Advertisement

Rock Mechanics and Rock Engineering

, Volume 51, Issue 12, pp 3737–3769 | Cite as

A Novel Contact Algorithm Based on a Distance Potential Function for the 3D Discrete-Element Method

  • Lanhao Zhao
  • Xunnan Liu
  • Jia Mao
  • Dong Xu
  • Antonio Munjiza
  • Eldad Avital
Original Paper
  • 336 Downloads

Abstract

The combined finite–discrete-element method (FDEM) has made a groundbreaking progress in the computation of contact interaction. However, FDEM has a strict requirement on the element type, and the simulation result may be inconsistent due to a deficiency of physical meaning of the potential function. To address this problem, a new 3D discrete-element method based on a distance potential is proposed for a system consisting of a large number of arbitrary convex polyhedral elements. In this approach, a well-defined distance potential is proposed as a function of the penetration between the contact pairs. It exhibits a clear physical meaning and a precise measurement of the embedding between the elements in contact. The newly presented method provides a holonomic and accurate contact interaction without being influenced by the element shape. Therefore, the restraint of the element type in FDEM is removed and the proposed method can be used for arbitrary convex polyhedrons. In addition, an improved contact detection algorithm for non-uniform block discrete elements is provided to overcome the constraint of elements with the same size in the Munjiza-No Binary Search contact detection method. The new approach retains the merits of the FDEM and avoids its deficiencies. It is validated with well-known benchmark examples including an impact simulation, a friction experiment, a joint structure of a sliding rock mass, pillar impact, block accumulation, and analysis for the failure process of wedge slope. The results of this proposed method are in excellent agreement with the existing experimental measurements and analytical solutions.

Keywords

Discrete-element modelling Arbitrary convex polyhedral element Distance potential function Tangential contact interaction Contact detection algorithm 

Abbreviations

CP

Common plane

DEM

Discrete-element method

DDA

Discontinuous deformation analysis

FDEM

The combined finite–discrete-element method

MMR

Multi-step Munjiza–Rougier algorithm

NBS

No binary search

NMM

Numerical manifold method

LWSP

Left structural weak surfaces

RWSP

Right structural weak surfaces

List of symbols

\(\varphi\)

Potential function

\(k\)

Penalty parameter

\(V\)

Volume

\({{\varvec{f}}_{\text{n}}}\)

Normal contact force

\({V_{{\text{t}} \cap {\text{c}}}}\)

Overlapping volume between the discrete elements \({\beta _{\text{t}}}\) and \({\beta _{\text{c}}}\)

\({\varphi _{\text{c}}}\)

Potential function in \({V_{{\text{t}} \cap {\text{c}}}}\) belonging to the elements \({\beta _{\text{c}}}\)

\({\varphi _{\text{t}}}\)

Potential function in \({V_{{\text{t}} \cap {\text{c}}}}\) belonging to the elements \({\beta _{\text{t}}}\)

\({S_{{\text{t}} \cap {\text{c}}}}\)

Boundary surface of \({V_{{\text{t}} \cap {\text{c}}}}\)

\({\varvec{n}}\)

Outward unit vector of the boundary surface \({S_{{\text{t}} \cap {\text{c}}}}\)

\({\varphi _{\text{d}}}\)

Distance potential function

\({h_{\rm I}}\)

Distance from the point p to the base \({\alpha _{\rm I}}\) of the sub-polyhedron

\(r\)

Radius of the maximum inscribed sphere of a polyhedral element

\(S\)

Intersection surface among the plane of the base of \({\beta _{\text{c}}}\) and the target sub-polyhedron of \({\beta _{\text{t}}}\)

\({S_1},{S_2} \ldots ,{S_n}\)

Nodes of the intersection surface \(S\)

\(B\)

Intersection polygon defined by the surface \(S\) and the base \(\alpha\) of \({\beta _{\text{c}}}\)

\({B_1},{B_2} \ldots ,{B_n}\)

Nodes of the polygon \(B\)

\({x_i},{y_i}\)

Local coordinates of the point on the polygonal surface \(B\)

\({A_1},{A_2} \ldots ,{A_n}\)

