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Mathematical Problems in Engineering
Volume 2016, Article ID 6217679, 25 pages
http://dx.doi.org/10.1155/2016/6217679
Research Article

Discontinuous Deformation Analysis Coupling with Discontinuous Galerkin Finite Element Methods for Contact Simulations

1School of Mathematics and Statistics, Xidian University, Xi’an 710126, China
2School of Engineering Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received 1 August 2016; Revised 18 October 2016; Accepted 19 October 2016

Academic Editor: Gerardo Severino

Copyright © 2016 Yue Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. G.-H. Shi, Discontinuous deformation analysis: a new numerical model for the statics and dynamics of block systems [Ph.D. thesis], University of California, Berkeley, Berkeley, Calif, USA, 1988.
  2. Gen-Hua Shi and R. E. Goodman, “Generalization of two-dimensional discontinuous deformation analysis for forward modelling,” International Journal for Numerical & Analytical Methods in Geomechanics, vol. 13, no. 4, pp. 359–380, 1989. View at Publisher · View at Google Scholar · View at Scopus
  3. J.-H. Wu and C.-H. Chen, “Application of DDA to simulate characteristics of the Tsaoling landslide,” Computers and Geotechnics, vol. 38, no. 5, pp. 741–750, 2011. View at Publisher · View at Google Scholar · View at Scopus
  4. Y. Zhang, X. Fu, and Q. Sheng, “Modification of the discontinuous deformation analysis method and its application to seismic response analysis of large underground caverns,” Tunnelling and Underground Space Technology, vol. 40, pp. 241–250, 2014. View at Publisher · View at Google Scholar · View at Scopus
  5. Y. X. Ben, Y. Wang, and G.-H. Shi, “Development of a model for simulating hydraulic fracturing with DDA,” in Proceedings of the 11th International Conference on Analysis of Discontinuous Deformation (ICADD '13), pp. 169–175, Fukuoka, Japan, August 2013. View at Scopus
  6. Y.-Y. Jiao, H.-Q. Zhang, X.-L. Zhang, H.-B. Li, and Q.-H. Jiang, “A two-dimensional coupled hydromechanical discontinuum model for simulating rock hydraulic fracturing,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 39, no. 5, pp. 457–481, 2015. View at Publisher · View at Google Scholar · View at Scopus
  7. W. E. Morgan and M. M. Aral, “An implicitly coupled hydro-geomechanical model for hydraulic fracture simulation with the discontinuous deformation analysis,” International Journal of Rock Mechanics and Mining Sciences, vol. 73, pp. 82–94, 2015. View at Publisher · View at Google Scholar · View at Scopus
  8. L. Q. Choo, Z. Zhao, H. Chen, and Q. Tian, “Hydraulic fracturing modeling using the discontinuous deformation analysis (DDA) method,” Computers and Geotechnics, vol. 76, pp. 12–22, 2016. View at Publisher · View at Google Scholar
  9. S. A. R. Beyabanaki, R. G. Mikola, S. O. R. Biabanaki, and S. Mohammadi, “New point-to-face contact algorithm for 3-D contact problems using the augmented Lagrangian method in 3-D DDA,” Geomechanics and Geoengineering, vol. 4, no. 3, pp. 221–236, 2009. View at Publisher · View at Google Scholar · View at Scopus
  10. S. A. R. Beyabanaki and A. C. Bagtzoglou, “Non-rigid disk-based DDA with a new contact model,” Computers and Geotechnics, vol. 49, pp. 25–35, 2013. View at Publisher · View at Google Scholar · View at Scopus
  11. Y. Ning, J. Yang, G. Ma, and P. Chen, “Contact algorithm modification of DDA and its verification,” in Analysis of Discontinuous Deformation: New Developments and Applications, G. Ma and Y. Zhou, Eds., pp. 73–81, 2010. View at Google Scholar
  12. H. Bao, Z. Zhao, and Q. Tian, “On the implementation of augmented lagrangian method in the two-dimensional discontinuous deformation analysis,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 38, no. 6, pp. 551–571, 2014. View at Publisher · View at Google Scholar · View at Scopus
  13. W. Wu, H. Zhu, X. Zhuang, G. Ma, and Y. Cai, “A multi-shell cover algorithm for contact detection in the three dimensional discontinuous deformation analysis,” Theoretical and Applied Fracture Mechanics, vol. 72, no. 1, pp. 136–149, 2014. View at Publisher · View at Google Scholar · View at Scopus
  14. G. H. Shi, “Contact theory,” Science China Technological Sciences, vol. 58, no. 9, pp. 1450–1496, 2015. View at Publisher · View at Google Scholar · View at Scopus
  15. H. Zheng and X. Li, “Mixed linear complementarity formulation of discontinuous deformation analysis,” International Journal of Rock Mechanics and Mining Sciences, vol. 75, pp. 23–32, 2015. View at Publisher · View at Google Scholar · View at Scopus
  16. H. Zheng, P. Zhang, and X. Du, “Dual form of discontinuous deformation analysis,” Computer Methods in Applied Mechanics & Engineering, vol. 305, pp. 196–216, 2016. View at Publisher · View at Google Scholar
  17. H. Zhang, S.-G. Liu, L. Zheng et al., “Extensions of edge-to-edge contact model in three-dimensional discontinuous deformation analysis for friction analysis,” Computers and Geotechnics, vol. 71, pp. 261–275, 2016. View at Publisher · View at Google Scholar · View at Scopus
  18. Y. Xiao, Q. Miao, M. Huang, Y. Wang, and J. Xue, “Parallel computing of discontinuous deformation analysis based on graphics processing unit,” International Journal of Geomechanics, 2016. View at Publisher · View at Google Scholar
  19. Y. Ning, J. Yang, X. An, and G. Ma, “Modelling rock fracturing and blast-induced rock mass failure via advanced discretisation within the discontinuous deformation analysis framework,” Computers and Geotechnics, vol. 38, no. 1, pp. 40–49, 2011. View at Publisher · View at Google Scholar · View at Scopus
