
Figure 2.7 Numerical damping versus sampling frequency in the NMM. ................... 2.26
Figure 2.8 Computational cost by the Newmark implicit scheme. .............................. 2.28
Figure 2.9 Computational cost by the Newmark explicit scheme. .............................. 2.29
Figure 2.10 Simplified model of n-DOF spring-mass-dashpot system........................ 2.30
Figure 2.11 Total displacement responses for 100-DOFs of system. .......................... 2.32
Figure 2.12 classification of instability and project examples for rock slopes. ........... 2.34
Figure 2.13 Equilibrium of forces on a sliding block (Chang et al. 1984). ................. 2.38
Figure 2.14 Displacement of rigid block on rigid base (Newmark 1965): (a) block on moving base; (b) acceleration plot; (c) velocity plot. ......................... 2.40
Figure 2.15 Integration of accelerograms to determine block movement (Goodman and Seed 1966). ....................................................................................... 2.42
Figure 3.1 A schematic of basic concepts in the NMM. (a) The physical domain and two MCs; (b) Overlapping of MCs and physical domain; (c) Corresponding PCs; (d) Six corresponding MEs. ................................... 3.48
Figure 3.2 The cover system in the NMM: (a) General cover system; (b) Generation of physical covers. ................................................................ 3.50
Figure 3.3 NMM model for the discontinuity problem. .............................................. 3.51
Figure 3.4 Construction of finite cover system in the NMM. ...................................... 3.53
Figure 3.5 Construction of PCs on the cover system. .................................................. 3.54
Figure 3.6 Structured mesh-based cover system in the NMM. .................................... 3.55
Figure 3.7 Construction of manifold elements on the cover system: (a) continuous elements; (b) discontinuous elements. ..................................................... 3.56
Figure 3.8 Three types of contacts: (a) angle-to-angle; (b) angle-to-edge; (c) edge-to-edge. .................................................................................................... 3.58
Figure 3.9 Entrance distance nd between a vertex and its entrance line. ..................... 3.59
Figure 3.10 Triangulate an element oij using coherent orientation. ............................ 3.64
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Figure 4.1 Two distinct contact force approaches: (a) point approach; (b) area approach. ................................................................................................. 4.74
Figure 4.2 Two schemes for contact problem: (a) Normal penetration method; (b) Direct penetration method. ...................................................................... 4.75
Figure 4.3 The proposed contact model: (a) contact elements; (b) contact points. ..... 4.78
Figure 4.4 Flowchart of the OCI in the ENMM. ......................................................... 4.80
Figure 4.5 Geometry of the simply supported beam bending problem. ...................... 4.81
Figure 4.6 Comparisons between simulated results and theoretical solution. ............. 4.82
Figure 4.7 Numerical model for an infinite plate with a traction free circular hole: (a) Geometry of model; (b) NMM meshing. ........................................... 4.83
Figure 4.8 Comparison of numerical results and analytical solution for infinite plate problem. .......................................................................................... 4.84
Figure 4.9 Comparison of simulated results and analytical solution. .......................... 4.85
Figure 4.10 Geometry of the slope modelling. ............................................................ 4.85
Figure 4.11 Simulation results by NMM and ENMM: (a) NMM, ∆t=2ms; (b) NMM, ∆t=1ms; (c) NMM, ∆t=0.1ms; (d) ENMM, ∆t=0.1ms................ 4.88
Figure 4.12 Real step-time used in NMM vs. ENMM. ................................................ 4.88
Figure 4.13 Displacements of measured point 1: (a) Horizontal; (b) Vertical. ........... 4.89
Figure 4.14 Displacements of measured point 3: (a) Horizontal; (b) Vertical. ........... 4.90
Figure 4.15 Displacements of measured point 6: (a) Horizontal; (b) Vertical. ........... 4.91
Figure 4.16 Geometry of the tunnel modelling. ........................................................... 4.92
Figure 4.17 Simulation results used by ENMM vs. NMM. ......................................... 4.92
Figure 4.18 Displacements of measured point 4. ......................................................... 4.93
Figure 4.19 Displacement of measured point 9. .......................................................... 4.94
Figure 5.1 Schematic of the rock bar model. ............................................................... 5.99
Figure 5.2 Eigenlength in the manifold mesh. ........................................................... 5.100
Figure 5.3 Percent errors at the end of first wavelength for different wave frequencies and element ratios. ............................................................. 5.101
Figure 5.4 Stress wave simulation using NMM and ENMM by two typical element ratios of 1/4 and 1/32 ( }0.1 st ). ....................................................... 5.102
Figure 5.5 Percent errors along the distance from wave source for different element ratios ( }0.1 st ). ................................................................... 5.103
Figure 5.6 Peak pressure attenuation for different time step. .................................... 5.106
Figure 5.7 Comparison between the simulated results and the Pyrak-Nolte’s analytical solution for a single joint. ..................................................... 5.109
Figure 5.8 Comparison between the simulated results of NMM and ENMM. .......... 5.110
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Figure 5.9 Simulated results at measure point by the NMM and ENMM. ................ 5.111
Figure 5.10 Schematic cross-section of the rock slope .............................................. 5.113
Figure 5.11 Record of acceleration of the seismic wave. .......................................... 5.113
Figure 5.12 Rock slope model in the NMM and ENMM. ......................................... 5.114
Figure 5.13 Simulated results of the failure of the rock slope using the NMM and ENMM (Time step in NMM =1ms, ENMM =0.2ms; total time =20s). 5.115
Figure 5.14 Measure point displacement simulated by the NMM and ENMM......... 5.115
Figure 6.1 A regularly-patterned triangular mesh in the NMM. ................................ 6.120
Figure 6.2 Transfer algorithm from the explicit to implicit integration. .................... 6.124
Figure 6.3 Geometry of the Newmark sliding modeling. .......................................... 6.129
Figure 6.4 Block displacement under horizontal ground acceleration. ...................... 6.129
Figure 6.5 Geology section of the open-pit mining. .................................................. 6.130
Figure 6.6 Study model of the layer 4#: a. Integrated Model; b. Refined Model 1; c. Refined Model 2. ............................................................................... 6.131
Figure 6.7 A stochastic horizontal seism acceleration. .............................................. 6.132
Figure 6.8 Simulation results for Refined Model 1 (Total time: 20s): (a) ϕ=100; (b) ϕ=150. .................................................................................................... 6.134
Figure 6.9 Simulation results for Refined Model 2 (Total time: 20s): (a) ϕ=100; (b) ϕ=150. .................................................................................................... 6.134
Figure 6.10 Measured point 1 with model 1: (a) ϕ=100; (b) ϕ=150. .......................... 6.135
Figure 6.11 Measured point 2 with model 1: (a) ϕ=100; (b) ϕ=150. .......................... 6.136
Figure 6.12 Measured point 3 with model 1: (a) ϕ=100; (b) ϕ=150. .......................... 6.137
Figure 6.13 Measured point 1 with model 2: (a) ϕ=100; (b) ϕ=150. .......................... 6.138
Figure 6.14 Measured point 2 with model 2: (a) ϕ=100; (b) ϕ=150. .......................... 6.139
Figure 6.15 Measured point with model 2: (a) ϕ=100; (b) ϕ=150. ............................. 6.140
Figure 7.1 An elastic body with a traction vector t. ................................................... 7.146
Figure 7.2 Sub-domain partition algorithm in the coupled E-I method: (a) Element partition method; (b) MCs partition method. ........................................ 7.150
Figure 7.3 Explicit and implicit covers for the contact problem in the NMM........... 7.152
Figure 7.4 Construction of onefold cover in the proposed NMM.............................. 7.153
Figure 7.5 Coefficient matrix of the coupled E-I algorithm: (a) Coupled E-I algorithm global coefficient matrix; (b) Implicit coefficient matrix; (c) Explicit coefficient matrix. .............................................................. 7.157
Figure 7.6 Flowchart of two alternative contact schemes: (a) shared contact algorithm; (b) separated contact algorithm. .......................................... 7.157
Figure 7.7 Contact representation in the coupled E-I algorithm. ............................... 7.158
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Figure 7.8 Domain profile suggested in stiffness estimation (Itasca 1993). .............. 7.164
Figure 7.9 Geometry of the multi-block system. ....................................................... 7.165
Figure 7.10 Displacement in the different cases: (a) case of ϕ=0; (b) case of ϕ=450. 7.166
Figure 7.11 Displacement of the top 2nd block: (a) case of ϕ=0; (b) case of ϕ=450. . 7.167
Figure 7.12 Rock slope stability analysis using the I-NMM, E-NMM and E-I NMM, respectively: (a) Initial modelling; (b) Explicit NMM; (c) Implicit NMM; (d) Explicit-implicit NMM. ......................................... 7.169
Figure 7.13 Displacement of the measured point 1 and 2. ......................................... 7.169
Figure 8.1 Location of the rock slope and aero-view from Google maps. ................ 8.172
Figure 8.2 Photograph of rock instability example in Cottesloe, Western Australia (photographed by X.L. Qu): a. right bank of the slope; b. left bank of the slope. ................................................................................................ 8.173
Figure 8.3 Continuum modeling of a rock slope by Abaqus/CAE 6.11: (a) slope meshing and 6-noded triangular element; (b) contour of the maximum strain ratio. ........................................................................... 8.176
Figure 8.4 Displacement nephogram at different times under seismic loading: (a) t=1.5s; (b) t=12.0s; (c) t=21.0s; (d) t=30.0s. ......................................... 8.177
Figure 8.5 Maximum displacement vectors and shear strain contours of the modelling in 2008 Wenchuan earthquake, China (Luo et al. 2012)...... 8.178
Figure 8.6 Simulation of a rock slope stability and failure under dynamic excavation using PFC technique (Wang et al. 2003). ........................... 8.179
Figure 8.7 Seismic simulation of Chiu-fen-erh-shan landslide by the Chi-Chi earthquake using DDA (Wu 2010). ....................................................... 8.180
Figure 8.8 Displacement distribution for each block after applying seismic loads (Miki et al. 2010). .................................................................................. 8.181
Figure 8.9 Modelling of rock fall failure under earthquake by NMM and DDA (Ning et al. 2012). ................................................................................. 8.181
Figure 8.10 Construction of onefold cover system from manifold cover system. ..... 8.185
Figure 8.11 Contact between explicit and implicit OEs based on onefold cover system. ................................................................................................... 8.186
Figure 8.12 Assembly of contact matrices in the coupled E-I algorithm. ................. 8.187
Figure 8.13 Parallel processing model: (a) UMA; (b) NUMA. ................................ 8.189
Figure 8.14 Construction of parallel computation of the NMM using OpenMP. ...... 8.190
Figure 8.15 Code segment of the parallel programming to the explicit NMM. ........ 8.190
Figure 8.16 Simulation results of the serial and parallel NMM codes. ..................... 8.192
Figure 8.17 CPU usage of the serial and multi-core NMM codes. ............................ 8.193
Figure 8.18 Computing time of the serial and parallel codes. ................................... 8.194
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Figure 8.19 Location of area of Lake Anderson slope in California USA (Keefer et al. (1980). .............................................................................................. 8.196
Figure 8.20 Numerical modelling of Lake Anderson slope. ...................................... 8.196
Figure 8.21 Record of acceleration of the earthquake. .............................................. 8.197
Figure 8.22 Simulated results of landslide under earthquake. ................................... 8.198
Figure 8.23 Scale map of geomechanical model (Zhou et al. 2008). ......................... 8.199
Figure 8.24 Modelling of the slope transfers from ENMM to DDA. ........................ 8.200
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LIST OF TABLES
Table 2.1 Values of based on different damping ratios. .......................................... 2.27
Table 2.2 CPU time for the proposed explicit and implicit scheme. ........................... 2.32
Table 4.1 Input parameters for the NMM simulation of rock slope. ........................... 4.86
Table 4.2 Maximum displacement of measure points in the NMM vs. ENMM. ........ 4.91
Table 5.1 Material properties of the rock bar. .............................................................. 5.99
Table 5.2 Comparison of CPU cost between the NMM and ENMM. ....................... 5.108
Table 5.3 CPU cost comparison between the NMM and ENMM. ............................ 5.112
Table 6.1 FoS using LEM by the integrated models. ................................................. 6.131
Table 6.2 Input parameters for the simulation of the modeling. ................................ 6.132
Table 6.3 CPU cost for the different study cases (hr.). .............................................. 6.141
Table 7.1 Contact types between two domains. ......................................................... 7.154
Table 8.1 Parameters of the used multi-core PCs. ..................................................... 8.191
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LIST OF SYMBOLS
iM = ith mathematical cover
iP = ith physical cover, reallocated with a single index i
jiP = jth physical cover generated from mathematical cover
iM
)(xi = Partition of unity function for mathematical cover iM
)(xi = Partition of unity function for mathematical cover iM
)(xui = local approximation function defined on physical cover
iP
)(xu = global approximation on the displacement field
= slope angle of an inclined plane
= a parameter of the Newmark method
= a parameter of the Newmark method
= Cauchy stress tensor
i = incident stress of P-wave
r = reflected stress of P-wave
t = transmitted stress of P-wave
= strain vector
= damping ratio
= algorithmic damping ratio
u = acceleration vector
u = velocity vector
u = displacement vector
= gradient operator
= density of material
b = body force
w = weight function
= arbitrary material domain
E = Young’s modulus
v = Poisson’s ratio
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= sampling frequency
= shear strength
c = shear strength
pc = P-wave velocity
= shear strength
T = shear strength
= natural frequency of the system
max = highest eigenfrequency of the system
t = time step size
ct = critical time step
Et = time step in the explicit algorithm
It = time step in the implicit algorithm
k = contact spring stiffness
nk = normal spring stiffness
sk = normal spring stiffness
d = penetration distance
nd = normal penetration distance
sd = shear penetration distance
u = prescribed displacement vector on u
t = traction vector on t
Nt = normal vector components
st = tangential vector components
= Lagrange multiplier vector
= shear modulus
= Kolosov constant
A = amplification matrix
B = strain displacement matrix
D = displacement matrix
D = velocity matrix
DEVELOPMENT OF THE NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY ANALYSIS OF ROCK SLOPE FAILURE
(xvii)
D = acceleration matrix
K = stiffness matrix
M = mass matrix
C = damping matrix
critC = critical damping matrix
F = loading matrix
L = differential operator matrix
T = deformation matrix
DEVELOPMENT OF THE NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY ANALYSIS OF ROCK SLOPE FAILURE
(xviii)
LIST OF ABBREVIATION
BEM = Boundary element method
CPU = Computer processing unit
DDA = Discontinuous deformation analysis
DEM = Distinct element method
EFGM = Element-free Galerkin method
ENMM= Explicit numerical manifold method
FCM = Finite cover method
FDM = Finite difference method
FEM = Finite element method
LEM = Limit equilibrium method
NMM = Numerical manifold method
PFC = Particle flow code
PSM = Pseudo static method
PUM = Partition of unity method
UDEC = Universal distinct element code
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
1.1
CHAPTER 1. INTRODUCTION
1.1 BACKGROUND
The stability of rock slope under dynamic effect is often significantly influenced by
the structural geology of the rock such as bedding planes, joints and faults (Wyllie and
Mah 2004). These properties are generally termed discontinuities. The significance of
discontinuities is that they are planes of weakness in the much stronger intact rock, so
failure tends to occur preferentially along these surfaces. For this reason, various efforts
for rock slope stability are often rated on how effectively they incorporate
discontinuities.
In rock engineering, pseudo-static methods (PSMs) (Baker et al. 2006; Seed 1979;
Loukidis, Bandini, and Salgado 2003; Li, Lyamin, and Merifield 2009) are treated as
limit equilibrium methods (LEMs)(Bishop 1955; Morgenstern and Price 1965) to
calculate a factor of safety (FoS) for a specified discontinuous surface, and then find a
critical failure surface that associated with the minimum safety factor. In PSMs,
earthquake effects are represented by an equivalent static force, the magnitude of which
is a product of a seismic coefficient k and the weight of the potential sliding mass. This
approach, however, is incapable of quantifying the extent to which rock slope has
displaced. The sliding block theory was firstly proposed by Newmark (1965) to evaluate
the permanent displacement of slopes on dams or embankments induced by
earthquakes. The principle of this method was assumed that the potential sliding block
is a rigid body on a yielding base. Lateral displacement of block was expected to take
place when the base acceleration exceeded the critical or yield acceleration of the block.
This method is a more realistic method of analysing seismic effects on rock slopes than
the PSM of analysis. However, this method is only applicable for a single, rigid block
analysis. Geological structures in jointed rock involve complexities associated with
geometry, material anisotropy, nonlinear behavior, in situ stresses and the presence of
coupled processes (e.g. pore pressure, thermal loading, etc.). PSMs and Newmark’s
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
1.2
sliding block theory cannot be used for such complex problems. Numerical modeling
techniques address most of the limitations of both methods, and can model different
failure mechanism of the rock slope instability.
For the seismic stability analysis of rock slopes, the numerical methods are more
suitable because the behaviour of a rock slope is much more dependent on characteristic
and integrity of the rock mass. In general, they can be classified into three types: (i)
continuum-based numerical methods; (ii) discontinuum-based numerical methods; and
(iii) hybrid continuum-discontinuum methods.
The continuum approach introduces discontinuous interfaces in the form of “joint
elements” (Goodman, Taylor, and Brekke 1968; Ghaboussi, Wilson, and Isenberg 1973)
or “displacement discontinuities” (Katona 1983) to model discontinuities explicitly.
During the past several decades, various continuum-based numerical methods have been
developed, such as the finite element method (FEM), the finite difference method
(FDM), the boundary element method (BEM), and various meshless methods (e.g.
Element-Free Galerkin Method (EFGM)) and have been used successfully in
applications where the rock mass does not undergo large deformations. However, when
the rock mass behaviour is governed by the geometry and strength characteristics of the
discontinuities, the interactions between the individual blocks defined by the
discontinuities must be considered.
Though great efforts have been made to the continuum-based numerical methods,
block rotations, complete detachment and large-scale opening still cannot be properly
treated, the number of discontinuities which can be dealt with is also limited. To solve
such problems, the discrete element method has been developed. The discrete element
method considers interaction between rock blocks by representing the rock mass as an
assembly of rigid or deformable discrete blocks, and is capable of capturing the
kinematics of the block system. It allows for opening/closing of discontinuities,
movement of blocks relative to each other, and sliding and toppling along the
discontinuities (Jing 2003). The discrete element method is especially suitable for
simulation of large-scale displacements of individual blocks, block rotations, and
complete detachment. Typical examples of the discrete element method include the
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
1.3
distinct element method (DEM) (Cundall 1971a, 1971b) and the discontinuous
deformation analysis (DDA) pioneered by Shi (1988). One of the major limitations of
the discrete element approaches is their high computational requirements when the
interaction of large number of discrete blocks with contacts is involved. Thus, it is
pivotal that the selected DEM not only solve the discontinuities problem but also with
acceptable computational efficiency.
The DEM adopts an explicit time integration scheme based on finite difference
principles (Cundall 1971a). It has been developed into the commercial 2D code UDEC,
3D code 3DEC and the particle versions PFC-2D and PFC-3D by Cundall and his
colleagues (Itasca 1993, 1994, 1995), which has been enjoying a wide application range
in rock engineering. The main benefit of the DEM is that its computational efficiency is
high due to its explicit time integration nature. However, it has also been argued that the
accuracy of simulated results may be sacrificed in some particular cases (O’Sullivan and
Bray 2001; Luccioni, Pestana, and Taylor 2001). To ensure numerical stability, a DEM
simulation requires that the time step must be small enough. Furthermore, artificial
damping is required to dissipate the energy in the DEM, but the selection of an
appropriate value of damping is difficult for different cases.
On the other hand, the DDA derived based on the variational method takes the
benefit of the implicit time integration method (Shi and Goodman 1985; Shi 1988). The
formulation of DDA is similar to that of the FEM. Due to its implicit time integration,
the DDA is inclined to be unconditionally stable and it is expected to accommodate
considerably large time steps. Additional features include simplex integration method
which is a closed-form integration for the element and block stiffness matrices and the
open-close iteration (OCI) contact algorithm. The DDA method has emerged as an
attractive model because its advantage in simulating a discrete system cannot be
replaced by continuum-based methods or explicit DEM formulations. Since the
initiation of the DDA, various developments and applications have been achieved
during the last two decades. Most of the publications are included in the series of
proceedings of the International Conference on Analysis of Discontinuous Deformation
(ICADD) symposia (Li, Wang, and Sheng 1995; Salami and Banks 1996; Ohnishi 1997;
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
1.4
Amadei 1999; Bicanic 2001; Hatzor 2002; Lu 2003; Maclaughlin and Sitar 2005; Ju,
Fang, and Bian 2007; Ma and Zhou 2009).
The major considerable drawback of the DDA is that much larger computational
cost will be required, especially when the system contains huge number of discrete
blocks and contacts. The convergence efficiency of the OCI for a complex discrete
system is also not clear. This has been long time the challenge for the development of
the 3D DDA method. A number of articles on DEM and DDA have been published over
the past several decades, detailed mathematical formulations and discussions between
the DEM and DDA can be found in the state-of-the-art articles organized by Jing (Jing
1990, 1998) and the book written by Jing and Stephansson (2007).
The recently developed numerical manifold method (NMM) (Shi 1991, 1992,
1995, 1996a, 1996b, 1997) is such a hybrid method. The NMM is an evolvement of the
DDA and combines the merits of FEM and DDA. It inherits all the attractive features of
the DDA, such as the implicit time integration scheme, the contact algorithm and the
minimum potential energy principle (Chen, Ohnishi, and Ito 1998). It adopts a dual
cover system, i.e. a mathematical cover system overlapping the domain of interest and a
physical cover system and considers the contained discontinuities in a united manner. In
the past two decades, many developments have been carried out to improve the
performance of the NMM. Review articles on the recent development of the NMM have
been published by Ma, An, and He (2010) and An, Ma, et al. (2011a). It has been
reorganized that the NMM has great potential to be further developed in simulating a
medium with massive discontinuities.
However, enjoying the benefits of the FEM and DDA, the NMM is also suffering
with the high computational costs arising from the inherent implicit time iteration
scheme and the OCIs for contacts since the DDA had no properly developed
constitutive models of rock fractures representing the contact surfaces between blocks.
The implicit time integration algorithm involves in the solution of a system of
equations, the computational cost increases dramatically with the degrees of freedom
(DOFs) of the system is increased since the large-scale simultaneous algebraic
equations must be solved in each time step (Newmark 1959, 1965). The OCI requires
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
1.5
zero-tension and zero-penetration at all contacts which additionally highs up the
computation costs in order to achieve a convergence state at each time instance.
Reducing the time step to a smaller one in the OCI when the tolerance of the penetration
depth is violated losses the benefit of using a large time step in an implicit time
integration algorithm.
It has been proved that an explicit time integration scheme derives as accurate
results as an implicit one if the time step is small enough (Belytschko, Yen, and Mullen
1979; Hughes 1983). For a discrete system, the zero-tension and zero-penetration
requirement at contacts determines the accuracy of the simulation results. Considering
that the computational accuracy is equally important with the computational efficiency
for engineering problems, a proper balance of the high computational efficiency based
on an explicit time integration scheme and the high accuracy based on an appropriate
OCI process is highly demanded.
1.2 OBJECTIVES OF RESEARCH
In the traditional NMM, simulations of the rock slope stability are computationally
expensive. Thus, it is essential to make a general investigation of the traditional NMM
not only in terms of its capability to handle the discontinuous nature of the problem but
also its computational efficiency. The purpose of this research is to develop an explicit
version of the NMM for dynamic stability analysis of rock slope. The specific targets
are outlined as follows:
To develop an explicit version of the NMM:
Investigate the traditional NMM in terms of computational accuracy and
efficiency;
Propose an explicit time integration scheme for the NMM and to verify it
with respect to the computational efficiency and accuracy
To combine the explicit and implicit algorithms for the NMM:
Couple the temporal explicit and implicit NMM;
Couple the spatial explicit and implicit NMM;
To extend the explicit NMM for the rock slope stability analysis:
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ANALYSIS OF ROCK SLOPE
1.6
Implement a seismic version of the explicit NMM for the dynamic stability
analysis of rock slope;
Apply the developed program to simulate the dynamic stability of rock
slope.
1.3 ORIGANIZATION OF THE THESIS
The thesis is organized into nine chapters. The contents of each chapter are briefly
summarized as follows:
Chapter 1 introduces the background and objectives of this research and the
organization of this thesis.
Chapter 2 presents a literature review on the original numerical manifold method
(NMM), time integration algorithms and numerical methods for dynamic stability
analysis of rock slope. The basic concepts of the NMM are briefly introduced firstly.
The current development of the NMM in the improvement of the accuracy, extension of
the NMM for discontinuity problems, development of 3D NMM and other
developments and applications of the NMM are reviewed. Comparison between the
NMM and other typical numerical methods, such as finite element method (FEM),
discontinuous deformation analysis (DDA) and discrete element method (DEM), is
conducted as well. Then, time integration algorithms for numerical methods and
methods for dynamic stability analysis of rock slope are reviewed, in which numerical
properties, numerical examples for time integration and the traditional methods for
dynamic stability analysis of rock slop are investigated, respectively.
In Chapter 3, the theory of the NMM and the integration schemes are introduced,
respectively. In the former part, the fundamentals of the NMM is presented, including
the finite cover system based on the finite element mesh and contact algorithm applied
in the NMM. In the later part, the integration schemes including the spatial simplex
integration and temporal time integration are introduced, respectively, to extend the
further understanding of the NMM.
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ANALYSIS OF ROCK SLOPE
1.7
The traditional NMM is extended to an explicit version in Chapter 4. The basic
concepts consist of the frame of the NMM is briefly introduced firstly. An explicit time
integration scheme is proposed, in which the mass matrix, internal force and damping
item based on the finite cover system are derived in detail, respectively. Contact force in
the NMM is presented, including contact force approach and calculation of the contact
force based on the explicit time integration. As the contact algorithm such as OCI used
in the developed explicit version of the NMM, this chapter gives a briefly review of the
OCI in the NMM. Several numerical simulations are carried out to calibrate the
developed explicit version of the NMM, including simply-supported beam subjected a
concentrated load, single block sliding along the inclined surface and highly fractured
rock slope stability analysis.
Chapter 5 verifies the developed explicit version of the NMM (ENMM) in terms of
computational efficiency and accuracy with wave propagation problems. The NMM and
its cover system and the explicit scheme are briefly reviewed firstly. Calibration of the
stress wave propagation using the ENMM are carried out, in which the effect of mesh
size, effect of the time step and computational efficiency are discussed, respectively.
stress wave propagation through fractured rock mass, including the homogeneous
medium and joints conditions are verified with respect to the efficiency and accuracy.
Finally, seismic wave effect in the fractured rock slope is verified as well.
The temporal and spatial coupled explicit-implicit (E-I) algorithms for dynamic
problems are presented to extend the developed the explicit NMM in Chapter 6 and 7,
respectively. The temporal coupled E-I algorithm, including transfer algorithm, contact
force calculation and damping algorithm, is developed to maximize the advantages of
the explicit and implicit schemes in terms of the temporal aspect. The spatial coupled E-
I algorithm, including the coupled algorithm and contact algorithm, is proposed to
enlarge the both merits in the spatial aspect. Then, an alternative approach for the spatial
E-I algorithm is developed, in which onefold cover system is introduced to simplify the
E-I algorithm. The contact algorithm and contact matrices based on the onfold cover
system are derived, respectively. Spring stiffness problems in the coupled E-I algorithm
is discussed as well in Chapter 7.
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ANALYSIS OF ROCK SLOPE
1.8
In Chapter 8, the dynamic stability analysis of rock slope failure using the
developed explicit NMM is carried out to extend the capability of the developed
ENMM. Traditional numerical methods on the dynamic stability of rock slope are
briefly introduced firstly, including different kinds of advanced numerical methods. The
developed explicit NMM (ENMM) and the coupled explicit-implicit (E-I) algorithm are
briefly introduced as well. The parallel computation of the NMM is carried out, in
which parallelization with OpenMP and speedup of it are discussed in this chapter. Two
typical examples of rock slopes are simulated using the developed methods, in which a
dynamic case study of rock slope stability analysis is conducted using the seismic NMM
code, and a project of Jinping hydropower station is simulated using the developed
ENMM coupled discontinuous deformation analysis (DDA).
Chapter 9 draws the conclusions and gives the recommendation for future study.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.9
CHAPTER 2. LITERATURE REVIEW
2.1 ORIGIN OF THE MANIFOLD
The numerical manifold method (NMM) is a newly developed numerical approach
appeared in a series of conference papers (Shi 1996a, 1991, 1992, 1995, 1997). It
provides a natural bridging to represent the physical change during numerical
simulations. It is also advantageous the over conventional numerical methods in
analyzing both continuous and discontinuous problems. The term of manifold comes
from the topological manifold and the differential manifold, the difference between
them lies in that the global functions of the differential manifold are highly
differentiable and defined irrelevant to the covers, while the global functions of the
numerical manifold here are defined based on covers and only piecewise differentiable.
In the traditional NMM, the manifolds connect many overlapped small patches
together to cover the entire problem domain. Each small patch is called a cover. A local
function is defined on each cover. The global behaviour is then determined by the
weighted average of local functions defined on each physical cover. Based on the finite
covers, the NMM combines the well developed analytical methods, widely used finite
element method (FEM) and the block-oriented discontinuous deformation analysis
(DDA) in a unified form.
Here, we give an example of natural world as represented in Fig. 2.1. Many
overlapped small patches scales connect together to cover the whole body of the fish.
Each piece of scale can be treated as one cover. The state of motion of fish body can be
represented as the connected scales motions. Manifold is the main subject of differential
geometry, algebraic topology, differential topology and modern algebra of mathematics,
even the natural world.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.10
Figure 2.1 The scales of fish and fish body.
2.2 BASIC CONCEPTS OF THE NMM
The NMM approximation is based on three basic concepts, i.e. the mathematical
cover (MC), the physical cover (PC) and the manifold element (ME).
The MCs are user-defined small patches. They may or may not overlap each other,
but their union must be large enough to cover the entire problem domain. One distinct
feature of the NMM is that its mathematical covers do not need to conform to neither
the external boundaries nor the internal discontinuities, thus can always be regular. The
meshing task in the NMM is very convenient. The re-meshing is totally avoided for
discontinuity propagation. On each MC, a partition of unity function is defined, which
satisfies non-zero value only on its corresponding MC, but zero elsewhere.
The PCs are the subdivision of the mathematical covers by the physical features
such as the external boundaries and the internal discontinuities. Each physical cover
inherits the partition of unity function from its associated mathematical cover as
The ME is defined as the common region of several physical covers. On each
manifold element, we use the partition of unity functions to paste all the local functions
together to give a global approximation as
In the NMM, the global approximation is first constructed in each ME. Element
stiffness matrices are constructed and then assembled into a global stiffness matrix. In
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.11
this respect, the ME is similar to the finite element in the FEM. Just due to this
similarity, the NMM retains all the advantageous features of the system equations in the
FEM such as the symmetry and sparsity.
As aforementioned descriptions, an example shown in Fig. 2.1 can be referred to
introduce the three basic concepts of the NMM. Each scale of fish can be regarded as
one MC, all scales of fish to be formed a finite cover system to cover the fish body and
represent the activity of it.
2.3 CURRENT DEVELOPMENTS OF THE NMM
Since the initiation of the NMM in 1991 (Shi 1991), various developments and
applications have been achieved during the last two decades. Most of the publications
are included in the series of proceedings of the International Conference on Analysis of
Discontinuous Deformation (ICADD) symposia (Li, Wang, and Sheng 1995; Salami
and Banks 1996; Ohnishi 1997; Amadei 1999; Bicanic 2001; Hatzor 2002; Lu 2003;
Maclaughlin and Sitar 2005; Ju, Fang, and Bian 2007; Ma and Zhou 2009). The main
developments and applications of the NMM are categorized into several groups as
follows:
2.3.1 Improvement of the accuracy of the NMM
The traditional NMM is based on triangular finite element covers and chooses
constants as the local approximation space for each physical cover. Shyu and Salami
(1995) implemented quadrilateral isoparametric element into the NMM. However, for
some particular problems such as the bending problem, the precision and efficiency of
the quadrilateral isoparametric elements are not adequate either. And Cheng et al. (2002)
incorporated Wilson non-conforming elements into the NMM for a cantilever slab
bending problem. In terms of developing high-order NMM, Chen et al. (1998)
introduced high-order polynomials into the local approximation space for each physical
cover; Su et al. (2003) carried out a simple method to automatically produce the
expressions and writing the subroutines with the software Mathematica; and Lin et al.
(2005) developed the formulations of three-dimensional NMM with high-order local
functions.
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ANALYSIS OF ROCK SLOPE
2.12
2.3.2 Extension of the NMM for discontinuity problems
The NMM can be used to explore the strong discontinuity problems such as crack
problems and shear failures. Tsay et al. (1999a) applied the NMM together with a local
mesh refinement and auto-remeshing schemes to predict the crack growth. Zhang et al.
(1999a, 1999b) coupled the NMM with the boundary element method (BEM) to
simulate the crack propagation problems. Chiou et al. (2002) proposed the NMM
combined with the virtual crack extension method to study the mixed-mode fracture
propagation. Li et al. (2005) and Gao and Cheng (2010) developed enriched meshless
manifold method for two-dimensional crack modeling. Ma et al. (2009) exhibited the
advantageous features of the NMM for multiple branched and intersecting cracks, in which
singular physical covers are enriched with asymptotic crack tip functions, the stress
intensity factors are evaluated by virtue of domain form of interaction integral. Zhang et al.
(2010) extended it to simulate the growth of complex cracks, adopting the maximum
circumferential stress criterion to determine the crack growth.
In terms of the weak discontinuity, the NMM describes the weak discontinuities by
splitting mathematical covers into physical covers attached with independent cover
functions. Terada et al. (2003) introduced the finite cover method (FCM) as an alias of
the NMM, presented the formulation for the static equilibrium state of a structure with
arbitrary physical boundaries including material interfaces, and extended the FCM to
analyze heterogeneous solids and structures involving the discontinuities in strains and
discontinuities in displacement. The more extensions of the FCM are involved modeling
of evolving discontinuities in heterogeneous media can be referred in (Terada and
Kurumatani 2004; Kurumatani and Terada 2005, 2009).
2.3.3 Development of 3-D NMM
Lin et al. (2005) developed the formulations of 3-D NMM with high-order cover
functions, and proposed a fast simplex integration method based on special matrix
operations. Terada and Kurumatani (2005) introduced an integrated procedure for three-
dimensional structural analysis utilizing the FCM. They provide the formulations of the
FCM with interface elements for the static equilibrium of a structure. Cheng and Zhang
(2008) proposed a three-dimensional numerical manifold method with tetrahedron and
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.13
hexahedron elements. Jiang et al. (2009) proposed a three-dimensional numerical
manifold method based on tetrahedral meshes and derived the matrices of
corresponding equilibrium equations from the minimum potential energy principle. Ma
and He (2009) also proposed a three-dimensional NMM based on tetrahedron elements.
Generation of mathematical covers, formulation of discrete equations, generating
discrete blocks from a given fracture pattern were discussed in details.
2.3.4 Other developments and applications of the NMM
Li et al. (2005) derived the governing equations of the NMM from the method of
weighted residual (MWR), which enriched the mathematical foundation of the NMM
and extended it to problems such as head conduction and potential flow, where the
governing equations cannot be obtained from the minimum potential energy principle or
other variational principles. A coupled discontinuous deformation analysis and
numerical manifold method (NMM-DDA) has been developed by Miki et al. (2010) to
take both methods’ advantages while avoiding their shortcomings.
The NMM has been applied to simulate various problems as well. For example,
saturated/ unsaturated unsteady groundwater flow analysis by Ohnishi et al. (1999),
rock masses containing joints of two different scales by Lin et al., (1999), data
compression by Fang et al. (2005), dynamic non-linear analysis of saturated porous
media by Zhang and Zhou (2006), the shear response of heterogeneous rock joints by
Ma et al. (2007), dynamic friction mechanism of blocky rock system Ma et al. (2007a),
fluid-solid interaction analysis by Su and Huang (2007) and plane micropolar elasticity
by Zhao et al. (2010), etc.
The NMM is also able to deal with the well-known example of multiple discrete
blocks well. (2010) Ma et al. (2010) conducted a toppling run process in a typical
domino problem and mineral separation process using the vibrating screen with a set of
coal blocks. As shown in Fig. 2.2, a vibrating screen and container are numerically
modeled by the NMM and the results are consistent with the experimentally observed
phenomenon.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.14
Figure 2.2 Size separation process with the vibrating screen modeled by the NMM (Ma et al. 2010).
2.4 COMPARISON WITH OTHER NUMERICAL METHODS
2.4.1 Comparison with FEM
In the FEM, the problem domain is divided into a collection of elements of smaller
sizes and standard shapes with fixed number of nodes at vertices and/or on the sides. On
the other hand, the approximation in the NMM is established based on covers. Under
such circumstances, the mathematical cover in the NMM is equivalent to the nodal
support in the FEM. The constant unknowns of each physical cover are equivalent to the
nodal unknowns in the FEM. The manifold elements in the NMM are equivalent to the
finite elements in the FEM.