Parameters of the formulation of distance potential function in local coordinate system

\(({x_1},{y_1})\)

Local coordinates of \({B_1}\)

\(({x_2},{y_2})\)

Local coordinates of \({B_2}\)

\(({x_3},{y_3})\)

Local coordinates of \({B_3}\)

\({{\varvec{f}}_{{\text{n}},B}}\)

Normal contact force over the polygonal surface \(B\)

\({{\varvec{n}}_B}\)

Outward unit vector of the polygonal surface \(B\)

\({k_{\text{n}}}\)

Normal contact stiffness

\({{\varvec{M}}_{x,B}},{{\varvec{M}}_{y,B}}\)

Moments contributed by the contact normal force \({{\varvec{f}}_{{\text{n}},B}}\) in the local coordinate system \((x,y)\)

\({N_i}(\eta ,\zeta )\)

Shape function

\(m\)

Number of the divided triangular surfaces of the polygonal surface \(B\)

\(\left| J \right|\)

Jacobi determinant of coordinate transformation

\({C_1},{C_2} \ldots ,{C_n}\)

Parameters of the formulations of the normal contact force and moments in the natural coordinate system

\({{\varvec{M}}_{\eta ,B}},{{\varvec{M}}_{\zeta ,B}}\)

Moments contributed by the contact normal force \({{\varvec{f}}_{{\text{n}},B}}\) in the natural coordinate system \((\eta ,\zeta )\)

\({\eta _{\text{n}}},{\zeta _{\text{n}}}\)

Coordinates of the action position of the normal contact force in the natural coordinate system

\({n_{\text{s}}}\)

Number of the boundary surfaces

\({\varvec{f}}_{{\text{s}}}^{i}\)

Tangential contact force at step i

\(\Delta \varvec{\delta}_{{{\text{s,t}}}}^{i}\)

Tangential increment displacement of each surface at step i

\({k_{\text{s}}}\)

Tangential contact stiffness

\({\varvec{v}^i}\)

Relative velocity of the contact element \({\beta _{\text{c}}}\) with respect to the target element \({\beta _{\text{t}}}\)

\(\varvec{v}_{c}^{i}\)

Translational velocity of block \({\beta _{\text{c}}}\) at step i

\(\varvec{v}_{t}^{i}\)

Translational velocity of block \({\beta _{\text{t}}}\) at step i

\(\varvec{\omega}_{{\text{c}}}^{i}\)

Angular velocity of block \({\beta _{\text{c}}}\) at step i

\(\varvec{\omega}_{{\text{t}}}^{i}\)

Angular velocity of block \({\beta _{\text{t}}}\) at step i

\(\Delta \varvec{\delta}_{{\text{s}}}^{i}\)

Incremental tangential displacement at step i

\(\Delta {{\varvec{s}}^i}\)

Incremental displacement between \({\beta _{\text{c}}}\) and \({\beta _{\text{t}}}\)

\({{\varvec{n}}_{\text{n}}}\)

Unit direction vector of total normal contact force

\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{{\varvec{r}}}\)

Rotation matrix that rotates the normal vector from step i − 1 to the normal vector at the current step i

\(\Delta {\varvec{f}}_{{\text{s}}}^{i}\)

Incremental tangential contact force

\({\varphi _\mu }\)

Maximum static friction angle

\(C\)

Cohesion force

\({({f_{\text{s}}})_{\rm{max} }}\)

Maximum possible value of the magnitude of the tangential force

\({\varvec{M}}_{{\text{s}}}^{i}\)

Tangential contact moment at step i

\({\varvec{\gamma}_{\text{s}}}\)

Vector from the force \({\varvec{f}}_{{\text{s}}}^{i}\) load position to the centroid of the element

\({d_i}\)

Maximum circumradius for each group block

\(D\)

Maximum circumradius among all the blocks

\({x_{{\text{cent}}}},{y_{{\text{cent}}}},{z_{{\text{cent}}}}\)

Current coordinates of the centroids in the global coordinate system

\({x_k},{y_k},{z_k}\)

Coordinates of the centroids when the elements are mapped into the cell

\(l\)

Length of the cell

\({l_{\text{c}}}\)

