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Detonation 2D Gif
Fully coupled fluid/solid interaction.



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This verification and validation manual presents a compilation of benchmark tests simulated using the Hybrid Optimization Software Suite.

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Books & Book Chapters

A. Munjiza, E. Rougier and E. E. Knight, Large Strain Finite Element Method: A Practical Course – First Edition, 2015 John Wiley and Sons, ISBN-10: 1118405307, ISBN-13: 978-1118405307.

A. Munjiza, E. E. Knight and E. Rougier, Computational Mechanics of Discontinua – First Edition, 2012 John Wiley and Sons, ISBN-10: 0470970804, ISBN-13: 978-0470970805.

Munjiza A. The combined finite-discrete element method. Wiley: New York, 2004. ISBN: 978-0-470-84199-0

Rougier, E., Knight, E.E. & Munjiza, A. Special issue titled “combined finite discrete element method and virtual experimentation”. Comp. Part. Mech. 7, 763 (2020).

A. Munjiza, E. Rougier, E. E. Knight, Z. Lei. Discrete Element and Particle Methods. Encyclopedia of Continuum Mechanics, 2018.

Chen, L., Hyman, J.D., Lei, Z., Min, T., Kang, Q., Rougier, E. and Viswanathan, H. (2018). Effect of Fracture Density on Effective Permeability of Matrix‐Fracture System in Shale Formations. In Geological Carbon Storage (eds S. Vialle, J. Ajo‐Franklin and J.W. Carey).

Peer Reviewed Journal Articles


Discontinua Scientific computing Artificial intelligence

E. Rougier, A. Munjiza, J. Munjiza. Distributed intelligence and the equivalence of matter and information. Computational Particle Mechanics, 7(5):1073–1080, 2020.

Fluid-solid interaction Contact, friction Fragmentation

E.E. Knight, E. Rougier, Z. Lei, B. Euser, V. Chau, S.H. Boyce, K. Gao, K. Okubo, M. Froment. HOSS: an implementation of the combined finite-discrete element method. Computational Particle Mechanics, 7(5):765–787, 2020.

Lagrangian model Granular media Mars regolith

Froment, M., Rougier, E., Larmat, C., Lei, Z., Euser, B., Kedar, S., et al. (2020). Lagrangian‐based simulations of hypervelocity impact experiments on Mars regolith proxy. Geophysical Research Letters, 47, e2020GL087393.

Stick-slip Fault friction Granular fault gouge

K. Gao, R.A. Guyer, E. Rougier, P.A. Johnson. Plate motion in sheared granular fault system. Earth and Planetary Science Letters, 548:116481, 2020.

Mixed-mode failure Carrara marble Confined extension

Boyce, S., Lei, Z., Euser, B. et al. Simulation of mixed-mode fracture using the combined finite–discrete element method. Comp. Part. Mech. 7, 1047–1055 (2020).

Earthquake rupture Fracture damage FDEM

Okubo, K., Rougier, E., Lei, Z. et al. Modeling earthquakes with off-fault damage using the combined finite-discrete element method. Comp. Part. Mech. 7, 1057–1072 (2020).

Finite element Discrete element Contact interaction

Lei, Z., Rougier, E., Euser, B. et al. A smooth contact algorithm for the combined finite discrete element method. Comp. Part. Mech. 7, 807–821 (2020).

Brittle fracture Shock loading Elastic moduli

K. Larkin, E. Rougier, V. Chau, G. Srinivasan, A. Abdelkefi, A. Hunter. Scale Bridging Damage Model for Quasi-Brittle Metals Informed with Crack Evolution Statistics. Journal of the Mechanics and Physics of Solids, 103921. 2020.

Fluid-structure interaction Fluid solver Immersed boundary

Munjiza, A., Rougier, E., Lei, Z. et al. FSIS: a novel fluid–solid interaction solver for fracturing and fragmenting solids. Comp. Part. Mech. 7, 789–805 (2020).

FDEM High strain rate Flyer plate, granite

Chau, V., Rougier, E., Lei, Z. et al. Numerical analysis of flyer plate experiments in granite via the combined finite–discrete element method. Comp. Part. Mech. 7, 1005–1016 (2020).

Hyperelastic Plastic flow rule Plasticity

E. Rougier, A. Munjiza, Z. Lei, V.T. Chau, E.E. Knight, A. Hunter, G. Srinivasan. 2020. The combined plastic and discrete fracture deformation framework for finite‐discrete element methods. International Journal for Numerical Methods in Engineering, 121(5), 1020-1035.


Earthquake rupture High-frequency radiation Energy budget

Okubo, K., Bhat, H.S., Rougier, E., Marty, S., Schubnel, A., Lei, Z., Knight, E.E., & Klinger, Y. 2019. Dynamics, Radiation, and Overall Energy Budget of Earthquake Rupture with Coseismic Off‐Fault Damage. Journal of Geophysical Research: Solid Earth, 124, 11771– 11801.