The NMM is more flexible than the FEM in discontinuity modeling. The
conventional FEM requires the finite element mesh to be consistent with the internal
(a) Step 0 (b) Step 6000
(c) Step 12000 (d) Step 18000
(e) Step 24000 (f) Step 30000
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.15
discontinuities, which often complicates the meshing task. When discontinuity
evolution needs to be modeled, remeshing is inevitable. Although having achieved great
success, the meshing and the subsequent remeshing process is absolutely not an easy
task. Moreover, additional inaccuracy may arise due to the state variable mapping in
such processes. In contrast, the mathematical covers in the NMM do not need to
conform to neither the external boundaries nor the internal discontinuities, which makes
the meshing task in the NMM very convenient and discontinuity evolution be modeled
without remeshing. The meshing task is very convenient in the NMM, which also
makes the NMM more suitable than the FEM for complex domain problems with
hundreds of inclusions, voids, and/or cracks.
The FEM is a continuum-based method, thus cannot well represent the block
rotations, complete detachment and large-scale opening. In addition, the number of
discontinuities, which can be handled by the FEM is also limited. In contrast, because of
its discrete root, the NMM can easily deal with the aforementioned problems
2.4.2 Comparison with DDA
The discontinuous deformation analysis (DDA) Shi (1988) was developed
originally to solve problems in which a rock mass is delimited into blocks by joints.
Thus, in addition to the contact detection and frictional contact modeling, the DDA also
presents algorithm to form blocks from joints. Since rock blocks generally undergo
small deformation, the DDA simulates each discrete body as a simple deformable block
with only six specific degrees of freedom (DOFs).
The DDA derived based on the variational method takes the benefit of the implicit
time integration method (Shi and Goodman 1985; Shi 1988). The formulation of DDA
is similar to that of the FEM. Due to its implicit time integration, the DDA is
unconditionally stable and it is expected to accommodate considerably large time steps.
Additional features include simplex integration method which is a closed-form
integration for the element and block stiffness matrices and the open-close iteration
(OCI) contact algorithm. The DDA method has emerged as an attractive model because
its advantage in simulating a discrete system cannot be replaced by continuum-based
methods or explicit DEM formulations. The NMM inherits all the attractive features of
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ANALYSIS OF ROCK SLOPE
2.16
the DDA, such as the implicit time integration scheme, the contact algorithm and the
minimum potential energy principle Chen et al. (1998). However, the NMM employs a
number of covers to raise the DOFs to more accurately describe the displacement field
and stress field in each block. Thus, we can say the NMM is the DDA with each block
discretized into finite covers. In other words, if one discrete block is one manifold
element with linear displacement field, then the NMM will be degraded to the DDA.
2.4.3 Comparison with DEM
There is another class of numerical methods called discrete element method
(DEM). Typical examples of the DEM include the distinct element code UDEC (Itasca
1993) and the DDA pioneered by Shi (1988). The DEM is an explicit version, while the
DDA is an implicit version. The DEM adopted an explicit time integration scheme
based on finite difference principles (Cundall 1971a). The DEM has been developed
into the commercial 2D code UDEC, 3D code 3DEC and the particle versions PFC2D
and PFC3D by Cundall and his colleagues (Itasca 1993, 1994, 1995), which has been
enjoying a wide application range in rock engineering. The main benefit of the DEM is
that its computational efficiency is high due to its explicit time integration nature. The
low computational cost of DEM is mainly due to that the explicit time integration
algorithm does not involve in the solution of coupled equations, so fewer computations
are needed per time step. However, it has also been argued that the accuracy of
simulated results may be sacrificed in some particular cases (O’Sullivan and Bray
2001). To ensure numerical stability, a DEM simulation requires that the time step must
be small enough. The NMM differs from the DEM (e.g. UDEC, 2-D version of DEM)
in the following aspects:
The UDEC is an explicit method. It calculates the state of a system at current
step from the states of previous steps. It requires small time step ∆ to keep the
error in the result bounded. The NMM is an implicit method. It finds a
solution by solving an equation involving both the previous step and the
current step. Much larger time step ∆ can be used. To achieve given accuracy,
the NMM takes much less computational time than the UDEC, even taking
into account that the NMM needs to solve the equilibrium equation;
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.17
The UDEC uses numerical integration techniques (e.g. Gauss quadrature)
while the NMM adopts closed-form integrations (i.e. simplex integration
method) to evaluate the weak form;
The joints in the UDEC are always assumed as persistent, failure occurs along
pre-existing persistent joints. However, it is highly unlikely that such a
network of fully persistent discontinuities exist in nature. The NMM allows
non-persistent joints. In the NMM, cracks propagate and coalesce with each
other to form a continuous failure path, thus seems more realistic;
The UDEC uses a finite difference mesh while the NMM uses a cover system
to resolve the stress / strain vibration within each block;
The UDEC sometimes uses artificial joints to connect all the blocks together
to represent an intact material, and then fracturing in intact material is realized
by changing artificial joints to real joints. The fracturing follows pre-defined
artificial joints, thus the results are sensitive to the block configuration. The
NMM allows the cracks arbitrarily align with the elements, thus mesh
dependency is avoided to some extent;
The UDEC is a relatively mature method. Various material models (e.g.
elastic, Mohr-Coulomb plasticity, double-yield, strain-softening, etc) are
available in the code. In addition, relative motion along the discontinuities can
be linear or non-linear. The UDEC has been applied to various engineering
applications. The NMM is a relatively new method. Only linear-elastic
material model is available, and only linear motion along the discontinuities
can be accounted for. Its application is also limited. Further developments are
required.
It is noted that the contact problems is a crucial factor affecting the corresponding
time integration algorithms. Thus, it is essential to investigate the contact methods in the
NMM and DEM. The NMM is an implicit method, which uses penalty method (Shi
1988; Jing 1998) to prevent interpenetration between the contact blocks. And the
behaviour of the contact can be classified three types: corner-to-edge, edge-to-edge and
corner-to-corner in two dimension, which are similar to the DEM (Cundall 1971a;
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.18
Cundall and Hart 1992; Itasca 1993) and different to the FEM such as ANSYS software,
in which the contact types are determined as single surface, nodes-to-surface and
surface-to-surface respectively by means of searching the corresponding slave and
master surfaces (ANSYS 2009). When the corner-to-corner contact is detected, the
NMM employs a direct scheme for the corner-to-corner contact to improve the contact
accuracy, which is referred as the shortest path method. If two potential contact edges
are passed by the corresponding corners simultaneously, a penetration takes place, as
represented in Fig. 2.3(a), where the normal penetrating distances between the reference
edges and the corresponding point are 1d and
2d , respectively. When the distances
satisfy 21 dd , then a contact spring is attached between the point and the reference edge
which is penetrated by distance 1d to assemble the global stiffness matrix. Otherwise,
the other reference edge is selected. The physical meaning of applying the stiff contact
spring is ‘‘to push the invaded angle out of the block along the shortest path” (Shi
1988). Resort to the implicit time integration algorithms and OCI criteria, the contact
problems can be solved successfully. However, it is a time-consuming work to obtain
contact convergence as the repeating iterations and solving global equation. Here, it is
noted that when the corner-to-corner contact is detected, the DEM (e.g. UDEC code)
uses rounded corners to avoid stress concentration in the explicit modeling (Itasca
1993). As shown in Fig. 2.3(b), a rounded corner is represented by an arc of a circle
tangent to the two adjacent edges, in which r is radius of the arc and d is distance to the
corner. It is the rounded corner makes contact blocks can smoothly slide past one
another when two opposing corners interaction. If the two corners are in contact, the
point of contact is the intersection between the line joining the two opposing centers of
the radius r and the circular arcs. The directions of normal and shear forces acting at
the contact are defined with respect to the direction of the contact normal. Combine the
central difference algorithm, the contact problems can be simulated in the DEM
explicitly. Thus, the efficiency of the UDEC is more efficient than that of the NMM.
However, it has also been argued that the accuracy of simulated results may be
sacrificed in some particular cases (O’Sullivan and Bray 2001). To ensure numerical
stability, a DEM simulation requires that the time step must be small enough.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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Figure 2.3 Corner-to-corner contact in the NMM and DEM: (a) The shortest path method in the
NMM; (b) The corner rounding technique in the DEM.
2.5 TIME INTEGRATION ALGORITHMS FOR NUMERICAL
METHODS
In general, there are two general classes of algorithms for dynamic problems:
implicit and explicit. Implicit algorithms tend to be numerical stable, permitting larger
steps, but the computational cost per step is high and storage requirements tend to
increase dramatically with the contacts between elements and these degrees of freedom
(DOFs), so they are suited to simulate the lower dynamics problems with less non-
linearities, resulting in more numerical stability and accuracy (Gelin, Boulmane, and
Boisse 1995; Yang et al. 1995; Sun, Lee, and Lee 2000). On the contrary, explicit
algorithms tend to be inexpensive per step and require less storage than implicit
algorithms, but numerical stability requires that small steps be employed, thus, they
generally used for highly non-linear problems with many DOFs (Dokainish and
Subbaraj 1989; Subbaraj and Dokainish 1989). Using the explicit algorithm, the
computational cost is proportional to the number of elements. On the other hand, using
d2 d1
d1 < d2
(a) The shortest path method in the NMM;
r
d
d = rd d >> r
r
(b) The corner rounding technique in the UDEC (Itasca 1993).
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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the implicit algorithm, experience shows that for many problems the computational cost
is roughly proportional to the square of the DOFs. To compare the explicit and implicit
algorithms in terms of efficiency, Fig. 2.4 illustrates the comparison of CPU time versus
DOFs using the explicit and implicit algorithms. When the DOFs increase up to some
extent, the explicit algorithm becomes more efficient than that of the implicit algorithm.
Figure 2.4 Efficiency versus DOFs between explicit and implicit algorithms.
It is noted that there are some dynamic problems, implicit algorithms are very
efficient and others explicit algorithms are very efficient. To take advantage of the
merits of implicit and explicit algorithms, many methods have been developed in
temporal and spatial discretizations, in which it is attempted to simultaneously achieve
the maximum contributions of both classes of algorithms. Belytschko and Mullen
(1976, 1978b) proposed an explicit-implicit (E-I) nodal partition and proved the
conditional stability of E-I partitions using energy methods and represented the time
step is limited strictly by the maximum frequency in the explicit partition of the mesh.
Hughes and Liu (1978) proposed an alternate element-by-element E-I partitions, in
which a similar stability condition is proven for the algorithm. Liu and Belytschko
(1982) proposed a general mixed time E-I partition procedure which permits different
time steps and different integration methods to be used in different parts of the semi-
discrete equations. Belytschko and Mullen (1978a) proposed a multi-time step
integration method involving different time steps in different zones of the model, in
which the nodal partition approach is employed for E-I systems and linearly interpolated
Explicit algorithm
Implicit algorithm
DOFs
CP
U t
ime
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.21
displacements at the interface. It differs from the referred mixed methods, which consist
of defined zones where different algorithms apply, but with a single time step defined
for the whole domain. Various other improvements in transient algorithms have also
been achieved referred in (Hughes, Pister, and Taylor 1979; Smolinski, Belytschko, and
Neal 1988; Miranda, Ferencz, and Hughes 1989; Belytschko and Lu 1992; Sotelino
1994; Smolinski, Sleith, and Belytschko 1996; Gravouil and Combescure 2001).
In many structural dynamics simulations, low mode response problems are usually
interested. For these cases, the use of implicit time integration scheme is generally
preferred due to its unconditionally stable algorithms over the explicit conditionally
stable scheme. The implicit scheme possesses the numerical property of numerical
dissipation to damp out any spurious participation of the higher modes, thus the larger
time-step size can be employed, but it is restricted in terms of considering
computational efficiency (Katona, Thompson, and Smith 1977), which is the point of
investigation of the time integration in the NMM. The purpose of investigation of
numerical dissipation is to reduce the spurious, non-physical oscillations and
computational effort to the most extent. Numerous improvements have been developed
while maintaining second-order stability and accuracy referred in (Wilson 1968; Hilber,
Hughes, and Taylor 1977; Hughes 1987; Miranda, Ferencz, and Hughes 1989; Chung
and Hulbert 1993; Zhai 1996). Moreover, Numerical dissipation is significant when
solving structural dynamic problems using the explicit integration scheme as well
(Hulbert and Chung 1996). The principal use of the explicit time integration scheme is
limited by time-step size with respect to the stability and accuracy. Newmark β method
when β=0, γ=1/2 is generally called explicit method which possessed no numerical
dissipation itself, results in the consequence is that oscillations occur in the solutions of
numerical simulations. The study of the proposed explicit scheme contrasts with the
implicit scheme in the NMM will be presented to investigate the numerical properties
for the time integration systematically. In addition, the computational efficiency of the
explicit scheme compared to the implicit scheme is explored as well.
2.5.1 Numerical properties for time integration
In order to investigate the time integration, the behaviour of an equivalent linear
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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2.22
system is considered. In such a system, the minimization of the system potential energy
will produce an equation of motion, which is similar to that in the FEM. Let
displacement term of and denote the approximation to the displacements
and ∆ for a time step ∆ , the discrete equation of motion can be expressed as
(2.1)
The component form of the mass , damping and stiffness terms are extensively
discussed in Shi (1991). A single step integration method can be used to solve Eq. (2.1)
for given initial conditions of and based on Newmark method, which can
be represented as
∙ ∆ ∆ ∙ ∙ ∆ ∙
1 ∆ ∙ ∆ ∙ (2.2)
where β and γ are velocity and acceleration weight parameters respectively.
In terms of numerical properties of the integration scheme, a single spring-mass
system is investigated by simplifying it into a single-degree-of-freedom (SDOF) system.
Eq. (2.1) can be degenerated in standard form with frequency and damping ratio as
follows:
fddd 22 (2.3)
where d ,d , and d are acceleration, velocity and displacement, respectively, MK/ is the
frequency of the undamped oscillator, M and K are the mass and stiffness of the
oscillator, KMCCC crit 2// is the damping ratio, KMCcrit 2 is the called critical damping,
and MFf / is the applied load.
At the n time step, extending Eq. (2.3) to be rewritten as
i
n
i
innn QAzAz
1
0 (2.4)
where A is the amplification matrix, iQ is the load vector;
0z is initial data; and
T
nnnn dtdtdz 2 . For the Newmark scheme, A is defined by (Bathe and Wilson
1972; Hilber and Hughes 1978; Hughes 1983) as:
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.23
333231
333231
333231
)1(11
)1(2/111
AAA
AAA
AAA
A
(2.5)
1
31
K
AQ
(2.6)
and
21
2/211
2
2
33
22
32
2
31
DD
A
DA
DA
(2.7)
where t , called sampling frequency. An integration method is stable if the spectral
radius of its amplification matrix remains bounded by 1, which determined by the
maximum modulus of eigenvalues denoted by i
iA max)( , 3,2,1i .
To find the spectral radius of A , the eigenvalues of A are required, which can be
determined by the characteristic equation
020)det( 322
13 AAAIA (2.8)
where1A , 2A and
3A are invariants of matrix A . tracAA2
11 , 2A =sum of principal
minors of A and AA det3 , respectively. Since 03 A , the eigenvalues of Eq. (2.8) are
given by
22
112,1 AAA (2.9)
In terms of the parameters and , the scheme is unconditional stable and
produces under-damped oscillatory response and spectral stability requirement when the
following conditions hold:
10 , 2
1 , 4/)
2
1( 2 (2.10)
At bif the complex conjugate roots bifurcate into two real distinct roots, and the
spectral radius of the amplification matrix A attains its minimum at this point, and
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.24
2)2/1(4
1
1bif
(2.11)
Bifurcation indicates more about the characteristic of the eigenvalues, which can be
evaluated by 4 bif.
The variation of spectral radius for the time integration is presented in Fig. 2.5, in
which the spectral radius 1 with time along. It satisfies the unconditional stability
conditions of Eq. (2.11). For best results from stability and numerical dissipation point
of view, a time-step size t select as(Doolin and Sitar 2004; Doolin 2005):
eet
maxmax
4,
1
(2.12)
where emax is element maximum angle frequency, which is related with minimum
value of element eigenlength.
Figure 2.5 Spectral radius versus sampling frequency for Newmark integration.
In the study, we study the accuracy property of Newmark method in terms of the
exact solution of the homogeneous SDOF model equation. As f is zero in Eq. (2.3), the
eigenvalues of A can be shown to take on a form as:
2/12 )1(
2,1 )( ieA (2.13)
where t and
2ln2
1A
Z
(2.14)
0.2
0.4
0.6
0.8
1
1.2
0.01 0.1 1 10 100 1000
ρ(A
)
Z
β=0.50, γ=1.0
β=0.25, γ=0.5 0.3333
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.25
As measures of the numerical dissipation, we take the algorithmic damping ratio (in
Eq. (2.14) and the period error TT / respectively, where /2T and /2T . The
following cases can be verified:
Case 1: 0 and 2/12 , then =0, which points no numerical dissipation;
Case 2: 0 and 2/12 , then >0, which is applied into NMM value of 1 is
associated with numerical dissipation.
Figure 2.6 Period errors for the undamped case of Newmark integration methods.
In Fig. 2.6, period errors are presented for undamped Newmark methods with case
1(i.e., 2/1 ) and case 2 (i.e., 1 ). We can find that for 4/1 , the methods presented
are conditionally stable, which is made evident by the abrupt decreases in period errors
with . And the central difference method (i.e. 0 ) tends to shorten periods whereas
the trapezoidal rule increases periods (i.e. 4/1 ). About the central difference method
will be investigated in the subsequently sections.
As we noted, the NMM time integration scheme possesses numerical dissipation,
which is shown in Fig. 2.7. It is can be found that the maximum value of 0.2845 of in
the NMM beyond the other parameter values of , which suggests that a time-step size
in this range will lead to faster convergence.
0
1
2
3
4
0 1 2 3 4 5 6
T
Z
β=0, γ=1/2
β=1/4, γ=1/2
β=1/2, γ=1
T=1.0
Z=4.0
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.26
Figure 2.7 Numerical damping versus sampling frequency in the NMM.
The accuracy of the solution depends on the numerical stability and conditioning of
the global stiffness matrix ce KKK as well, where eK is element stiffness matrix,
and cK is contact stiffness matrix, respectively. For the SDOF problem above, one
lower limit of time-step size t can be estimated in terms of element mass and spring
stiffness as KMt /2 . If t is smaller than required value, the characteristics of K are
loosened and the solution becomes unstable. Whereas one larger t will lead to small
ratios of the inertia term of 2/ tM , the effect of M and numerical damping will be
insignificant.
In the Newmark integration schemes, when the parameters are 2/1,0 , recall
Eq. (2.2), we can get an explicit forward central different scheme used by velocity and
acceleration at step n to )1( n . It can be expressed as
∆∆ ∙
/ /
∆∆ ∙ / /
(2.15)
where subscripts of n+1/2 and n-1/2 denotes the central at the step n to )1( n and
)1( n to n , respectively.
Comparing with the implicit time scheme in the NMM, the explicit scheme is
conditionally stable. From the discussion above, the range of bif can be classified by
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4
ξ
Z
β=0.50
β=0.52
β=0.54
β=0.60
β=1.00
0.28450.2659
0.25190.2222
0.1457
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.27
around two general cases: 0 and 0 . In the first case, the spectral radius is equal to
unit throughout the range of 2 , when 2 the oscillation happen even approach
infinite. It is conditionally stable and time-step size t is considered as
eet
maxmax
2,
0
(2.16)
It is noted that there is no numerical dissipation when 0 . In the second case, the
spectral radius is at the range of 1)(0 A , in which the minimum appears at the point
of bif . As discussed on the implicit time integration scheme, numerical dissipation in
the proposed explicit scheme can be improved resort to the damping. In general, smaller
damping ratio of the cover system corresponds to larger time-step size t as presented
in Table. 2.1. Here, we propose )1,0(,/2 max et , in which is one coefficient to
determine the time-step size based on different damping ratios. Compared to implicit
scheme in the NMM, the proposed explicit is more suitable to solve the high frequency
problems as the numerical property of conditionally stable. When the damping is taken
into account in the explicit scheme, the time-step size is sensitive to damping and the
proposed time-step size can be selected in accordance with the coefficient of in Table
2.1. Further more, an appropriate value of should be considered against different
damping effect in terms of solution convergence.
Table 2.1 Values of based on different damping ratios.
In terms of efficiency, the solution convergence is taken into consideration between
the Newmark implicit and explicit schemes. Convergence of Newmark implicit scheme
has been proven in the above discussion. Here, it is demonstrated through solving the
SDOF problem using smaller and larger time-step sizes, respectively. To track CPU
time, a high resolution timer function which can measure up to 1/100000th of a second is
added to keep track of CPU time for each time step. All analyses are run on computer
ξ 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
δ ≤ 1 0.89 0.79 0.70 0.64 0.60 0.54 0.49
ξ 0.4 0.45 0.5 0.6 0.7 0.8 0.9 1
δ ≤ 0.44 0.40 0.39 0.35 0.29 0.28 0.25 0.23
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.28
with the system configuration: processor speed = 3.0 GHz and RAM = 3.0 GB. As
shown in Fig. 2.8, it is clearly that there is a range of time-step sizes, which produces
the shortest computational cost, larger and smaller time-step sizes produce more
computational cost. Comparing the convergence with the spectral radius plotted in Fig.
2.5, the convergence trend closely accords with which of the spectral radius .
Figure 2.8 Computational cost by the Newmark implicit scheme.
Due to the proposed explicit integration algorithm is conditionally stable, the
spectral radius for the scheme is constant and equal to unit throughout the stable
range of )2,0( . Comparing to the implicit scheme, the explicit scheme uses viscous
damping to improve numerical dissipation in order to achieve faster solution
convergence. Fig. 2.9 gives one special case of 1.0 for computing the SDOF problem
using a variety of step-step sizes. When the value of t is near that of critt (critical value
of t ), the computational cost taken is less than the level of 0.1 second to achieve
convergence. If the selected value of t is smaller or larger than that of critt , more
computational cost happens even over 1.0 second or more. Furthermore, it is noted that
the convergence orientation is consistent with the spectral radius .
0
0.2
0.4
0.6
0.8
1
1.2
0.0001 0.001 0.01 0.1 1 10
Com
pu
tati
onal
cos
t (s
)
Time-step size (s)
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.29
Figure 2.9 Computational cost by the Newmark explicit scheme.
2.5.2 Numerical examples for time integration
To further investigate the time integration properties, such as numerical stability
and numerical dissipation, one numerical example of n -DOF problem constituted by
multiply mass-spring-dashpot systems is considered here. The analysis is run on the
same computer with the system configuration: processor speed = 3.17 GHz and RAM =
4.0 GB.
In the present analysis, a simplified model of n -DOF mass-spring-dashpot system,
as shown in Fig. 2.10, is studied to investigate the proposed explicit and implicit
schemes in terms of computational accuracy and efficiency. The structural properties of
the model are assumed to be im = 1000kg and
ik = 10MPa, in which im and
ik ( i =1, 2,
…, n) are ith mass and stiffness, respectively. A harmonic excitation of )(tF =1.0e4
)sin( , in which 100/ and is the number of time-step, is applied to five different
systems by taking n =10, 50, 100, 200 and 500, respectively. The initial conditions of
the systems are assumed )0(iu = )0(iu =0, where )0(iu and )0(iu are the displacement
and velocity of the ith mass, respectively. Different time-step sizes are selected for both
integrations based on stability and accuracy considerations.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.0E-07 1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02
Com
pu
tati
onal
cos
t (s
)
Time-step size (s)
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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2.30
Figure 2.10 Simplified model of n-DOF spring-mass-dashpot system.
The system displacements responses time histories of a typical case of n =100
under damping and no-damping are studied, respectively. The simulated results are
presented in Fig. 2.11. When no-damping case is taken into account, a proper t is
selected to investigate the explicit and implicit schemes, although a larger time-step size
can be used in the implicit scheme as it is unconditionally stable. The system obtained
displacements under no-damping reveal that the solutions of the explicit scheme are
very close to the implicit scheme in Figs. 2.11(a) and (b). On the other hand, since the
damping factor in the explicit scheme can improve the numerical dissipation and
solution convergence effectively, smaller time-step sizes are applied into the under
damping case. In Fig. 2.11(c), damping ratio of 1.0i is used in the explicit scheme,
the amplitude of the solution is stable and identical compared with the implicit scheme.
And critical damping of 0.1i is applied in the Fig. 2.11(d), the system displacements
converge to zero fleetly both in explicit and implicit schemes, which indicates that the
explicit scheme exhibits no overshooting in terms of numerical accuracy when the
appropriate t is selected. It is noted that under damping case, where accuracy
requirements restrict the t to be very small, the implicit scheme uses a dynamic
coefficient (i.e. dd in the code) of 0.999 to achieve a stable solution, which means the
velocity is reduced or damped by 0.1% before it is set as the initial velocity for the next
step. This is similar to damping item in the explicit scheme.
nu 1u2uiu
nk
nc
nm
ic
ik 2k
2c
1k
1c
im2m 1m
)(tF
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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2.31
(a) no-damping, t =10ms;
(b) no-damping, t =1ms;
(c) 1.0i , t =0.1ms;
-0.12
-0.06
0
0.06
0.12
0 0.5 1 1.5 2D
isp
l. (m
)
Time (s)
Newmark implicit
Newmark explicit
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 0.2 0.4 0.6 0.8 1
Dis
pl.
(m)
Time (s)
Newmark implicitNewmark explicit
-0.04-0.03-0.02-0.01
00.010.020.030.04
0 0.2 0.4 0.6 0.8 1
Dis
pl.
(m)
Time (s)
Newmark implicit
Newmark explicit
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.32
(d) 0.1i , t =0.01ms.
Figure 2.11 Total displacement responses for 100-DOFs of system.
In order to compare the computational efficiency between the explicit and implicit
schemes conveniently, mst 1.0 is employed in both schemes. The CPU time involved
for the simulation is recorded and assembled in Table 2.2, in which CPU time consumed
by employing the implicit scheme is denoted by ICPU1 and ICPU 2
, while that involved by
the proposed explicit scheme is represented by ECPU1 and ECPU 2
based on two cases of
i =0 and 0.1 respectively. From sixth to eighth columns of Table 2.2, where the values
of I
E
CPU
CPU
1
1 , I
E
CPU
CPU
2
2 and E
E
CPU
CPU
2
1 reveals the CPU time consumed by the explicit scheme is less
than one fifth of that in the implicit scheme, and which is decreased sharply as the
number of degree of freedom (i.e. n) growing in both damping and no-damping
conditions. It is clear that the explicit scheme is efficient to solve multiply DOF
problem as it doesn’t involve iterations of equations such as in the implicit scheme.
Table 2.2 CPU time for the proposed explicit and implicit scheme.
n-DOF 0i 1.0i
I
E
CPU
CPU
1
1 I
E
CPU
CPU
2
2 E
E
CPU
CPU
2
1
ICPU1 ECPU1
ICPU 2 ECPU2
10 0.266 0.063 0.266 0.063 0.236842 0.236842 1
50 0.969 0.094 0.969 0.094 0.097007 0.097007 1
100 1.766 0.110 1.813 0.110 0.062288 0.060673 1
200 3.516 0.156 3.485 0.141 0.044369 0.040459 1.106383
500 8.672 0.266 8.485 0.282 0.030673 0.033235 0.943262
-0.0004
0
0.0004
0.0008
0.0012
0 0.2 0.4 0.6 0.8 1
Dis
pl.
(m)
Time (s)
Newmark implicit
Newmark explicit
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.33
The numerical results present good accuracy and stability of the Newmark explicit
scheme compared to the implicit scheme. The Newmark explicit scheme is more
efficient in solving the nonlinear systems and such problems compared to implicit
scheme with respect to computational efficiency.
2.6 METHODS FOR DYNAMIC STABILITY ANALYSIS OF
ROCK SLOPE
Rock slope stability analyses are routinely performed and directed towards
assessing the safe and equilibrium conditions of slopes. The analysis technique chosen
depends on both in-site conditions and potential type of rock failure, with careful
consideration being given to the varying strength, weaknesses and limitations inherent
in each methodology. In general, the primary objectives of rock slope stability analyses
can be summarized as:
Determination of the factor of safety (FoS) of the rock slope to investigate potential
failure mechanisms;
Determination of sensitivity to different triggering mechanism (i.e. seismic loading,
blasting effect and pore pressures, etc.);
Test and comparison of different support and stabilization schemes; and
Optimization design for the slope to cover safety, reliability and economics.
To properly conduct such investigations, and analyse and evaluate the potential
hazard relating to an unstable rock slope, it is essential to understand the processes and
mechanisms pushing the failure of slopes. Basically, the instability of rock slope can be
categorized into planar failure, wedge failure, toppling failure, circular failure and rock
fall failure, and they are summarized and presented in Fig. 2.12. To study these five
kinds of typical failure modes conveniently, the corresponding simplified models are
plotted, respectively. In practices, many performances have been carried out to study the
stability of rock slope. There are the analytical methods: i.e. Limit Equilibrium Methods
(LEMs) (Bishop 1955; Morgenstern and Price 1965) and Newmark methods (Newmark
1959, 1965), and numerical methods: i.e. Shear Strength Reduction Method of Finite
Element Method (SSRM-FEM) (Zienkiewicz, Humpheson, and Lewis 1975; Griffiths
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.34
and Lane 1999), Distinct Element Method (DEM-UDEC/3DEC) (Itasca 1999b, 1999a),
Discontinuous Deformation Analysis (DDA) (Shi 1988, 1993) and Numerical Manifold
Method (NMM) (Shi 1991, 1992), etc.).
Figure 2.12 classification of instability and project examples for rock slopes.
Further more, some real examples photographs are attached to represent these
typical modes of rock slope failure in order to illustrate these rock slope failure visually.
1. ref. technology.infomine.com; 2. ref. www.rocscience.com; 3. ref. www.rocscience.com;
4. ref. elkorose.schopine.com; 5. ref. www.capetownskies.com.
Type Simplified mode Project example
Planar failure containing persistent joints striking parallel to the rock face for jointed rock slopes.
Wedge failure on two intersecting discontinuities for jointed rock slopes.
Toppling failure of columns separated from the rock mass by steeply dipping parallels or near parallels to the slope.
Circular failure in weak rock mass or heavily jointed rock slope with a spoon-shaped surface.
Rock fall failure contains sliding, rotating, falling and bouncing of loose rocks and boulders on the slope.
2.
1.
3.
4.
5.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.35
For the category of rock slope instability, critical parameters, methods of analysis and
acceptability criteria, more details can be referred as (Hoek and Bray 1981; Hoek 1991;
Cruden and Varnes 1996; Wyllie and Mah 2004).
In rock slope engineering, the analysis of seismic stability of rock slope is one of
important issues attracted the attention by geotechnical engineering and earthquake
engineering. Many researchers have attempted to develop and elaborate on the methods
for slope stability analysis. Exiting methods for evaluating the stability of rock slope
subjected to seismic effect can be classified into three categories: (1) force-based
pseudo-static methods; (2) displacement-based sliding block methods; (3) numerical
methods.
Conventional pseudo-static approach is a widely used based on the LEM (Bishop
1955; Seed 1979; Chang, Chen, and Yao 1984) to evaluate slope stability where the
dynamic effects are major from earthquake can be simplified as horizontal and/or
vertical dynamic coefficients ( and ), in which the magnitude of the coefficients is
expressed in terms of a percentage of gravity acceleration. Due to the simplicity of the
pseudo-static approach, it has drawn the attention of a number of investigators (Seed
1979; Chang, Chen, and Yao 1984; Ling, Leshchinsky, and Mohri 1997; Baker et al.
2006; Li, Merifield, and Lyamin 2008). Li et al. (2008) investigated the seismic effects
on rock slope stability coupled with Hoek-Brown failure criterion using the pseudo-
static method (PSM), in which the advantage of limit theorems were exploited to
bracket the true solutions for rock slope stability numbers and to provide a range of
seismic stability charts for rock slopes (Lyamin and Sloan 2002a, 2002b). However, the
pseudo-static approach has certain limitations, since it cannot simulate the transient
dynamic effects of earthquake shaking, because it assumes a constant unidirectional
pseudo-static acceleration (Cotecchia 1987; Kramer 1996).
Dynamic analysis has also been employed using a simplified manner. Huang et al.
(2001) conducted Newmark’s sliding block analysis considering the whole failure rock
mass as one single block, where they demonstrated that the surface-normal acceleration
played a vital factor in the initiation of the Chiu-fen-erh-shan landslide. Jibson (1993)
and Jibson et al. (1998) have developed procedures for estimating the probability of
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.36
landslide occurrence as a function of Newmark displacement based on observations of
landslides caused by the 1994 Northridge earthquake in California. In this section, we
give a brief overview of the developed methods in the past several decades.
2.6.1 LEM
The LEM of determining the FoS of a sliding block can be modified to incorporate
the effect on stability of seismic ground motions. The analysis procedure, known as the
pseudo-static method, involves simulating the ground motions as static horizontal forces
acting in a direction out of the face. The magnitude of this force is the product of a
seismic coefficient Hk and the weight of the sliding block w . The value of Hk may be
taken as equal to the design peak ground acceleration (PGA), which is expressed as a
fraction of the gravity acceleration (i.e. Hk = 0.1 if the PGA is 10% of gravity).
However, this is a conservative assumption, the actual transient ground motion is
controlled by constant forces acting on the entire design of the slope.
In the design of soil slopes, it is common that Hk is fraction of the PGA, provided
that there is no loss of shear strength during cyclic loading (Seed 1979; Pyke 1999).
Study of slopes using Newmark analysis with a yield acceleration yk equal to 50% of
the PGA (i.e. gaky /5.0 max ) showed that permanent seismic displacement would be less
than 1m (Hynes-Griffin and Franklin 1984). Based on these studies, the California
Department of Mines and Geology (CDMG 1997) suggests that it is reasonable to use a
value of Hk equal to 50% of the design PGA, in combination with a pseudo-static FoS of
1.0 -1.2. With respect to rock slopes where the rock mass contains no distinct sliding
surface and some movement can be tolerated, it may be reasonable to use the CDMG
procedure to determine a value for Hk . However, for rock slopes there are two
conditions for which it may be advisable to use Hk values somewhat greater than 0.5
times the PGA. First, where the slope contains a distinct sliding surface for which there
is likely to be a significant decrease in shear strength with limited displacement; sliding
planes on which the strength would be sensitive to movement include smooth, planar
joints or bedding planes with no infilling. Second, where the slope is a topographic high
point and some amplification of the ground motions may be expected. In critical
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.37
situations, it may also be advisable to check the sensitivity of the slope to seismic
deformations using Newmark analysis. The FoS of a plane failure using the pseudo-
static method is given by modifying as follows (assuming the slope is drained):
)cos(sin
tan))sin(cos(
pHp
pHp
kW
kWcAFoS
(2.17)
The equation demonstrates that the effect of the horizontal force is to diminish the FoS
because the shear resistance is reduced and the displacing force is increased.
Under circumstances where it is considered that the vertical component of the
ground motion will be in phase with, and have the same frequency, as the horizontal
component, it may be appropriate to use both horizontal and vertical seismic
coefficients in stability analysis. If the vertical coefficient is Vk and the ratio of the
vertical to the horizontal components is Kr (i.e. HVK kkr / ), then the resultant seismic
coefficient Tr is
2/12 )1( kHT rkr (2.18)
Acting at an angle )/( HVk kkatn above the horizontal, and FoS is given by
))cos((sin
tan)))sin((cos(
kpTp
kpTp
kW
kWcAFoS
(2.19)
Study of the effect of the vertical component on the FoS has shown that incorporating
the vertical component will not change the FoS by more than about 10%, provided that
HV kk ((NHI) 1998). Furthermore, Eq. (2.19) will only apply when the vertical and
horizontal components are exactly in phase. Based on these results, it may be acceptable
to ignore the vertical component of the ground motion.
A model presented by Chang et al. (1984) developed for the evaluation of the
critical condition and the subsequent response to earthquakes, is here represented. The
model applies the pseudo-static limit equilibrium analysis for the determination of the
critical condition of the slope and the Newmark analytical procedure to assess the
displacement of the rigid block.
The computation procedure of the model can be subdivided into the following
steps:
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.38
1). A designed earthquake is selected and the seismic accelerations,gk are determined
for all the time steps considered during the earthquake shaking. The choice of the
constant time interval of 0.001s was suggested by the model authors to designate the
time and to subsequently estimate all the corresponding accelerations. The acceleration
within a time interval is assumed to be linear, but not necessarily constant.
2). The motion acceleration 1x acting on the sliding block can be calculated for the
case of a block on a plane (where Ckk and 1x ) as described in Fig.2.13.
Figure 2.13 Equilibrium of forces on a sliding block (Chang et al. 1984).
According to equilibrium equations of forces on the block, Newton second law
can be employed to express the seismic force as
xg
WF 1
1 (2.20)
where
2cos
)cos()(
gkkx ci .
3). By using the results obtained in step 2 and starting from the beginning of the
seismic event, the first positive motion accelerationix , which corresponds to the
starting of the sliding motion at the timeit , is determined. If
ix is the first positive
motion acceleration, then 1ix at time
1it must be negative, except that particular case in
which 01 ix . Time t , at which x =0 must then be computed. The motion velocity x
will start to increase from zero from this time. By linear interpolation one obtains:
11
11 )(
i
ii
iii txx
ttxt
(2.21)
(a) (b)
1W
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.39
At the time in which the acceleration induced by the seismic event exceeds the yield
acceleration, the sliding block velocity increases from zero and the motion displacement
occurs.