Distance between the centroids of the contact pairs

\({l_{\rm{max} }}\)

Contact distance

\({l_{1\rm{max} }},{l_{2\rm{max} }}\)

The longest distances between the vertexes and centroids of the two contact elements

\({{\varvec{u}}_0}\)

Initial velocity of the sliding block

\({{\varvec{s}}_0}\)

Initial displacement of the sliding block

\(g\)

Gravity acceleration

\(\theta\)

Angle of the inclined plane

\(\mu\)

Friction coefficient

\(t\)

Time

\({V_{\text{w}}}\)

Wave velocity

\(E\)

Elastic modulus

\(\rho\)

Density of the block

\({\varvec{F}}\)

Normal contact force between the blocks in impact simulation of a pillar

\(\Delta \varvec{\delta}\)

Relative displacement between the contact blocks in impact simulation of a pillar

\(\xi\)

Damping radio

Notes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant no. 51279050), the 15th Fok Ying-Tong Education Foundation for Young Teachers in the Higher Education Institutions of China (Grant no. 151073), the National Key R&D Program of China (Grant no. 2016YFC0401601), the project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (Grant YS11001), and the 111 Project.

References

  1. Albusaidi A, Hazzard JF, Young RP (2005) Distinct element modeling of hydraulically fractured Lac du Bonnet granite. J Geophys Res Solid Earth 110:B06032Google Scholar
  2. Bao H, Zhao Z (2012) The vertex-to-vertex contact analysis in the two-dimensional discontinuous deformation analysis. Adv Eng Softw 45:1–10CrossRefGoogle Scholar
  3. Boon CW, Houlsby GT, Utili S (2014) New insights into the 1963 Vajont slide using 2D and 3D distinct-element method analyses. Geotechnique 64:800–816CrossRefGoogle Scholar
  4. Boon CW, Houlsby GT, Utili S (2015) A new rock slicing method based on linear programming. Comput Geotech 65:12–29CrossRefGoogle Scholar
  5. Cai Y, He T, Wang R (2000) Numerical simulation of dynamic process of the Tangshan earthquake by a new method—LDDA. Pure Appl Geophys 157:2083–2104CrossRefGoogle Scholar
  6. Cundall PA (1971) A computer model for simulating progressive, large-scale movements in blocky rock systems. In: Proceedings of symposium of international society of rock mechanics, Nancy, France, pp II-8Google Scholar
  7. Cundall PA (1988) Formulation of a three-dimensional distinct element model—Part I. A scheme to detect and represent contacts in a system composed of many polyhedral blocks. Int J Rock Mech Min Sci Geomech Abstr 25:107–116CrossRefGoogle Scholar
  8. Cundall PA, Hart RD (1985) Development of generalized 2-D and 3-D distinct element programs for modeling jointed rock. Itasca Consulting Group Misc. Paper SL-85-1. U.S. Army Corps of Engineering, VicksburgGoogle Scholar
  9. Cundall PA, Strack OD (1979) A discrete numerical model for granular assemblies. Geotechnique 29:47–65CrossRefGoogle Scholar
  10. Feng YT, Han K, Owen DRJ (2012) Energy-conserving contact interaction models for arbitrarily shaped discrete elements. Comput Methods Appl Mech Eng 205:169–177CrossRefGoogle Scholar
  11. Garcia X, Latham J, Xiang J, Harrison JP (2009) A clustered overlapping sphere algorithm to represent real particles in discrete element modelling. Geotechnique 59:779–784CrossRefGoogle Scholar
  12. Hart R, Cundall PA, Lemos J (1988) Formulation of a three-dimensional distinct element model—Part II. Mechanical calculations for motion and interaction of a system composed of many polyhedral blocks. Int J Rock Mech Min Sci Geomech Abstr 25:117–125CrossRefGoogle Scholar
  13. He L, An X, Ma G, Zhao Z (2013) Development of three-dimensional numerical manifold method for jointed rock slope stability analysis. Int J Rock Mech Min Sci 64:22–35CrossRefGoogle Scholar