Brittle fracture Elastic moduli Crack statistics

N. Vaughn, A. Kononov, B. Moore, E. Rougier, H. Viswanathan, A. Hunter. 2019. Statistically informed upscaling of damage evolution in brittle materials. Theoretical and Applied Fracture Mechanics. 102, 210-221.

Time integration Hydraulic fracture Incompressible fluid

Lei, Z., Rougier, E., Munjiza, A., Viswanathan, H., & Knight, E.E. 2019. Simulation of discrete cracks driven by nearly incompressible fluid via 2D combined finite‐discrete element method. Int J Numer Anal Methods Geomech. 43, 1724-1743.

Crack interaction Propagation FDEM

Euser, B., Rougier, E., Lei, Z., Knight, E.E., Frash, L., Carey, J.W., Viswanathan, H., & Munjiza, 2019. Simulation of fracture coalescence in granite via the combined finite-discrete element method. Rock Mech Rock Eng. 52(9), 3213-3227.

Earthquakes Stress chains Granular materials

Gao, K., Guyer, R.A., Rougier, E., Ren, C.X., & Johnson, P. 2019. From Stress Chains to Acoustic Emission. Physical Review Letters. 123(4), 048003.

Nonlinear elasticity Wave propagation Non-destructive testing

Gao, K., Rougier, E., Guyer, R.A., Lei, Z., Johnson, P. 2019. Simulation of crack induced nonlinear elasticity using the combined finite-discrete element method. Ultrasonics. 98, 51-61.

Graph theory Machine learning Fracture process zone

Mudunuru, M.K.; Panda, N.; Karra, S.; Srinivasan, G.; Chau, V.T.; Rougier, E.; Hunter, A.; Viswanathan, H.S. Surrogate Models for Estimating Failure in Brittle and Quasi-Brittle Materials. Appl. Sci. 2019, 9, 2706.

Fracking Poromechanics Biot coeffecient

S. Rahimi-Aghdam, V.-T. Chau, H. Lee, H. Nguyen, W. Li, S. Karra, E. Rougier, H. Viswanathan, G. Srinivasan, and Z.P. Bažant. Branching of Hydraulic Cracks in Gas or Oil Shale with Closed Natural Fractures: How to Master Permeability. Proceedings of the National Academy of Sciences, 116(5):1532-1537, 2019.

Brittle failure Deep learning Neural networks

M. Schwarzer, B. Rogan, Y. Ruan, Z. Song, D.Y. Lee, A.G. Percus, V.T. Chau, B.A. Moore, E. Rougier, H.S. Viswanathan, G. Srinivasan. 2019. Learning to fail: Predicting fracture evolution in brittle material models using recurrent graph convolutional neural networks. Computational Materials Science. 162, 322-332.

Brittle cracks Machine learning Graph theory

A. Hunter, B.A. Moore, M. Mudunuru, V. Chau, R. Tchoua, C. Nyshadham, S. Karra, D. O’Malley, E. Rougier, H. Viswanathan, G. Srinivasan. 2019. Reduced-order modeling through machine learning and graph-theoretic approaches for brittle fracture applications. Computational Materials Science. 157, 87-98

FAST FDEM Sensitivity analysis

H.C. Godinez, E. Rougier, D. Osthus, Z. Lei, E. Knight, G. Srinivasan. 2019 Fourier amplitude sensitivity test applied to dynamic combined finite‐discrete element methods–based simulations. International Journal for Numerical and Analytical Methods in Geomechanics. 43, 30–44.


Hydrology Materials Science

G. Srinivasan, J.D. Hyman, D.A. Osthus, B.A. Moore, D. O’Malley, S. Karra, E. Rougier, A.A. Hagberg, A. Hunter, H.S. Viswanathan. 2018. Quantifying topological uncertainty in fractured systems using graph theory and machine learning. Scientific reports. 8(1).

Stick-slip Granular material Fault gouge

Gao, K., Euser, B., Rougier, E., Guyer, R.A., Lei, Z., Knight, E.E., Carmeliet, J., & Johnson, P. 2018. Modeling of stick‐slip behavior in sheared granular fault gouge using the combined finite‐ discrete element method. Journal of Geophysical Research: Solid Earth. 123(7), 5774-5792.

Off-fault damage Satellite image Kaikoura earthquake

Klinger, Y., Okubo, K., Vallage, A., Champenois, J., Delorme, A., Rougier, E., Lei, Z., Knight, E.E., Munjiza, A., Satriano, C., Baize, S., Langridge, R., & Bhat, Harsha S. 2018. Earthquake Damage Patterns Resolve Complex Rupture Processes. Geophysical Research Letters, 45(19), 10,279-10,287.