4). The motion velocity ix , at the time t , can be computed by assuming a linear
variation of the acceleration as )(2/ ttxx i . By knowing ix , the motion velocity
1ix
can similarly be calculated as:
]2/))([( 111 iiiiii ttxxxx (2.22)
The value of 1ix , in this equation, is obtained from Eq. (2.21). All the velocities, in the
selected time, can be calculated by using the same procedure. The resistance to the
uphill movement can be assumed as being indefinitely large without causing serious
errors, as Newmark pointed out. The time 2nt can be expressed as:
112
1212
)(
iii
iiin t
xx
ttxt
(2.23)
Two non-consecutive displacements can be computed during the time which passes
from 1it to 2it . However, the calculations for these two separated displacements
are required only in the case in which the motion acceleration is negative at the time
1it and positive at the time 2it . Otherwise, only the time 1nt (i.e.
2211 inni tttt )
will be required and the movement will cease at time 2it .
5). The displacement 1ix , between time
it and time 1it can be calculated as:
6/]))(2[()( 211111 iiiiiiiii ttxxttxxx (2.24)
Thus the block overall displacement can be determined to all the times of the seismic
effect.
2.6.2 Newmark method
When a rock slope is subject to seismic shaking, failure does not necessarily occur
when the dynamic transient stress reaches the shear strength of the rock. Furthermore, if
the FoS on a potential sliding surface drops below one at some time during the ground
motion it does not necessarily imply a serious problem. What really matters is that
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.40
magnitude of permanent displacement caused at the times that the FoS is less than 1.0
(Lin and Whitman 1986). The permanent displacement of rock and soil slopes as the
result of earthquake motions can be calculated using a method developed by Newmark
(1965). This is a more realistic method of analysing seismic effects on rock slopes than
the pseudo-static method of analysis.
Figure 2.14 Displacement of rigid block on rigid base (Newmark 1965): (a) block on moving base;
(b) acceleration plot; (c) velocity plot.
The principle of Newmark’s method is illustrated in Fig. 2.14. It is assumed that
the potential sliding block is a rigid body on a yielding base. Displacement of a block
occurs when the base is subjected to an uniform horizontal acceleration pulse of
magnitude ag with duration 0t . The velocity of block is a function of the time t and is
designated )(ty , and its velocity at time t is y .Assuming a frictional contact between
the block and the base, the velocity of the block will be x , and the relative velocity
between the block and the base will be u where yxu .
The resistance to motion is accounted for by the inertia of the block. The maximum
force that can be used to accelerate the block is the shearing resistance on the base of
the block, which has a friction angle . This limiting force is proportional to the weight
of the block W and magnitude of tanW , corresponding to a yield acceleration ya of
tang , as shown in Fig. 2.13(b) by the dashed line on the acceleration plot. The shaded
area shows that the ground acceleration pulse exceeds the acceleration of the block,
resulting in slippage. Fig. 2.13(c) shows the velocity as a function of time for both the
ground and the block accelerating forces. The maximum velocity for the ground
accelerating force has a magnitude v , which remains constant after an elapsed time of
Time
Acceleration Velocity
Time (b) (c)(a)
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.41
0t . The magnitude of the ground velocity gv is given by
0agtvg , while the velocity of
the block bv is tangtvb . After time mt , the two velocities are equal and the block
comes to rest with respect to the base, the relative velocity, 0u . The value of mt is
calculated by equating the ground velocity to the velocity of the block to give the
following expression for the time mt :
tang
vt b
m (2.25)
The displacement m of the block relative to the ground at time
mt is obtained by
computing the area of the shaded region as follows:
)tan
1(tan22
1
2
1 2
0
ag
vvtvtmm
(2.26)
Eq. (2.26) gives the displacement of the block in response to a single acceleration pulse
0t and ag that exceeds the yield acceleration tang , assuming infinite ground
displacement. The equation shows that the displacement is proportional to the square of
the ground velocity. While Eq. (2.26) applies to a block on a horizontal plane, a block
on a sloping plane will slip at a lower yield acceleration and show greater displacement,
depending on the direction of the acceleration pulse. For a cohesionless surface where
the FoS of the block is equal to (p tan/tan ) and the applied acceleration is horizontal,
Newmark shows that the yield acceleration ya can be given as
py gFoSa sin)1( (2.27)
where is the friction angle of sliding surface, and p is the dip angle of this surface.
Note that for 0p , tangay . Also Eq. (2.27) shows that for a block on a sloping
surface, the yield acceleration is higher when the acceleration pulse is in the down-dip
direction compared to the pulse in the up-dip direction.
The displacement of a block on an inclined plane can be calculated by combining
Eqs. (2.26) and (2.27) as follows:
)1(2
)( 2
a
a
ga
agt y
ym (2.28)
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.42
In an actual earthquake, the pulse would be followed by a number of pulses of
carrying magnitude, some positive and some negative, which will produce a series of
displacement pulses. This method of displacement analysis can be applied to the case of
a transient sinusoidal acceleration gta )( illustrated in Fig. 8.6(Goodman and Seed 1966)
. If during some period of the acceleration pulse the shear stress on the sliding surface
exceeds the shear strength, displacement will take place. Displacement will take place
more readily in a downslope direction; this is illustrated in Fig. 2.14 where the shaded
areas are the portion of each pulse in which movement takes place. For the conditions
illustrated in Fig. 2.15, it is assumed that the yield acceleration diminishes with
displacement, that is, 321 yyy aaa due to shearing of the asperities in the manner.
Figure 2.15 Integration of accelerograms to determine block movement (Goodman and Seed 1966).
Integration of the yield portions of the acceleration pulses gives the velocity of the
block. It will start to move at time 1t when the yield acceleration is exceeded, and the
velocity will increase up to time 2t when the acceleration drops to zero at time 3t as the
acceleration direction begins to change from down slope to up slope. Integration of the
velocity pulses gives the displacement of the block, with the duration of each
displacement pulse being (13 tt ). The simple displacement models shown in Fig. 2.14
has since been developed to more accurate model displacement due to actual earthquake
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.43
motions, with much of this work being related to rock slopes (Jibson 1993; Jibson et al.
1998).
The Newmark method transcends the LEM to provide an estimate the displacement
of a landslide block subjected to seismic motion. On this method, that portion of the
design accelerogram above a critical acceleration ca , is intergrated twice to obtain the
displacement, the critical acceleration is defined using Eq. (2.29)
sin)1( gFoSac (2.29)
where ca is critical acceleration, in meters per-second squared, FoS is static factor for
safety and is slope angle, respectively. A rigorous Newmark analysis involves
subtracting ca , from the accelerogram and integrating the difference twice to compute
the total displacement. When the earthquake acceleration does not overcome the initial
limit equilibrium of the slope, and the rigorous Newmark analysis predicts no slope
displacement. In Newmark’s approximation, the total slope displacementnD , is a
function of the peak particle velocity (ppv) and the critical acceleration ca , as shown in
Eq. (2.30)
),6(2
2
Maxa
ppvD
cn (2.30)
where ppv is peak partical velocity and repsents a dimensionless constant equal to the
duration of strong motion, respectively.
Another approximation to the Newmark method by mapping landslide hazard in
southern California is presented by Jibson et al. (1998). The development of this
approximation was motivated by the difficulty of using a rigorous Newmark approach
within the geographic information system (GIS) framework, commonly adopted for
regional hazard mapping. The Arias intensity has recently found use in representing
earthquake shaking likely to cause landslides. It is given by
dttag
I a
0
2)]([2
(2.31)
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.44
where aI is the Arias intensity and )(ta is the ground acceleration as a function of time,
respectively.
2.6.3 Numerical methods
With respect to the numerical methods, Zhang et al. (1997) carried out studies on
the dynamic behaviour of a 120-m high rock slope and a blocky arch structure of the
Three Gorges Shiplock using DEM, which demonstrate that the effects of different
seismic input and parameters on the dynamic response behaviours and failure mode of
the slope are significant. Hatzor et al. (2004) carried out dynamic two-dimensional
stability analysis of a highly discontinuous rock slope, which is upper terrace of King
Herod’s Palace in Masada using a fully dynamic version of DDA where a 2% kinetic
damping is introduced to predict the realistic damage of the slope. Bhasin and Kaynia
(2004) performed static and dynamic rock slope stability analyses for a 700-m high rock
slope in western Norway using a numerical discontinuum modelling technique, in which
three cases have been simulated for predicting the behaviour of the rock slope under
existing environmental and earthquake conditions. Liu et al. (2004) studied the dynamic
response of Huangmailin Phosphorite rock slope in China under explosion using
DEM/UDEC and compared with field measurements, the numerical results show that
UDEC is efficient to simulate the dynamic response of jointed rock slope. Crosta et al.
(2007) performed 3D dynamic analysis of the thurwieser rock Avalanche, Italian Alps,
where the propagation of rock avalanche and runout were studied. Latha and Garaga
(2010) carried out a comprehensive study on seismic slope stability of a natural slope in
jointed rock mass using FLAC (Itasca 2002) through a case study in the Himalayan
region of India. Wu (2010) and Wu et al. (2009) carried out a seismic landslide
simulation in DDA and dynamic discrete analysis of an earthquake-induced large-scale
landslide, respectively. In the simulations, three available algorithms incorporate
seismic impacts into DDA simulations for earthquake-induced slope failure are
investigated. Chiu-fen-erh-shan landslide, triggered by 6.7wM Taiwan Chi-Chi
earthquake is studied using the DDA, in which the main objectives of the study were to
investigate if it was possible to numerically model the landslide progression, including
slope disintegration, and to reproduce the post-failure configuration. Miki et al.(2010)
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
2.45
studied earthquake response analysis of rock slope using the coupled DDA-NMM, the
simulated results of the earthquake response analysis indicated that the failure modes
and travelling distance of the collapsed rock blocks are affected by the joint strength
between blocks significantly. An et al. (2012) investigated the seismic stability of rock
slope using the NMM, in which the validity of the NMM in predicting the ground
acceleration induced permanent displacement is verified by comparing its results with
the analytical solutions and the Newmark- numerical integration solutions. Ning et al.
(2012) studied rock fall of earthquake-induced failure with pre-existing non-persistent
joints located on the crest of a rock slope, in which the failure of rock mass is modelled
by the NMM coupled with a fracturing algorithm based on the Mohr-Coulomb criterion,
and DDA model based on a strain and kinematic energy conservation transition
technique, respectively.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
3.46
CHAPTER 3. THEORY OF THE NUMERICAL MANIFOLD
METHOD AND ITS INTEGRATION SCHEMES
3.1 INTRODUCTION
Since the birth of the NMM in 1991 (Shi 1991), its proposed concepts and
perspectives are distinct from the definitions of traditional numerical method. The
NMM employs a dual cover system to represent physical domain, which increases its
difficulty in terms of understanding and further development to some extent. To
recognize these distinct characteristics of the NMM clearly, the finite cover system, the
cover-based contact algorithm and integration scheme are introduced in this chapter.
The basic concept and constructed finite cover system are firstly illustrated in Section
3.2. Then, the integration schemes based on the cover system are explored with respect
to the spatial and temporal domain in Section 3.3 as well.
3.2 FUNDAMENTALS OF THE NMM
NMM is one of the newest developed numerical methods in recent decades. It
merges the FEM and DDA nicely and thus it reconciles the Continuum-based methods
with Discontinuum-based methods. The discretization involves two domains: Physical
domain is used to describe geometry property and Mathematical domain is used to build
global displacement function. The mathematical treatment to these two domains is
relatively independent, but these two domains interact with each other according to
special integration and interpolation scheme. Because its geometrical model can be
prepared independently without considering the meshing issue, NMM becomes very
suitable for the numerical simulations of crushing-like material (rock mass) or relevant
complex engineering applications.
As present in Section 2.2, mathematical cover (MC) is the essential component in
the mathematical domain. Each MC possesses the separate cover function and been
represented on its nodal star. The nodal stars are connected by weight function in a
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
3.47
mathematical sense to describe the mathematical domain. As a result, the global
displacement function is generated by the weighted average of local independent cover
functions on the common part of corresponding MCs. Physical cover (PC) is generated
as an inter-media between the physical domain and the corresponding MC. The PCs are
overlapping to each other by the nodal stars. Manifold element (ME) is the final product
following the PCs of the Physical domain. It includes the geometrical information
mapping back to the corresponding PCs. In a sense, it is equivalent to the integration
field of element in FEM.
3.2.1 Finite cover system in the NMM
The finite cover system is defined as a combination of the mathematical domain
and physical domain. The mathematical mesh defines the fine or rough approximation
of unknown functions, which is used to build MCs that present small regions of the
whole field and can be any of shape and size. They can overlap each other and do not
need to coincide with the PC as long as they are large enough to cover the physical
domain.
To visualize these concepts, an example illustrated in Fig. 3.1 is used. There are
two MCs in total, a regular hexagon and a circle . The thick lines define the
physical domain ]2,1[ . Intersected with the physical domain, and . are divided,
respectively, into two PCs, i.e. , and , , as shown in Fig. 3.1(b). Here, notation
represents the jth PC generated from the Ith MC.
On each MC , a weight function is defined, which satisfies
Ii
Ii
Mxx
MxCCx
,0)(
),10()( 00
(3.1)
With
J
JMifx
J x 1)( (3.2)
Eq. (3.1) indicates that the weight function has non-zero value only on its
corresponding MC, but zero otherwise, whereas Eq. (3.2) is just the partition of unity
property to assure a conforming approximation. The weight function )(xi associated
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
3.48
with IM will be accordingly transferred to
iP , any of the PCs ][ jiP in
IM , which is
expressed as )(xi on iP hereafter.
Figure 3.1 A schematic of basic concepts in the NMM. (a) The physical domain and two MCs; (b) Overlapping of MCs and physical domain; (c) Corresponding PCs; (d) Six corresponding MEs.
So far, each MC is associated with several PCs, and each PC has two indices, i.e. I
and j. Considering the simple example, for instance, the four PCs can be renumbered by
1 11 , , and in light of different MCs. These four PCs
finally form six MEs as shown in Fig. 3.1(d), i.e. )( ]1[11 PEE , ),( ]2[
1]1[
12 PPEE ,
)( ]2[13 PEE , )( ]1[
24 PEE , ),( ]2[2
]1[25 PPEE and )( ]2[
26 PEE .
(c)
Overlapping of MCs and
physical domain
Physical domainMCs
PC PC
PC
(a)
(b)
(d)
PC
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
3.49
With these concepts, the interpolation approximation can be constructed. First, a
cover function )(xui is defined individually as a local approximation on the PC
iP for
the displacement field, which can be constant, linear, high order polynomials or other
functions with unknowns (also termed DOFs) to be determined. Then, the global
displacement )(xu on a certain CE e is approximated to be
i
iPe
ii xuxxu )()()( (3.3)
Here, we give another example as shown in Fig. 3.2(a), the mathematical cover
system, which is united by six rectangle patches denoted by , , , , and
respectively. The overlapping patches cover the whole material domain without
considering any physical properties, so any arbitrary shape of mathematical cover can
be chosen. And then, physical covers can be obtained from these mathematical covers
intersect with the physical domain , a manifold element can be produced as the
common area of physical covers. Each small rectangle patch is termed as a
mathematical cover (MC), denoted by iM (i= 1, 2, 3, …, 6). External boundary and
internal joints or cracks may intersect one MC into several separate sub-patches, then
each one within the material domain is termed as a physical cover (PC), denoted by jiP (
Nj ). As can be seen in Fig. 3.2(b), material domain is intersected by patch to
generate one PC within its material domain, denoted by 11P . When the internal
discontinuities (i.e. cracks or joints) are taken into accounted in the NMM, each
discontinuous boundary is considered as one special material domain to form a new PC.
If the crack passes through the whole patch within the material domain, two isolated
PCs form by the crack surface just as 4M and
6M , two separated PCs, denoted by 14P ,
24P based on
4M and 16P , 2
6P based on 6M , respectively. On the other case, when the
crack cuts MC partially, only one PC forms within the material domain, which can be
seen by 2M ,
3M and 5M , only one PC generates denoted by 1
2P , 13P and 1
5P respectively.
Furthermore, the common area of several overlapping PCs is termed as a manifold
element (ME).
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3.50
(a) General cover system in the NMM;
(b) Generation of physical covers for the NMM.
Figure 3.2 The cover system in the NMM: (a) General cover system; (b) Generation of physical
covers.
For a two-dimensional problem, the MCs can be generated from a regularly-
patterned triangular finite element mesh or a regularly-patterned rectangular finite
element mesh. Fig. 3.3 shows a NMM model for discontinuity problem, we can use a
finite element mesh composed of equilateral triangles to construct the MCs. Each node
in the triangular finite element mesh is termed as a nodal star. The union of six triangles
sharing a common star forms a hexagonal MC. When one MC is interacted by two
physical domains, to simplify procedure for construction of PCs and MEs, the MC will
24P
15P
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3.51
be divided into a pair MCs automatically with neighboured indices, i.e. star 17 and 18,
25 and 26, 27 and 28, 29 and 30, 33 and 34, 35 and 36, 40 and 41, respectively.
Figure 3.3 NMM model for the discontinuity problem.
Here, each physical cover has two indices, i.e. I and j, in which I and j are the index
number of the MCs and physical domains (2 domains in the example), respectively. To
simplify an implementation, a reallocated single index to each PC can be referred,
which can be expressed as ≜ with i calculated by
jmjIiI
ll
1
1
),( (3.4)
Continuing the above discussions, each PC is assigned a local function. In the
NMM, the local functions are usually taken as the polynomials as
iT
i xpxu )()( (3.5)
where i is an array of constant coefficients, and )(xpT is the matrix of polynomial bases
as
pppp
ppppT
yxyyxxyx
yxyyxxyxxp
00000010
00000001)(
11
11
(3.6)
Though the polynomials can approximate smooth functions well, for strong
singularity problems, the smooth basis polynomial local approximations cannot
Nodal star
1 PC
2 PCs
MC
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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3.52
properly capture the high gradient solutions. Therefore, various singular functions may
need to be used to enrich the approximation space for capturing the singularities without
refining the meshes.
Consider the NMM model in Fig. 3.3 for an example. The common area of MCs
and the two physical domains forms total 42 PCs, denoted as )42,2,1( iPi, where the
subscript represents the number of the MCs. Then, the overlapping of the PCs and finite
element mesh (represents the MCs) generates MEs, denoted as )42,2,1( iE ji
, where the
subscript represents the number of the associated PCs and superscript represents the cut
element number by the associated finite element mesh. Here, we choose three typical
nodal stars, marked as 9, 25 and 26, 35 and 36, respectively, to illustrate the
construction of the MEs based on the finite cover system. Taking stars 25 and 26 for
instance, two complete overlapped MCs, 25M and
26M form two associated PCs, 25P and
26P , respectively. The intersection of the PCs and finite element mesh generates 9 MEs,
denoted as 125E , 2
25E , 325E , 4
25E , 525E , 6
25E , 725E and 1
26E , 226E , respectively. The other two
constructions of MEs and the associated PCs can be obtained in a similar way, as
illustrated in Fig. 3.4. Then, the NMM adopts the partition of unity functions to paste all
the local functions together to give the global approximation.
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Figure 3.4 Construction of finite cover system in the NMM.
As shown in Fig.3.5, the structured mesh-based cover system is built on the
triangular finite element mesh, in which each node is termed as a star. The union of six
triangles sharing a common star forms a hexagonal MC. Each PC coincides with the
corresponding MC and physical domain of the block, thus each MC generates one or
more PCs, in which a local function is assigned to each PC, such as that is allocated an
index of ① generates a PC denoted by 11P , ② created two PCs for the joint in the block
denoted by 12P and 2
2P , respectively. Similarly, following the criterion we have indices of
④, ⑤ and ⑥ generate one PC denoted by 14P , 1
5P and 16P ; ③ generates two PCs denoted
by 13P and 2
3P , respectively. Each triangular element e is constructed by the associated
three PCs starred at its three nodes.
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its three nodes.
Figure 3.5 Construction of PCs on the cover system.
When internal discontinuities such as joints and cracks present in the physical
domain as shown in Fig. 3.6, the structured mesh-based cover system is built on the
triangular finite element mesh, in which each node is termed as a star. The union of six
triangles sharing a common star forms a hexagonal MC. To the continuous media, each
PC coincides with the corresponding MC, thus each MC generates a PC, in which a
local function is assigned. Each triangular element e is constructed by the associated
three PCs starred at its three nodes. When the discontinuities (i.e. ① and ②) are taken
into account in the problem domain, a MC can be sub-divided two and more PCs shared
the original star (i.e. 2 PCs and 4 PCs). If one MC is not or partly cut by the
discontinuities, only one PC is constructed. In this case, we usually apply refining mesh
technique and cutting off the discontinuity tips by the element edges to avoid singular
matrices occurrence at utmost extend. Since each cover has two degrees of freedom,
thus each element formed by the overlapping of the three PCs has six degrees of
freedom. For the discontinuities and physical boundaries are considered in the cover
system, many new generated PCs will be reallocated updated indices to give the global
approximations.
MC
①
②
②
①
③
④
⑤
⑥
block
MC Overlap PCs Joint
1 PC
2 PCs
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Figure 3.6 Structured mesh-based cover system in the NMM.
Here, we give a simple example to illustrate the constructions of the PCs and MEs
on the cover system as plotted in Fig. 3.7. To the continuum as shown in left of the
figure, two interconnected MEs ie and je sharing two PCs indexed )2( and )3( , in
which ie is constructed by the associated three PCs indexed by )1( , )2( and )3( , and je
is constructed by the PCs indexed by )2( , )4( and )3( , accordingly. Since the global
approximation involves the assembling of a global stiffness matrix, the interactions
between ie and je can be offset in the implicit time integration of the NMM. On the
other hand, when a discontinuity passes through the PCs coving the MEs as presented in
right of the figure, each MC is divided into two PCs and each ME is separated into two
MEs subsequently. A formula of single index is used to reallocate the new generated PC
and the new generated MEs are rerecorded as follows: 1ie is built by )1( , )3( and )5( ; 2
ie
is built by )2( , )4( and )6( ; 1je is built by )3( , )7( and )5( ; 2
je is built by )4( , )8( and )6( ;
respectively. The contacts between MEs 1re and 2
re ( jir , ) occur and the interactions
between the MEs are calculated by adding and removing the linear stiff springs on
contact points in the normal and shear directions.
① ②
①, ②: discontinuities
1 PC
2 PCs 4 PCs
1 PC
①
①
②
①
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(a) continuous elements;
(b) discontinuous elements.
Figure 3.7 Construction of manifold elements on the cover system: (a) continuous elements; (b)
discontinuous elements.
On each ME, the NMM adopts the partition of unity functions to paste the three
associated local functions together to give the global approximation as
),(
),(),(
).(
).( 3
1 yxv
yxuyxw
yxv
yxu
i
i
ii
e
e (3.7)
Where ),( yxwi is the weight function defined on the associated PC iP with the
expression as
ii
ii
Pyxyxw
Pyxyxw
),(,0),(
),(,0),( (3.8)
The weight function is a partition of unity and satisfies
discontinuity
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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3.57
ePyx
i yxyxwi
),(,1),(),(
(3.9)
where e is the problem domain. When the constant local function are considered, the
global displacement on each manifold element can be rewritten as
)3(
)2(
)(
)3()2()1( ),(),(),(),(),(
),(
e
e
ie
eeeeee
e
D
D
D
yxTyxTyxTDyxTyxv
yxu (3.10)
in which
),(0
0),(),(
)(
)(
)( yxw
yxwyxT
ie
ie
ie and 3,2,1,
),(
),(
)(
)(
)(
iyxv
yxuD
ie
ie
ie.
3.2.2 Contact algorithm
Contact algorithm is a requisite tool for further developing the numerical
simulation technique. The NMM aims at the discontinuous problems, even with
movements. When intersected with physical features like cracks and material interfaces,
each MC forms several independent PCs associated with different local functions. Thus,
the adjacent MEs formed by these PCs are independent on each other in the framework
of NMM. An example is the problem with discrete bodies, in which the displacement
across each body boundary is discontinuous; however, one body cannot penetrate into
another body. Such constraints are normally termed as non-penetration or unilateral
condition, and attributed to a contact problem in physics.
Since the frictional contact problems are inherently nonlinear and irreversible, for
the sake of generality, an incremental approach is adopted in the NMM. It is assumed
that:
The contact state at the initiation of the time step, ntt , is known and the contact
state at 1 ntt , the end of the step, after time interval
nn ttt 1 is to be solved;
The time incremental for each time step is chosen small enough so that the
displacements of all the contact points within the problem domain are less than a
predefined maximum displacement limit .
In the NMM, contact detection at the beginning of each time step can be seperated
into two steps:
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A global search is first done to find the bodies which might possible come into
contact by representing the bodies by bounding boxes, i.e. if the distance between
the bounding boxes of any two bodies is less than 2, they might contact within
this time step;
A local search is then conducted to figure out all the vertex-edge contact pairs.
There are generally three types of contacts: angle-to-edge, angle-to-angle and edge-
to-edge, as shown in Fig. 3.8. Among them, edge-to-edge contact can be treated as two
angle-to-edge contacts. The contact will be of angle-to-angle if both of the following
conditions are satisfied: (1) minimum distance of the two angle vertices is less than 2;
(2) the maximum overlapping angle of the two angles is less than 2 when one angle
vertex translates to the vertex of the other angle without rotation. The contact will be of
angle-to-edge if both of the following conditions are satisfied: (1) minimum distance
between the vertex and the edge is less than 2; (2) maximum overlapping angle of the
angle and the edge is less than 2 when the angle vertex translates to the edge without
rotation.
(a) angle-to-angle; (b) angle-to-edge; (c) edge-to-edge.
Figure 3.8 Three types of contacts: (a) angle-to-angle; (b) angle-to-edge; (c) edge-to-edge.
For each contact pair, there are three possible contact modes: open, sliding and
sticking. At the beginning of each time step, the contact modes for all contact pairs are
assumed as sticking except those contact pairs inherited from the last time step. For a
sticking contact pair, a normal spring is applied to push the vertex away from the
entrance line in the normal direction and another shear spring is applied to avoid the
tangential displacement between the vertex and the entrance line.
1P 2P 3P 2P
3P 1P
4P
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To calculate the penetration, an entrance distance nd , as shown in Fig. 3.9, is
introduced, which is defined as the distance from the vertex 1P of the angle to the
entrance line 32 PP at the end of the step. Contacts are pre-determined according to the
entrance distance. On this basis, the stiff springs are applied to push the vertex 1P away
from32 PP .
Assume 1P is the vertex, 32 PP is the entrance line and ),( qq yx and ),( qq vu are the
coordinates and displacement of )3,2,1( qPq, respectively; point 1P is in element i, and
points 2P , 3P are in element j, stiffness of the normal and shear springs are
nk and sk . If
three points 1P , 2P , 3P rotate anticlockwise, then the normal distance
nd from vertex 1P
to 32 PP is
3333
2222
1111
1
1
11
vyux
vyux
vyux
ldn
(3.11)
where l is the distance between points 2P and 3P .
Figure 3.9 Entrance distance nd between a vertex and its entrance line.
Since the contact distance nd is such small, the second-order infinite small terms in
Eq. (3.7) can be omitted. The equation can be re-expressed as
l
SDGDHd j
Ti
Tn
0)()( (3.12)
where
1P
2P 3P 0P nd
i
j
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ANALYSIS OF ROCK SLOPE
3.60
33
22
11
0
1
1
1
yx
yx
yx
S (3.13)
23
3211)( ),(
1
xx
yyyxT
lH T
i (3.14)
12
2133)(
31
1322)( ),(
1),(
1
xx
yyyxT
lxx
yyyxT
lG T
jT
j (3.15)
The potential energy due to the normal spring stiffness nk can be expressed as
2
0)(
0)(
0
)()()()()()(
2
0)()(
2
22
22
2
2
l
SGD
l
SHD
l
S
DGHDDGGDDHHDk
l
SDGDH
k
dk
Tj
Ti
jTT
ijTT
jiTT
in
jT
iTn
nn
n
(3.16)
Thus, the sub-matrices due to the normal spring can be obtained and assembled
into global stiffness matrix and global loading vector as
3,2,1,
3,2,1,
3,2,1,
3,2,1,
)()()()(
)()()()(
)()()()(
)()()()(
srKGGk
srKHGk
srKGHk
srKHHk
sjrjT
sjrjn
sirjT
sirjn
sjriT
sjrin
siriT
sirin
(3.17)
3,2,1
3,2,1
)()(0
)()(0
rFGl
Sk
rFHl
Sk
rjrjn
ririn (3.18)
The point ),( 000 yxP is the projection point of vertex 1P on entrance line 32 PP , which
is also the hypothetical contact points, with its coordinates given as
30200
30200
)1(
)1(
ytyty
xtxtx
(3.19)
The shear displacement of point 1P within the current step can be expressed as
2233
2233001100113210 )(
11
vyvy
uxuxvyvyuxux
lPPPP
lds (3.20)
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3.61
Omitting the second-order infinite small terms, Eq. (3.20) can be rewritten as
l
SDGDHd j
T
i
T
s0
)()(
~~~
(3.21)
where
23
2301010
~yy
xxyyxxS (3.22)
23
2311)( ),(
1~yy
xxyxT
lH T
i
30201
3020133)(
30201
3020122)(
32
3200)(
2)12(
2)12(),(
1
)21()1(2
)21()1(2),(
1
),(1~
ytyty
xtxtxyxT
lytyty
xtxtxyxT
l
yy
xxyxT
lG
Tj
Tj
Tj (3.23)
The potential energy induced by the shear spring sk can be obtained as
2
0)(
0)(
0
)()()()()()(
2
0)()(
2
~~
~2
~~
2
~~2
~~~~2
~~~
2
2
l
SGD
l
SHD
l
S
DGHDDGGDDHHDk
l
SDGDH
k
dk
Tj
Ti
j
TTij
TTji
TTi
s
j
T
i
Ts
ss
n
(3.24)
Then, the sub-matrices due to the shear spring are gained and assembled to the
global stiffness matrix and global loading vector as
3,2,1,
~~
3,2,1,~~
3,2,1,~~
3,2,1,~~
)()()()(
)()()()(
)()()()(
)()()()(
srKGGk
srKHGk
srKGHk
srKHHk
sjrj
T
sjrjs
sirj
T
sirjs
sjri
T
sjris
siri
T
siris
(3.25)
3,2,1~
~
3,2,1~
~
)()(0
)()(0
rFGl
Sk
rFHl
Sk
rjrjs
riris (3.26)
For the sliding case, besides the normal spring, a pair of frictional forces instead of
a shear spring will be implemented. Based on the Coulomb’s friction law, the frictional
force can be calculated as
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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3.62
tan nn dksignF (3.27)
where ‘sign’ is assigned as ‘+’ or ‘-’ according to the direction of relative sliding, nndk
is the normal force and is the friction angle, respectively.
Then, the potential energy due to frictional force F on the element i is
HDFyy
xxvu
l
F Tif
ˆ)(
23
2311
(3.28)
where
23
2311)( ),(
1ˆyy
xxyxT
lH T
i (3.29)
The potential energy due to frictional force F on the element j is
GDFyy
xxvu
l
F Tjf
ˆ)(
23
2300
(3.30)
where
23
2300)( ),(
1ˆyy
xxyxT
lG T
j (3.31)
Then, the sub-matrix due to the frictional force are obtained and assembled to the
global loading vector as
3,2,1ˆ
3,2,1ˆ
)()(
)()(
sFGF
rFHF
sjsj
riri (3.32)
After the load increment of the current time step is applied, the equilibrium
equation is solved and the state quantities such as displacements, stresses, etc, are
obtained.
3.3 INTEGRATION SCHEMES IN THE NMM
In terms of spatial aspect in the NMM, MCs are usually formed by a regularly-
patterned triangular or rectangular finite element mesh, the MEs may have arbitrary
shapes because of the intersection with the external boundaries and/or the internal
discontinuities. Thus, direction integration scheme over the whole element like that in
the conventional FEM is not applicable. Simplex integration method can be adopted to
evaluate the integrations over an arbitrary ME. On the other hand, in the temporal aspect,
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the discrete equations are based on a dynamic analysis. For a static analysis, discrete
equations can be given by removing the inertia item from the governing equation. In the
NMM, the Newmark time integration scheme is used to solve the governing equation by
single time step.
3.3.1 Simplex integration
In both the ph cloud method and the meshless method, the integration is carried
out using Gauss quadrature. In the NMM, integrations are evaluated analytically. The
domain of integration is represented as a simplicial complex via a simplicial chain
regardless of the convexity of the domain. Analytical integration maybe carried out
using the generalized Stoke’s equation (Flanders 2012). Shi (1996b), on the other hand,
suggests direct integration over each simplex in the chain, because many functions, in
particular polynomials, can be integrated analytically on a simplex. With a coordinate
transformation, analytical results can be obtained for integration over an arbitrary n-
dimensional Euclidean space.
This scheme is explained with an 2R example. A physical domain ),,,,( mlkji and a
triangulation are shown in Fig. 3.10. For both, the oriented boundaries are considered of
the same sequence ordered edges: ),(),(),(),(),( immllkkjji . A boundary preserved
triangulation can easily be achieved by connecting each pair to one single point. The
coordinate origin, )0,0(o is a desirable choice of the three ordered vertices is a two-
simplex by Li et al. (2005) .
With a simplex chain representation of a ME e , the integration becomes a sum of
simplex integration as follows
iS ie
e
Eie
dKdKdKA (3.33)
Each of the integration on a simplex can be evaluated analytically. This is carried
out in two steps. First, an integration in terms of area coordinates, 1L , 2L and 3L , over a
coordinate simplex, i.e., simplex with vertices )0,0(0U , )0,1(1U , )1,0(2U , can be evaluated
analytically. Namely,
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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)!2(
!!!
210
21021210
21
210
0
nnn
nnnLdLLLL nn
UUU
n (3.34)
Second, a general integration in terms of x and y over a simples with vertices
),( 00 yxo , ),( 11 yxi and ),( 22 yxj is evaluated by the coordinate transformation
221100
221100
2101
LyLyLyy
LxLxLxx
LLL (3.35)
Such that
21221100221100 )()()( LdLLyLyLyLxLxLxJsigndxdyyx ba
oij
ba (3.36)
where, sign(J) is a signal Jacobian. The more details on the simplex integration method
can be found in Shi (1996b).
Figure 3.10 Triangulate an element oij using coherent orientation.
3.3.2 Time integration
In order to investigate the time integration in the NMM, the behaviour of an
equivalent linear system is considered. In such a system, the minimization of the system
potential energy will produce an equation of motion, which is similar to that in the
FEM.
When a single step integration method is used to solve Eq. (2.1) for given initial
conditions of 00 dD and
00 VD based on Newmark method as expressed in sub-section
2.5.1 , taking the parameters of 2/1 and 1 , Eq. (2.1) can be simplified as
j
i
k
m
l
o
i
j
o
o - Coordinate origin
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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FDK
(3.37)
where nn DDD 1,
Kt
MK 2
2 and
nn D
t
MFF
21
, respectively.
3.4 SUMMARY
This chapter presents a brief review on the fundamentals of the NMM, and
emphatically introduces the finite cover system and cover-based contact algorithm,
including the constructions of the MC, PC and ME using the finite element mesh, the
builds of local and global approximation for the NMM, and the derivations of the
contact sub-matrices based on the finite cover system.
Then, the integration schemes are reviewed with respect to spatial and temporal
aspects in this chapter to build up the foundation of the thesis work. The simplex
integration scheme in terms of the spatial aspect is illustrated firstly, and the Newmark
integration scheme is brief introduced and derived in detail to further deepen the
understanding the NMM and its implementations.
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CHAPTER 4. AN EXPLICIT TIME INTEGRATION
SCHEME FOR THE NUMERICAL MANIFOLD METHOD
4.1 INTRODUCTION
Discrete element methods have the advantage in simulating discrete block systems
under different actions, which has been widely used applied in rock engineering. There
are two major representatives in the discrete element method family. The distinct
element method (DEM hereafter) adopted an explicit time integration scheme based on
finite difference principles (Cundall 1971a, 1971b). The discontinuous deformation
analysis (DDA) derived based on the variational method takes the benefit of the implicit
time integration method (Shi and Goodman 1985; Shi 1988). Since the initiation of the
DDA, many developments and applications have been implemented by Ohnishi et al.
(1995), Hatzor et al. (2004) and Zhao et al. (2011), etc.
The numerical manifold method (NMM) involved in this chapter is an evolvement
of the DDA, which combines the merits of FEM and DDA. The NMM inherits all the
attractive features of the DDA, such as the implicit time integration scheme, the contact
algorithm and the minimum potential energy principle (Chen et al. 1998). It adopts a
dual cover system, i.e. a mathematical cover system overlapping the domain of interest
and a physical cover system, which considers the contained discontinuities, such as
material joints, voids, interfaces and aggregates, in a united manner. In the past two
decades, many efforts have been carried out to improve the performance of the NMM,
which are stress intensity factors (SIFs) problems (Ma et al. 2010; Zhang et al. 2010),
crack propagation problems (Tsay et al. 1999; Zhang et al. 2010), high order NMM
theory (Chen et al. 1998; Lin et al. 2005), and extensions of the NMM (Terada et al.