  14. Hohner D, Wirtz S, Kruggelemden H, Scherer V (2011) Comparison of the multi-sphere and polyhedral approach to simulate non-spherical particles within the discrete element method: influence on temporal force evolution for multiple contacts. Powder Technol 208:643–656CrossRefGoogle Scholar
  15. Ikegawa Y, Hudson JA (1992) Novel automatic identification system for three dimensional multi-block systems. Eng Comput 9:169–179CrossRefGoogle Scholar
  16. Itasca (2014a) PFC. 2D (Particle flow code in 2 dimensions), 5.0 edn. Itasca Consulting Group, MinneapolisGoogle Scholar
  17. Itasca (2014b) PFC. 3D (Particle flow code in 3 dimensions), 5.0 edn. Itasca Consulting Group, MinneapolisGoogle Scholar
  18. Itasca (2016a) 3DEC-3-D distinct element code, 6.0 edn. Itasca Consulting Group, MinneapolisGoogle Scholar
  19. Itasca (2016b) UDEC-universal distinct element code, 6.0. edn. Itasca Consulting Group, MinneapolisGoogle Scholar
  20. Jiang Q, Zhou C, Li D (2009) A three-dimensional numerical manifold method based on tetrahedral meshes. Comput Struct 87:880–889CrossRefGoogle Scholar
  21. Jiang M, Shen Z, Wang J (2015) A novel three-dimensional contact model for granulates incorporating rolling and twisting resistances. Comput Geotech 65:147–163CrossRefGoogle Scholar
  22. Jin F, Zhang C, Hu W, Wang J (2011) 3D mode discrete element method: elastic model. Int J Rock Mech Min Sci 48:59–66CrossRefGoogle Scholar
  23. Jing L (2000) Block system construction for three-dimensional discrete element models of fractured rocks. Int J Rock Mech Min Sci 37:645–659CrossRefGoogle Scholar
  24. Jing L (2003) A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering. Int J Rock Mech Min Sci 40:283–353CrossRefGoogle Scholar
  25. Jung JW, Santamarina JC, Soga K (2012) Stress-strain response of hydrate-bearing sands: numerical study using discrete element method simulations. J Geophys Res Solid Earth 117:B04202Google Scholar
  26. Kawamoto R, Ando E, Viggiani G, Andrade JE (2016) Level set discrete element method for three-dimensional computations with triaxial case study. J Mech Phys Solids 91:1–13CrossRefGoogle Scholar
  27. Kodam M, Bharadwaj R, Curtis JS, Hancock BC, Wassgren C (2010a) Cylindrical object contact detection for use in discrete element method simulations. Part I—Contact detection algorithms. Chem Eng Sci 65:5852–5862CrossRefGoogle Scholar
  28. Kodam M, Bharadwaj R, Curtis JS, Hancock BC, Wassgren C (2010b) Cylindrical object contact detection for use in discrete element method simulations, Part II—Experimental validation. Chem Eng Sci 65:5863–5871CrossRefGoogle Scholar
  29. Latham JP, Munjiza A (2004) The modelling of particle systems with real shapes. Philos Trans R Soc A 362:1953–1972CrossRefGoogle Scholar
  30. Li X, Zheng H (2015) Condensed form of complementarity formulation for discontinuous deformation analysis. Sci China Technol Sci 58:1509–1519CrossRefGoogle Scholar
  31. Li S, Zhao M, Wang Y, Rao Y (2004) A new numerical method for DEM-block and particle model. Int J Rock Mech Min Sci 41:414–418CrossRefGoogle Scholar
  32. Li SH, Wang J, Liu B, Dong DP (2007) Analysis of critical excavation depth for a jointed rock slope using a face-to-face discrete element method. Rock Mech Rock Eng 40:331–348CrossRefGoogle Scholar
  33. Lin CT, Amadei B, Jung J, Dwyer JF (1996) Extensions of discontinuous deformation analysis for jointed rock masses. Int J Rock Mech Min Sci Geomech Abstr 33:671–694CrossRefGoogle Scholar
  34. Lu G, Third JR, Muller CR (2015) Discrete element models for non-spherical particle systems: from theoretical developments to applications. Chem Eng Sci 127:425–465CrossRefGoogle Scholar