Machine learning Fracture networks Multi-scale

B.A. Moore, E. Rougier, D. O’Malley, G. Srinivasan, A. Hunter and H. Viswanathan. 2018. Predictive modeling of dynamic fracture growth in brittle materials with machine learning. Computational Materials Science. 148, 46-53.

Porous media Flow, percolation Discrete fracture

J.D. Hyman, S. Karra, C.W. Gable, H. Viswanathan, J.W. Carey, E. Rougier, Z. Lei. 2018. Discontinuities in effective permeability due to fracture percolation. Mechanics of Materials. 119, 25-33.

FDEM Split-Hopkinson Probabilistic calibration

D. Osthus, H.C. Godinez, E. Rougier, G. Srinivasan. Calibrating the stress-time curve of a combined finite-discrete element method to a Split Hopkinson Pressure bar experiment. International Journal of Rock Mechanics and Mining Sciences, 106:278-288, 2018.

2017 & Older

Composite tetrahedron Incompressible elasticity Volumetric locking

Lei, Z., Rougier, E., Knight, E.E., Frash, L., Carey, J.W., & Viswanathan, H. 2016. A non- locking composite tetrahedron element for the combined finite discrete element method. Engineering Computations. 33(7), 1929-1956.

Material deformation Anisotropy Finite Elements

Lei, Z., Rougier, E., Knight, E.E., Munjiza, A., & Viswanathan, H. 2016. A generalized anisotropic deformation formulation for geomaterials. Comp. Part. Mech 3, 215-228.

Hydraulic fracturing Shale gas Geomechanics

Carey, J.W., Lei, Z., Rougier, E., Mori, H. & Viswanathan, H.S. 2015. Fracture-permeability behavior of shale. J. Unconventional Oil and Gas Resources, 11, 27-43.

Environmental sci. Geophysics Hydrology

Jordan, A., Stauffer, P., Knight, E. et al. Radionuclide Gas Transport through Nuclear Explosion-Generated Fracture Networks. Sci Rep 5, 18383 (2015).

Dynamic fracture Rate dependency Anisotropy

A. Zubelewicz, E. Rougier, M. Ostoja-Starzewski, E.E. Knight, C. Bradley, H.S. Viswanathan. A mechanisms-based model for dynamic behavior and fracture of geomaterials. Intl. J. Rock Mech. & Min. Sci. 72, 277-282 (2014).

Parallelization FDEM Virtual Engine

Lei, Z., Rougier, E., Knight, E.E., & Munjiza, A. 2014. A Framework for Grand Scale Parallelization of the Combined Finite Discrete Element Method in 2D. Comp. Part. Mech. 1, 307-319.

FDEM Granite High strain rate

Rougier, E., Knight, E.E., Broome, S.T., Sussman, A.J. & Munjiza, A. 2014. Validation of a three-dimensional Finite-Discrete Element Method using experimental results of the Split Hopkinson Pressure Bar test. Intl. J. of Rock Mechanics and Mining Sciences, 70, 101-108.

Discrete elements Finite elements Shells

Munjiza, A., Lei, Z., Divic, V. & Peros, B. 2013. Fracture and fragmentation of thin shells using the combined finite–discrete element method. Int. J. Numer. Meth. Eng. 95(6), 478–498.

Rock blasting Mass fracturing Numerical modeling

Ning, Y.J., Yang, J., Ma, G.W. & Chen, P.W. 2011. Modeling Rock Blasting Considering Explosion Gas Penetration using Discontinuous Deformation analysis. Rock Mech. Rock Eng., 44, 483-490.

Hydrodynamic modeling Source scaling laws Yield estimation

Rougier, E., Patton, H. J., Knight, E. E., and Bradley, C. R. (2011), Constraints on burial depth and yield of the 25 May 2009 North Korean test from hydrodynamic simulations in a granite medium, Geophys. Res. Lett., 38, L16316.

Particle physics Modeling

Rougier, E., Munjiza, A. and Latham, J.P. (2004), "Shape selection menu for grand scale discontinua systems", Engineering Computations, Vol. 21 No. 2/3/4, pp. 343-359.

Finite elements Discrete elements Contact, interaction

Munjiza, A. & Andrews, K.R.F. 2000. Penalty function method for in combined finite-discrete element systems comprising large number of separate. Int. J. Num. Meth. Eng., 49, 1377-96.<1377::AID-NME6>3.0.CO;2-B

FEA Deformable elements Progressive fracture

Munjiza, A., Owen, D.R.J. & Bicanic, N. 1995. A combined finite-discrete element method in transient dynamics of fracturing solids. Int. J. of Engineering Computation, 12, 145-174.