2003; Miki et al. 2010), etc. A review article on the recent development of the NMM
has been published by Ma et al. (2010) and An et al. (2011). It has been reorganized that
the NMM has great potential to be further developed in simulating a medium with
massive discontinuities.
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In this chapter, a modified version of the NMM based on an explicit time
integration algorithm is derived. The original NMM based on displacement method is
revised into an explicit formulation of a force method. The governing equations are built
up on the dual cover system and the global stiffness matrices used in the traditional
NMM are no longer necessary. A diagonal mass matrix is derived for the dual cover
system which makes the solution highly efficient at each time step. The OCI is still
employed, however, the relative cost is much lower because of the explicit time
integration scheme without solving simultaneous algebraic equations in each step and
the smaller penetration incurred due to a smaller time step used. The developed method
is validated by two examples, one static problem of a continuous simply supported
beam, and one dynamic problem of a single block sliding down on a slope. Results
showed that the accuracy of the explicit numerical manifold method (ENMM) can be
ensured when the time step is small for both the continuous and the contact problems. A
highly fractured rock slope is subsequently simulated. It is shown that the computational
efficiency of the proposed ENMM can be significantly improved, while without losing
the accuracy, comparing to the implicit version of the NMM. The ENMM is more
suitable for large-scale rock mass stability analysis and it deserves to be further
developed for engineering computations of practical rock engineering problems.
4.2 BRIEF DESCRPTIONS OF THE NMM
The traditional NMM is based on the dual cover system, which consists of
mathematical covers (MCs), physical covers (PCs) and manifold elements (MEs). The
MCs are user-defined small patches, and their union covers the entire problem domain.
The PCs are the subdivision of the MCs by the physical features such as the external
boundaries and the internal discontinuities, and each PC inherits the partition of unity
function from its associated MC. The ME is defined as the common region of several
PCs. On each ME, partition of unity functions is used as well to assemble all the local
functions associated the PCs to offer a global approximation.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
4.68
For a two-dimensional problem, a regularly structured mesh is employed in the
NMM to form the cover system, which is similar as that in the FEM. More details of the
descriptions of the NMM can be referred in Section 3.2 of the previous chapter.
4.3 EXPLICIT TIME INTEGRATION FOR THE NMM
In order to investigate the time integration in the NMM, the behaviour of an
equivalent linear system is considered. In such a system, the minimization of the system
potential energy will produce an equation of motion, which is similar to that in the
FEM. Let displacement term of nD and 1nD denote the approximation to the
displacements )(tD and )( ttD for a time step t , the discrete equation of motion can
be expressed as
1111 nnnn FDKDCDM (4.1)
The component form of the mass M , damping C and stiffness K terms are
extensively discussed in (Shi 1991). A single step integration method can be used to
solve Eq. (4.1) for given initial conditions of 00 dD and 00 VD based on
Newmark method (described in Section 3.3). And a variety of well-known members of
the Newmark family methods are developed with reference in (Hughes 1983). In the
NMM, the implicit scheme is carried out by minimizing the potential energy associated
with an increment of time t , which can be represented as:
FDK ˆˆ (4.2)
where nn DDD 1,
Kt
MK 2
2ˆ and
nn D
t
MFF 2ˆ
1.
Since the NMM uses an implicit time integration scheme which provides numerical
damping, the explicit damping term C is assumed to be zero in Eq. (4.1). Eq. (4.2)
requires assembling the global stiffness matrix and solving the coupled system of
equations using successive over relaxation iteration method.
The DEM uses an explicit scheme for the discontinuous problem, this motivates us
to develop an explicit version for the NMM. When the parameters are 2/1,0 , we
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
4.69
can get an explicit forward central different scheme, which can be modified using the
Verlet algorithm, where the velocity is calculated at each half time step (Langston
1995), it can be expressed as:
2/11
2/11
2/12/1
2
1
nnn
nnn
nnn
DtDD
DtDD
DtDD
(4.3)
Eq. (4.3) is required as the calculation of the damping force depends on the velocity at
the next step )1( n . Then, substituting Eq. (4.3) into Eq. (4.1), we can get
FDM n
~1
(4.4)
where }~
{}~
{}{}~
{ 1 IFFF dn is the force item, in which }]{[}
~{ 1 nd DCF is the damping
item and the direction is such that energy is always dissipated, and }~
{I is the internal
force item assembled in the PC, respectively. When the explicit scheme is used, the
mass matrix on the PC can be diagonalizable as an equivalent lumped matrix ]~
[M , which
is different from that in the implicit scheme of the traditional NMM. Eq. (4.4) is
essentially the Newton’s second law of motion, and it is used as the main equation of
motion in the proposed explicit scheme. In contrast to the implicit scheme in the NMM,
the proposed explicit scheme eliminates the assembly of global stiffness matrix and
inversion of the global matrix, which uncoupling of the equation of motion is one of
major advantage than the implicit scheme. Since ]~
[M is diagonal, the }{ 1nD on the PCs
can be solved explicitly without assembling the global stiffness matrix. Thus, it is more
efficient than the implicit scheme.
4.3.1 Mass matrix
The mass matrix in Eq. (4.4) is traditionally called consistent mass matrix, which
uses the same weight functions as that used for displacement. Given an element e, we
can get a mass consistent matrix as
3,2,1,)()( srdxdyTTM se
T
A ree (4.5)
in which is the element mass density. Since the element generated by the three
associated PCs, eM is allocated to the PCs to form M~ on each PC. A common
procedure to obtain the lumped mass matrix M~ uses the row-sum lumping technique.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
4.70
And many other strategies of the mass matrix diagonalization can be referred in (Wu
and Qiu 2009).
In this chapter, we can gain the M~ based on the displacement matrix ),( yxTe
.
Considering the matrix ),(),( yxTyxT eT
e for the contributions of element e, we have
L
yxwyxwyxwyxwyxw
yxwyxwyxwyxwyxw
yxwyxwyxwyxwyxw
yxwyxwyxwyxwyxw
yxwyxwyxwyxwyxw
yxwyxwyxwyxwyxw
yxTyxTyxT
yxT
yxT
yxT
TT
eeeee
eeeee
eeeee
eeeee
eeeee
eeeee
eeeT
e
Te
Te
eT
e
2)3()3()2()3()1(
2)3()3()2()3()1(
)3()2(2
)2()2()1(
)3()2(2
)2()2()1(
)3()1()2()1(2
)1(
)3()1()2()1(2
)1(
)3()2()1(
)3(
)2(
)1(
),(0),(),(0),(),(0
0),(0),(),(0),(),(
),(),(0),(0),(),(0
0),(),(0),(0),(),(
),(),(0),(),(0),(0
0),(),(0),(),(0),(
)],([)],([)],([
)],([
)],([
)],([
(4.6)
Then, the lumped mass matrix M~ can be obtained using the row-sum lumping
technique. Decompose matrix L , we can get
2,1),,()1(
6
1
iyxwL ej
ij (4.7)
4,3),,()2(
6
1
iyxwL ej
ij (4.8)
6,5,),(6
1)3(
iyxwLj
eij (4.9)
Eq. (4.7) maps the first PC associated with the element, the subsequent Eqs. (4.8) and
(4.9) map the second and third PCs, respectively. Thus, the allocated mass at the each
PC can be expressed as
3,2,1),(~
)()( rdxdyyxTMA rere (4.10)
Since the element shape in the NMM is generally polygon, )(
~reM can be calculated
using the analytical method such as simplex integration method. Then, the lumped
matrix M~ at each PC can be assembled by the associated )(
~reM on the cover system. We
can find that when all the PCs use the lumped mass matrices, the globally assembly
mass matrix is diagonal as well.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
4.71
4.3.2 Internal force
In this chapter, internal force item I~ in Eq. (4.4) can be formulated by element
stiffness eK and contact element stiffness cK . To the continuous problem, each element
internal force eI~ is generated by the eK , which satisfies the equation of motion as
}{}~
{}]{[}]{[}]{[ 11111e
nnne
nne ffDKDCDM (4.11)
in which }~
{ 1nf is a unknown force vector from the neighbour elements and }{ 1e
nf is the
external force vector but the neighbour elements, respectively. When the whole system
is assembled into a global lumped mass matrix on the cover system, the item of }~
{ 1nf
can be offset, which satisfies
e
enn
en
fF
f
}{}{
0}~
{
11
1 (4.12)
Thus, only ][ eK is considered. Since the mass matrix is lumped, ][ eM of each element
can be uncoupled to calculate }{ 1nD explicitly. The internal force }~
{ eI in the element can
be expressed as
}]{[}~
{ 1 ne
e DKI (4.13)
On the other hand, to discontinuous problems, as the PCs associated with the contact
elements have no overlap with each other, the contact elements are taken into account to
form contact matrix ][ cK . In each contact pair, a linear stiff spring is added to generate
force item and cause the stiffness change of the contact elements. To the contact
element, }~
{ 1nf can not be neglected and is determined by the contact matrix, which
satisfies
en
e
enn
en
ffF
f
}~
{}{}{
0}~
{
111
1 (4.14)
Then, the internal force }~
{ eI in the element can be rewritten as
}]{[}]{[}~
{ 11 nc
ne
e DKDKI (4.15)
For a discrete block system involving m elements, there are N contact pairs have
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
4.72
been detected to an element i ( mi ,2,1 ). Here, we assume one element denoted by j (
Nj ,2,1 ) and i are detected in contact state, ][ cK between i and j can be expressed as
][][
][][cjj
cji
cij
ciic
KK
KKK (4.16)
in which ][ cijK , ji and mji ,2,1, , is defined by the contact spring between the
contact elements i and j, and the value is zero if the elements i and j have no contact.
Since each element is consisted by three associated PCs, thus the matrix ][ cijK is a 6×6
sub-matrix. It is noted that the displacement 3,2,1,)( rD ri on the PCs can be predicted by
the previous step n. Then contact forces associated with ][ cijK on the contact element i
are assembled as
NjDKIN
jn
cij
ci ,,2,1,][
11
(4.17)
The total internal forces on the element i can be represented as
NjDKKIDKI n
N
j
cij
eii
cin
eiii ,,2,1},{][][}{}{][}
~{ 1
11
(4.18)
in which iI~ is the element internal force vectors and e
iiK is the stiffness matrix of
element. Since each element is formed by the three associated PCs, thus iI~ can be
rewritten as
)3(
)2(
)1(
~
~
~
~
i
i
i
i
I
I
I
I (4.19)
in which )1(
~iI maps the first PC associated the element, the subsequent )2(
~iI and )3(
~iI
map the second and third PCs, respectively. Then, I~ at each PC can be assembled by
the associated iI~ on the cover system.
4.3.3 Damping algorithm
Damping algorithm is used to dissipate the excessive energy in the contact
problems due to the use of linear springs between contact elements. As referred in
(Cundall 1982), we suggest an alternative scheme to simulate the damping, in which the
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
4.73
damping force with the inertial force is in direction proportion. The total potential
energy from the damping item is summered in an element e, which can be written as
3,2,1,}]{~
[}{ )()()( rDMDd
erere
Tre
(4.20)
where is damping ratio. Substituting Eq. (4.16) using the variational principle, the
equivalent damping force assembled on each PC can be expressed as
}]{~
[~
1 nd DMF (4.21)
which is then added to Eq. (4.4) to form the matrix item F~ . To the discrete systems,
there are usually some domains in stable conditions and the others in motion statuses, in
which the different damping force can be considered in the different domains as an
appropriate method. The damping force described in Eq. (4.17) is proportional to the
inertial force and varied in the whole system, thus it is more adaptive even the system
approached the stable conditions.
In terms of the numerical stability, since the applied explicit time integration as
same as the DEM (Cundall 1971a) is conditionally stability, the size of the selected
time-step t is usually smaller than that of the implicit scheme. When the damping item
is taken into account in the system motions, smaller damping ratio of the PCs system
corresponds to larger time-step size t . Here, we propose
max
2
t (4.22)
where λ ∈ 0,1 is a coefficient associated with damping ratio ; is the highest
eigenfrequency of the system. For no damping case, the value of λ can be considered as
1. More details of discussions about the selection of t can be referred in (Cundall 1982;
Bath 1982; Hughes 1983). It is noted that the explicit scheme employs dynamics
method to solve the uncoupled equations, in which the generated kinetic energy can not
be neglected. It is noted that the time-step t for dynamic problems depends not only on
damping ratio and , but contact stiffness, mesh resolution, deformation stiffness
of rock mass.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
4.74
4.4 CONTACT FORCE IN THE ENMM
4.4.1 Contact force calculation approach
In general, there are two different types of techniques to treat the contact forces
acting on the contact elements in the numerical methods, as shown in Fig. 4.1: (i) point
contact forces or (ii) area contact forces. The point approach usually assumes vertex
contact force is a function of penetration of an individual contact element vertex into
another contact element, while area method is evaluated from the shape and area of the
overlap between two contact elements. A thorough discussion and formulations of these
approaches can be found in (Munjiza 2004).
(a) Point contact force approach; (b) Area contact force approach.
Figure 4.1 Two distinct contact force approaches: (a) point approach; (b) area approach.
In the present paper, we apply the point contact force approach combining the
contact point and entry area to simulate the contact interaction problems and calculate
the contact forces. As shown in Fig. 4.2, two different strategies of normal penetration
method and direct penetration method can be employed to determine the penetration
distance d. Firstly, the entry area n can be determine by the penetration vertex
),( iii yxP and entry element vertices ),( jjj yxP and ),( kkk yxP , which is represented as
kk
jj
ii
n
yx
yx
yx
1
1
1
2
1 (4.23)
The penetration distance d in the normal penetration method can be determined as
ld n
2 (4.24)
j
i
ifjf
i jj
i
);,( jiNji
i1i
f
3if
1jf
3jf
2jf
ajf
j
;,( Nba );, jiNji
2if
bif
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
4.75
in which 22 )()( kjkj yyxxl is the entry element boundary length. On the other
hand, the direct penetration method calculates d directly as
22 )()( nini yyxxd (4.25)
in which ),( nnn yxP is assumed as contact point between the contact elements. In the
present study, the direct penetration method is employed into the calculation of contact
force in the ENMM.
(a) Normal penetration method; (b) Direct penetration method.
Figure 4.2 Two schemes for contact problem: (a) Normal penetration method; (b) Direct
penetration method.
4.4.2 Calculation of contact force
As the explicit equations are not coupled, contact force is calculated explicitly. Fig.
4.3 describes the proposed contact model for the ENMM. The proposed contact force
approach treats contact problem explicitly by adding and subtracting the normal and
shear spring when contact is detected. Here we choose the contact block pair of block I
and J with the contact vertex of P1 to study the proposed contact algorithm in the present
paper. The contact elements i and j , penetration vertex of P1 i can be found by the
associated with partition of domain qU (i.e. q is the index number in the programming
code) as can be seen in Fig. 4.3(a). Then the contact pair is constructed between contact
element i and j , in which the contact point P1 i , the contact position is represented by
P0, and entry line P2 P3 (i.e. (P0, P2, P3) j ) are presented in Fig. 4.3(b). The penetration
distance of element i can be presented as d, which can be expressed as two components
of nd andsd satisfying equation of 22
sn ddd . The penetration angle can be expressed
d
Pn
Pi
Pk Pj
Entry area ∆n
Pn
Pi
Pk Pj
Entry area ∆n
d
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
4.76
as ddn /sin or dd s /cos .The penetration distance d between contact elements can be
written as
20011
20011 )()( vyvyuxuxd (4.26)
where ),( 11 yx , ),( 11 vu and ),( 00 yx , ),( 00 vu are the coordinates and displacement increments
of contact point P1 and penetration point P0 in one step time. Once the contact elements
in the loops are detected, the normal and shear stiffness springs and damper are applied
at each penetration point. The stiffness of contact springs and damper can be expressed
as nk ,
sk and nc , sc respectively. The strain energy stored in the contact springs c is
minimized with respect to the displacement variable vector ][ D and added to the contact
element stiffness sub-matrix to produce internal force vector. Then, the contact block
displacement is expressed as
DyxTyxv
yxu),(
),(
),(
(4.27)
where ),( yxT is displacement matrix and D is displacement vectors on the PC system
respectively. The strain energy of stiffness springs can be written as
222 )cossin(2
1dkk snC (4.28)
Substituting d in Eq. (4.26), Eq. (4.28) can be rewritten as
))(
][}{2][}{2}]{[][}{
}]{[][}{2}]{[][})({cossin(2
1
01
010101
01
01
01
01
22
yy
xxyyxx
yy
xxTD
yy
xxTDDTTD
DTTDDTTDkk
Tj
Tj
Ti
Tijj
Tj
Tj
jjT
iT
iiiT
iT
isnC (4.29)
The contribution of the contact normal and shear stiffness to the equilibrium
coefficient matrix ck can be given by
)][}{2}]{[][}{2
}]{[][}({)cossin(2
1
01
01
222
2
yy
xxTDDTTD
DTTDdd
kkdd
k
Ti
Tijj
Ti
Ti
iiT
iT
iirit
snirit
ccii , 6,2,1, rt (4.30)
forms a 66 sub-matrix
][])[])([cossin( 22 ciii
Tisn KTTkk (4.31)
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
4.77
which is then added to the sub-matrix ][ ciiK in Eq. (4.16).
)][}{2}]{[][}{2
}]{[][}({)cossin(2
1
01
01
222
2
yy
xxTDDTTD
DTTDdd
kkdd
k
Tj
Tjjj
Ti
Ti
jjT
jT
jirit
snjrjt
ccjj
, 6,2,1, rt (4.32)
forms a 66 sub-matrix
][])[])([cossin( 22 cjjj
Tjsn KTTkk (4.33)
which is then added to the sub-matrix ][ cjjK in Eq. (4.16).
})]{[][}{2()cossin(2
1 222
2
jjT
iT
ijrit
snjrit
ccij DTTD
ddkk
ddk
, 6,2,1, rt (4.34)
forms a 22 sub-matrix
][])[])([cossin( 22 cijj
Tisn KTTkk (4.35)
which is then added to the sub-matrix ][ cijK in Eq. (4.16).
})]{[][}{2()cossin(2
1 222
2
jjT
iT
iirjt
snirjt
ccji DTTD
ddkk
ddk
, 6,2,1, rt (4.36)
forms a 66 sub-matrix
][])[])([cossin( 22 cjii
Tjsn KTTkk (4.37)
which is then added to the sub-matrix ][ cjiK in Eq. (4.16). And the equivalent force
matrix can be described as
)][}{2()cossin(2
1)0(
01
0122
yy
xxTDd
kkd
F Ti
Ti
irsn
ir
ci
, 6,2,1, rt (4.38)
forms a 16 sub-matrix
iTisn Fyy
xxTkk
)])([cossin(
01
0122 (4.39)
which is then added to the force item F in Eq. (4.4).
)][}{2()cossin(2
1)0(
01
0122
yy
xxTDd
kkd
F Tj
Tj
jtsn
jt
cj , 6,2,1, rt (4.40)
forms a 16 sub-matrix
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
4.78
jT
jsn Fyy
xxTkk
)])([cossin(01
0122 (4.41)
which is then added to the force item F in Eq. (4.4).
Figure 4.3 The proposed contact model: (a) contact elements; (b) contact points.
4.5 OPEN-CLOSE ALGORITHM IN THE ENMM
The NMM uses a penalty-constrain approach to treat the contact problems in which
the contact is assumed to be rigid. When two elements boundaries overlap, an
impenetrability constraint is employed by applying a numerical penalty function
analogous to stiff springs at the contact points with the direction of the penetrating
vertex to arrest interpenetration. Numerically, it can be carried out by means of adding
or subtracting penalty springs values to the contact elements to produce contact stiffness
and contact forces, then assembling them into the coupled global equation (i.e. equation
in the dotted box in Fig. 4.4) to get the global displacements of the whole cover system.
Assuming n contact pairs have been detected, the terms are selected by checking the
previous and current statuses of the contact, each controlled by a vector of values i =
-1, 0 and 1, ni ,,2,1 . Accordingly, within each time step, the assembled global
equations are solved iteratively by repeatedly adding and removing contact springs until
each of the contacts converge to a constant state, which is known as OCIs proposed by
Shi (1991).
Here, we use the well-known open-close algorithm as well to obtain contact
convergences in the ENMM. As we can see in Fig. 4.4, from the term of initial contact
condition to the end of contact computation in each step, the vital process concentrates
a.
Block J
Block I qU
i
j
b.Contact element i
dn
ds
d
P0
P1
P2 P3
Contact element j
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
4.79
on achieving the contact statuses identically as the merged in the dotted box. In the
implement, the contact statues don’t stop change until that is constant in one step as
presented with blue diamond shadow: open-open, lock-lock and slide-slide, then the
open-close processor completes in this time step. In the OCI, the no-penetration and no-
tension contact constraint is imposed at each detected contact point. To satisfy this
constraint, the open-close iterative solver proceeds until this is no penetration at any
contact point. Number of OCIs (i.e. OCIN ) is required for contact convergence in each
time step. When it is over the set value of limN (i.e. value of 6 in the code) , the time-step
will decrease by 02 t even
03 t , in which 2 is prescribed as one third and 3 is
the reciprocal of the maximum displacement ratio (which represents the tolerance
criteria of maximum penetration distance nd is less than the prescribed allowable value
0d ) in the OCIs. The value OCIN dramatically increases for problems involving more
contact elements, which increases the computational cost as well. Since the small time-
step is used in the ENMM, the large penetration between the contact elements can be
avoided to achieve contact convergence. Thus, the unnecessary OCIs procedures can be
removed to improve efficiency. Further more, the explicit equations are uncoupled
without assembling the global stiffness as in the NMM, the ENMM is more efficient in
terms of solving equations even the OCIs. An alternative scheme for simplifying open-
close algorithm in the ENMM can be considered, in which only the maximum
displacement ratio 3 (i.e. 23 in the code) is regarded as the judgement criteria to
obtain contact convergences. Since the size of t used in the ENMM is very small,
contact convergence can be obtained within one time of iteration. Then the penetrations
are beyond the prescribed displacement ratio, t will decrease by 03 t directly. The
OCI runs until the system penetration is less than the prescribed ratio. As the small
time-step is employed, the accuracy of contact normally can be satisfied in the OCIs as
well. On the other hand, the efficiency is improved dramatically.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
4.80
Figure 4.4 Flowchart of the OCI in the ENMM.
4.6 NUMERICAL SIMULATIONS
In order to further investigate the proposed explicit scheme for the NMM, two
calibration examples are simulated firstly, in which one static problem of a continuous
simple-supported beam and dynamic one of a single block sliding along on a slope are
validated using the ENMM, respectively. Then a discrete rock slope modelling is
studied, in which the proposed ENMM using the open-close algorithm is investigated
and compared with the NMM in terms of computational accuracy and efficiency. In
Previous status Open Lock Slide
Open-open Lock-lock Slide-slide
Current status Open Lock Slide
Contact check
Tolerance criteria
0ddn
no
yes
Initial contact condition Contact detection Contact transfer
Set ntt ;
0tt
START
Step(n)
Calculate contact stiffnesses and contact forces
Solve explicit equation
Solve implicit equation
0OCIN
yes 1 OCIOCI NN
limNNOCI
no
Next Step
limNNOCI
Storage and update geometry and physical data
ntt no
yes
ntt ;03 tt
ntt ;02 tt
Resume contact status and other variables
0tt
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
4.81
order to track CPU time for each time step, a high resolution timer function which can
measure up to 1/100000th of a second is added to keep track of CPU time for each time
step. All analyses are run on the same computer with the system configuration:
processor speed = 3.17 GHz and RAM = 4.0 GB.
4.6.1 Simply-supported beam subjected a concentrated load
This simulation studies a simply supported beam subjected to a concentrated load
at its center. As shown in Fig. 4.5, the dimension of the beam is 10 m long, 1 m deep
and 1m wide. The point loading is P = 200 N. The material of the beam is assumed
elastic without damage with properties of E = 1×105 Pa and ν= 0.24. There are totally 11
measure points shown in Fig.4.5.
Figure 4.5 Geometry of the simply supported beam bending problem.
The theoretical solutions for simply supported beam under central point loading
can be expressed as (Chen et al. 1998):
)2/0(1612
)(2
3 lxxEI
Plx
EI
Pxv (4.42)
where E is the Young’s modulus, I is the moment of inertia and l is the length of the
beam. The simulated results of the proposed ENMM, NMM and the theoretical
solutions are compared in Fig. 4.6. It can be found that the numerical result converges
to the theoretical solutions using the proposed ENMM.
5 m 5 mP=200 N
x
y
1 m
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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4.82
Figure 4.6 Comparisons between simulated results and theoretical solution.
4.6.2 Numerical simulation of plane stress field problem
To verify the accuracy of the proposed explicit NMM code in accurately describing
the displacement field and stress field with the refinement of the mathematical covers.
An infinite plate with a traction free circular hole subjected to a unidirectional tension is
considered here. The geometry of model is presented in Fig. 4.7(a), in which the radius
of the hole is prescribed by 0.1m, 8 prescribed measuring points distant the top of
circular hole in vertical direction 0.05m, 0.1m, 0.2m, 0.3m, 0.4m, 0.5m, 0.6m, 0.7m and
0.8m respectively. The material properties of the model, i.e., Mass density, Young’s
modulus and Possion’s ratio, are assumed to be =2000 kg/m3, E =25 GPa and
=0.25 accordingly. The far-field uniaxial tension is prescribed as 4KPa. The manifold
element mesh topology representing the problem is shown in Fig. 4.7(b). All elements
consist of divided regular-patterned triangular mathematical mesh and physical
boundary. A refinement scheme of the mesh is employed by gradually refining the mesh
from level of coarseness to fineness in this study.
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0 1 2 3 4 5 6 7 8 9 10x (m)
Dis
plem
ent (
m) Analytical
NMMENMM
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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Figure 4.7 Numerical model for an infinite plate with a traction free circular hole: (a) Geometry of
model; (b) NMM meshing.
The theorical solution of stress in x direction for the model can be expressed as:
4
2
342
2
31
4
4
2
2
cos)coscos(r
d
r
dTx
(4.43)
4
2
342
2
14
4
2
2
cos)coscos(r
d
r
dTy
(4.44)
where T is the uniaxial tensile stress applied to the plate, d is the radius of the circular
hole, r and are polar coordinates associated with its origin located at the center of
the hole.
The refined element mesh gives satisfactory results for the present analysis. Fig.4.8
gives a plot of computed x versus the analytical solution and the NMM. A good
agreement has been observed.
Measuring points (a) Geometry of model;
y
xTT
(b) NMM meshing.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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Figure 4.8 Comparison of numerical results and analytical solution for infinite plate problem.
4.6.3 Single block sliding along the inclined surface
In this simulation, one single block sliding along the inclined surface is studied
here using the implicit and explicit NMM respectively. In the numerical model, one
square block with the size of 1 m×1 m locates on the top of the inclined surface, the
bottom triangle block with the size of 5 m×8.66 m is fixed by the fixed point. The angle
of the inclined surface is 300.
In theory, one block slides along the inclined surface by itself gravity. The
displacement along the inclined surface by time history can be derived based on
Newton’s second law as
2)tancos(sin
2
10
tgd g
(4.45)
where dg is the displacement of the block, g is the acceleration of gravity, is the angle
of inclined surface, is the friction angle, t is the time. The case with friction angle of
5° is simulated using the NMM and proposed ENMM. The displacement time history is
compared with the analytical solution as shown in Fig. 4.9, and good agreements have
been found. In the simulation, the two blocks are divided into 292 MEs to calibrate the
proposed ENMM. To obtain the computational accuracy of the proposed ENMM, a
damping ratio value of 0.001 is used to simulate the damping item. Comparing to the
residual error of -1.2883% using the NMM, the explicit NMM has approximate
1.6391% to the analytical solution. This calibration example shows that the proposed
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
5 10 15 20 25 30 35 40 45 50
σ x/ K
Pa
r /m
Analytical
NMM
ENMM
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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ENMM can be accepted with respect to the computational accuracy. Furthermore, the
CPU time by the ENMM and NMM methods taken are 192.05s and 293.46s as the
larger step-time of 2ms is employed and the scale of the model is not large.
Figure 4.9 Comparison of simulated results and analytical solution.
4.6.4 Highly fractured rock slope stability analysis
In this simulation, one highly fractured rock slope modeling is taken into account
as shown in Fig. 4.10. 6 measure points are prescribed to investigate the computational
accuracy and efficiency of the proposed ENMM compared with that of the NMM. The
modeling is divided into 390 discontinuous blocks including 627 elements in total. In
the present study, selections of t are taken as 2ms, 1ms and 0.1ms in the NMM, 0.1ms
in the proposed ENMM, respectively. The input parameters for the simulation of the
rock slope can be seen in Table 4.1.
Figure 4.10 Geometry of the slope modelling.
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.1 0.2 0.3 0.4 0.5Time (s)
Dis
plac
emen
t (m
) Analytical NMM
ENMM
110m
20m
80m Measured point 5
Measured point 1, 2, 3
Measured point 4
Measured point 6
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Table 4.1 Input parameters for the NMM simulation of rock slope.
Physical property Parameter
Unit weight (kN/ m3) 26.0
Young’s modulus (GPa) 1.0
Possion’s ratio 0.2
Internal friction angle(deg.) 20.0
Cohesion (MPa) 5.0
Tensile strength (MPa) 0.0
NMM
Joint normal stiffness (GPa) 1.0
ENMM
Joint normal stiffness (GPa) 0.5
Joint shear stiffness (GPa) 0.25
In order to represent the discontinuity of the rock slope, the cut blocks are
displayed explicitly, and the corresponding subdivided elements in the NMM and
ENMM are concealed, which can be seen in Fig. 4.11. The total time of one second is
prescribed to simulate the rock slope stability, the different selections of t result in
different computational steps and CPU time, which are shown that when t =2ms is
applied in the NMM, the code runs 650 steps to reach 1 second with 1.28 hours. As the
decline of the step-time applied, both computational steps and CPU time increase from
1090 steps with 1.83 hours to 10000 steps with 5.41 hours, respectively (see Fig. 4.11
(b) and (c)). Alternatively, the proposed ENMM gains a well agreement with the
simulation results of the NMM in terms of computational accuracy, and the most
importance is the CPU time of the ENMM code falls to 0.28 hour dramatically, which is
shown in Fig. 4.11(d).
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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(a) NMM, ∆t=2ms (1.28 hrs);
(b) NMM, ∆t=1ms (1.83 hrs);
(c) NMM, ∆t=0.1ms (5.41 hrs);
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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(d) ENMM, ∆t=0.1ms (0.28hrs).
Figure 4.11 Simulation results by NMM and ENMM: (a) NMM, ∆t=2ms; (b) NMM, ∆t=1ms; (c)
NMM, ∆t=0.1ms; (d) ENMM, ∆t=0.1ms.
It is noted that the NMM employs implicit scheme and OCI to treat the contact
problem, which cut down the computational velocity and reduce the computational
efficiency. Fig. 4.12 shows that the real used step-time decreases dramatically as the
increase of t . When large t can not achieve contact convergence, t will be cut down
until which satisfies the OCI convergence. This increases the CPU cost and declines the
computational efficiency. On the other hand, t used in the proposed EMM is stable
and efficient.
Figure 4.12 Real step-time used in NMM vs. ENMM.
In terms of computational accuracy, the horizontal and vertical displacements of
the measure points 1, 3 and 6 are plotted in Figs. 4.13, 4.14 and 4.15, respectively. It is
0
0.001
0.002
0.003
0.004
0 0.2 0.4 0.6 0.8 1Total time (s)
Rea
l ∆t (
s)
NMM, ∆t=2msNMM, ∆t=1msNMM, ∆t=0.1msENMM, ∆t=0.1ms
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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noted that a damping ratio value of 0.001 is used to simulate the damping item in this
simulation. The simulated results using the ENMM are well agreement with that of the
NMM. The maximum relative error is under 0.27%, which is shown in Table 4.2. It can
be concluded that the ENMM can be taken into account in the application to rock
engineering, especially involving the computational efficiency of the modelling for
large scale engineering project.
(a) horizontal displacement;
(b) vertical displacement.
Figure 4.13 Displacements of measured point 1: (a) Horizontal; (b) Vertical.
-0.012
-0.01
-0.008
-0.006
-0.004
-0.0020
0.002
0 0.2 0.4 0.6 0.8 1Time (s)
Dis
pl. (
m)
NMM, Δt=2ms
NMM, Δt=1ms
NMM, Δt=0.1ms
ENMM, Δt=0.1ms
-1.8
-1.5
-1.2
-0.9
-0.6
-0.3
0
0 0.2 0.4 0.6 0.8 1Time (s)
Dis
pl. (
m)
NMM, Δt=2ms
NMM, Δt=1msNMM, Δt=0.1ms
ENMM, Δt=0.1ms
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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(a) horizontal displacement;
(b) vertical displacement.
Figure 4.14 Displacements of measured point 3: (a) Horizontal; (b) Vertical.
(a) horizontal displacement;
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1Time (s)
Dis
pl. (
m)
NMM, Δt=2ms
NMM, Δt=1ms
NMM, Δt=0.1ms
ENMM, Δt=0.1ms
-3
-2.5
-2
-1.5
-1
-0.5
0
0 0.2 0.4 0.6 0.8 1Time (s)
Dis
pl. (
m)
NMM, Δt=2ms
NMM, Δt=1ms
NMM, Δt=0.1ms
ENMM, Δt=0.1ms
-0.25
-0.2
-0.15
-0.1
-0.05
0
0 0.2 0.4 0.6 0.8 1Time (s)
Dis
pl. (
m)
NMM, Δt=2ms
NMM, Δt=1ms
NMM, Δt=0.1ms
ENMM, Δt=0.1ms
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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(b) vertical displacement.
Figure 4.15 Displacements of measured point 6: (a) Horizontal; (b) Vertical.
Table 4.2 Maximum displacement of measure points in the NMM vs. ENMM.
Measure point 1 3 6
Max. Displ. (m) Vertical Horizontal Vertical Horizontal Vertical Horizontal
NMM, ∆t=2ms -0.01179 -1.61859 1.3911 -2.73225 -0.21954 -0.62965NMM, ∆t=1ms -0.00966 -1.49954 1.36399 -2.71774 -0.23232 -0.63504
NMM, ∆t=0.1ms -0.00613 -1.58604 1.40826 -2.76374 -0.23629 -0.66333
ENMM, ∆t=0.1ms -0.01053 -1.58071 1.73911 -2.60442 -0.21174 -0.55114
Max. rel. error (%) 0.718 0.054 0.275 -0.057 -0.104 -0.169
4.6.5 Rock tunnel stability analysis
In this study, a tunnel modelling is taken consideration to extend the capability of
the proposed explicit version of the NMM. The tunnel consists of two sets of
intersecting joints by the orientation of 600 and 1300 as shown in Fig. 4.16. Two
neighbouring tunnels with same sectional dimension are surrounded by the fractured
rock blocks. The width of the tunnel a is 7.8 m, the interval distance between the two
tunnels b is 4.2 m, the high of the arch h is 2.7 m, and the high of the tunnel H is 3.0 m.
The whole cross section is with the length of 66.0 m and high of 33.0 m, and it is
symmetrical. 8 measure points locate on the top of the tunnels, one measure point is at
the centre of the modelling. The input physical parameters are identical with the last
rock slope modelling, which can be referred as Table 4.1.
-0.8
-0.6
-0.4
-0.2
0
0 0.2 0.4 0.6 0.8 1Time (s)
Dis
pl. (
m)
NMM, Δt=2ms
NMM, Δt=1ms
NMM, Δt=0.1ms
ENMM, Δt=0.1ms
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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4.92
Figure 4.16 Geometry of the tunnel modelling.
To compare the NMM and ENMM conveniently, the same time step is chosen to
simulate the stability of the tunnel modelling. The simulated results are presented in Fig.
4.17, in which the total time is set as 0.5 second. It is noted that CPU time taken by the
ENMM (i.e. 0.381 hour) is less than that of the NMM (i.e. 1.174 hours). Thus, the
ENMM is obviously more efficient than the NMM in terms of efficiency.
Figure 4.17 Simulation results used by ENMM vs. NMM.
66.0m
h
H
a ab33.0m
h
H
15.0
m
18.0
m
Measured point: 1, 2, 3, 5, 6, 7, 8.9,
(a) ENMM (CPU time 0.381hr);
(b) NMM (CPU time 1.174hr).
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Meanwhile, the horizontal and vertical displacements of the measure point at the
top of the tunnels are plotted in Figs. 4.18, respectively. It is noted that a damping ratio
value of 0.002 is used to simulate the damping item in this simulation. The simulated
results using the ENMM are well agreement with that of the NMM. The simulation of
the tunnel stability further reveals the capability of the developed ENMM in terms of
efficiency, and accepted level of accuracy. Especially, the ENMM is promising in
simulations of the large scale under-ground engineering such as tunnel stability analysis.
(a) horizontal displacement;
(b) vertical displacement.
Figure 4.18 Displacements of measured point 4.