  35. Mahabadi OK, Grasselli G, Munjiza A (2010) Y-GUI: a graphical user interface and pre-processor for the combined finite-discrete element code, Y2D, incorporating material heterogeneity. Comput Geosci 36:241–252CrossRefGoogle Scholar
  36. Mahabadi OK, Lisjak A, Munjiza A, Grasselli G (2012) Y-Geo: new combined finite-discrete element numerical code for geomechanical applications. Int J Geomech 12:676–688CrossRefGoogle Scholar
  37. Mcdowell GR, Harireche O (2002) Discrete element modelling of soil particle fracture. Geotechnique 52:131–135CrossRefGoogle Scholar
  38. Morgan WE, Aral MM (2015) An implicitly coupled hydro-geomechanical model for hydraulic fracture simulation with the discontinuous deformation analysis. Int J Rock Mech Min Sci 73:82–94CrossRefGoogle Scholar
  39. Munjiza A (2004) The Combined finite-discrete element method. Wiley, ChichesterCrossRefGoogle Scholar
  40. Munjiza A, Andrews KRF (1998) NBS contact detection algorithm for bodies of similar size. Int J Numer Methods Eng 43:131–149CrossRefGoogle Scholar
  41. Munjiza A, John NWM (2002) Mesh size sensitivity of the combined FEM/DEM fracture and fragmentation algorithms. Eng Fract Mech 69:281–295CrossRefGoogle Scholar
  42. Munjiza A, Owen DRJ, Bicanic N (1995) A combined finite-discrete element method in transient dynamics of fracturing solids. Eng Comput 12:145–174CrossRefGoogle Scholar
  43. Munjiza A, Bangash T, John NWM (2004) The combined finite-discrete element method for structural failure and collapse. Eng Fract Mech 71:469–483CrossRefGoogle Scholar
  44. Munjiza A, Rougier E, John NWM (2006) MR linear contact detection algorithm. Int J Numer Methods Eng 66:46–71CrossRefGoogle Scholar
  45. Nezami EG, Hashash YMA, Zhao D, Ghaboussi J (2004) A fast contact detection algorithm for 3-D discrete element method. Comput Geotech 31:575–587CrossRefGoogle Scholar
  46. Nezami EG, Hashash YMA, Zhao D, Ghaboussi J (2006) Shortest link method for contact detection in discrete element method. Int J Numer Anal Methods Geomech 30:783–801CrossRefGoogle Scholar
  47. Nie W, Zhao ZY, Ning Y, Sun JP (2014) Development of rock bolt elements in two-dimensional discontinuous deformation analysis. Rock Mech Rock Eng 47:2157–2170CrossRefGoogle Scholar
  48. Ning Y, An X, Ma G (2011) Footwall slope stability analysis with the numerical manifold method. Int J Rock Mech Min Sci 48:964–975CrossRefGoogle Scholar
  49. Rougier E, Bradley CR, Broom ST, Knight EE, Munjiza A, Sussman AJ, Swift RP (2011) The combined finite-discrete element method applied to the study of rock fracturing behavior in 3D. In: American Rock Mechanics Association 45th U.S. rock mechanics/geomechanics symposium, San Francisco, USA, 26–29 June. No. ARMA 11-517Google Scholar
  50. Shi GH (1991) Manifold method of material analysis. In: Transaction of the 9th army conference on applied mathematics and computing, Minneapolis, Minnesota, US Army Research Office, pp 57–76Google Scholar
  51. Shi GH (2001) Three-dimensional discontinuous deformation analysis. In: Proceedings of the forth international conference on analysis of discontinuous deformation Glasgow, Scotland, UK, 6–8 June, pp 1–21Google Scholar
  52. Shi GH, Goodman RE (1985) Two dimensional discontinuous deformation analysis. Int J Numer Anal Methods Geomech 9:541–556CrossRefGoogle Scholar
  53. Shimizu H, Murata S, Ishida T (2011) The distinct element analysis for hydraulic fracturing in hard rock considering fluid viscosity and particle size distribution. Int J Rock Mech Min Sci 48:712–727CrossRefGoogle Scholar
