Development of a 3D Hybrid Finite-Discrete Element Simulator Based on GPGPU-Parallelized Computation for Modelling Rock Fracturing Under Quasi-Static and Dynamic Loading Conditions

  • Daisuke Fukuda
  • Mojtaba Mohammadnejad
  • Hongyuan LiuEmail author
  • Qianbing Zhang
  • Jian Zhao
  • Sevda Dehkhoda
  • Andrew Chan
  • Jun-ichi Kodama
  • Yoshiaki Fujii
Original Paper


As a state-of-the-art computational method for simulating rock fracturing and fragmentation, the combined finite-discrete element method (FDEM) has become widely accepted since Munjiza (2004) published his comprehensive book of FDEM. This study developed a general-purpose graphic-processing-unit (GPGPU)-parallelized FDEM using the compute unified device architecture C/C ++ based on the authors’ former sequential two-dimensional (2D) and three-dimensional (3D) Y-HFDEM IDE (integrated development environment) code. The theory and algorithm of the GPGPU-parallelized 3D Y-HFDEM IDE code are first introduced by focusing on the implementation of the contact detection algorithm, which is different from that in the sequential code, contact damping and contact friction. 3D modelling of the failure process of limestone under quasi-static loading conditions in uniaxial compressive strength (UCS) tests and Brazilian tensile strength (BTS) tests are then conducted using the GPGPU-parallelized 3D Y-HFDEM IDE code. The 3D FDEM modelling results show that mixed-mode I–II failures are the dominant failure mechanisms along the shear and splitting failure planes in the UCS and BTS models, respectively, with unstructured meshes. Pure mode I splitting failure planes and pure mode II shear failure planes are only possible in the UCS and BTS models, respectively, with structured meshes. Subsequently, 3D modelling of the dynamic fracturing of marble in dynamic Brazilian tests with a split Hopkinson pressure bar (SHPB) apparatus is conducted using the GPGPU-parallelized 3D HFDEM IDE code considering the entire SHPB testing system. The modelled failure process, final fracture pattern and time histories of the dynamic compressive wave, reflective tensile wave and transmitted compressive wave are compared quantitatively and qualitatively with those from experiments, and good agreements are achieved between them. The computing performance analysis shows the GPGPU-parallelized 3D HFDEM IDE code is 284 times faster than its sequential version and can achieve the computational complexity of O(N). The results demonstrate that the GPGPU-parallelized 3D Y-HFDEM IDE code is a valuable and powerful numerical tool for investigating rock fracturing under quasi-static and dynamic loading conditions in rock engineering applications although very fine elements with maximum element size no bigger than the length of the fracture process zone must be used in the area where fracturing process is modelled.


Rocks 3D fracture process analysis FDEM Quasi-static loading Dynamic loading Parallel computation GPGPU CUDA C/C++ 

List of Symbols


Shape parameter of softening curve

b, B and Bij

Exponent, shape parameter of softening curve, and left Cauchy–Green strain, respectively

η and ηcrit

Viscous damping coefficient, and critical viscous damping coefficient, respectively

c, crock, C, CKLMN

Cohesion, cohesion of rock, shape parameter of softening curve, and effective elastic stiffness tensor, respectively

Dij, D, f(D)

Rate of deformation tensor, damage variable, and characteristic function, respectively


Kronecker delta


Relative displacement vector

E and Erock

Young’s modulus, and Young’s modulus of rock, respectively


Green–Lagrange strain tensor

εinci, εrefl and εtans

Axial strains in the incident, reflection and transmission bars, respectively

ϕ and ϕrock

Internal friction angle, and internal friction angle of rock, respectively

fIB and fTB

Axial compressive forces in incident and transmission bars, respectively

fint, fcoh, fcon, fext and fcon_tan

Equivalent nodal forces corresponding to internal load, cohesive force, contact force, external load, and contact tangential force, respectively


Deformation gradient


Gravitational acceleration

GfI, GfI_rock, GfII and GfII_rock

Mode I fracture energy, mode I fracture energy of rock, mode II fracture energy, and model II fracture energy of rock, respectively


Friction coefficient

h and have

Element length, and average element length, respectively


Determinant of the deformation gradient


Mode I fracture toughness


Lumped nodal mass vector

N, NBpG, Nthread and NTpB

Number of elements, number of “blocks” per “grid”, number of threads, and number of “threads” per “block”, respectively,

λ and µ

Lame constants

ρ and ρrock

Density, and density of rock, respectively

o, op, ot, omax, ooverlap and on

Crack opening displacement, “artificial” elastic limit of o, critical value of o, maximum value of o, normal overlap, and nominal normal overlap, respectively

Popen, Ptan, Poverlap and Pn_con

Artificial penalty terms for opening in the normal direction, sliding in the tangential direction, and overlapping in the normal direction and normal “contact penalty”, respectively

s, sp, st and smax

Crack sliding displacement, “artificial” elastic limit of s, critical values of s, and maximum values of s respectively


Second Piola–Kirchhoff stress tensor


Cauchy stress tensor


Normal cohesive traction

σinci, σrefl and σtans

Axial stresses in the incident, reflection, and transmission bars, respectively

Δt and t

Time step increment and current time step, respectively

T, Ts and Ts_rock

Transition force, tensile strength, and tensile strength of rock, respectively


Shear cohesive traction


Nodal displacement vector


Initial velocity


Poisson’s ratio of rock



Boundary element method


Brazilian tensile strength


6-Node initially zero-thickness cohesive element




Compute unified device architecture


Cohesive zone model


Distinct element method


Discontinuous deformation analysis


Dynamic fracture process analysis


Efficient contact detection activation


Extrinsic cohesive zone model


Combined finite-discrete element method


Finite difference method


Finite element method


Fracture process zone


General-purpose graphic-processing-unit




Intrinsic cohesive zone model


Incident bar


International Society for Rock Mechanics


Message-passing interface




Open computing language


Open multi-processing


Personal computer


Scaled boundary finite element method


Split Hopkinson pressure bar


Smoothed particle hydrodynamics


4-Node tetrahedral finite element


Transmission bar




TET4 to point—contact interaction between the contactor 4-node tetrahedral finite element and the target point


TET4 to TET4—contact interaction between the contactor 4-node tetrahedral finite element and the target 4-node tetrahedral finite element




Uniaxial compressive strength



This work was supported by JSPS KAKENHI for Grant-in-Aid for Young Scientists (Grant numbers 18K14165) for the first author. The corresponding author would like to acknowledge the support of Australia-Japan Foundation, Department of Foreign Affairs and Trade, Australian Government (AJF Grant no. 17/20470), Australia Academy of Science (AAS Grant no. RI8) and Institution Research Grant Scheme (IRGS Grant no. L0018929) and Natural Science Foundation of China (NSFC Grant nos. 51574060 and 51079017), in which the corresponding author is the international collaborator. Moreover, all authors would like to thank the editor-in-chief, i.e. Prof G. Barla, and the three anonymous reviewers for their helpful and constructive comments that significantly contributed to improving the final version of the paper.

Compliance with Ethical Standards

Conflict of interest

The first author and the corresponding author declare that the received funds do not lead to any conflicts of interest regarding the publication of this manuscript.


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of EngineeringHokkaido UniversityHokkaidoJapan
  2. 2.College of Sciences and EngineeringUniversity of TasmaniaHobartAustralia
  3. 3.CSIRO Minerals Resources Business UnitQueensland Centre for Advanced TechnologiesBrisbaneAustralia
  4. 4.Department of Civil EngineeringMonash UniversityMelbourneAustralia

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