‐0.2‐0.18‐0.16‐0.14‐0.12‐0.1
‐0.08‐0.06‐0.04‐0.02
00.02
0 0.1 0.2 0.3 0.4 0.5
Displement (m
)
Time (s)
NMM
ENMM
‐0.4
‐0.35
‐0.3
‐0.25
‐0.2
‐0.15
‐0.1
‐0.05
0
0 0.1 0.2 0.3 0.4 0.5
Displacement (m
)
Time (s)
NMM
ENMM
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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4.94
(a) horizontal displacement;
(b) vertical displacement.
Figure 4.19 Displacement of measured point 9.
4.7 SUMMARY
In this chapter, an explicit time integration scheme for the NMM is proposed to
improve the computational efficiency, in which a modified version of the NMM based
on an explicit time integration algorithm is derived on the dual cover system. The
original NMM based on displacement method is revised into an explicit formulation of
a force method. Although the ENMM requires small time-step due to numerical stability
of the scheme, it is efficient without assembling the stiffness equations. Compared to
the OCI used in the NMM, the open-close algorithm is more efficient in the ENMM
because of the explicit time integration scheme without solving simultaneous algebraic
0
0.002
0.004
0.006
0.008
0 0.1 0.2 0.3 0.4 0.5
Displacement (m
)
Time (s)
NMM
ENMM
‐0.7
‐0.6
‐0.5
‐0.4
‐0.3
‐0.2
‐0.1
0
0 0.1 0.2 0.3 0.4 0.5
Displacement (m
)
Time (s)
NMM
ENMM
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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equations in each step and the smaller penetration incurred due to a smaller time step
used. The developed method is validated by three examples, two static problems of a
continuous simple-supported beam and plan stress field problem, the other dynamic one
of a single block sliding down on a slope. Results showed that the accuracy of the
ENMM can be ensured when the time step is small for both the continuous and the
contact problems. A highly fractured rock slope and tunnel modelling are subsequently
simulated. It is shown that the computational efficiency of the proposed ENMM can be
significantly improved, while without losing the accuracy, comparing to the original
implicit version of the NMM. The ENMM is more suitable for large-scale rock mass
stability analysis and it deserves to be further developed for engineering computations
of practical rock engineering problems.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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CHAPTER 5. VERIFICATION OF COMPUTATIONAL
EFFICIENCY AND ACCURACY OF THE EXPLICIT
NUMERICAL MANIFOLD METHOD WITH WAVE
PROPAGATION PROBLEMS
5.1 INTRODUCTION
Wave propagation problems are always hot issues in the engineering analysis
drawing researchers’ attention. For rock engineering, the damage criteria of rock mass
under dynamic loads are generally governed by the threshold values of wave amplitudes
(Zhao et al. 2006). Therefore, the prediction of wave attenuation across the fractured
rock mass is important on assessing the stability and damage of rock mass under
dynamic loads. In the past several decades, many analytical approaches and numerical
methods are developed for the solution of wave propagation problems in both the
continuous and discontinuous media.
The numerical manifold method (NMM), originally proposed by Shi (1991, 1992),
is based on topological manifold and differential manifold, which combines both the
continuum-based finite element method (FEM) and the discontinuum-based
discontinuous deformation analysis (DDA) (Shi and Goodman 1985; Shi 1988) in a
unified form. Since the NMM uses a mesh-based partition of unity method (PUM)
(Babuska and Melenk 1995; Melenk and Babuška 1996; Babuška and Melenk 1997) to
combine all the local approximations together to give a global approximation, thus
which can also be viewed as a PU-based extension to the conventional FEM. One of the
most innovative features of the method is that it employs a dual cover system, i.e.
mathematical covers (MCs) and physical covers (PCs) to formulate the physical
problem. These two covers are interrelated through the application of the PUM. More
importantly, due to the cover-based property, the NMM is in essence different from the
classical FEM, and particularly suitable for modelling arbitrary discontinuities. With
respect to the discontinuum, the NMM inherits all the attractive features of the DDA,
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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5.97
such as the implicit time integration scheme, the contact algorithm and the minimum
potential energy principle (Chen et al. 1998; Jing 1998).
On the other hand, when the implicit time integration algorithm involves in the
solution of a system of equations, the computational cost increases dramatically with the
DOFs of the system is increased since the large-scale simultaneous algebraic equations
must be solved in each time step (Newmark 1959; Wilson, Farhoomand, and Bathe
1972). The OCI requires non-tension and non-penetration at all contacts which
additionally highs up the computation costs in order to obtain a convergence state at
each time instance (Doolin and Sitar 2004). On the contrary, the finite difference
method (FDM) (Kaczkowski and Tribillo 1975; Mitchell and Griffiths 1980) and
distinct element method (DEM) (Cundall 1971a) employ the explicit time integration
schemes based on finite difference principles. To the DEM, the main benefit of the
DEM is that its computational efficiency is high due to its explicit time integration
nature. However, it has also been argued that the accuracy of simulated results may be
sacrificed in some particular cases (O’Sullivan and Bray 2001). To ensure numerical
stability, a DEM simulation requires that the time step must be small enough. In the past
several decades, a great mount of discussions have been carried out between the implicit
and explicit time integrations (Bathe and Wilson 1972; Belytschko and Hughes 1983;
Dokainish and Subbaraj 1989a, 1989b).
In chapter 4, a modified version of the NMM based on an explicit time integration
algorithm is derived. The original NMM based on displacement method is revised into
an explicit formulation of a force method. The governing equations are built up on the
dual cover system and the global consistent mass, damping and stiffness matrices are no
longer necessary. A diagonal mass matrix is derived for the dual cover system which
makes the solution is highly efficient at each time step. OCI is still employed, however,
the relative cost is much lower because of the explicit time integration scheme without
solving simultaneous algebraic equations in each step and the smaller penetration
incurred due to a smaller time step used. To validate the proposed scheme, the stress
wave propagation problems through rock mass are simulated with continuous and
discontinuous considerations. The wave propagation problems depend on many factors,
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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5.98
such as the computational model and the algorithm used, the model size, the mesh size,
the initial condition, the boundary condition, the material model and the damping and
time step size, etc. (Lysmer and Kuemyer 1969; Chen 1999; Zhao et al. 2008; Gu and
Zhao 2009; Ma, Fan, and Li 2011).
In this chapter, the calibration study of the explicit numerical manifold method
(ENMM) on the P-wave propagation along a one-dimensional elastic rock bar is
conducted to investigate the accuracy of the ENMM on wave propagation problems.
Parametric studies are carried out to obtain an insight into the influencing factors of the
ENMM model on wave propagation problems. The reflected and transmitted waves
through the fractured rock mass are also numerically simulated. Furthermore, to verify
the capability of the proposed ENMM in modelling of seismic wave effect in fractured
rock mass, a dynamic stability assessment for fractured rock slope under seismic effect
is analysed as well. It is shown that the computational efficiency of the proposed
ENMM can be significantly improved, while without losing the accuracy, comparing to
the original implicit version of the NMM.
5.2 THE BRIEF OVERVIEW OF THE NMM
5.2.1 The NMM and its cover system
In the NMM, the manifolds connect many overlapped small patches together to
cover the entire problem domain. Each small patch is called a cover. A local function is
defined on each cover. One manifold element is generated through a set of overlapping
covers, and the behaviour is then determined by the weighted average of local functions
defined on the associated physical covers. It is the cover system, distinguishes the
NMM from other numerical methods and it equips the NMM as a robust tool for both
continuous and discontinuous problems. More details can be referred in Section 3.2 of
Chapter 3.
5.2.2 The explicit scheme of the NMM
More details of derivation of the explicit scheme of the NMM can be referred in the
Section 4.3 of Chapter 4.
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5.3 STRESS WAVE PROPAGATION IN A CONTINOUS BAR
To further investigate the proposed ENMM, stress wave propagation problems
through the rock mass are studied using the proposed explicit scheme to compare with
the implicit version with respect to computational efficiency and accuracy. In order to
track CPU time for each time step, a high resolution timer function which can measure
up to 1/100000th of a second is added to keep track of CPU time during the analysis.
Both analyses are run on the same computer with the system configuration: processor
speed = 3.17 GHz and RAM = 4.0 GB.
In the present simulation, a slender rock bar model subjected to impact loading is
studied as shown in Fig. 5.1. For the sake of decreasing transverse effect caused by
wave propagation, the one-dimensional elastic rock bar with width of 0.05 m, the length
of 2.00 m. The rock bar model consists of divided regular-patterned triangular manifold
element. A half sinusoidal impacting loading with the amplitude of 1MPa and different
frequencies are employed at the left edge of the model. The physical material properties
of the model can be seen in Table 5.1. The manifold element mesh topology
representing the model is shown in Fig. 5.1.
Figure 5.1 Schematic of the rock bar model.
Table 5.1 Material properties of the rock bar.
Property Value
Density(kg/m3) 2650
Young's Modulus(GPa) 66.25
Poisson's Ratio 0.25
P-wave velocity (m/s) 5000
0.05m
2.0m
P-wave Measuring point y
xo
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5.100
5.3.1 Effect of mesh size
It is commonly recognized that the mesh size of the numerical model significantly
influences the accuracy of numerical results for wave propagation problems whether the
model is based on the continuum or the discontinuum approach. This is because the
dynamic problems may lead to a computational instability if the following condition is
violated: xvt max, where maxv is the maximum particle velocity and x is the length
interval. One key parameter in the design of the mesh size is the mesh ratio ( rl ), defined
as the ratio between the eigenlength ( eigenl ) (see Fig. 5.2) of the largest element along the
wave propagation direction and the smallest wavelength ( min ).
mineigen
r
ll (5.1)
Based on the study on the mesh size limitation in the FEM, Kuhlemeyer and
Lysmer (1973) proposed that the mesh ratio must be smaller than 1/8~1/12 for the
accurate modeling of one-dimensional wave propagation through a semi-infinite
continuous medium. Theoretically, a numerical model with a refined mesh can produce
more accurate results, because the model is hardened by predefining a function to
represent the displacement or stress field in the elements. An element size larger than a
certain critical value may result in the numerical oscillation of wave presentation.
Figure 5.2 Eigenlength in the manifold mesh.
In the numerical model, the mesh ratio varies as 4
1 , 8
1 , 12
1 , 16
1 , 32
1 and 64
1 in the
longitudinal direction. Three cases corresponding to the same one half sinusoidal P-
wave with pressure amplitude of 1MPa and frequencies of 2500, 5000 and 10000 Hz,
respectively, are studied in this modeling. The corresponding wavelengths are 2.0, 1.0,
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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5.101
and 0.5m, respectively. The relative error, defined as the ratio of peak pressure
difference between the ENMM modeling results and the theoretical solution is used to
assess the accuracy of the numerical results, which can be repressed as
%
T
TE
P
PP (5.2)
As the rock material in the computational model is treated as a linear elastic medium,
there is no geometrical damping occurring in the one-dimensional wave propagation
problem. Thus, an elastic wave propagates through the rock with virtually no
attenuation and the theoretical solution is the same as the incident wave.
Figure 5.3 Percent errors at the end of first wavelength for different wave frequencies and element
ratios.
The relative error at the end of the first wavelength for different wave frequencies
and mesh ratio in the middle of the bar is shown in Fig. 5.3. It can be seen that with the
decrease of the mesh ratio, the percent error will decline evidently. When the mesh ratio
is equal to or less than one-eighth of the element dimension along the wave propagation
direction, the errors are less than 5%. Furthermore, the higher incident wave frequencies
will generate less accurate simulation result. The result is consistent with the previous
results obtained by the other researcher using the DEM and DDA (Chen 1999; Gu and
Zhao 2009). To further describe the differences of accuracy between the explicit and
implicit time integrations, lumped and consistent mass matrices used in the ENMM and
NMM, respectively. Two typical stress waves by the mesh ratios 1/4 and 1/32 in the
0
5
10
15
20
1/4 1/8 1/12 1/16 1/32 1/64
Rel
ativ
eer
ror
(%)
Mesh ratio
2500Hz
5000Hz
10000Hz
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
5.102
case of wave frequency 5000 Hz are presented in Fig. 5.4. It is found that the refined
mesh brings more accurate result to the theory solutions, but on the other hand, the
computational cost is more expensive whether using the NMM or ENMM.
Figure 5.4 Stress wave simulation using NMM and ENMM by two typical element ratios of 1/4 and 1/32 ( }0.1 st ).
Fig. 5.5 shows the error along the distance from wave source for different mesh
ratios with the wave frequency of 5000 Hz. It can be seen that the error increases with
the increasing distance whether in the NMM or ENMM, and the mesh ratio equal to or
less than one-sixteenth provides stable results with errors less than 2.5% even at the
furthest distance. It can also be seen that the fluctuation of percent errors occurs along
the distance from the wave source for the smaller mesh ratio in which the error is less
than 5%. This indicates that too finer a mesh may not necessarily produce more accurate
results in which the error may be dominated by other factors such as time step,
boundary condition and convergence error. Since the original NMM using the implicit
algorithm, the percent errors are less than the ENMM when the same time step is used.
On the other hand, the solution of the equations in the NMM is uneconomical as the
assembly of the coupled global stiffness matrix and the repeated over-relaxation
iterations, even as the increase of the DOFs in the large scale simulations, the efficiency
of the NMM is declined dramatically.
-1.2
-0.8
-0.4
0
0.4
0.8
1.2
0 0.0002 0.0004 0.0006 0.0008 0.001
Str
ess
(MP
a)
Time (s)
NMM, lr=1/4ENMM,lr=1/4NMM, lr=1/32ENMM, lr=1/32
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
5.103
(a) NMM;
(b) ENMM.
Figure 5.5 Percent errors along the distance from wave source for different element ratios ( }0.1 st ).
5.3.2 Effect of time step
The time step selection in a dynamic analysis is crucial to ensure the accuracy and
stability of the numerical results. In the absence of damping, both the ENMM and the
NMM employ the same discrete element framework to solve the equilibrium equations
with inter-block constraints, which are similar as the DEM and DDA. One of the main
differences between them is the time integration algorithm used to implement the time-
marching equations. O’Sullivan and Bray (2001) discussed the relative merits of
implicit and explicit schemes for discrete element modeling. Cundall and Strack (1979)
0
5
10
15
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Per
cen
t er
ror
(%)
Distance from wave source (m)
1/4
1/8
1/12
1/16
1/32
0
5
10
15
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Per
cen
t er
ror
(%)
Distance from wave source (m)
1/41/81/121/161/32
·
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
5.104
used the explicit, central difference time integration scheme in the DEM. A limitation of
this approach is that it is only conditionally stable and the critical time increment needs
to be calculated additionally. The implicit time integration scheme avoids the stability
issues arising from the explicit time integrator. However, the implicit scheme is
computationally more expensive because it needs a significant number of iterations to
form the stiffness matrix that is compatible with the contact state at the end of each time
step. The modified explicit time integration algorithm applied in the present ENMM
program constructs the lumped mass matrices and force vectors based on the state of
contact between the blocks at the end of the previous time step. Resorting to the solved
acceleration in each cover, the displacement increments are determined and the related
stiffness matrix and force item are updated, depending on whether contacts are open or
close over the course of the time step. For a discontinuous problem, the time step
required for the initial assumption is determined by the minimum of two time steps: one
is for the internal element system calculation and the other one is for the calculation of
contacts (de Lemos 1997).
The internal mesh calculations of the cover system assume that no information is
transmitted from a cover to another cover in the same zone in less than one time step.
The time step required for the stability of the zone of computations is proportional to the
mesh size and is estimated as
}/min{ min pa chat (5.3)
where pc is the P-wave velocity and
minh is the minimum height of the element, a is a
user-supplied factor intended to account for the increase of apparent stiffness due to the
contact springs attached to the boundary zones.
For the calculation of contacts of a rigid block system, the time step is calculated
by analogy to a single DOF system as (Last and Harper 1990)
maxmin /2 KMbtb (5.4)
where minM is the mass of the smallest block in the system;
maxK is the maximum contact
stiffness; b is user-supplied factor in order to account for that a block may be in contact
with several blocks.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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5.105
When damping item is used, the following formula is used to adjust the time step to
maintain the calculation stability (Belytschko and Hughes 1983):
}/))1((2min{ 2iiict (5.5)
where i is the critical damping ratio for mode with angular frequency
i . The
representation of the critical damping ratio can be found from Bathe and Wilson (1972):
)/(2
1iii (5.6)
where and are the mass-proportional and stiffness-proportional damping constant,
respectively.
The time step for a discontinuous problem is thus chosen as:
},,min{ cba tttt (5.7)
The maximum time increment will be set based on the value smaller than this
selected t that can be used in a time step. The initial calculation will be processed to
check whether this value satisfies the assumption of infinitesimal displacements and
equilibrium state for all the blocks with appropriate contact conditions within a time
step. Within a specified number of iterations, the procedure of contact lock selection so
called OCI will be repeated for solving the motion equations until there is no
penetration and no tension occurrence at each contact point. The program will
automatically adjust a smaller time step if iteration number becomes too large.
The simulation model is taken account stress wave with frequency of 5000 Hz and
the mesh ratio 1/32, thus the minimum size of the element is 0.03125 m. All the other
parameters used are the same as in the previous study. From Eq. (5.7), we can obtain the
time step required for both mesh ratio and contacts as: at =3.61 μs (here, a is selected
as 3/1 ), bt =54.27 μs (assumed one single joint in the middle of the bar, b is selected as
0.5, maxK is 45 GPa) and
ct =22.48 μs (here, critical damping ratio i selected as 1.0).
Therefore, t is selected to be 3.61μs. In this simulation, five different time increments
have been set to study the influence of the time step, the time increments are 0.1, 0.5,
1.0, 2.0 and 5.0μs, respectively.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
5.106
As shown in Fig. 5.6, it can be found that all the results are stable by both the
NMM and ENMM simulations when the maximum time increment is lower than the
selected t . With the decrease of the time step, the wave attenuation along the bar
becomes smaller and the results are more accurate for the peak pressure. But on the
other hand, the computational cost will be lengthened. In addition, the fluctuation
phenomena before the arriving wave become larger when larger time increment is used,
especially at the case of the ENMM modelling. Considering both accuracy and
efficiency, the time increment 0.5μs is a good choice for the ENMM model.
(a) NMM;
(b) ENMM.
Figure 5.6 Peak pressure attenuation for different time step.
0.4
0.6
0.8
1
0 0.4 0.8 1.2 1.6Distance from wave source (m)
Pea
k pr
essu
re (M
Pa)
0.1μs
0.5μs
1.0μs2.0μs
5.0μs
0.4
0.6
0.8
1
0 0.4 0.8 1.2 1.6Distance from wave source (m)
Pea
k pr
essu
re (M
Pa)
0.1μs
0.5μs1.0μs2.0μs5.0μs
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
5.107
5.3.3 Computational efficiency
Since the ENMM applies the explicit time integration scheme, the motion
equations can be solved explicitly without assembling the coupled global stiffness
matrix and repeating iterative solvers. In the ENMM, the lumped mass matrix is
employed instead of the consistent mass matrix used in the NMM. Thus, the
computational efficiency of the ENMM is significantly higher than that of the NMM
when the selected time step satisfies the numerical stability of the explicit integration
algorithm. To further investigate the efficiency of the ENMM comparing with the
NMM, five different time increments of 0.1, 0.5, 1.0 and 2.0 are used to simulate the
stress wave propagation in the bar by the different mesh ratios of 1/4, 1/8, 1/12, 1/16
and 1/32, respectively. In order to track CPU time for each time step, a high resolution
timer function which can measure up to 1/100000th of a second is added to keep track
of CPU time for each time step. All analyses are run on the same computer with the
system configuration: processor speed = 3.17 GHz and RAM = 4.0 GB.
The CPU time used by the ENMM and NMM are listed in Table 5.2, in which the
total time is set 1.2 ms and the stress wave with frequency of 5000 Hz is employed to
study the wave propagation in the modelling. We can find that the efficiency of the
ENMM is over four times of the NMM when the same time step size is used. Even
though the different time steps are applied in the NMM and ENMM, such as 0.5μs used
in the ENMM and 1.0μs in the NMM with the mesh ratio 1/32, the ENMM is more
efficient than that of the NMM as well.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
5.108
Table 5.2 Comparison of CPU cost between the NMM and ENMM.
Mesh ratio 1/4 1/8 1/12 1/16 1/32
Δt=0.1μs NMM(hr) 0.2731 0.2971 0.3236 0.3406 0.4295
ENMM(hr) 0.0436 0.0434 0.0547 0.0572 0.1047
Efficiency 6.264 6.846 5.916 5.955 4.103
Δt=0.5μs NMM(hr) 0.0603 0.0565 0.0576 0.0606 0.0817
ENMM(hr) 0.0079 0.0098 0.0116 0.0156 0.0218
Efficiency 7.6317 5.766 4.961 3.890 3.752
Δt=1.0μs NMM(hr) 0.0274 0.03 0.028 0.0294 0.0393
ENMM(hr) 0.0041 0.0045 0.005 0.0056 0.0099
Efficiency 6.683 6.667 5.600 5.250 3.970
Δt=2.0μs NMM(hr) 0.014 0.0144 0.0148 0.0156 0.0179
ENMM(hr) 0.0022 0.0023 0.0025 0.0028 0.0051
Efficiency 6.364 6.261 5.920 5.572 3.510
5.4 STRESS WAVE PROPAGATION THROUGH FRACTURED
ROCK MASS
When the wave propagates through fractured rock mass, both the reflection and the
transmission waves will be generated. In this section, two situations are considered, one
is to verify the transmission/reflection coefficients for the wave propagation through a
single joint in a homogeneous rock bar and the other one is the wave propagating
through the joint between different materials. Furthermore, the stress wave propagation
through the multiple parallel joints is investigated using the NMM and ENMM,
respectively.
5.4.1 P-wave propagation through homogeneous medium
When a wave transmits through a joint, the wave transmission, wave reflection and
energy dissipation occur at the joint. Pyrak-Nolte (1988) proposed a complete solution
for the effect of a joint on seismic waves based on a linear joint stress-deformation
relationship. In the case of a wave normal to the joint, the absolute values of the
reflection and transmission coefficients are written as
2/1
2 1)//2(
1
ZkR (5.8)
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
5.109
2/1
2
2
1)//2(
)//2(
Zk
ZkT (5.9)
where k is the joint stiffness; is the wave angular frequency; Z is the wave
impedance ( c );c is wave propagation velocity; is the density.
Fig. 5.7 shows the NMM and ENMM modelling results comparing with the Pyrak-
Nolte’s analytical solution in Eqs. (5.8) and (5.9). A normalized joint stiffness, )//( Zkn,
is used to represent the x -value where nk is joint normal stiffness. It indicates that both
the NMM and ENMM modelling results are consistent with the analytical solution.
Figure 5.7 Comparison between the simulated results and the Pyrak-Nolte’s analytical solution for
a single joint.
5.4.2 P-wave propagation through joint between different mediums
When the wave propagates through the joint between two mediums, conditions
should be met at the interface for both equilibrium and compatibility. Those two
conditions can be further expressed as (Bedford and Drumheller 1994)
irEE
EE
1122
1122 (5.10)
itEE
E
1122
222 (5.11)
where i ,
r and t are the incident, reflected, and transmitted stress form, respectively.
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12Normalised joint stiffness, kn/Z/ω
Coe
ffic
ient
s, R
, T
Analytical solution, RAnalytical solution, TNMM simulation, RNMM simulation, TENMM simulation, RENMM simulation, T
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
5.110
In the numerical simulation, the rock bar with 2.0m is divided into two half with
different materials. All the related parameters are same with the mesh ratio 1/32 and
only Young’s modulus of the second 1.0m part is modified to be 4 times that of the first
part. The theoretical solution for the reflected r is 0.33
i with a negative sign while the
transmitted t is 1.33
i .
In Fig. 5.8, the wave forms at two measured points (0.2m and 1.01m away from the
wave source) are compared with the theoretical solutions. It can be seen that the
numerical results on both the reflected and transmitted waves agree well with the
analytical solutions.
Figure 5.8 Comparison between the simulated results of NMM and ENMM.
5.4.3 Stress wave propagation through the multiple parallel joints
The wave propagation through multiple joints is more complicated than that at a
single joint. When a stress wave propagates through multiple joints, each joint transmits
and reflects the wave, and causes time delay, thus multiple transmissions and reflections
occur. Based on the previous studies, the wave propagation through 5 parallel joints
with the same spacing is investigated for the half sine P-wave with frequency of 5000
Hz using the NMM and ENMM, respectively. The joint normal and shear stiffness are
assumed as 45 and 10 GPa, respectively. Joint properties such as friction angle,
-1.6
-1.2
-0.8
-0.4
0
0.4
0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012
Pre
ssu
re (
MP
a)
Time (s)
NMM, D=0.2m
NMM, D=1.01m
ENMM, D=0.2m
ENMM, D=1.01m
Analytic solution
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
5.111
cohesion and tensile strength are taken account as 15 degree, 0 GPa and 0 GPa,
respectively. Five joints uniformly locate at 1.0, 1.2, 1.4, 1.6 and 1.8 m of the bar. A
measure point is set behind the fourth joint (at 1.8 m) to record the transmitted wave.
Fig. 5.9 shows the simulated results at the measure points by the NMM and ENMM. It
can be seen that based on both the NMM and ENMM, the stress wave is superposed
from the transmitted and reflected waves with different time delay through the 5 joints.
Figure 5.9 Simulated results at measure point by the NMM and ENMM.
To investigate the efficiency of the ENMM comparing with the NMM, the time
steps of 0.1μs and 0.5μs with mesh ratios of 1/32 and 1/64 are employed in the
simulations, respectively. When the joints in the modelling are taken into account, the
contact computation is a time-consuming job as the repeated OCI solvers to achieve the
no-penetration and no-tension requirement. The CPU time of the simulations by the
NMM and ENMM is presented in Table 5.3. It is shown that the ENMM is more
efficient than that of the NMM to solve the discontinuous problems comparing with the
continuous problems. In particular, the more joints locate in the modelling, the higher
efficiency to the ENMM comparing with the NMM.
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.0002 0.0004 0.0006 0.0008 0.001 0.0012
Pre
ssu
re (
MP
a)
Time (s)
NMM ENMM
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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5.112
Table 5.3 CPU cost comparison between the NMM and ENMM.
Mesh ratio 1/32 1/64
Δt=0.1μs NMM (hr) 0.4382 0.9623
ENMM (hr) 0.0841 0.1755
Efficiency 5.211 5.484
Δt=0.5μs NMM (hr) 0.0889 0.1748
ENMM (hr) 0.0167 0.0376
Efficiency 5.324 4.649
5.5 SEISMIC WAVE EFFECT FOR A FRACTURED ROCK
SLOPE
To further verify the capability of the proposed ENMM in modelling of wave
propagation in fractured rock mass, a dynamic stability assessment for a fractured rock
slope under seismic effect is analysed in this section.
Fig. 5.10 shows the schematic cross-section of the rock slope. The asperity of the
slope is denoted by abcdefghi , and the fractured zone locates at the crest of the slope.
The span of the slope is 100 m, the height of the left and right edges are 30 m and 9 m,
respectively. A horizontal seismic acceleration history as shown in Fig. 5.11 is applied
on the bottom of the slope to simulate a seismic effect. The physical material properties
of the rock, such as Young’s modulus, Poisson’s ratio and mass density are assumed as
10 GPa, 0.2 and 2200 kg/m3, respectively. To get access to computational accuracy by
the seismic wave effect, the fractured joints are simulated with the normal contact
stiffness of 20 GPa and 12 GPa in the NMM and ENMM, respectively. The friction
angle, cohesion and tensile strength of the joint are assumed as 20 degree, 0 GPa and 0
GPa, respectively.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
5.113
Figure 5.10 Schematic cross-section of the rock slope
Figure 5.11 Record of acceleration of the seismic wave.
To obtain the effective accuracy of the computations, the simulation for the
fractured slope with the mesh, as shown in Fig. 5.12, is performed. The discrete blocks
in the dotted box are represented by two sets joints by the dips of 90 and 32 degrees as
the fractured zone. The input seismic wave acts on the fixed points at the bottom of the
slope. Fig. 5.13 shows the simulated results of final state of the discrete blocks by the
NMM and ENMM. The measure point displacement simulated by the ENMM and
NMM are plotted in Fig. 5.14. The final displacement of the measure point under the
seismic effect is 6.437 m in the ENMM, which is close to that of 6.212 m in the NMM.
-0.4
-0.2
0
0.2
0.4
0 5 10 15 20 25
Acc
eler
atio
n (
g)
Time (s)
100m
9m
30m
a b c d
e f
g h i Fractured Zone
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
5.114
Figure 5.12 Rock slope model in the NMM and ENMM.
In terms of the efficiency, the ENMM is obvious efficient compared with the
NMM since the explicit time integration is used. In the NMM, the open-close algorithm
is employed to obtain contact convergence, in which the no-penetration and no-tension
contact constraint is imposed at each detected contact point. To satisfy this constraint,
the open-close iterative solver proceeds until zero penetration at any contact point. This
procedure is time-consuming and cockamamie in the computations. As the increase of
scale of the discrete blocks, the efficiency is declined dramatically, even the size of time
step decreases less than the explicit one to obtain the effective contact accuracy. In the
present simulation, although time step in the NMM (=1.0 ms) is five times of the
ENMM used (=0.2 ms), the CPU cost of the ENMM (=0.729 hour) is more efficient
than that of the NMM (=1.387 hours).
Measured point
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
5.115
Figure 5.13 Simulated results of the failure of the rock slope using the NMM and ENMM (Time
step in NMM =1ms, ENMM =0.2ms; total time =20s).
Figure 5.14 Measure point displacement simulated by the NMM and ENMM.
5.6 SUMMARY
In this chapter, a modified version of the NMM based on an explicit time
integration algorithm is proposed. The calibration study of the ENMM on P-wave
propagation across a rock bar has been conducted. Various considerations in the
numerical simulations are discussed and parametric studies have been carried out to
obtain an insight into the influencing factors in wave propagation simulation. The
0
2
4
6
8
0 4 8 12 16 20
Dis
pla
cem
ent
(m)
Time (s)
NMM ENMM
NMM, CPU time= 1.387 hrs
ENMM, CPU time= 0.729 hrs
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
5.116
numerical results from the ENMM and NMM modelling are accordant well with the
theoretical solutions. The mesh ratio is regarded as one of the major factors influencing
the simulation accuracy. With the consideration of both calculation accuracy and
efficiency, a mesh ratio of 1/16 is recommended for one dimension ENMM analysis.
Furthermore, the selection of a suitable time step depends on the internal element
system and the contact transfer between the interfaces. With the decrease of the time
step increment, the results become more accurate for the incident wave. In terms of
efficiency, the ENMM is more efficient than that of the NMM, even though the
different time steps are used. To further verify the capability of the proposed ENMM in
modelling of seismic wave effect in fractured rock mass, a dynamic stability assessment
for fractured rock slope under seismic effect is analysed as well. The simulated results
show that the computational efficiency of the proposed ENMM can be significantly
improved, while without losing the accuracy, comparing to the original implicit version
of the NMM. The various studies presented in this paper demonstrate a promising future
for the ENMM method in modelling stress wave propagation and other dynamic
problems for rock engineering.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
6.117
CHAPTER 6. THE TEMPORAL COUPLED EXPLICIT-
IMPLICIT ALGORITHM FOR DYNAMIC PROBLEMS
USING THE NUMERICAL MANIFOLD METHOD
6.1 INTRODUCTION
The efficiency and accuracy are usually regarded as two indices to check the
capability of a numerical method in terms of time integration for dynamic problems. In
general, there are two classes of time integration algorithms for dynamic problems:
implicit and explicit (Gelin, Boulmane, and Boisse 1995). Implicit algorithms methods,
such as the continuum-based finite element method (FEM) and discontinuum-based
discontinuous deformation analysis (DDA), tend to be numerical stable permitting
larger steps, but the computational cost is high and storage requirements incline to
increase dramatically with the contacts between elements and these degrees of freedom
(DOFs). Thus, the algorithms are suited to simulate the lower dynamics problems with
less non-linearities, resulting in more numerical stability and accuracy (Gelin et al. 1995;
Yang et al. 1995; Sun et al. 2000). On the contrary, explicit algorithms such as finite
difference method (FDM) and discrete element method (DEM) attempt to be
inexpensive per step and require less storage than implicit algorithms, but numerical
stability requires that small steps be employed, thus, the algorithms generally are used
for highly non-linear problems with many DOFs (Subbaraj and Dokainish 1989a, 1989b;
Yang et al. 1995). To take advantage of the merits of implicit and explicit algorithms,
many methods have been developed in temporal and spatial discretizations, in which it
is attempted to simultaneously achieve the maximum contributions of both classes of
algorithms. These studies normally concentrated on the continuum-based methods with
prescribed contact states.
When more contact problems are involved in the discontinuum-based methods
such as NMM and DEM, the efficiency is significantly declined. Thus, how to treat the
contact problems balancing the efficiency and accuracy, an appropriate time integration
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
6.118
algorithm is required. In general, low-frequency motion states, less-nonlinear contact
states and contact without external forces can be treated as quasistatic problems to apply
the implicit time integration in the NMM. On the other hand, when high non-linear
contact problems such as high velocity impact problems, blasting problems and
earthquake problems take places, the implicit schemes can not calculate and simulate
the phases change during the contact efficiently and accurately as the large time steps
are used. The explicit algorithms can be employed to solve the above non-linear contact
cases efficiently. On account of the different phases change in the conatct, the
corresponding algorithms can be used to maximize the merits of both algorithms. Thus,
the corresponding temporal coupled algorithm is required to combine the explicit and
implicit algorithms effectively.
In this chapter, the numerical manifold method (NMM) is taken into account to
combine the two time integration algorithms. During the previous two decades, the
NMM has been widely carried out for solving various structural dynamic problems. The
traditional NMM is originally proposed by Shi (1991, 1992) based on topological
manifold and unifies both FEM and DDA. It employs the implicit time integration and
open-close contact iteration for the simulations of complicated dynamic problems in
rock engineering and rock mechanics. Since the implicit scheme requires the assembling
of the coupled global stiffness matrix for the governing equations, which may involve
many thousand DOFs, especially when such more contact problems and nonlinear
problems are encountered, the computational cost can be increased dramatically. This
has motivated us toward developing more efficient computational algorithm for the
NMM. Thus, the choice of an appropriate algorithm is essential to ensure efficiency and
robustness of the numerical simulations, but the difficulty resides in being able to
combine robustness, accuracy, stability and efficiency of the algorithms. Furthermore,
Newmark-β family methods (Newmark 1959) are employed. When the different
parameters of β and γ are taken into account, the different algorithms can be selected to
simulate the corresponding dynamic problems. The distinction between explicit and
implicit where we have considered is that the explicit uses a diagonal mass matrix and
the implicit applies a consistent inertia matrix (Liu and Belytschko 1982). Then, a
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
ANALYSIS OF ROCK SLOPE
6.119
temporal coupled E-I algorithm for the NMM based on the dual cover system is
proposed, in which different time steps, time integration schemes and contact
algorithms are applied in temporal discretization. Then, some calibration examples and
numerical simulations are studied to validate the coupled E-I algorithms.
6.2 THE NMM AND ITS COVER SYSTEM
In the traditional NMM, one manifold element is generated through a set of
overlapping covers, which is the distinct characteristic differs from other numerical
methods. The cover system in the NMM provides a robust tool for both continuous and
discontinuous problems. In this section, the fundamentals of the NMM and its cover
system are described respectively.
6.2.1 Dual cover system in the NMM
Since the NMM doesn’t require MC sides to coincide with the material boundaries
and internal cracks, arbitrary shapes of MCs can be employed in the NMM simulation.
For convenience, a regularly structured mesh is employed in the NMM which is similar
as that in the FEM. As is shown in Fig. 6.1, in which a regularly-patterned triangular
mesh is presented, in which each MC is defined through six triangular elements sharing
a common node (i.e. nodal star). Each cover has two degree of freedom is similar as
node property in the FEM, each element formed by the overlapping of three
neighbouring hexagonal covers has six degree of freedom for the second order time
integration. The mathematical mesh covers the whole physical domains to form PC
system, the common areas denoted by ie , je and
ke are formed by the neighbouring
three hexagonal MCs combined with the material domains. When the linear triangular
element weight function is applied based on cover system, the global displacement
function over a ME can be expressed as
ei
iie yx,yxUyxyxu
),(),(),(),(3
1
(6.1)
where ),( yxi is the weight function over the three associated MCs, ),( yxUi is the
displacement function on the three associated PCs. Here, it is the cover system makes
the solution for both continuous and discontinuous problems practicability without any
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re-meshing technique used in the FEM. Moreover, three separate material domains
constituted by smooth curved boundaries actually is approximated as polygons
composed of straight edges in order to apply the simply integration over the ME.
Figure 6.1 A regularly-patterned triangular mesh in the NMM.
6.2.2 Contact problems in the NMM
The traditional NMM uses a penalty-constrain approach in which the contact is
assumed to be rigid. Numerically, it can be carried out by means of adding or
subtracting penalty value to the contact terms into the global equation to produce
contact stiffness matrix. The terms are selected by checking the previous and current
states of the contact, each controlled by a vector of values i = -1, 0 and 1, i =1, 2, ...
n , respectively. Accordingly, within each time step, the assembled global equations are
solved iteratively by repeatedly adding and removing contact springs until each of the
contacts converges to a constant state, which is known as OCIs proposed by Shi (1988).