  54. Smeets B, Odenthal T, Vanmaercke S, Ramon H (2015) Polygon-based contact description for modeling arbitrary polyhedra in the discrete element method. Comput Methods Appl Mech Eng 290:277–289CrossRefGoogle Scholar
  55. Terada K, Asai M, Yamagishi M (2003) Finite cover method for linear and non-linear analyses of heterogeneous solids. Int J Numer Methods Eng 58:1321–1346CrossRefGoogle Scholar
  56. Wang J, Yan H (2013) On the role of particle breakage in the shear failure behavior of granular soils by DEM. Int J Numer Anal Methods Geomech 37:832–854CrossRefGoogle Scholar
  57. Yan CZ, Zheng H (2017) A new potential function for the calculation of contact forces in the combined finite–discrete element method. Int J Numer Anal Methods Geomech 41:265–283CrossRefGoogle Scholar
  58. Yan CZ, Zheng H, Ge XR (2015) Unified calibration based potential contact force in discrete element method. Rock Soil Mechanics 36:249–256 (in Chinese) Google Scholar
  59. Yeung MR, Jiang Q, Sun N (2003) Validation of block theory and three-dimensional discontinuous deformation analysis as wedge stability analysis methods. Int J Rock Mech Min Sci 40:265–275CrossRefGoogle Scholar
  60. Yeung MR, Jiang Q, Sun N (2007) A model of edge-to-edge contact for three-dimensional discontinuous deformation analysis. Comput Geotech 34:175–186CrossRefGoogle Scholar
  61. Yoon JS, Zang A, Stephansson O (2014) Numerical investigation on optimized stimulation of intact and naturally fractured deep geothermal reservoirs using hydro-mechanical coupled discrete particles joints model. Geothermics 52:165–184CrossRefGoogle Scholar
  62. Yu Q, Ohnishi Y, Xue G, Chen D (2009) A generalized procedure to identify three dimensional rock blocks around complex excavations. Int J Numer Anal Methods Geomech 33:355–375CrossRefGoogle Scholar
  63. Zhang Y, Xu Q, Chen G, Zhao JX, Zheng L (2014) Extension of discontinuous deformation analysis and application in cohesive-frictional slope analysis. Int J Rock Mech Min Sci 70:533–545CrossRefGoogle Scholar
  64. Zheng H, Jiang W (2009) Discontinuous deformation analysis based on complementary theory. Sci China Technol Sci 52:2547–2554CrossRefGoogle Scholar
  65. Zheng H, Li X (2015) Mixed linear complementarity formulation of discontinuous deformation analysis. Int J Rock Mech Min Sci 75:23–32CrossRefGoogle Scholar
  66. Zheng H, Xu D (2014) New strategies for some issues of numerical manifold method in simulation of crack propagation. Int J Numer Methods Eng 97:986–1010CrossRefGoogle Scholar
  67. Zheng H, Zhang P, Du X (2016) Dual form of discontinuous deformation analysis Computer. Methods Appl Mech Eng 305:196–216CrossRefGoogle Scholar
  68. Zhou B, Huang R, Wang H, Wang J (2013) DEM investigation of particle anti-rotation effects on the micromechanical response of granular materials. Granul Matter 15:315–326CrossRefGoogle Scholar
  69. Zhu JB, Deng X, Zhao XB, Zhao J (2013) A numerical study on wave transmission across multiple intersecting joint sets in rock masses with UDEC. Rock Mech Rock Eng 46:1429–1442CrossRefGoogle Scholar
  70. Zhu H, Wu W, Chen J, Ma G, Liu X, Zhuang X (2016) Integration of three dimensional discontinuous deformation analysis (DDA) with binocular photogrammetry for stability analysis of tunnels in blocky rockmass. Tunn Undergr Space Technol 51:30–40CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Water Conservancy and HydropowerHohai UniversityNanjingChina
  2. 2.State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin UniversityTianjinChina
  3. 3.Faculty of Civil EngineeringUniversity of SplitSplitCroatia
  4. 4.School of Engineering and Materials ScienceQueen Mary University of LondonLondonUK

Personalised recommendations