The purpose of the OCI is to achieve the contact modes identically. In the NMM,
the no-penetration and no-tension contact constraint is imposed at each detected contact
point. To satisfy this constraint, the open-close iterative solver proceeds until this is no
penetration at any contact point, and number of OCIs required for contact convergence
in each time step dramatically increases for problems involving more contact elements,
which increases the computational cost as well. To reveal the efficiency of the OCI, a
numerical example of rock slope referred in (Khan, Riahi, and Curran 2010) is
investigated, in which the OCI used by the DDA compares with contact algorithm
MC
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applied in the DEM. The simulated results show that the OCI is one of the most
important factors affecting its computational efficiency. Since several times of iterations
are required in each time step, the computational cost will become more expensive.
There are two major issues of this approach; (i) coming up with an appropriate
penalty value for an arbitrary problem is difficult, and (ii) the open-close iterative
algorithm used to enforce penalty constraint is a time consuming process. These two
issues make NMM penalty method computationally expensive. Moreover, the
assumption that contacts are infinitely rigid is less realistic because blocks undergo
some local deformation at contact points that must be accounted for in computing
contact forces. By definition, this approach assumes zero thickness for the filling
material between the contact interfaces.
6.3 TEMPORAL COUPLED EXPLICIT-IMPLICIT ALGORITHM
IN THE NMM
6.3.1 Summary of equations of motion and time integration
Generally, the equation of motion can be derived in the strong form as
vb (6.2)
in which is the symmetric Cauchy stress tensor, is the gradient operator, is the
density of material, b is the body forces and v is the velocity, respectively. Constitutive
equations are required to couple the Cauchy stress tensor and the density to the
kinematics of the deformation, Eq. (6.2) can be written as the well-known discrete
equation
}{}]{[}]{[ extFuKuM (6.3)
where ][M is mass matrix, ][ K is stiffness matrix and }{ extF is the external force vector,
respectively; }{u and }{u are acceleration and displacement vectors, respectively. It is
noted that ][M and ][ K are symmetric, whereas they are also banded and sparse as the
local property of deformation matrix ][T of the MCs in the NMM, and it requires more
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computer memory with the assembly of global stiffness matrices for solving the
governing equations.
In general, two classified methods for direct integration of the equations of motion
can be considered: explicit methods, in which the accelerations are taken into account
and then integrated to obtain the displacements; and implicit methods, in which the
equations of the motion combine with the time integration operator in order to gain
displacements directly. Both methods are developed from different formulas based on
the Newmark- methods (Newmark 1959) using an increment of step time t . In the
explicit algorithm of the integration, Eq. (6.3) can be written as
}){}({][}{ int1 FFMu ext (6.4)
where }{ intF is the internal force vector. We apply the explicit central difference
scheme, i.e. 0 and 2/1 , the velocity and displacement can be calculated as
2/11
2/11
2/12/1
2
1
nnn
nnn
nnn
utuu
utuu
utuu
(6.5)
where 2/1nu is the central velocity by half a time step. A big advantage of it is the use of
a diagonal mass matrix (see Section 4.3) to improve the computational efficiency,
whereas it is conditionally stable. Thus, the increment of step time t must be satisfied
by less than a critical step time crt , which can be expressed as
ecrt
max
2
(6.6)
where emax is element maximum angle frequency, which is related with minimum value
of element eigenlength, )1,0( is the coefficient to determine t based on different
damping ratios.
When the implicit time integration is carried out by minimizing the potential
energy associated with an increment of time t , which is similar to the Newmark-
methods with parameters 2/1 and 1 correspond to the constant acceleration
scheme. Then, substituting the parameters into Eq. (6.6), we can have
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}{][2
}{}{)(
][2][
2 next ut
MFu
t
MK
(6.7)
where nn uuu 1 is the displacement increment from the current time step t to
)( tt corresponds to the step increment of n to )1( n . Eq. (6.7) requires assembling
the global stiffness matrix and solving the coupled system of equations using successive
over relaxation iteration, which increases the computational cost even the OCI is
applied into the contact problems. Since the implicit scheme which provides numerical
damping, the explicit damping term C is assumed to be zero. It is noted that the implicit
scheme is stable for larger time step, while the appropriate selection of t is made in
accordance with the required level of numerical damping. For best results from stability
and numerical dissipation point of view, a time-step t is selected as (Doolin and Sitar
2004; Doolin 2005):
et
max
4
(6.8)
6.3.2 The coupled explicit-implicit algorithm
In the computations, time integration discretized in temporal algorithm for the
dynamic problems combines the explicit and implicit algorithms as a coupled method to
expose both advantages at utmost. For the different problems, there are two types of
coupled approaches can be considered: implicit-explicit (E-I) algorithm and explicit-
implicit (I-E) algorithm. When different approaches are employed, the different step
time scale can be applied into the corresponding time integration scheme. In order to
investigate the temporal couples algorithm, the E-I algorithm is taken into account in the
present study. Furthermore, the Newmark- methods with two characteristic
parameters and for all sub-domains are assumed here.
As is shown in Fig. 6.2, the initial diagonal mass matrix and force vector are
constructed for the explicit algorithm, then the explicit central difference method, i.e.
the Newmark method with the parameters 01 and 2/11 , is employed from the
initial step time 0t to
nt at the step number n to simulate the high frequency part of the
dynamic problems. And then, the explicit integration algorithm switches to the implicit,
in which the transfer algorithm is proposed in order to achieve the conservation of the
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kinematic energy from the explicit to implicit integration, and IE DD ,
IE vv and
IE are satisfied for the coupled E-I integration without the element and node
partition. Thus, it is convenient to achieve in the programming code. In the part of the
implicit integration from step time 1nt to rnt , the constant acceleration method with the
parameters 2/12 and 12 is used in the implicit integration by the end of step
)( rn for the low frequency and quasi-static problems. Continuing the explicit
procedure, the initial inertial, stiffness matrix and force vector for the implicit
integration are require to construct again as the difference items in the equations of
motion. It is noted that different step time sizes are adopted before and after the
transition in the couples E-I algorithm in terms of the numerical stability and accuracy,
respectively. Normally, the step time size It in the implicit algorithm is larger than
Et
in the explicit algorithm, which denotes EI tt , 1 is the coefficient to describe
the scale between the implicit and explicit integrations. Sequentially, the transfer
algorithm of the coupled E-I integration is exposed and discussed in the following
section.
Figure 6.2 Transfer algorithm from the explicit to implicit integration.
6.3.3 Transfer algorithm for the E-I algorithm
In the coupled E-I method, we employ the explicit algorithm to model motion of
the system at the early stage, followed by the implicit algorithm to simulate the
subsequent motions of the system. Thus, an explicit physical model in the NMM will be
transferred to the implicit one at a certain time so that the coupled method is more
Initial diagonal mass matrix and force matrix for explicit
Transfer explicit to implicit
Initial inertia, stiffness matrix and force matrix for implicit
Newmark method
Newmark method
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efficient. In the transition, the geometric configurations, physical and mechanical
parameters, and status, including stress state and velocities, are consistent and continue.
Therefore, the transfer algorithm is required to satisfy the kinetic energy and potential
energy conservation from the explicit integration to the implicit one, which can be
represented as
n
i
Iy
Ix
Ii
n
i
Ey
Ex
Ei vvMvvM
1
22
1
22)(
2
1)(
2
1 (6.9)
),,(),,( I
e
Ixy
Iy
Ix
E
e
Exy
Ey
Ex (6.10)
where EiM and I
iM are the i-th explicit and implicit element mass, respectively; Exv ,
Eyv and I
xv , Iyv are the velocity components of an explicit element and implicit element
in the x and y directions, respectively; E
e
Exy
Ey
Ex ),,( and I
e
Ixy
Iy
Ix ),,( are the
explicit and implicit element potential energy, in which Exy
Ey
Ex ,, and I
xyIy
Ix ,, are
stress components of the explicit and implicit elements respectively. Furthermore,
equations of Ii
Ei MM , I
xEx vv , I
yEy vv , I
xEx , I
yEy , I
xyExy are satisfied in the
transfer algorithm to ensure the parameters of the terms are consistent and the
computation is continuous.
When the explicit integration transfers to the implicit one, the time step size control
is required as the difference between them with respect to the numerical stability. In
general case, we choose the coefficient more than ten times is efficient for the
coupled E-I method. In theory, the implicit step time size It is near the certain value
as described in Eq. (6.8) leads to the maximum numerical dissipation and more efficient
computation cost, and if the explicit step time Et is beyond the limited by Eq.(6.6) will
result in the numerical oscillations as the numerical dissipation property of the central
difference method. Even if larger magnitude of is given, the iterative solution and
open-close criteria in the NMM make the used It is close to a certain critical value.
Thus, the large value of is not always a good thing in the coupled E-I method,
especially more contacts occur in the simulations. In the present work, we suggest the
value of is taken the range from 10 to 20 according to the large or small scale of the
studied problems. Since the different styles of equations of the motion between the
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explicit and implicit algorithms, the initial conditions are constructed before and after
the transfer from the explicit to implicit integrations, respectively. It is noted that the
contact detection and contact condition judgment are shared between the explicit and
the implicit algorithms in the overall simulations.
6.4 CONTACT ALGORITHM OF THE COUPLED ALGORITHM
6.4.1 Contact force calculation
As previously mentioned, contact detection and contact force calculations are done
by the NMM. Once contacts have been detected, a contact interaction algorithm is
employed to evaluate contact forces between the contact elements. Contact interaction
between two neighbouring contact elements occurs along the discontinuity loops. At
low contact condition, the discontinuity boundaries may touch only at a few points.
With increasing of penetration distance and elastic deformation of individual interface
asperities occurs, resulting in an increase in the real contact area and hence an increase
in number of contact points. A thorough discussion and formulations of these
approaches can be found in (Munjiza 2004).
For a discrete block system involving m elements, when element i and j have
contacts, cK between i and j can be expressed as
][][
][][cjj
cji
cij
ciic
KK
KKK (6.11)
in which ][ cijK , ji and mji ,2,1, , is defined by the contact spring between the
contact elements i and j, and the value is zero if the elements i and j have no contact.
Since each element is consisted by three associated PCs, thus the matrix ][ cijK is a 6×6
sub-matrix. It is noted that the displacement 3,2,1,)( rD ri on the PCs can be predicted by
the previous step n. Then contact forces associated with ][ cijK on the contact element i
are assembled as
3,2,1;,,2,1,][1
)(
rmjDKIm
jri
cij
ci
(6.12)
The total internal forces on the element i can be represented as
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3,2,1;,,2,1,][~
)(1
rmjDKKI ri
m
j
cij
eiii
(6.13)
in which iI~ is the element internal force vectors and e
iiK is the stiffness matrix of
element. Since each element is formed by the three associated PCs, thus iI~ can be
rewritten as
)3(
)2(
)1(
~
~
~
~
i
i
i
i
I
I
I
I (6.14)
in which )1(
~iI maps the first PC associated the element, the subsequent )2(
~iI and )3(
~iI
map the second and third PCs, respectively. Then, intF at each PC can be assembled by
the associated iI~ on the cover system.
6.4.2 Damping algorithm
Damping algorithm is used to dissipate the excessive energy in the contact
problems due to the use of linear springs between contact elements. Two types of
damping are used: mass-proportional damping and stiffness-proportional damping. The
combined use of two damping forms is usually termed Rayleigh damping, which can be
proved as effective way of considering damping for analysis of structures. Rayleigh
damping can be given by (Bath 1982)
][][][ KbMaC (6.15)
where ][C is the damping matrix; a and b are the given Rayleigh constants, which are
also called as mass proportional and stiffness proportional respectively.
Since the explicit algorithm is conditional stability, it can be shown that the
stability limit for damping item is given in terms of the highest eigenvalue in the
system:
)1(2 2
max
t (6.16)
It is noted that the explicit scheme employs dynamics method to solve the
uncoupled equations, in which the generated kinetic energy can not be neglected. To the
static or quasi-static problems, it requires the physical damping to adsorb the kinetic
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energy of the systems so that the systems achieve stable condition. More details of the
derivation of the damping force can be seen in Section 4.3.
6.5 NUMERICAL EXAMPLES
In order to investigate the validity of the proposed algorithms, some numerical
examples are simulated in terms of temporal respect. In the simulations, the proposed
coupled temporal algorithms for the NMM are calibrated between the simulation results
and analytical solutions, firstly. Then, an examples of open-pit mining slope stability
analysis are taken into account using the proposed E-I algorithms to further extend the
current NMM, which represent the potential of the proposed E-I algorithms for
simulating the larger scale projects in the further research.
6.5.1 Calibration of the temporal coupled E-I algorithm
In order to calibrate the proposed coupled E-I algorithm for the temporal problems,
one Newmark sliding modelling of block sliding under input horizontal acceleration Ha
is studied here. The geometry of the modelling is shown in Fig. 6.3, in which a block
rests on an inclined plane is taken into account as a first approximation of the Newmark
sliding model. The angle of the plane is 31.470. A sinusoidal seismic acceleration Ha is
employed to impose to excitation point as expressed in Eq. (6.17), where g is the gravity
acceleration, t is the simulation duration for the simulation. In this study, we assume the
frictional angle 030 , the total displacement of analytical solutions can be referred in
(Newmark 1965; An, Ning, et al. 2011), then the simulated results of the proposed E-I
NMM can be obtained as shown in Fig. 6.4. It is noted that when the E-I algorithm is
considered, the proposed transfer algorithm is employed from the explicit to implicit
algorithm at the time of st 1 , and the final results of the simulations are well
agreement with the analytical solution.
st
sttga H 10
14sin1.0 (6.17)
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Figure 6.3 Geometry of the Newmark sliding modeling.
Figure 6.4 Block displacement under horizontal ground acceleration.
6.5.2 Open-pit mining stability analysis
An open-pit mine often encounters a fault or fracture zone or weak discontinuous
planes consisted by jointed rock masses. Joints usually occur in sets which are more or
less parallel and regularly spaced. There are usually several sets in very different
directions so that the rock mass is broken up into a blocky system (Jaeger and Cook
1979). Such geology to an open-pit mining slope have a great influence on the slope
stability. The general approach, when investigating the deformation and failure
characteristics of a slope, is to carry out numerical analysis, varying the input
parameters. The results of the numerical analysis can be verified by comparing them
with the measured displacement data at the mine site or by carrying out a test on a
laboratory-scale model. When every parameter is properly scaled down in accordance
with the scaled-down geometry, it is possible to represent the geological conditions of
0
0.02
0.04
0.06
0.08
0.1
0 0.5 1 1.5 2
Displ. (m)
Time (s)
Analytical
E-INMM
Transfer algorithm
Explicit algorithm Implicit algorithm
Sliding block
Excitation points
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the operation site with the scaled model (Jeon et al. 2004). To investigate the stability of
open-pit mining, the proposed E-I algorithm is employed in the present study.
In this simulation, one open-pit mine slope modeling is assumed to study the
stability using the proposed E-I algorithm. As shown in Fig. 6.5, there are 9 layers
denoted by 1# to 9# separate the whole modeling, in which we assume the layer 4# is
the fracture zone constituted by many discontinuous joints. The inclined angle of the
slope is 42° and drop is 120 m. In order to investigate the effect of the fracture zone to
stability of the slope, two models of layer 4# are represented in Fig. 6.6. Integrated
Model considers the whole layer as one domain, on the other hand, Refined Model adds
more joints into the layer to approach the realistic condition, in which two sets of joints
with nearly perpendicular orientation (i.e. 32.470 and 139.090) are constructed as seen
Figs. 6.6(b) and (c) to simulate the fractured zone of the open-pit slope.
Figure 6.5 Geology section of the open-pit mining.
Traditional methods apply to the slope stability analysis is to determine the factor
of stability (FoS) of the slope using the limit equilibrium method (LEM) without
considering the effect of the dynamic loading with time history (i.e. seismic loading,
blasting loading, etc). Here, the FoS is computed using the LEM to the integated model,
the results can be seen in Table 6.1. We can find that the determined values of FoS of
the Integrated Model (IM) is over one and the slope approaches to the stability. It is
noted that FoS may be less than 1 when the RM is employed, since the IM is simplified
into a block without considering the fracture joints. When the model with joints are
taken into consideration, the slope tends to be more instable, such as the case of RM2 is
failure when the joint frictional angle is up to ϕ=150 .Thus the RM is closer to the
Layer 2#
Layer 8# Layer 7# Layer 6#
Layer 5#
Layer 3# Layer 4#
Layer 9#
Layer 1# Altitude (m)
160
80
10 10
40 40
120
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realistic property of the fracture zone 4#. The above discussion presents that the RM
determines the low limit of the FoS and should be take into account to investigate and
design the slope of open-pit mining
Figure 6.6 Study model of the layer 4#: a. Integrated Model; b. Refined Model 1; c. Refined Model
2.
Table 6.1 FoS using LEM by the integrated models.
Study Model Integrated model
ϕ=10° ϕ=15°
FoS 1.436 2.182
Continue the above study, the RM is used into the simulation of slope stability
analysis for the seismic loading using the proposed E-I algorithm. In order to further
investigate the stability of the slope under the earthquake, a stochastic horizontal
seismic acceleration (Ml = 6.4) with maximum value of 0.2g is applied into this
simulation by the cases of ϕ=100 and ϕ=150 as shown in Fig. 6.7.The detailed of
physical parameters in this simulation are list in Table 6.2.
In this study, total time assumed as 20s. The proposed E-algorithm is used into the
simulation at the first 10s of seismic loading, then, the following I-algorithm is applied
Layer 4#
a.
Integrated Model
Refined Model 2
Measured point 3
c.
Refined Model 1
Measured point 3
b.
Measured point 1
Measured point 1
Measured point 2
Measured point 2
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into the simulation in order to simulate the slope efficiently. The simulation results for
RM1 and RM2 using the E-I algorithm with the case of ϕ=100 and ϕ=150 can be seen in
Figs. 6.8 and 6.9. The fracture zone (i.e. layer 4#) in the RM2 is unstable and slides
along the interface between fracture zone and layer 5# for the both cases of ϕ=100 and
ϕ=150 when the seismic loading is employed, but the RM1 is nearly stable at the case of
ϕ=150. It is shown that the simulated results are different with the traditional LEM and
easily to approach the realistic conditions of the slopes. The instability focuses on the
fracture zone and grows at the beginning of the dynamic loading, then the other blocks
slide following the movement of the fracture zone. We can find that the fracture zone is
the key effect on the slope stability in the open-pit mining engineering and should be
paid attention more seriously. Furthermore, field investigation is important to find the
location of the fracture zone and the properties of the joints of the fracture zone in order
to study open-pit mining slope truly and correctly.
Figure 6.7 A stochastic horizontal seism acceleration.
Table 6.2 Input parameters for the simulation of the modeling.
Physical property Parameter
Unit weight (kN/ m3) 26.0
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 5 10 15 20
Acc
eler
atio
n (
g)
Time (s)
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Young’s modulus (GPa) 1.0
Possion’s ratio 0.2
Internal friction angle(deg.) 20.0
Cohesion (MPa) 5.0
Tensile strength (MPa) 0.0
Joint normal stiffness (GPa) 1.0
Joint shear stiffness (GPa) 0.5
To investigate the instability of the fracture zone under earthquake loading, the
measure points selected in the fracture zone and the displacements of them at both cases
of ϕ=100 and ϕ=150 are presented in Figs 6.10 to 6.15, respectively, in which the
original NMM, proposed ENMM and E-I NMM are used to simulate the open-pit
modeling. It is noted that the different time steps are chosen to the explicit and implicit
algorithms with respect to the numerical properties of the time integrations. We select
0.5 ms to simulate the slope in the ENMM, 5.0 ms in the NMM and 0.5 ms in the first
explicit part of the E-INMM, 5.0 ms in the second implicit part of the E-INMM. The
simulated results of the measure point are nearly identical in the three approaches with
both case of ϕ=100 and ϕ=150. It is verified that the proposed E-INMM satisfies the
computational accuracy comparing with the original NMM. We can find that the slope
is instable at the case of ϕ=100 whether static or dynamic states, but the slope
approaches to be stable after the seismic loading at the case of ϕ=150. Thus, the
fractured zone should be taken into account to the design of the open-pit slope to
improve the stability of slope.
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Figure 6.8 Simulation results for Refined Model 1 (Total time: 20s): (a) ϕ=100; (b) ϕ=150.
Figure 6.9 Simulation results for Refined Model 2 (Total time: 20s): (a) ϕ=100; (b) ϕ=150.
(a) ϕ=100
(a) ϕ=100
(b) ϕ=150
(b) ϕ=150
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(a) ϕ=100;
(b) ϕ=150.
Figure 6.10 Measured point 1 with model 1: (a) ϕ=100; (b) ϕ=150.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 5 10 15 20
Displacement (m
)
Time (s)
NMM
ENMM
E‐INMM
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20
Displacement (m
)
Time (s)
NMM
ENMM
E‐INMM
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(a) ϕ=100;
(b) ϕ=150.
Figure 6.11 Measured point 2 with model 1: (a) ϕ=100; (b) ϕ=150.
0
2
4
6
8
10
12
0 5 10 15 20
Displacement (m
)
Time (s)
NMM
ENMM
E‐INMM
0
0.2
0.4
0.6
0.8
0 5 10 15 20
Displacement (m
)
Time (s)
NMM
ENMM
E‐INMM
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(a) ϕ=100;
(b) ϕ=150.
Figure 6.12 Measured point 3 with model 1: (a) ϕ=100; (b) ϕ=150.
0
50
100
150
200
0 5 10 15 20
Displacement (m
)
Time (s)
NMM
ENMM
E‐INMM
0
1
2
3
0 5 10 15 20
Displacement (m
)
Time (s)
NMM
ENMM
E‐INMM
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(a) ϕ=100;
(b) ϕ=150.
Figure 6.13 Measured point 1 with model 2: (a) ϕ=100; (b) ϕ=150.
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20
Displacement (m
)
Time (s)
NMM
ENMM
E‐INMM
0
0.2
0.4
0.6
0.8
0 5 10 15 20
Displacement (m
)
Time (s)
NMM
ENMM
E‐INMM
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(a) ϕ=100;
(b) ϕ=150.
Figure 6.14 Measured point 2 with model 2: (a) ϕ=100; (b) ϕ=150.
0
5
10
15
20
0 5 10 15 20
Displacement (m
)
Time (s)
NMM
ENMM
E‐INMM
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
Displacement (m
)
Time (s)
NMM
ENMM
E‐INMM
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(a) ϕ=100;
(b) ϕ=150.
Figure 6.15 Measured point with model 2: (a) ϕ=100; (b) ϕ=150.
With respect to the efficiency of the proposed algorithms, CPU time is taken into
consideration to check the computational cost of the algorithms. In order to track CPU
time for each time step, a high resolution timer function which can measure up to
1/100000th of a second is added to keep track of CPU time for each time step during the
analysis. All three algorithms are run on the same computer with the system
configuration: processor speed = 4.0 GHz and RAM = 4.0 GB. As represented in Table
6.3, the proposed E-I algorithm is more efficient comparing the explicit and implicit
algorithms in the model of RM1 and RM2 with both cases of ϕ=100 and ϕ=150. In
particular, E-I algorithm can be considered as one computational criteria for the large
0
100
200
300
0 5 10 15 20
Displacement (m
)
Time (s)
NMM
ENMM
E‐INMM
0
40
80
120
160
0 5 10 15 20
Displacement (m
)
Time (s)
ENMM
ENMM
E‐INMM
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scale engineering as it combines the merits of both the explicit and implicit algorithms
in terms of accuracy and efficiency of the computations dramatically.
Table 6.3 CPU cost for the different study cases (hr.).
6.6 SUMMARY
The temporal coupled explicit and implicit algorithm for the numerical manifold
method (NMM) is proposed in this chapter. The time integration schemes, transfer
algorithm, contact algorithm and damping algorithm are studied in the temporal coupled
E-I algorithm to merge both merits of the explicit and implicit algorithms in terms of
efficiency and accuracy. Then, some numerical examples are simulated using the
proposed coupled algorithms, in which one calibration example is studied with respect
to the coupled temporal based on the dual cover system. One numerical example of
open-pit slope seismic stability analysis using the coupled E-I algorithm is investigated
as well. The simulated results are well agreement with the implicit and explicit
algorithms simulations, but the efficiency is improved evidently. It is predicted that the
couple E-I algorithm proposed in the present paper can be applied into larger scales
engineering systems to combine the merits of both the implicit and explicit algorithms
in the NMM.
Study
Case
10° 15° ICPU ECPU IECPU ICPU ECPU
IECPU
Model 1 1.467 1.376 0.882 1.816 1.515 0.911
Model 2 1.485 1.501 0.875 1.271 1.326 0.809
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CHAPTER 7. THE SPATIAL COUPLED EXPLICIT-
IMPLICIT ALGORITHM FOR DYNAMIC PROBLEMS
USING THE NUMERICAL MANIFOLD METHOD
7.1 INTRODUCTION
In dynamic analysis, there are two general classes of algorithms can be referred:
explicit and implicit. The explicit algorithm in spatial discretization using finite or
discrete elements is very widespread, especially for contact or impact problems. Since
an explicit algorithm allows the implementation of large-scale models with a relatively
affordable computational cost, and the conditional stability property is not a matter if
the time step satisfies the equation of the critical time step. However, the smallest
element size in the meshes determines the time step for the whole system, thus it is
necessary to develop an approach, in which the explicit algorithm can be employed
involving different time steps in different sub-domains for the system. On the other
hand, the implicit algorithm can use larger time steps in terms of numerical
unconditionally stable, but for the complex large-scale problems, the solution of system
equations involving too many thousand degrees of freedom (DOFs) can be costly using
the implicit algorithm. Therefore, more efficient computational algorithm is required to
develop for the dynamic analysis, in which some problems applying implicit algorithms
are very efficient and others employing explicit algorithms are very efficient.
In the computations, the explicit algorithms tend to be inexpensive per step and
require less storage than implicit algorithms, but numerical stability requires that small
steps be employed, thus, they generally used for highly non-linear problems with many
DOFs (Subbaraj and Dokainish 1989a, 1989b; Yang et al. 1995). On the contrary, the
implicit algorithms tend to be numerical stable, but the computational cost per step is
high, and storage requirements tend to increase dramatically with the contacts between
elements and these DOFs. Thus, they are suited to simulate the lower dynamics
problems with less non-linearities, resulting in more numerical stability and accuracy
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(Yang et al. 1995; Gelin et al. 1995; Sun et al. 2000). To take advantage of the merits of
implicit and explicit algorithms, many methods have been developed in temporal and
spatial discretization, in which it is attempted to simultaneously achieve the maximum
contributions of both classes of algorithms. Belytschko and Mullen (Belytschko and
Mullen 1976, 1978b) proposed an explicit-implicit (E-I) nodal partition and proved the
conditional stability of E-I partitions using energy methods and represented the time
step is limited strictly by the maximum frequency in the explicit partition of the mesh.
Hughes and Liu (1978) proposed an alternate element-by-element E-I partitions, in
which a similar stability condition is proven for the algorithm. Liu and Belytschko
(1982) proposed a general mixed time E-I partition procedure which permits different
time steps and different integration methods to be used in different parts of the semi-
discrete equations. Belytschko and Mullen (1978a) proposed a multi-time step
integration method involving different time steps in different zones of the model, in
which the nodal partition approach is employed for E-I systems and linearly interpolated
displacements at the interface. It differs from the referred mixed methods, which consist
of defined zones where different algorithms apply, but with a single time step defined
for the whole domain. Various other improvements in transient algorithms have also
been achieved referred in (Hughes, Pister, and Taylor 1979; Muller and Hughes 1984;
Smolinski, Belytschko, and Neal 1988; Miranda, Ferencz, and Hughes 1989)
(Belytschko and Lu 1992; Sotelino 1994; Smolinski, Sleith, and Belytschko 1996;
Gravouil and Combescure 2001).
To the time integration schemes, Newmark-β family methods (Newmark 1959,
1965) are adopted. When different parameters of β and γ are taken into account, the
different algorithms between the explicit and implicit schemes can be selected to
simulate the corresponding dynamic problems. When the two parameters of β and γ are
identical for all sub-domains, the distinction between explicit and implicit where we
have considered is that the explicit uses a diagonal mass matrix and the implicit applies
a consistent inertia matrix (Liu and Belytschko 1982), respectively. These algorithms
are used for first-order systems (Belytschko, Smolinski, and Liu 1984) and second-order
systems (Belytschko et al. 1979). In the former case, a stability study of the algorithm
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with mixed method is proposed for the E-I algorithm, but no extension to this case with
element partition is proposed; for the latter case, there is no general stability analysis in
the E-I algorithm with the mixed method. Moreover, other techniques can be found to
couple arbitrary numerical schemes of the Newmark family in each sub-domain with
different time-steps (Combescure et al. 1998; Combescure and Gravouil 1999).
In the present study, the numerical manifold method (NMM) is taken into account
to couple the both algorithms. The traditional NMM is original proposed by Shi (1991,
1992, 1996, 1997) based on topological manifold and unifies both finite element method
(FEM) and discontinuous deformation analysis (DDA) (Shi 1988), which employs the
implicit time integration and open-close contact iteration for the modelling of
complicated dynamic problems in rock engineering and rock mechanics. However, the
coupled global equations may involve many thousand DOFs, especially such more
discontinuous contact problems that nonlinear problems encountered, the computational
cost can be increased dramatically. This has motivated research toward developing more
efficient computer algorithms for the NMM analysis. Thus, the choice of an appropriate
algorithm is an essential criterion to ensure efficiency and robustness of the numerical
simulations, in which the difficulty resides in being able to combine robustness,
accuracy, stability and efficiency of the algorithms. In this study, we present a coupled
E-I algorithm for the NMM based on the onefold cover system, in which different time
steps, time integration schemes and contact algorithms are applied in spatial
discretization. To illustrate the proposed algorithm systematically, this chapter is
basically organized by 5 sections. In section 7.2, the explicit and implicit algorithms are
investigated based on the dual cover system of the NMM. In section 7.3, an alternative
coupled E-I algorithms is proposed with respect to the spatial discretization, in which
onefold cover system is proposed and derived in detail, respectively. Then, in section
7.4, some calibration examples and numerical simulations are studied to validate the
coupled E-I algorithms in terms of the accuracy and efficiency. Section 7.5 concludes
and summaries the spatial coupled E-I algorithms.
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7.2 COUPLED ALGORITHM
7.2.1 Summary of equations of motion and time integration
Generally, the equation of motion can be derived in the strong form as
vb (7.1)
in which is the symmetric Cauchy stress tensor, is the gradient operator, is the
density of material, b is the body forces and v is the velocity, respectively. Constitutive
equations are required to couple the Cauchy stress tensor and the density to the
kinematics of the deformation. Eq. (7.1) can be multiplied by weight function w and be
integrated over an arbitrary material domain . We employ the principle of virtual
work and rewrite Eq. (7.1), the corresponding weak form can be obtained as
t
tdwbdwdwdvw : (7.2)
where nt is the traction vector, n is the outward normal vector of the boundary, t
is the traction boundary. As is shown in Fig. 7.1, an elastic body with a traction vector
t is taken into account. The boundary ctu is constituted by the prescribed
displacement boundary u , the traction boundary
t and discontinuous surface c ,
respectively. And the boundary needs to satisfy
TN ttt on t (7.3)
uu on u (7.4)
where Nt and
Tt are the normal and tangential vector components of t along the
boundary t when the contact problem is considered, u and u are the displacement
vector and the prescribed displacement vector on u , respectively.
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Figure 7.1 An elastic body with a traction vector t.
Thus, using the variational principle (Washizu 1982), Eq. (7.2) can be rewritten as
0)( ut
duuutdubdVudVdVuu TTTTT (7.5)
where is the Green strain tensor of the second order, is the first order variation, and
is the penalty value of the penalty meth (Zienkiewicz, Taylor, and Zhu 2005) to treat
the contact problem, respectively. As the MCs are not necessarily consistent with the
boundaries and the DOFs are defined on the PCs rather than nodes as in the FEM, thus
the penalty value can not be treated by prescribing DOFs of the FEM.
As similar as the principle in FEM, the whole material domain can be discretized
into a number of elements in the NMM. Following the approach by Galerkin, the shape
function matrix ][T is applied to substitute Eq. (7.5), the governing equations for a
discrete model can be repressed in the form
}{}{}]{[ intFFuM ext (7.6)
where
dxdyTTdxdyTTMM T
eV e
Te
ee
e
][][)][][(][][ (7.7)
tdTbdxdyTfF TT
eextext ][][}{}{ (7.8)
dxdyBfF T
e
][}{}{ intint (7.9)
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in which ][M is the mass matrix, }{ extF the external force vector, }{ intF the internal force
vector, ][B the strain displacement matrix and }{u is the acceleration vector,
respectively. In the dynamic analysis, an increment of step-time is applied into the
continuous and discontinuous deformation analyses, linear elastic problems can be
considered as
}]{[}{ int uKF (7.10)
where ][K is the stiffness matrix, in which
dxdyBEBKK T
ee ]][[][][][ (7.11)
where ][E is the elastic matrix, thus, we can obtain the well-known equation
}{}]{[}]{[ extFuKuM (7.12)
From Eqs. (7.7) and (7.11), it is noted that ][M and ][K are symmetric, whereas they are
also banded and sparse as the local property of matrix ][T of the MCs in the NMM, and
it requires more computer memory with the assembly of global stiffness matrices for
solving the governing equations. These properties inherit that of the FEM as the
implement of the element in the computation.
7.2.2 Coupled explicit-implicit algorithm in the NMM
In general, there are two classes of algorithms for dynamic problems: implicit and
explicit. Discussions of the advantages and disadvantages of the implicit and explicit
algorithms can be referred (Belytschko and Mullen 1977, 1978; Hughes et al. 1979;
Muller and Hughes 1984; Miranda et al. 1989), in which the elements are partitioned
into three types, explicit-explicit (E-E), implicit-implicit (I-I) and explicit-implicit (E-I)
algorithms; the nodes are partitioned into two types, explicit and implicit. We can
consider two sub-domains partitioned by the elements or the covers (i.e. nodal stars),
which distinction is vital for the interface since each of the methods is specific in the
way the interface is treated. Besides, we assume Newmark- methods with two
characteristic parameters and differently for the different sub-domains between the
explicit and implicit algorithms.
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Sub-domain partition algorithm in the coupled E-I method
In the present study, we assume the element partition method and MCs partition
method to couple the E-I algorithm for the dynamic problems. As shown in Fig. 7.2,
when the element partition method is taken into account for the modeling of the
systems, the whole domain is divided into two classes elements: explicit elements,
denoted by 1Ee , 2
Ee , 3Ee , 4
Ee , …, 21Ee , 23
Ee ; and implicit elements, denoted by 1Ie , 2
Ie , 3Ie ,
4Ie , …, 19
Ie , 20Ie , respectively. In Fig. 7.2(a), we can find interface between the Explicit
and the implicit elements, e.g. 4Ie and 1
Ee , 5Ie and 2
Ee , 11Ie and 6
Ee , 10Ie and 5
Ee , 16Ie and 11
Ee , 15Ie
and 17Ee , 19
Ie and 10Ee , 20
Ie and 18Ee , and ach interface is in conjunction with a pair of explicit
and implicit elements. As the ME is based on the MCs system, we suggest each MC is
split into explicit MC and implicit MC accordingly. Taking an example of interface 16Ie
and 11Ee , we assume the original indexes of the MCs are
iC , jC and
kC to form a ME, then
the coupled E-I algorithm based on element partition method is used to generate a new
pair MEs: explicit element 11Ee and implicit element 16
Ie . Furthermore, the new updated
indexes of the MCs accompany the generation of the new MEs, iC separates into E
iC and
IiC ,
jC becomes EjC and I
jC , kC develops into E
kC and IkC , respectively. Following the
above procedure, the explicit and implicit MCs are formed to apply the corresponding
algorithms, respectively. When the MCs partition method is considered, one domain is
covered by the separated explicit MCs and implicit MCs, which becomes more efficient
to solve the equations of systems explicitly.
Let notation of be the set of all MEs, 46,...,3,2,1e . The assembly of the mass
matrix ][M can be represented as
e
eMM ][][ (7.13)
where ][ eM is the element contribution matrix. Then, let I and
E be the subsets of
corresponding to the implicit and explicit elements. We can get
EI (7.14)
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EI (7.15)
Consequently, substituting into Eq. (7.13) can be rewritten as
EI e
ee
e MMM
][][][ (7.16)
Accordingly, the other items of the equations can be expressed as
EI e
ee
e CCC
][][][ (7.17)
EI e
ee
e KKK
][][][ (7.18)
EI e
ee
e FFF
(7.19)
then, the explicit and implicit terms can be separated and solved based on different time
integration algorithms.
When the MCs partition method is taken into account, as can be seen in Fig. 7.2(b),
the whole domain is covered by the explicit MCs, denoted by 1En , 2
En , 3En , 4
En , …, 11En ,
12En , and the implicit MCs, denoted by 1
In , 2In , 3
In , 4In , …, 13
In , 14In , respectively. The
MCs partition method is more convenient to rearrange the index number of the original
MCs in contrast to that of the element partition method, and is efficient to solve the
coupled and uncoupled equations of motions in the system. Thus, the coupled E-I
algorithm is inclined to the MCs partition to simulate the continuous problems in the
NMM, so that the computational cost can be efficient saved. However, to the
discontinuous problems, the contact positions change along with the different contact
conditions. Neither the element partition method nor the MCs partition method can treat
the contact problems efficiently. Thus, the more efficient method is required to develop
to solve the discontinuous problems using the coupled E-I algorithm in the NMM.
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(a) Element partition method
(b) MCs partition method
Figure 7.2 Sub-domain partition algorithm in the coupled E-I method: (a) Element partition
method; (b) MCs partition method.
Contact algorithm in the coupled E-I algorithm
The NMM treating the contact problems inherits the DDA with the same
methodology, details of the discussion can be referred in (Shi 1988, 1991). In general,
the contract problems are inherently nonlinear and irreversible, and then a maximum
Implicit MCs
Explicit MCs
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displacement ratio is prescribed in the NMM so that the displacements of all the points
within the problem domain are less than the prescribed limit value. Meanwhile, the
contact zones may change considerably within a time step, in which a surface point of a
domain may contact any other section of the surface of another domain, and such a
point can even contact itself surface of the domain. Thus, the algorithm for detecting
contact is of the utmost importance.
In the traditional NMM, there are three possible contact conditions are assumed:
open, slide and lock. At the beginning of each time step, the contact conditions for all
the contact pairs are supposed as lock except which carries forward the previous time
step. When the contact condition of lock is taken into account using the implicit
algorithm, the normal and shear springs are applied to resist the normal penetration of
the vertex and tangential displacement between the vertex and the surface, respectively.
Consequently, all the possible contact pairs satisfy the contact conditions convergence
by means of the OCI technique, the computation then proceeds to the next time step. On
the other hand, the explicit algorithm employs the normal and shear springs explicitly to
compute the contact forces when the contact pairs are detected within each time step.
This approach saves the computational cost to some extent, whereas the explicit
algorithm is conditionally stable, the step time t is required to satisfy the
corresponding numerical property. As shown in Fig. 7.3, we assume two domains:
explicit domain and implicit domain are in contact condition, the explicit and implicit
MCs overlap the explicit and implicit domains, respectively. When the contact is
detected, the contact zone is formed in junction with both the explicit MCs and implicit
MCs. There are five common MCs in the box of the contact zone, which brings the
trouble to determine which algorithm applies to the common MCs. One alternative
scheme can be considered to rearrange the index number of the five common MCs to
generate the new explicit and implicit MCs as the above discussions.
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Figure 7.3 Explicit and implicit covers for the contact problem in the NMM.
7.3 AN ALTERNATIVE APPROACH FOR THE COUPLED E-I
ALGORITHM
Since the coupled E-I algorithm involves the selection and optimization of the
partition methods as presented in the previous section, neither the element nor MCs
partition methods can be considered as one efficient method to deal with the complex
and nonlinear contact problems due to the distinguish characteristic of the MC cover
system in the NMM. Each MC (denoted by “star”) is not always corresponding to the
node as in the FEM when the interface occurs between the explicit and implicit
algorithms. Thus, it is time-consuming to rearrange the index number of the MCs no
matter the element or MCs partition method is employed in the coupled E-I algorithm.
In the study, we give one alternative approach to simulate the contact problems of the
coupled E-I algorithm.
7.3.1 Onefold cover system
In order to solve contact problems efficiently, we propose onefold cover system to
describe the contact between the explicit and implicit domains. As shown in Fig. 7.4, a
Implicit cover Explicit cover
Implicit & explicit cover
Contact domain
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onefold cover system is constructed to couple the explicit and implicit algorithms, in
which one upper triangle cover system is built to combine the explicit and implicit
covers, and the constructed upper triangle is required to cover the whole domains. In
order to represent the whole domains, one box approach is assumed by searching the
maximum and minimum coordinates in x and y directions, and then the four vertices
coordinates can be determined as ),( minmin yx , ),( minmax yx , ),( maxmax yx and ),( minmin yx . Thus,
each ME represents the domain itself, and which defined by a group of index numbers
of MCs. In Fig. 7.4, there are r MEs are taken into account in the system based on
onefold cover system, in which n implicit MEs and )( nr explicit MEs are formed
based on the area algorithm as discussed in the previous section. The implicit elements
constituted by the MCs )1(Ii , )2(
Ii , )3(Ii , …, )(n
Ii ; )1(Ij , )2(
Ij , )3(Ij , …, )(n
Ij ; and )1(Ik , )2(
Ik ,
)3(Ik ,…, )(n
Ik , respectively. Accordingly, the explicit elements are built by MCs )1(Ei , )2(
Ei ,
)3(Ei , …, )( nr
Ei ; )1(
Ej , )2(Ej , )3(
Ej , …, )( nrEj ; and )1(
Ek , )2(Ek , )3(
Ek ,…, )( nrEk , respectively.
Based on the constructed onefold cover system, the displacements and deformations of
the explicit and implicit elements can be obtained using the corresponding algorithms
respectively. Furthermore, the contact problems between the explicit and implicit
elements can be solved efficiently.
Figure 7.4 Construction of onefold cover in the proposed NMM.
, , ...,
,
, , ...,
,
, , ..., ,
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7.3.2 Contact algorithm on the onefold cover system
Basically, there are three kinds of contact types can be detected using the area
algorithm in the onefold system: EE , II and IE . As can be seen in Table 7.1,
when the different contact types are detected, the different algorithms with the
corresponding step time t will be applied to solve the equations of the system. The
explicit and implicit algorithms can be employed to deal with EE and II ,
respectively. EE is proposed to extend the contact type of EE with different step
time in the different domains using the explicit algorithms only, in which the whole
domains are divide into two sub-domains using the element partition method. One
partition is integrated with a time step /t , where is an augmenter greater or equal
to 1, the other with a time step t . This mixed method of the explicit algorithm is more
efficient for the linear system equations, but the interface interpolation for the
integration does not guarantee that all of the difference equations are satisfied at all
nodes. Thus, the numerical stability is required to investigate once more, which increase
the computational cost on the other hand, and the details of EE scheme can be seen
in reference (Belytschko et al. 1979). Liu and Belytschko (1982) have proposed one
mixed algorithm of IE using finite elements for transient analysis, in which an
effective implicit integration procedure with t is used in the implicit domains and an
explicit algorithm with t is applied in the explicit partitions to simulate fluid-structure
system, respectively.
Table 7.1 Contact types between two domains.
Denotation Contact type Explicit Implicit
EE Explicit-Explicit with t --
II Implicit-Implicit -- with t
IE Explicit-Implicit with t with t
EE Explicit-Explicit with /t , t --
IE Explicit-Implicit with t with t Notes: referred in (Belytschko et al. 1979); referred in (Liu and Belytschko 1982).
After selection of the coupled E-I algorithms, the corresponding integration scheme
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will be adopted to form the global coefficient matrix, in which the stiffness sub-matrix,
damping item and mass item are assembled to satisfy the corresponding algorithm for
the equations of system. To illustrate these properties of the coupled E-I algorithm
based on onefold cover system, the coupled E-I coefficient matrix with r elements as
shown Fig. 7.5 is considered here. Since the assembled stiffness matrix and damping
matrix both have a sparse band-profile framework in the implicit algorithm, the
coefficient sub-matrix ][K is formed to the implicit algorithm. Accordingly, the diagonal
coefficient sub-matrix ][M is generated to the explicit algorithm, and the whole matrix
structure is symmetric and positive definite when the coupled E-I algorithm is used
based on onefold cover system. We give an example of global coefficient matrix of
)( rr , in which each element is overlapped by three MCs of sub-matrix )66( . From the
index number of elements 1 to 1s and )1( 2 s to 3s are composed of implicit elements, the
I-algorithm is used to form sub-matrix ][K for the equations of system; on the other
hand, from )1( 1 s to 2s and )1( 3 s to r are constituted by explicit elements, the E-
algorithm is applied to generate sub-matrix ][M into the global coupled coefficient
matrix. In order to improve the computational efficiency, one alternative technique is
applied to divide the global coupled coefficient matrix into implicit coefficient matrix
and explicit one, which can be re-assembled in Figs. 7.5(b) and (c). For each case, the
corresponding algorithm with time step t is applied to solve the equations directly, in
which the formation of the implicit coefficient matrix is minimized by optimizing the
MCs index. In addition, non-zero storage and SOR iteration technique are applied to
save the computational cost; the explicit algorithm employs the central different method
to solve the equations explicitly.
For the contact problems in the onefold cover system, there are two alternative
contact schemes can be considered as shown in Fig. 7.6 to the contact type of IE :
uniform contact algorithm and separated contact algorithm. In the initiation, each
domain area is calculated using the simplex integration, and the judge criteria of area
algorithm is applied to partition the whole domains to explicit domains and implicit
domains. Then, the corresponding contact algorithms and integration schemes are used
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to form the matrix items of the equations of systems. The contact algorithm in the
original NMM uses the penalty technique to restrict the elements penetration at the
normal and shear directions, and the normal and shear springs are added to form tangent
stiffness matrices for the implicit Newmark- method time integration. Resort to the
OCI, the contact convergence is achieved finally, which is a time-consuming work.
Here, we define/assume the explicit contact algorithm, in which the contact springs are
added explicitly when the contact is detected. The contact forces are formed for the
explicit Newmark- method time integration, which is more efficient to solve the
contact problems without considering the OCI problems. When the uniform contact
algorithm is used (see Fig. 7.6(a)), implicit or explicit contact algorithm is chosen as the
uniform algorithm to fo6m the contact matrix and contact force, then solve the
equations of system until the contact convergence. On the other hand, the separated
contact algorithm (see Fig. 7.6(b)) is seemed to be more efficient to simulate the
coupled E-I contact problems under te demand of computational accuracy. Therefore,
the separated contact algorithm is assumed as the contact algorithm to apply into the
contact computations.
(a) Global coefficient matrix for the coupled E-I algorithm;
12 3 .
Symmetric
I-algorithm
E-algorithm
I-algorithm
E-algorithm
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(b) Implicit coefficient matrix; (c) Explicit coefficient matrix.
Figure 7.5 Coefficient matrix of the coupled E-I algorithm: (a) Coupled E-I algorithm global
coefficient matrix; (b) Implicit coefficient matrix; (c) Explicit coefficient matrix.
(a) Shared contact algorithm; (b) Separated contact algorithm.
Figure 7.6 Flowchart of two alternative contact schemes: (a) shared contact algorithm; (b)
separated contact algorithm.
7.3.3 Contact matrices of the coupled E-I algorithm
In the study, the separated contact algorithm is employed as discussed in the
previous section. Once the penetration is detected, the contact matrices are added to the
12 3.
Symmetric
I-algorithm
E-algorithm
yes no
Solve equations
Calculate domain area
Judge criteria
Explicit domain Implicit domain
Uniform algorithm
yes no
Solve equations
Calculate domain area
Judge criteria
Explicit domain Implicit domain
Explicit algorithm Implicit algorithm
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corresponding contact elements. Here, we focus on the contact matrices of the coupled
E-I algorithm. As illustrated in Fig. 7.7, one penetration is searched between the explicit
domain E and the implicit domain I from the time step
0t to )( 0 tt , the
corresponding contact pairs of Ee and Ie are recognized as well. The vertex of Ee is
assumed as 1P ),( 11 yx ; and the penetration edge
32 PP in the implicit element Ie is defined
as well, in which the coordinates of the vertices 2P and 3P are assumed as ),( 22 yx and
),( 33 yx . Consequently, the normal contact matrix, shear contact matrix and friction
matrix can be derived as follows
Figure 7.7 Contact representation in the coupled E-I algorithm.
Normal Contact Matrix
It is assumed that vertices 1P , 2P and 3P rotate in the same sense as the rotation of
ox to oycoordinate axis, the normal penetration distance is assumed as nd , which can be
written as
3333
2222
1111
1
1
11
vyux
vyux
vyux
lldn
(7.20)
where
223
223 )()( yyxxl (7.21)
at at
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3
31221
2
23113
1
123320 )()()(
v
uxxyy
v
uxxyy
v
uxxyyS (7.22)
33
22
11
0
1
1
1
yx
yx
yx
S (7.23)
in which the step displacements ),( ii vu , 3,2,1i is small as the results of small time
step, thus contact distance nd is small from the definition of the contacts. And,
})}{,({ 111
1 EE DyxTv
u
(7.24)
})}{,({ 222
2 II DyxTv
u
(7.25)
})}{,({ 333
3 II DyxTv
u
(7.26)
Then, the normal contact distance nd is rewritten as
}{}{}{}{0 ITIETEn DGDH
l
Sd (7.27)
where
23
3211
)3(
)2(
)1(
)},({1
}{xx
yyyxT
lH
H
H
H TEi
E
E
E
E (7.28)
12
2133
31
1322
)3(
)2(
)1(
)},({1
)},({1
}{xx
yyyxT
lxx
yyyxT
lG
G
G
G TITI
I
I
I
I (7.29)
Then, the normal contact spring with the stiffness of nk is introduced between the
contact point and the implicit element. The potential energy of the normal spring is
2
02 }{}{}{}{22
I
jTIE
iTEn
nn
n DGDHl
Skd
k (7.30)
Minimising n by the derivatives in terms of }{D , the four 66 sub-matrices of ][K
and two 16 sub-matrices of load matrices can be obtained, which are
][}}{{ ))(()()(E
srTE
sEr KHHp (7.31)
][}}{{ ))(()()(IEsr
TIs
Er KGHp (7.32)
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][}}{{ ))(()()(EIsr
TEs
Ir KHGp (7.33)
][}}{{ ))(()()(I
srTI
sIr KGGp (7.34)
}{}{ )()(0 E
rErn FH
l
Sk
(7.35)
}{}{ )()(0 I
sIsn FG
l
Sk
(7.36)
where 3,2,1, sr are the order number of MCs of the corresponding explicit and
implicit elements, respectively.
Shear contact matrix
As shown in Fig. 7.7, the assumed contact point 0P ),( 00 yx is on the edge
32 PP in the
explicit element. The shear spring is introduced along the edge on the direction of32 PP ,
and connects vertices 1P and
0P . Then, the shear displacement sd of 0P and
1P along the
edge32 PP can be repressed using inner product form as
0
03232
1
12323
03210 )(
1)(
1ˆ1
v
uyyxx
lv
uyyxx
ll
SPPPP
ld s
(7.37)
where
23
2301010 )(ˆ
yy
xxyyxxS (7.38)
As assumed that point 1P belongs to the explicit element and point 0P belongs to the
implicit element, so Eq. (7.37) can be rewritten as
}{}ˆ{}{}ˆ{ˆ
0 ITIETEs DGDH
l
Sd (7.39)
where
23
2311
)3(
)2(
)1(
)},({1
ˆ
ˆ
ˆ
}ˆ{yy
xxyxT
lH
H
H
H TE
E
E
E
E (7.40)
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32
3200
)3(
)2(
)1(
)},({1
ˆ
ˆ
ˆ
}ˆ{yy
xxyxT
lG
G
G
G TI
I
I
I
I (7.41)
Then, the shear contact spring with the stiffness of sk is introduced on the direction
32 PP connects vertices 1P and
0P . The potential energy of the shear spring is
2
02 }{}ˆ{}{}ˆ{ˆ
22
ITIETEs
ss
s DGDEl
Skd
k (7.42)
Minimising s by the derivatives in terms of }{D , the four 66 sub-matrices of ][ K
and two 16 sub-matrices of load matrices can be obtained, which are
][}ˆ}{ˆ{ ))(()()(E
srTE
sEr KHHp (7.43)
][}ˆ}{ˆ{ ))(()()(IEsr
TIs
Er KGHp (7.44)
][}ˆ}{ˆ{ ))(()()(EIsr
TEs
Ir KHGp (7.45)
][}ˆ}{ˆ{ ))(()()(I
srTI
sIr KGGp (7.46)
}{}ˆ{ˆ
)()(0 E
rErs FH
l
Sk
(7.47)
}{}ˆ{ˆ
)()(0 I
sIss FG
l
Sk
(7.48)
where 3,2,1, sr are the order number of MCs of the corresponding explicit and
implicit elements, respectively.
Friction Matrix
When the vertex 1P relative to
0P slides along the edge32 PP , the friction force can
be calculated based on the Coulomb’s law in the case of friction angle is not zero.
Then, we can get
)tan( gnnn sdkF (7.49)
where gns is a Sign function to represent the movement of 1P relative to
0P along the
vector of 32 PP .Then, the potential energy E
f and If of friction force F on the explicit
and implicit elements can be obtained, respectively, which are
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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23
2311
23
2311 )},({}{)(
yy
xxyxTD
l
Fyy
xxvu
l
F TETEEf
(7.50)
23
2300
23
2300 )},({}{)(
yy
xxyxTD
l
Fyy
xxvu
l
F TITIIf
(7.51)
Minimising Ef and I
f by the derivatives in terms of }{D , two 16 sub-matrices of
friction load matrices can be obtained, respectively, which can be repressed as
}{}~
{ )()(Er
Er FHF (7.52)
}{}~
{ )()(Is
Is FGF (7.53)
in which
23
23
11)(
11)()( ),(0
0),(1}
~{
yy
xx
yxw
yxw
lH E
r
ErE
r (7.54)
23
23
00)(
00)()( ),(0
0),(1}
~{
yy
xx
yxw
yxw
lG I
s
IsE
s (7.55)
where 3,2,1, sr are the order number of MCs of the corresponding explicit and
implicit elements, respectively; ),( 11)( yxw Er
and ),( 00)( yxw Is
are the weight functions in
the explicit and implicit contact elements, respectively.
7.3.4 Spring stiffness problems
The selection of spring stiffness k is important to the accuracy of the solutions in
the contact problems. Cheng (1998) observed that the choice of contact spring stiffness
nk and/or sk or time-step size t has a significant effect on convergence. Thus, the
stiffness of the spring plays an important role in the solution and the quality of solution
depends on the range of stiffness values chosen. If the value of k is too small, the
penetration distance becomes large so that the constraints are poorly satisfied, which
may result in the following conditions (Shi 1992):
The closed contacts can not be detected and transferred to the next step;
The stresses in the material may be reduced; and
The deformation along the contact edges can be incorrect.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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For large values of k , the diagonal items of the global stiffness matrix making it
numerically ill-conditioned. Furthermore, the OCI may not converge and the calculated
contact forces and displacements may not be realistic. Lin et al. (1996) has proposed the
augmented Lagrangian method, in which the value of k can be variable and does not
have to be a very large number, but the solution requires more iterations to reach contact
convergence increasing the computational cost.
For the explicit algorithm, it is noted that the spring stiffness of nk and/or sk are
difficult to obtain. Cundall emphasized the importance of using realistic values for the
sk by demonstrating how the ratio of sk to nk dramatically affects the Poisson
response of a rock mass (Itasca 1993). Moreover, some guidelines are suggested
estimate the contact spring stiffness for a reasonable analysis, in which nk and sk
should be kept smaller than ten times the equivalent stiffness of the stiffest
neighbouring zone of elements adjoining the contact interface, and it can be written as
min
3/4max0.10
dk
(7.56)
where and are the bulk and shear modulus, respectively; and mind is the smallest
width of the domain adjoining the contact interface in the normal direction as shown in
Fig. 7.8. If the stiffness value is greater than 10 times the equivalent stiffness, the
solution time of the model will be significantly longer, which increases the
computational cost on the side.
Contact interface
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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7.164
Figure 7.8 Domain profile suggested in stiffness estimation (Itasca 1993).
7.4 NUMERICAL EXAMPLES
In order to investigate the validity of the proposed algorithms, some numerical
examples are simulated in terms of temporal and spatial respects, respectively. In the
simulations, the proposed coupled temporal and spatial algorithms for the NMM are
calibrated between the simulation results and analytical solutions, firstly. Then, a
examples of open-pit slope stability analysis are taken into account using the proposed
E-I algorithms to further extend the current NMM, which represent the potential of the
proposed E-I algorithms for simulating the larger scale projects in the further research.
7.4.1 Calibration of the spatial coupled E-I algorithm
To calibrate the proposed coupled spatial E-I algorithm in the NMM, one
numerical simulation of dynamic friction mechanism of multi-block system as shown in
Fig. 7.9 is considered. In the present study, the multi-block system with four identical
properties blocks sequentially connect with each other, which system is subjected to a
dynamic pulse loading )(tFv on the top of the block in the vertical direction and a
constant loading )(tFh on the next block in the horizontal direction. In the simulation,
the multi-block system is classified into two kinds of blocks: the top two blocks are
taken into account as explicit blocks, which are denoted by 1E and 2E , simulated using
the proposed explicit algorithm, including the explicit time integration and explicit
contact algorithm. The bottom two blocks are considered as implicit blocks, which are
denoted by 1I and 2I , simulated using the original implicit algorithm, including the
implicit time integration and open-close contact iteration, respectively.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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Figure 7.9 Geometry of the multi-block system.
In the current study, a half-sine pulse loading with amplitude of 1MPa and duration
of 20 ms is considered. The analytical solution of the total vertical displacement can be
referred in (Chopra 2001; Shu et al. 2007), which can be expressed as
TtTtTv
TtTu
TttDtCtBtA
tu
DD
D
DD
)(sin)(
)(cos)(
)sinsin()sincos(
)(
(7.57)
where mkD is the natural frequency of the multi-block system; is the loading
frequency; T =20ms is the duration of pulse loading; A and B are coefficient items
related to the initial condition, C and D are coefficient items related the frequency
aspects, which are can be referred in (Ma, An, and Wang 2009). Once the vertical
displacement is obtained, the friction between the blocks can be expressed as
tan)( tukF nH (7.58)
E1
E2
I1
I2
2m
2m
2m
2m
4m 1m 1m
Fv(t)
Fh(t)
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in which HF is the dynamic friction force in the horizontal direction, nk is the normal
contact spring stiffness of the blocks, and .is the friction angle.
In the simulations, two cases of friction angles are studied: 0 and 045
respectively. The density, Young’s modulus and Possion’s ratio are 2500 Kg/m3, 10
GPa and 0.25, respectively. The simulation results of horizontal displacements between
the two cases can be seen in Fig. 7.10. It can be found that the simulated results as
plotted in Fig. 7.11 of the proposed coupled spatial E-I algorithm of the NMM are well
agreement with the analytical solution. In terms of efficiency, the coupled E-I algorithm
CPU cost is 141s, t is 0.01 ms) is obviously efficient than that of the original NMM
version (CPU cost is 317s, t is 0.01 ms).
(a) case of ϕ=0;
(b) case of ϕ=450.
Figure 7.10 Displacement in the different cases: (a) case of ϕ=0; (b) case of ϕ=450.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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(a) case of ϕ=0;
(b) case of ϕ=450.
Figure 7.11 Displacement of the top 2nd block: (a) case of ϕ=0; (b) case of ϕ=450.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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7.4.2 Simulation of discrete blocks sliding on an inclined surface
In the present simulation, a rock slope stability analysis using the coupled E-I
algorithm is taken into account to investigate validity and applicability of the algorithm.
In the study, three models of the rock slope are considered as shown in Fig. 7.12, in
which explicit algorithm, implicit algorithm and coupled explicit-implicit algorithm are
employed to simulate the rock slope failure process, respectively. It is noted that area
algorithm is employed in the coupled explicit-implicit algorithm, base of the slope is
regarded as implicit block and blocks on the slope are assumed as explicit blocks as can
be seen in Fig. 7.12(d). The displacements of the measured point 1 and 2 are plotted in
Fig. 7.13, respectively. We can find that the coupled explicit-implicit algorithm based
on onefold cover system is workable and satisfies the numerical accuracy comparing the
implicit and explicit algorithms. The efficiency of the coupled algorithm (CPU cost is
336s, t is 0.05 ms) can be improved contrast to the original implicit version of the
NMM (CPU cost is 1161s, t is 0.05 ms) significantly under the requirement of the
computational accuracy.
In addition, selection of the contact spring stiffness is sensitive to the
computational accuracy, how to determine an appreciate value of the spring stiffness is
still current hot spot in the simulations using the NMM, which will be paid more
attention in the further research.
(a) Initial modelling;
ρ = 25 KN/m3
E = 1.0 GPa ν = 0.2 ϕ = 50, c = 0, T = 0
Measured point 1, 2
410
(b) Explicit NMM;
Kn = 0.25 GPa Ks = 0.1 GPa Δt = 0.02ms umax = 0.02mm
Explicit blocks
Explicit block
Kn = 5.0 GPa Ks = 1.0 GPa Δt = 0.02ms umax = 0.02mm
Implicit blocks
Implicit block
(c) Implicit NMM;
Kn = 0.03 GPa Ks = 0.01 GPa Δt = 0.02ms umax = 0.02mm
Explicit blocks
Implicit block
(d) Explicit-implicit NMM.
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Figure 7.12 Rock slope stability analysis using the I-NMM, E-NMM and E-I NMM, respectively: (a)
Initial modelling; (b) Explicit NMM; (c) Implicit NMM; (d) Explicit-implicit NMM.
(a) Displacement of measure point 1; (b) Displacement of measure point 2.
Figure 7.13 Displacement of the measured point 1 and 2.
7.5 SUMMARY
The coupled E-I algorithm for the numerical manifold method (NMM) is proposed
in this chapter. The time integration schemes in the E-I algorithm, transfer algorithm of
the coupled E-I algorithm, the implicit contact algorithms based on the implicit
integration scheme and explicit contact algorithm based on the explicit integration are
studied in terms of accuracy and efficiency, respectively. In particular, onefold cover
system to the coupled E-I algorithm is proposed and drawn into the coupled spatial E-I
algorithm, in which the contact algorithm based on the onefold cover system is
discussed and derived in detail. Finally, some numerical examples are simulated using
the proposed coupled E-I algorithms, in which one calibration example is studied using
the proposed E-I algorithms relied on the onefold cover system; One numerical example
of rock slope seismic stability analysis using the coupled E-I algorithm is studied as
well. The simulated results are well agreement with the implicit and explicit algorithms,
0
1
2
3
0 0.2 0.4 0.6 0.8 1
Dis
pl.
(m)
Time (s)
I-NMM
E-NMM
E-I NMM
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1D
isp
l. (m
)
Time (s)
I-NMM
E-NMM
E-I NMM
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but the efficiency of the coupled algorithm is obviously higher than that of the original
version of the NMM. It is predicted that the couple E-I algorithm proposed in the
present paper can be applied into larger scales engineering systems to combine the
merits of both the implicit and explicit algorithms in the NMM.
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CHAPTER 8. DYNAMIC STABILITY ANALYSIS OF
ROCK SLOPE FAILURE USING THE EXPLICIT
NUMERICAL MANIFOLD METHOD
8.1 INTRODUCTION
Comparing with the static stability analysis of rock slope, the dynamic stability is
more difficult as the complexity of the triggered mechanism of dynamics, such as
earthquake load and blast effect (Hoek and Bray 1981). During the previous a few
decades, the dynamic stability analysis of rock slope is still hot issue to motivate
researchers conduct different techniques.
It is noted that when the complex dynamics is taken into account, the traditional
implicit version of the NMM will become inefficient especially in the simulations of
large scale modelling. Thus, how to simulate the dynamic stability of rock slope
efficiently draws our attentions; how to improve the efficiency of the current version of
the NMM within the appropriate accuracy motivates us to develop the new version of
the code. In this paper, the traditional NMM is further extended for earthquake-induced
rock slope stability analysis using the proposed explicit scheme. The validity of the
explicit NMM (ENMM) has been investigated by comparing its results with available
reference solutions in the previous sections. Here, we put the proposed ENMM and
other techniques into the dynamic stability analysis of rock slope failure.
To illustrate the dynamic stability of rock slope, we give one example of rock slope
failure in Perth of Western Australia as shown in Fig. 8.1. A slim north-south orientated
rock slope (described by shallow red shadow zone from Google map ref.
http://maps.google.com.au/ ) sites close to beach of Cottesloe WA, this hazard zone is
instable and danger to passengers, particularly the effect of impact from ocean tide,
typhoon and tsunamis. Furthermore, this district belongs to a classic Mediterranean
climate (Sturman et al. 1996; Linacre and Geerts 1997). Annual rainfall falling between
May and September is heavy, which is one of factors affect the slope stability. It is
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noted that this is a very normal phenomenon exists around coast in Western Australia as
the broad coastline. Here, the slope example is taken into account as shown in Fig. 8.2
to investigate the stability and potential failure types. In Fig. 8.2a, right bank of the
slope is a slim slope bordered on the India Ocean. Area B is smooth and consists by
intact rock mass, this area is regarded stable. On the other hand, area A is consisted by
loose and discrete rock masses, especially rock blocks denoted by and can be
considered as potential instability such as rock sliding even rock fall failure under
dynamic conditions, thus this slope is required to investigate the stability rely on in-situ
measurement technique and other methods such as LEM and numerical methods. At the
left bank of the slope as shown Fig. 8.2b, the major type of rock failure focuses on the
rock fall from the shadow denoted by , and , respectively. It is considered
potential instable and hazard area, and a yellow warning board saying “ROCKFALL
RISK AREA NO ENTRY” and wooden fence locate in the lower left of the photo to
remind the passengers keep personal safety.
Figure 8.1 Location of the rock slope and aero-view from Google maps.
N
Location
WA
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8.173
Figure 8.2 Photograph of rock instability example in Cottesloe, Western Australia (photographed
by X.L. Qu): a. right bank of the slope; b. left bank of the slope.
From the above referred examples, we can find that rock slope stability analysis
should be taken into account at the both cases of static and dynamic conditions. These
motivate researcher to study these failure mechanisms by generation to generation in the
past several decades. In general, dynamic stability analysis of rock slope concentrates
on the triggering factor such as earthquake, blasting even tsunamis respects.
The dynamic stability analysis of rock slope is studied using different techniques
by several earlier researchers. Here, a comprehensive study is carried out with an
emphasis on dynamic stability analysis of rock slope in jointed rock mass using the
proposed ENMM codes, coupled algorithms and parallel computation.
Area A
Area B
a
b
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8.2 NUMERICAL METHODS FOR ROCK SLOPE DYNAMIC
STABILITY
For the seismic stability analysis of rock slopes, the numerical methods are more
suitable because the behaviour of a rock slope is much more dependent on characteristic
and integrity of the rock mass. At present, dynamic FEM techniques (Zienkiewicz et al.
1975; Griffiths and Lane 1999) have became one of the important tools in seismic
stability analysis for rock slopes, yet it still has difficulties in the simulation of
numerous rock discontinuities (such as faults, joints, etc.) which are discreted by special
elements. To overcome the disadvantage of FEM in seismic stability analysis of rock
slope, new numerical methods have been developed for the deformation based on
multiple blocks system containing a large quantity of discontinuities. Among these
methods, distinct element method (DEM) (Cundall 1971a, 1971b; Itasca 1993, 1994,
1995), discontinuous deformation analysis (DDA) (Shi 1988, 1993), and the numerical
manifold method (NMM) (Shi 1991, 1992) are typical. The NMM was developed to
integrate the DDA and the FEM. The distinct feature of NMM is which employs dual
cover system to describe a problem domain. The advantages of the NMM are releasing
the task of meshing and combining continuum and discontinuum problems into one
framework. The NMM has been successfully applied to strong discontinuity problems,
weak discontinuity problems and rock failure, etc (Tsay, Chiou, and Chuang 1999b; Ma
et al. 2008; Ma and He 2009; Zhang et al. 2010; Ning, An, and Ma 2011; An, Ma, et al.
2011b; Wu and Wong 2012; An et al. 2013).
Traditional methods of rock slope stability analysis are limited to simplified
problems. They employ simple slope geometries and basic loading conditions, which
provide little insight into slope failure mechanisms without considering complexities
relating to geometry, material anisotropy, non-linear behaviour, in situ stresses and the
realistic loading conditions, such as pore pressures, seismic and blasting loading, etc. To
address these limitations, numerical modeling techniques have been forwarded to
provide approximate solutions to problems over the traditional techniques. In general,
numerical methods for rock slope stability analysis can be classified into three
approaches: (1) continuum modeling; (2) discontinuum modeling; (3) hybrid modeling.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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Continuum method is best suited for the analysis of rock slopes that are comprised of
massive intact rock, weak rocks or heavily fractured rock masses; discontinuum
modelling is appropriate for slopes governed by discontinuity behaviour; hybrid
approach involves the coupling of above both techniques to maximize their critical
advantages.
8.2.1 Continuum methods
Continuum approaches applied into the dynamic stability analysis of rock slope
contain the finite difference methods (FDM) and FEM. Both methods divide the
problems domain into a set of sub-domains or elements. The difference is the former
relies on numerical approximations of the governing equations, on the other hand, the
latter resorts to the continuity of displacements and stresses of the conjoint elements.
Both advantages and limitations of these two methods are discussed in (Hoek,
Grabinsky, and Diederichs 1991). To slope stability analysis, earlier studies are often
limited to static and elastic analysis, which is restrained in the application. Nowadays,
most continuum based codes incorporate a facility for discrete fractures such as faults
and bedding planes and dynamic input parameters analysis such as FLAC 2D/3D and
Abaqus/CAE. Fig. 8.3 illustrates an application of an elasto-plastic constitutive model
based on a mohr-Coulomb yield criterion using the general commercial software
Abaqus/CAE 6.11 (Abaqus 2011). The input geometry parameters: section size is
100*40 m, angle of slope is 450; material parameters: ρ= 20 kN/m3, c = 38.2 kPa, φ =
17o, E = 100 MPa, ν = 0.3), which results can be simulated by FALC 2D and the same
criterion is employed to model translational slide movements of Frank Slide, Canada by
(Stead et al. 2000).
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(a) slope meshing and 6-noded triangular element;
(b) contour nephogram of the maximum strain ratio.
Figure 8.3 Continuum modeling of a rock slope by Abaqus/CAE 6.11: (a) slope meshing and 6-
noded triangular element; (b) contour of the maximum strain ratio.
In terms of seismic stability analysis of the slope modelling, the El-Centro
earthquake ( 0.7agM and 4.6lM ) acceleration in 1940, U.S.A. is taken into account to
simulate the slope displacement using software of Abaqus/CAE. In the simulation, the
horizontal seismic acceleration with time depended 25s is exerted in the bottom of the
slope base, the displacements depend on time history of seismic acceleration can be
obtained as shown Fig. 8.4, in which the horizontal displacements nephogram at
different times are plotted respectively. It is noted that the displacement is changed
smoothly at the initial stage of the earthquake, and then the maximum displacement is
achieved following the peak seismic acceleration. Thus, the peak seismic acceleration
plays virtual role in the rock slope failure under seismic effects.
1
2
3
4 5
6
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Figure 8.4 Displacement nephogram at different times under seismic loading: (a) t=1.5s; (b)
t=12.0s; (c) t=21.0s; (d) t=30.0s.
8.2.2 Discontinuum methods
When a rock slope comprises multiple joint sets, which control the mechanism of
failure, then a discontinuum modelling technique may be considered more appropriate.
In the discontinuum approaches, problems domain is treated as an assemblage of
distinct, interacting bodies or blocks that subjected to external dynamic loads such as
earthquake, blasting even tsunamis, and is expected to undergo significant motion with
time-history. The underlying basis of the discontinuum method is that the dynamic
equation of equilibrium for each block in the motion system is repeatedly solved until
the boundary conditions and laws of contact and motion are required. Nowadays, the
discontinuum technique constitutes the most commonly applied numerical approach to
rock slope analysis and generally three alternative variations of the methodology exist:
(1) DEM; (2) DDA; (3) particle flow codes (PFC).
The DEM is originally developed by Cundall (1971a, 1971b) and further described
in detail by Hart (1991), in which the algorithm is based on a force-displacement law
and a law of motion. In the respect of the dynamic stability analysis of rock slope, the
explicit solution of the DEM in the time domain used by the method is ideal for
following the time propagation of a stress wave. Fig. 8.5 provides an example of a
typical high-speed landslide hosted on consequent bedding rock induced by Wenchuan
(a) t = 1.5s; (b) t = 12.0s;
(c) t = 21.0s; (d) t = 30.0s.
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earthquake at a medium-steep hill slope (Luo et al. 2012), in which the maximum
displacement vectors and shear strain contours at time of t=10s and t=25s are studied
using UDEC, respectively. The model show that it is helpful for understanding seismic
dynamic responses of consequent bedding rock slopes, where the slope stability could
be governed by earthquakes. In addition, Eberhardt and Stead (1998) carried out
dynamic stability analysis of a natural rock slope using the UDEC, in which an initially
stable slope subjected to an earthquake, resulting in yielding and tensile failure of intact
rock at the slope toe and rotational type movements.
(a) maximum displacement vectors at time of t=10s and t=25s;
(b) shear strain contours at time of t=10s and t=25s.
Figure 8.5 Maximum displacement vectors and shear strain contours of the modelling in 2008
Wenchuan earthquake, China (Luo et al. 2012).
One more recent development in discontinuum modelling techniques is the
application of distinct-element methodologies by a new pattern of Particle Flow Code,
PFC2D/3D (Itasca 1995), in which clusters of particles can be bonded together to form
joint-bounded blocks. This code is capable of simulating fracture of the intact rock
blocks through the stress-induced breaking of bonds between the particles. This is a
significant development as it allows the influence of internal slope deformation to be
investigated both due to yield and intact rock fracture of jointed rock. Wang et al.
(2003) demonstrate the application of PFC in the analysis of heavily jointed rock slopes
unit: mm
t =10s
unit: mm
t =25s
t =10s t =25s
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under dynamic excavation as shown Fig. 8.6, which simulation results reveal an intact
rock with certain mechanical properties, such as joints (discontinuities) play a
determinant role in slope stability.
Figure 8.6 Simulation of a rock slope stability and failure under dynamic excavation using PFC
technique (Wang et al. 2003).
The DDA (Shi 1988, 1993) has also been applied in the modelling of dynamic
stability analysis with some success, both in terms of landslides (Wu et al. 2009; Wu
2010) and jointed rock slope (Hatzor et al. 2004). This approach differs from the
distinct-element method in that the unknowns in the equilibrium equations are
displacements as opposed to forces, by which the equilibrium equations can be solved in
the same manner as the matrix analysis used in FEM based on minimum potential
energy principle. With respect to slope dynamic stability analysis, the method has the
advantage of being able to model large deformations and rigid body movements, and
can simulate coupling or failure state between contacted blocks. In addition, SASAKI et
al. (2006) carried out the contact damper to controlling the surplus penetration in high
speed velocity to simulate the seismic response analysis of Myo-ken slope in Niigata,
Japan. Fig. 8.7 demonstrates a dynamic DDA version to simulate landslide by the Chi-
Chi earthquake in Taiwan, in which time-dependent accelerations and constraining
seismic displacements of the base rock are studied, respectively; and a novel algorithm
is carried out to diminish the velocity of the base rock in the seismic analysis. Further
more, seismic DDA analysis results coincide well with the topography of the Chiu-fen-
erh-shan landslide slope. Thus, DDA is a useful tool for simulating the response of a
block assembly under the impact of an earthquake as we expected.
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Figure 8.7 Seismic simulation of Chiu-fen-erh-shan landslide by the Chi-Chi earthquake using
DDA (Wu 2010).
8.2.3 Hybrid methods
Hybrid methods are increasingly being adopted in rock slope stability analysis. An
alternative coupled method using LEM and finite-element and stress analysis such as
adopted in the GEO-SLOPE suite of software (Geo-Slope 2000). In particular, a recent
new developed method of NMM is performed by Shi (1991, 1992) based on topological
manifold, unifies both the FEM (Zienkiewicz et al. 1975) and the DDA (Shi 1998,
1993), and applied into the dynamic stability analysis of rock slope. For the dynamic
response analysis of discontinuous rock slopes, seismic forces are commonly applied to
the basement block modelled using a single DDA block. However, it is necessary to
consider the local variation of seismic forces and stress conditions, especially when the
size of slopes is large and/or the slope geometry becomes complicated. There is
difficulty in DDA to consider the local displacements of the single block for the
basement due to the fact that the strain in the single block is uniform and displacement
function is defined at the gravity center. On the other hand, the NMM can simulate both
continuous and discontinuous deformation of blocks with contact and separation.
However, the rigid body rotation of blocks, which is one of the typical behaviors for
rock slope failure, cannot be treated properly because NMM does not deal with the rigid
body rotation in explicit form. Fig. 8.8 presents an application for the discontinuous
rock slope behaviour during earthquake successfully, in which the mechanical
behaviour of falling rock blocks is simulated by DDA with the basement block covered
Time =0 sec
653.1m
Time =30 sec
Time =60 sec
874.8m
Time =300 sec
1013.1m
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by the NMM mesh, where seismic forces are given. Further more, the formulation for
the coupled NMM and DDA (NMM-DDA) is presented with the programming code
developments. Ning et al (2012) carried out a numerical modelling of earthquake-
induced failure of a rock slab with pre-existing non-persistent joints using the couple
NMM and DDA, as shown in Fig. 8.9, in which the complete rock failure process
including the fracturing of the intact rock and the motions of the generated rock blocks
is wholly reproduced.
Figure 8.8 Displacement distribution for each block after applying seismic loads (Miki et al. 2010).
Figure 8.9 Modelling of rock fall failure under earthquake by NMM and DDA (Ning et al. 2012).
8.3 THE NEW DEVELOPMENT OF THE NMM FOR DYNAMIC
STABILITY ANALYSIS OF ROCK SLOPE
In the present study, a new development of the NMM is introduced to simulate
dynamic stability analysis of rock slope. In order to investigation of the performance of
geo-structures such as rock slopes and/or tunnels subjected to seismic loading, the
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NMM is further extended for earthquake-induced rock slope stability analysis. Ning et
al. (2012) proposed the seismic NMM code combining DDA to simulate the seismic
stability of rock slope, in which the earthquake loading is applied in the form of
constraining seismic displacement time history. Seismic acceleration time histories
recorded in earthquakes is transferred into displacement time histories to be loaded by
the following formulas:
2/21
1
tatvdd
tavv
nnnn
nnn (8.1)
where 1nv and
nv are the seismic velocity at step n+1 and n, respectively, and 00 v ;
1nd and nd are the constraining seismic displacement at step n+1 and n, respectively,
and 00 d ; na is the seismic acceleration at step n; t is the time interval between two
adjacent steps. To improve the transformation accuracy, especially when the sampling
frequency of the data is low, several steps can also be interpolated between two adjacent
records, where the acceleration at each step is obtained by linear interpolation between
the two adjacent acceleration records. With such seismic NMM code, the earthquake
acceleration amplitude that can lead to the initiation of the failure could be derived. The
complete rock failure process including the fracturing of the intact rock and the motions
of the generated rock blocks are wholly reproduced. Continuing the research, An et al.
(2012) investigated the seismic stability of rock slopes using the developed seismic
NMM code, in which the instability mechanisms of rock slopes under horizontal and
vertical ground accelerations are revealed, respectively. And the validity of the NMM in
predicting the ground acceleration induced permanent displacement has been verified by
comparing its results with the analytical solutions and the Newmark- numerical
integration solutions. Furthermore, the proper values of control parameters for NMM
calculations are suggested in the study. Thus, the seismic NMM adopted is promising
for the modelling of earthquake-induced rock failures and deserves to be further studied.
Following the previous study on the dynamic stability analysis of rock slope using
the NMM, here we present an alternative algorithm of time integration for the NMM.
As the explicit central different technique applied in the DEM, the proposed explicit
algorithm in the NMM can be considered for modelling rock slope to seismic events
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relating to earthquake and/or blasting as well. In addition, the explicit algorithm in the
time domain is particularly well suited for the time propagation of a stress wave and is
efficient in terms of computational efficiency. Comparing to the implicit algorithm used
in the current NMM code, the proposed explicit algorithm owns the following
advantages except the term of conditionally stability:
☑ Few computations are required per time step, low computational cost;
☑ It requires little computer memory without the assembly of stiffness matrices;
☑ Size of the explicit NMM code is shortened, especially cancelling iterative solvers;
☑ It is reliable with regards to accuracy and completion of the computation.
8.3.1 Explicit NMM
The explicit algorithm of the NMM owns distinct advantage in contrast to the
implicit algorithm with respect to the computation efficiency. To further improve the
efficiency of simulation under the accuracy, the coupled explicit NMM and DDA can be
taken into account to simulate dynamic stability of large scale rock slope, in which a
transfer algorithm is required to carry out the conversion from the NMM to DDA. Since
both NMM and DDA employ the same methodologies of global formula and contact
treatment technique, Ning et al. (2012) proposed one possible approach to transfer
NMM to DDA, in which the geometric configurations, physical and mechanical
parameters, and status, including stress state and velocities, are inherited from an NMM
model. It is noted that material area change problem and the unnatural deformation
problem can not be solved easily using the NMM as the rigid body rotation is not
represented in an explicit form (Miki et al. 2009). Thus, an alternative approach to
transfer the rotational velocity based on the principle of conservation of the kinematic
energy is applied and derived in the form as
DDANMM
n
i
ei
ei JJ 2
1
2
2
1
2
1
(8.2)
where eiJ and e
i are the moment of inertia and rotational velocity of the i-th element
with respect to its centre of the NMM, respectively; J and are the moment of inertia
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and rotational velocity of the block with respect to the block centre of the DDA,
respectively. Continuing the discussion of the explicit NMM and DDA, a numerical
example will be investigated in the following section to verify the proposed numerical
algorithm in detail.
8.3.2 Coupled E-I NMM
The explicit and implicit algorithms have their own limitations and advantages in
terms of numerical properties and computational efficiency, respectively. The implicit
algorithm can use the large time step whereas more computational cost produces
especially when the contact problems occurrence; on the other hand, the explicit
algorithm has reverse points in contrast to the implicit algorithm. The more details of
the studies between explicit and implicit algorithms can be referred in Chapter 6 and 7.
In addition, the NMM is hybrid numerical method combining the continuum and
discontinuum approaches relied on the mathematical and physical covers to simulate the
dynamic problems. In the present study, the failure process of crack propagation is not
considered in the coupled explicit-implicit NMM. Therefore, the study focuses on the
sub-domain partition of the coupled explicit-implicit NMM.
Construction of onefold cover system
In the NMM, the coupled E-I algorithm is inclined to the MCs partition to simulate
the continuous problems, so that the computational cost can be efficient saved.
However, to the discontinuous problems, the contact positions change along with the
different contact conditions, neither the element partition method nor the MCs partition
method can treat the contact problems efficiently. Further more, each MC (denoted by
“star”) is not always corresponding to the node as in the FEM when the interface occurs
between the explicit and implicit algorithms. Thus, it is time-consuming to rearrange the
index number of the MCs no matter the element or MCs partition method is employed.
Therefore, the more efficient method is required to develop to solve the contact
problems. Here, we present an alternative approach based on onefold cover system to
couple the explicit and implicit algorithms, in which a onefold cover system is built and
the contact problems is solved efficiently. To illustrate the proposed onefold cover
system, one example is given as shown Fig. 8.10, in which there are five partly
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overlapped mathematical covers denoted by 1MC , 2MC , 3MC , 4MC and 5MC to cover
the whole domain based on the manifold cover system in the current NMM, and five
overall overlapped mathematical covers denoted by 1OC , 2OC , 3OC , 4OC and 5OC based on
the onefold cover system, respectively. Comparing manifold with onefold system, the
distinct difference is 1MC to 5MC use different nodal coordinates while 1OC to 5OC share
the common nodal coordinates, which improve the computational efficiency and save
computation cost in simulations.
Figure 8.10 Construction of onefold cover system from manifold cover system.
When the onefold cover system is considered to simulate the contact problems in
the coupled E-I algorithm, the partition of explicit and implicit element (denoted by OE)
become clear based on the onefold system. Here, we give one simple example to expose
the partition technique based on onefold system as shown Fig. 8.11. When two OEs
contact is searched using the contact detect criteria, the explicit and implicit OEs can be
determined relying on certain judgement algorithm (i.e. area algorithm, stress and strain
algorithms and/or displacement algorithm etc.), then the corresponding onefold covers
coordinates can be found. Here, we present one upper triangular cover to cover the
whole contact domain, the stars of 1Ii , 1
Ij and 1Ik describe the implicit OE, the stars of
1Ei , 1
Ej and 1Ek describe the explicit OE, respectively.
5MC
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Figure 8.11 Contact between explicit and implicit OEs based on onefold cover system.
Contact matrices between the explicit and implicit elements
In the coupled explicit-implicit NMM, once the contact is detected using the search
algorithm, the corresponding contact types can be determined based on certain
judgement criteria and the contact matrices are added to the contact elements. Basically,
there are three kinds of contact types can be detected using the area algorithm in the
onefold cover system: explicit to explicit (E-E), explicit to implicit (E-I) and implicit-
implicit (I-I). After this, the corresponding integration scheme will be adopted to form
the global coefficient matrix, in which the stiffness sub-matrix, damping item and mass
item are assembled to satisfy the corresponding algorithm for the equations of
equations. Since the assembled stiffness matrix and damping matrix both have a sparse
band-profile framework in the implicit algorithm, the coefficient sub-matrix 66][ K is
formed to the implicit algorithm. Accordingly, the diagonal coefficient sub-matrix
66][ M is generated to the explicit algorithm, and the whole matrix structure is
symmetric and positive definite when the coupled E-I algorithm is used based on
onefold cover system. To illustrate these properties of the coupled E-I algorithm, one
example of assembly of contact matrices involving r contact elements as shown in Fig.
8.12 is considered here. When two implicit contact elements (i.e. i and j ) participate in
, ,
,
Boundary point
Onefold cover star Contact domain
Explicit OE Implicit OE
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contact, contact matrix can be expressed as language C++ format of IIjj
IIji
IIij
IIii kkkk ,
and two explicit (i.e. k and l ) elements contact matrix is written as EEll
EElk
EEkl
EEkk kkkk
, respectively. Contact between one explicit element (i.e. k ) and one implicit element
(i.e. l ) will produce contact matrix 22
IEjkk to the equilibrium equation of the dynamic
system.
Figure 8.12 Assembly of contact matrices in the coupled E-I algorithm.
8.4 THE PARALLEL COMPUTATION OF THE NMM
Nowadays, parallelization has become the most important way to accelerate
engineering computations and simulations, in which several processors are distributed
to execute the computations simultaneously. With development of High Performance
Computing (HPC) (Chang 2006), a variety of parallel processors have been used in
different situations. These processors can be classified into three types: CMP (Chip
Multi-Processors) (2010), GPGPU (General Purpose GPU) (Owens et al. 2008), and
Heterogeneous Multiprocessor (Baker 2000). These processors are further organized as
SMP (Shared Memory Processors), MPP (Massively Parallel Processors) or DSM
(Distributed Shared Memory) style to form cluster systems. Each type of processor has
its own features. CMP, such as Intel Core2 Duo, Xeon and AMD Phenom, has the most
rr
Symmetric
I-I contact matrixII
iik IIijk
IIjik
EEkkk
EEllk
EEklk
EElkk
IEjkk
E-E contact matrix
E-I contact
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market share. In fact, CMP with 2 or 4 cores has been used for nearly all laptops,
desktops and workstations. GPGPU based computing, proposed by NVidia and AMD,
is now a hot topic in fast developing. It requires cooperation between GPU and CPU
hardware. Heterogeneous Multiprocessor, such as IBM Cell used in the Top One HPC
cluster Roadrunner, is powerful but not easily available for common customers.
To personal computer, more multi-core processors such as quad-core CPU, 8×core
CPU, even 80-core CPU (Chang 2006) are developed to obtain the best parallel
efficiency. Parallel programming environments such as OpenMP (OpenMP 2010),
pThreads (Nichols, Buttlar, and Farrell 1996) and TBB (TBB 2010) can be used to
implement the multi-core version of an existing code. Normally, the parallelization of a
code on multi-core PC is relatively simple as it only needs to deal with the shared
memory environment. It does not need to consider the task distribution and
communication between different processors. However, there also exist some
disadvantages of multi-core processor (Merritt 2008; ). Miao et al. (2009) carried out a
parallel implementation of NMM based on multiprocessor platforms, in which parallel
Jacobi's iterative method with OpenMP is implemented to improve computing
performance for such class of engineering problems. In the present chapter, we focus on
the parallel implementation of the NMM with Open MP.
8.4.1 Parallelization with openMP
There are general two models to do parallel processing: OpenMP and Message
Passing Interface (MPI) (MPI 2010), which is presented in Fig. 8.13. OpenMP is based
on Uniform Memory Access (UMA), is suited for shared memory systems like we have
on our desktop computers. Shared memory systems are systems with multiple
processors but each are sharing a single memory subsystem. Using OpenMP is just like
writing your own smaller threads but letting the compiler do it. On the other hand, MPI
is based on Non Uniform Memory Access (NUMA), is most suited for a system with
multiple processors and multiple memory such as a cluster of computers with their own
local memory, relying on MPI to divide workload across this cluster, and merge the
result when it is finished. Both support C/C++ and Fortran languages packed in
commercial software platform of Microsoft visual studio 2010. As the NMM employs
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implicit solution, which involves more computations of matrices to assemble global
matrix of system and contact treatments to the discontinuous problems, the
computational cost always draws our attentions when the large scale engineering is
considered. These motivate us to develop more efficient programming codes to reduce
the computational cost.
(a) UMA; (b) NUMA.
Figure 8.13 Parallel processing model: (a) UMA; (b) NUMA.
One alternative method to improve the efficiency is the explicit scheme of the
NMM, in which the explicit time integration is used and the code is shortened without
the assembly of global matrix of current NMM code. Since OpenMP is an explicit
programming based on fork-join model, parallelization can be as simple as taking a
serial program and inserting compiler directives. To further improve the computational
efficiency, a hypothesis construction scheme of parallel computation of the explicit
NMM using OpenMP is considered based on Duo-core CPU (Intel E8500 @ 3.16GHz,
3.17GHz) as shown in Fig. 8.14. We can find that the current NMM code is serial and
only one master thread through out the whole computation. The parallel code uses the
fork-joint model to let one parallel region being calculated by more than one thread such
as four threads in region Ⅰand two threads in region Ⅱ. In each parallel region, the
forked NMM thread is allocated to different threads to carry out parallel computing
based on shared memory (i.e. 4GB) and the value of environment variables,
OMP_NUM_THREADS is set 4 to obtain the most computational efficiency.
System Interconnect
P1 P2 Pn
SM1 Shared Memory
I/O SMn
Processors
Message-passsing Interconnection network
(Mesh, ring, torus, hypercube, cube-connected cycle, etc.)
PM
P
M
P
M
PM
P
M
P
M
PM
PM
P M
P M
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Figure 8.14 Construction of parallel computation of the NMM using OpenMP.
It is noted that, in the case of the explicit NMM, force the item and displacement
item can be paralleled directly by dividing into two parallel regions using the OpenMP,
which solves the motion equations conveniently and efficiently. A code segment of the
parallel programming to the explicit NMM is attached as shown in Fig. 8.15, in which
the computational performance of the code can be fully increased under the original
code structure. On the other hand, as the current NMM code is an implicit solution of
the equilibrium equation by principle of minimum potential energy, the assembly of the
global matrix in the computations reduces the efficiency when the OpenMP is taken into
account in the simulations.
Figure 8.15 Code segment of the parallel programming to the explicit NMM.
T1
T2
T3
T1
T4
T2
Parallel region Ⅰ Parallel region Ⅱ
NMM thread
T1 T2 T1 T4 T2 T3
NMM thread
int i=0; #pragma omp parallel for private (j) for (i=1; i<= m1; i++) { for (j=1; j<= 2; j++) { r[i][j] = f[i][j] + c0[i][j]; } /* j */ } /* i */
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8.4.2 Speedup
Speedup is the most important factor to evaluate the performance of parallel
algorithms, which is defined as the computing time ratio between the parallel runtime
for a given number of CPUs and the serial runtime (Kumar et al. 1994). The formula
can be expressed as
s
p
t
tS (8.3)
in which pt is the runtime of the serial code using the best optimization and st is the
runtime of the parallel code for the same problem, respectively.
To illustrate the speedup of parallel computing, a simply rock slope is considered
to compare the serial and parallel computational efficiency. The parameters of the two
used multi-core PCs in the parallel computations are presented in Table 8.1. The
simulated results using the serial and parallel NMM codes (total time 20s), as shown in
Fig. 8.16, indicates they are identical between two codes, which reveals that the parallel
NMM code is succeed to implement into the simulations. Fig. 8.17 shows the CPU
usage of the serial and multi-core NMM codes. It can be seen that the serial NMM has
not taken full advantage of the Multi-core CPU, and only 54% computing resource is
used for the serial NMM, but which increases up to almost 100% for the multi-core
NMM (both in the case of 2 CPUs and 4 CPUs). It means the OpenMP implementation
is effective and the computing resources can be fully used.
Table 8.1 Parameters of the used multi-core PCs.
CPU Core Hyper Threading Speed Memory
Intel E 8500 2 No 3.16 GHz 4 GB
Intel i5 760 4 No 2.80 GHz 4 GB
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Figure 8.16 Simulation results of the serial and parallel NMM codes.
In the present study, the rock slope modelling with 357 elements and 54 blocks is
simulated by the current NMM code and the explicit NMM code, in which both serial
and parallel codes are employed, respectively. The simulated results obtained by the two
types of codes are just the same. The computing time of the parallel codes with
OpenMP comparing the serial codes of the NMM and ENMM are presented in Fig.
8.18. It can be seen that speed of the parallel code with OpenMP is obvious faster than
the serial one both the cases of NMM and explicit NMM. The speedup of the 2-core and
4-core NMM and ENMM codes have been tested as shown in Fig. 8.18 (b), in which the
maximum value is up to 1.533 and 1.414 by the case of NMM and ENMM,
respectively. Furthermore, explicit NMM is more efficient than that of the current
NMM as the explicit code removes the assembly of the global matrix of the implicit
solutions.
(a) Serial NMM;
(b) Parallel NMM (2 CPUs).
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Figure 8.17 CPU usage of the serial and multi-core NMM codes.
(a) Serial NMM; (b) 2-core NMM;
(c) 4-core NMM.
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(a) CPU time;
(b) Speedup.
Figure 8.18 Computing time of the serial and parallel codes.
8.5 NUMERICAL EXAMPLES
In recent years, numerical methods have been widely implemented to the rock
engineering, which motivates researchers seek varied numerical technologies for the
analysis of the rock engineering. In this section, to reveal the validity and applicability
of the proposed NMM, some project examples of rock slope stability are investigated.
The first one is a dynamic case of Lake Anderson slope failure from the earthquake of 6
August 1979 Coyote Lake, California USA. Simulated results of the seismic NMM
code will compare with the field measurements to illustrate the applicability of the
0
1000
2000
3000
4000
5000
6000
7000
NMM ENMM
CP
U t
ime
(s)
Serial 2 CPUs 4 CPUs
0
0.5
1
1.5
2
1 2 3 4
Sp
eed
up
CPUs
NMM ENMM
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seismic NMM code. The other example of JinpingⅠHydropower Station dam left
abutment slope is taken into account to present the coupled ENMM with DDA in terms
of efficiency. Furthermore, the coupled explicit and implicit NMM are presented to
simulate the stability of rock slope as well.
8.5.1 A dynamic case study of rock slope stability analysis
The Coyote Lake, California, earthquake of 6 August 1979 (ML 5.7) provided a
rare opportunity to perform a dynamic numerical analysis of a seismically induced slope
failure using the NMM. This earthquake was recorded by several strong-motion
instruments located in the epicentral region (Porcella et al. 1979). The earthquake
caused a slope failure near the east shore of Lake Anderson, 9 km northwest of the
epicentre. Location of the slope can be seen in Fig. 8.19, in which denotation of dot
presents the earthquake epicentre, pentacle indicates area of Lake Anderson slope
failure and triangle expresses the location of strong-motion instruments. Wilson et al
(Wilson and Keefer 1983) carried out a extend Newmark analysis to calculate
displacements of the landslide located strong-motion area under the action of seismic
ground motion, in which an expression for critical acceleration in terms of static FoS is
determined as
sin)1( gFoSac (8.4)
where FoS is the static FoS and is the angle of slope, respectively. The Newmark
analysis satisfactorily predicts the occurrence of the slope failure and the amount of
displacement of the landslide as actually measured.
In the simulation, the Lake Anderson slope is considered to investigate the stability
under the Coyote Lake earthquake, which of modelling can be seen in Fig. 8.20, in
which the landslide block of AFGHIJ is investigated using the proposed seismic NMM.
The results of conversional Pseudo-static analysis for the slope are 492.1FoS and
gac 22.0 as referred in (Wilson and Keefer 1983), and dynamic analysis parameters of
the NMM can be seen in Fig. 8.20. Fig. 8.21 shows a record of acceleration of the
earthquake by this region, in which the peak acceleration value of ap = 0.345g is
presented as well.
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Figure 8.19 Location of area of Lake Anderson slope in California USA (Keefer et al. (1980).
Figure 8.20 Numerical modelling of Lake Anderson slope.
N
B G A F
H I
J K
+
Pseudo-static analysis (Wilson and Keefer 1983),: Strength values c=300psf ϕ=25° Static FoS = 1.492 β=26°
Critical acceleration ac = 0.22g Dynamic analysis: Mass density ρ = 22 KN/m3 Youngs modulus E = 1.0 GPa Passion ration ν = 0.2 Frictional angle ϕ = 25° Cohesion c = 0.143 MPa
×
β
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Figure 8.21 Record of acceleration of the earthquake.
The simulated results can be seen in Fig. 8.22, in which the displacement of the
landslide block is simulated using proposed seismic NMM and explicit NMM,
respectively. Since the acceleration is low at the beginning stage of the earthquake, the
displacement is small, and then it increases up to 23.50mm (seismic-ENMM) and
20.71mm (seismic-NMM) around 4 seconds, respectively. The results are in excellent
agreement with the estimated displacement of 21mm from field measurements of the
slope and the Newmark analysis solution of 27mm (Wilson and Keefer 1983).
Therefore, the proposed seismic NMM and explicit NMM are applicable to analysis the
stability of rock slope, which can be considered as a convenient numerical tool to apply
into other rock engineering. It is noted that the simulated results are closely related with
the physical parameters of the modelling, so more in situ measurements are necessary
and required to build realistic modelling of the slope.
-0.4
-0.2
0
0.2
0.4
0 5 10 15 20 25
Acc
eler
atio
n (
g)
Time (s)
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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Figure 8.22 Simulated results of landslide under earthquake.
8.5.2 Dynamic stability analysis of Jinping I hydropower station
Jinping I hydropower station is located at the border of Yanyuan and Muli counties
in Sichuan Province, China. It is built on the Yalong River as a controlling cascade
hydropower station in the middle and downstream of the main stem. In this simulation,
the proposed explicit NMM-DDA method described in this paper is used to analyse the
left abutment slope stability of JingpingⅠ hydropower station. The scale map of
geomechanical model taken for analysis in 2D is shown in Fig. 8.23. We can find that
the slope is cut by bedding planes that have a stride towards the hillside, with
inclinations of 55°-70°. Since rocks in the left bank mainly consist of metasand stone
and slate of group 332 zT , plus complex geological structure, together with variablestrata
and stress-relief disturbance has affected the stability of rock masses on both sides of
the river (Song et al. 2011).
0
5
10
15
20
25
30
0 2 4 6 8 10
Dis
pl.
(mm
)
Time (s)
Seismic-NMM
Seismic-ENMM
Displ. =27
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Figure 8.23 Scale map of geomechanical model (Zhou et al. 2008).
When excavation is involved in the slope, such as blasting technique is employed
in the excavation, dynamic stability analysis of the slope is required to keep slope still
stable under dynamic effect. In this case, there is one alternative scheme of coupled
NMM and DDA can be considered in terms of computational efficiency. In order to
save computational time, we can use the proposed explicit NMM to simulate the
transient analysis of the basting effect and the DDA to study the slope stability after
blasting. To illustrate the efficiency of the proposed ENMM and DDA, the modelling of
the slope is constructed using the ENMM and DDA as shown in Fig. 8.24. The first
stage of simulation uses the proposed explicit NMM as the blasting duration is transient,
it is very suited for the explicit than the implicit algorithm, and then following the DDA
simulation. Here, we choice first one tenth of the total time to use explicit NMM code
and the other left time to simplify the modeling using the DDA code. There are total
1018 NMM elements and 32 blocks both in NMM and DDA. To compare the
efficiency, the implicit NMM ( st 001.0 ) is employed to simulate the stability of this
slope, the cost of which is 763.84s, is more than twice time of the proposed explicit
NMM and DDA ( st 0005.0 ). It is clear that the proposed method is more efficient to
simulate the stability of the slope when more element and blocks involving in the study.
It can be predicated that the proposed method can be extend applied to larger scale
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project by simplifying the computation model using the proposed explicit NMM-DAA
method.
Figure 8.24 Modelling of the slope transfers from ENMM to DDA.
8.6 SUMMARY
The dynamic stability analysis of rock slope failure using the numerical manifold
method (NMM) is studied in the present chapter. Firstly, conservational pseudo-static
methods, Newmark Method and numerical methods applying into the seismic stability
analyses are investigated, the advantages and limitations of which are studied by
contrast of the NMM. Then, an alternative explicit algorithm of the NMM and coupled
explicit-implicit NMM are proposed to study the seismic stability of rock slope.
Furthermore, parallel computing with openMP is evaluated to improve efficiency of the
NMM. To reveal the validity and applicability of the proposed NMM, some numerical
examples of rock slope stability analysis are investigated. The first one is a dynamic
case of Lake Anderson slope failure from the earthquake of 6 August 1979 Coyote
Lake, California USA. Simulated results of the NMM will compare with the field
measurements to illustrate the applicability of the NMM. The other example of Jinping
ⅠHydropower Station dam left abutment slope is taken into account to present the
coupled explicit NMM with DDA in terms of efficiency. Furthermore, the coupled
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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explicit and implicit NMM are presented to simulate the stability of rock slope as well.
Therefore, it can be predicted that the proposed method is promising and can be extend
applied to larger scale project of rock slope with respect to dynamic stability analysis.
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CHAPTER 9. CONCLUSIONS AND
RECOMMENDATIONS
9.1 SUMMARIES
The stability of rock slope under dynamic effect is often significantly influenced by
the discontinuities of the rock masses. The traditional NMM employs implicit time
integration and OCI algorithm to solve the discontinuous problems in rock slope. Based
on the finite covers system, the NMM combines the well developed analytical methods,
widely used FEM and the block-oriented DDA in a unified form. However, it is less
efficient to simulate the rock slope stability under dynamic effect when many contacts
involved.
This thesis focuses on the development of the explicit version of the NMM for
dynamic stability analysis of rock masses. A thorough investigation of the traditional
NMM is made in terms of the time integration and contact mechanics. The specific
works are outlined as follows:
Developed an explicit version of the NMM:
Investigated the traditional NMM in terms of computational accuracy and
efficiency;
Proposed an explicit time integration scheme for the NMM and to verify it
with respect to the computational efficiency and accuracy;
Combined the explicit and implicit algorithms for the NMM:
Couple the temporal explicit and implicit NMM;
Coupled the spatial explicit and implicit NMM;
Extended the explicit NMM for the rock slope stability analysis:
Implemented a seismic version of the explicit NMM for the dynamic
stability analysis of rock slope;
Applied the developed programming code to simulate the dynamic
stability of rock slope.
DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY
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9.2 CONCLUSIONS
The major contribution of this research is concentrated on the development of the
explicit numerical manifold method (ENMM) and its implementation on the stability
analysis of rock slope under dynamic effect. Based on the work in this thesis, the
following conclusions can be drawn:
1). Newmark integration scheme used in the NMM is investigated. The numerical
results present good accuracy and stability of the Newmark explicit scheme
compared to the implicit scheme. The Newmark explicit scheme is more
efficient in solving the nonlinear dynamic systems and such problems
compared to implicit scheme with respect to computational efficiency.
2). An explicit time integration scheme for the NMM is proposed to improve the
computational efficiency, in which a modified version of the NMM based on
an explicit time integration algorithm is derived on the dual cover system. The
original NMM based on displacement method is revised into an explicit
formulation of a force method. Although the ENMM requires small time-step
due to numerical stability of the scheme, it is efficient without assembling the
stiffness equations. Compared to the OCI used in the NMM, the open-close
algorithm is more efficient in the ENMM because of the explicit time
integration scheme without solving simultaneous algebraic equations in each
step and the smaller penetration incurred due to a smaller time step used. The
developed method is validated by three examples, two static problems of a
continuous simple-supported beam and plane stress field problem, the other
dynamic one of a single block sliding down on a slope. Results showed that
the accuracy of the ENMM can be ensured when the time step is small for
both the continuous and the contact problems. A highly fractured rock slope
and tunnel modelling are subsequently simulated. It is shown that the
computational efficiency of the proposed ENMM can be significantly
improved, while without losing the accuracy, comparing to the original
implicit version of the NMM. The ENMM is more suitable for large-scale
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rock mass stability analysis and it deserves to be further developed for
engineering computations of practical rock engineering problems.
3). A modified version of the NMM based on an explicit time integration
algorithm is proposed. The calibration study of the ENMM on P-wave
propagation across a rock bar has been conducted. Various considerations in
the numerical simulations are discussed and parametric studies have been
carried out to obtain an insight into the influencing factors in wave
propagation simulation. The numerical results from the ENMM and NMM
modelling are accordant well with the theoretical solutions. The mesh ratio is
regarded as one of the major factors influencing the simulation accuracy. With
the consideration of both calculation accuracy and efficiency, a mesh ratio of
1/16 is recommended for one dimensional ENMM analysis. Furthermore, the
selection of a suitable time step depends on the internal element system and
the contact transfer between the interfaces. With the decrease of the time step
increment, the results become more accurate for the incident wave. In terms of
efficiency, the ENMM is more efficient than that of the NMM, even though
the different time steps are used. To further verify the capability of the
proposed ENMM in modelling of seismic wave effect in fractured rock mass,
a dynamic stability assessment for fractured rock slope under seismic effect is
analysed as well. The simulated results show that the computational efficiency
of the proposed ENMM can be significantly improved for the simulation of
stress wave propagation problems.
4). A temporal coupled explicit and implicit algorithm for the numerical manifold
method (NMM) is proposed. The time integration schemes, transfer algorithm,
contact algorithm and damping algorithm are studied in the temporal coupled
E-I algorithm to combine both merits of the explicit and implicit algorithms in
terms of efficiency and accuracy. Then, some numerical examples are
simulated using the proposed coupled algorithms, in which one calibration
example is studied with respect to the coupled temporal based on the dual
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cover system. One numerical example of open-pit slope seismic stability
analysis using the coupled E-I algorithm is investigated as well. The simulated
results are well agreement with the implicit and explicit algorithms
simulations, but the efficiency is improved evidently. It is predicted that the
couple E-I algorithm proposed in the present paper can be applied into larger
scales engineering systems to combine the merits of both the implicit and
explicit algorithms in the NMM.
5). A spatial coupled E-I algorithm for the NMM is proposed. The time
integration schemes in the E-I algorithm, transfer algorithm of the coupled E-I
algorithm, the implicit contact algorithms based on the implicit integration
scheme and explicit contact algorithm based on the explicit integration are
studied in terms of accuracy and efficiency, respectively. In particular, onefold
cover system to the coupled E-I algorithm is proposed and drawn into the
coupled spatial E-I algorithm, in which the contact algorithm based on the
onefold cover system is discussed and derived in detail. Finally, some
numerical examples are simulated using the proposed coupled E-I algorithms,
in which one calibration example is studied using the proposed E-I algorithms
relied on the onefold cover system; One numerical example of rock slope
seismic stability analysis using the coupled E-I algorithm is studied as well.
The simulated results are well agreement with the implicit and explicit
algorithms, but the efficiency of the coupled algorithm is obviously higher
than that of original version of the NMM. It is predicted that the couple E-I
algorithm proposed in the present paper can be applied into larger scales
engineering systems to combine the merits of both the implicit and explicit
algorithms in the NMM.
6). The dynamic stability analysis of rock slope failure using the numerical
manifold method (NMM) is studied. Firstly, conservational pseudo-static
methods, Newmark Method and numerical methods applying into the seismic
stability analyses are investigated, the advantages and limitations of which are
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studied by contrast of the NMM. Then, an alternative explicit algorithm of the
NMM and coupled explicit-implicit NMM are proposed to study the seismic
stability of rock slope. Furthermore, parallel computing with OpenMP is
evaluated to improve efficiency of the NMM. To reveal the validity and
applicability of the proposed NMM, some numerical examples of rock slope
stability analysis are investigated. The first one is a dynamic case of Lake
Anderson slope failure from the earthquake of 6 August 1979 Coyote Lake,
California USA. Simulated results of the NMM will compare with the field
measurements to illustrate the applicability of the NMM. The other example of
JinpingⅠHydropower Station dam left abutment slope is taken into account to
present the coupled ENMM with DDA in terms of efficiency. Furthermore, the
coupled explicit and implicit NMM are presented to simulate the stability of
rock slope as well. Therefore, it can be predicted that the proposed method is
promising and can be extend applied to larger scale project of rock slope with
respect to dynamic stability analysis.
9.3 RECOMMENDATIONS
The present study has validated the efficiency and robustness of the developed
ENMM to account for the dynamic problems such as stress wave propagation and
seismic stability analysis of rock slope. It is demanded a long way to further improve its
capability and applications in the project. Thus, there are still lots of work to be done in
the future research.
As demonstrated in Chapter 4 and 5, the accuracy and efficiency the ENMM is
more easily apt to be controlled by the finite element mesh size than that of the NMM.
The research can be carried out as the following:
Development of the adaptive finite mesh technique to avoid the tiny and
singular elements to the most degree;
Modification of the contact mechanics of OCI to further improve the contact
accuracy of the ENMM, including the selection of the appropriate contact
spring stiffness and the optimization of the current contact algorithm;
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Development of the more efficient explicit scheme to further improve the
efficiency the current ENMM.
The coupled E-I algorithms in Chapter 6 to 8 have demonstrated the viability of the
developed ENMM combining the NMM and other numerical methods. Thus, the
following work can be done:
Development of other coupled algorithms such as E-I and E-E with different
time step sizes in spatial and temporal aspects;
Development of the coupled method of ENMM and DDA;
Development of the explicit scheme of the 3D NMM to further improve the
capability of the current 3D code;
Development of the parallelization computation such as MPI parallel algorithm
and so on.
The preliminary studies have demonstrated the great potential of the ENMM
enabling simulating for stability analysis of rock slope under seismic effect with
different conditions. Thus, the more influent factors can be taken into account such as
seepage, liquefaction and mechanical effects in the future research.
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