  20. Y.-Y. Jiao, X.-L. Zhang, and J. Zhao, “Two-dimensional DDA contact constitutive model for simulating rock fragmentation,” Journal of Engineering Mechanics, vol. 138, no. 2, pp. 199–209, 2012. View at Publisher · View at Google Scholar · View at Scopus
  21. S. A. R. Beyabanaki, A. Jafari, and M. R. Yeung, “High-order three-dimensional discontinuous deformation analysis (3-D DDA),” International Journal for Numerical Methods in Biomedical Engineering, vol. 26, no. 12, pp. 1522–1547, 2010. View at Publisher · View at Google Scholar · View at Scopus
  22. R. Grayeli and A. Mortazavi, “Discontinuous deformation analysis with second-order finite element meshed block,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 30, no. 15, pp. 1545–1561, 2006. View at Publisher · View at Google Scholar · View at Scopus
  23. R. Grayeli and K. Hatami, “Implementation of the finite element method in the three-dimensional discontinuous deformation analysis (3D-DDA),” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 32, no. 15, pp. 1883–1902, 2008. View at Publisher · View at Google Scholar · View at Scopus
  24. S. A. R. Beyabanaki, A. Jafari, S. O. R. Biabanaki, and M. R. Yeung, “Nodal-based three-dimensional discontinuous deformation analysis (3-D DDA),” Computers and Geotechnics, vol. 36, no. 3, pp. 359–372, 2009. View at Publisher · View at Google Scholar · View at Scopus
  25. H. Bao and Z. Zhao, “Modeling brittle fracture with the nodal-based discontinuous deformation analysis,” International Journal of Computational Methods, vol. 10, no. 6, Article ID 1350040, 26 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. Q. Tian, Z. Zhao, and H. Bao, “Block fracturing analysis using nodal-based discontinuous deformation analysis with the double minimization procedure,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 38, no. 9, pp. 881–902, 2014. View at Publisher · View at Google Scholar · View at Scopus
  27. Y. M. Cheng and Y. H. Zhang, “Coupling of FEM and DDA methods,” International Journal of Geomechanics, vol. 2, no. 4, pp. 503–517, 2002. View at Publisher · View at Google Scholar · View at Scopus
  28. M. Zhang, H. Yang, and Z. Li, “Coupling model of the discontinuous deformation analysis method and the finite element method,” Tsinghua Science and Technology, vol. 10, no. 2, pp. 221–226, 2005. View at Publisher · View at Google Scholar · View at Scopus
  29. M. S. Khan, Investigation of discontinuous deformation analysis for application in jointed rock masses [Ph.D. thesis], Department of Civil Engineering of University of Toronto, 2010.
  30. Z. Zhao and J. Gu, “Stress recovery procedure for discontinuous deformation analysis,” Advances in Engineering Software, vol. 40, no. 1, pp. 52–57, 2009. View at Publisher · View at Google Scholar · View at Scopus
  31. F. M. De Sciarra, “A nonlocal finite element approach to nanobeams,” Advances in Mechanical Engineering, vol. 2013, Article ID 720406, 8 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus
  32. F. M. de Sciarra and R. Barretta, “A new nonlocal bending model for Euler-Bernoulli nanobeams,” Mechanics Research Communications, vol. 62, no. 1, pp. 25–30, 2014. View at Publisher · View at Google Scholar · View at Scopus
  33. B. Rivi{\`e}re, S. Shaw, M. F. Wheeler, and J. R. Whiteman, “Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity,” Numerische Mathematik, vol. 95, no. 2, pp. 347–376, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. B. M. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, Frontiers in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 2008.
  35. D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69, Springer, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  36. R. Liu, M. F. Wheeler, and C. N. Dawson, “A three-dimensional nodal-based implementation of a family of discontinuous Galerkin methods for elasticity problems,” Computers & Structures, vol. 87, no. 3-4, pp. 141–150, 2009. View at Publisher · View at Google Scholar · View at Scopus
  37. P. Kaufmann, S. Martin, M. Botsch, and M. Gross, “Flexible simulation of deformable models using discontinuous Galerkin FEM,” Graphical Models, vol. 71, no. 4, pp. 153–167, 2009. View at Publisher · View at Google Scholar · View at Scopus
  38. J. D. D. Basabe, M. K. Sen, and M. F. Wheeler, “Elastic wave propagation in fractured media using the discontinuous Galerkin method,” Geophysics, vol. 81, no. 4, pp. T163–T174, 2016. View at Publisher · View at Google Scholar
  39. G.-H. Shi, “Modeling dynamic rock failure by discontinuous deformation analysis with simplex integrations,” in Proceedings of the 1st North American Rock Mechanics Symposium, pp. 591–598, Austin, Tex, USA, 1994.
  40. S. Lin and Z. Xie, “Performance of DDA time integration,” Science China Technological Sciences, vol. 58, no. 9, pp. 1558–1566, 2015. View at Publisher · View at Google Scholar · View at Scopus
  41. X. Fu, Q. Sheng, Y. Zhang, and J. Chen, “Investigation of highly efficient algorithms for solving linear equations in the discontinuous deformation analysis method,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 40, no. 4, pp. 469–486, 2016. View at Publisher · View at Google Scholar · View at Scopus
  42. K. L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1985.
  43. M. M. MacLaughlin and D. M. Doolin, “Review of validation of the discontinuous deformation analysis (DDA) method,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 30, no. 4, pp. 271–305, 2006. View at Publisher · View at Google Scholar · View at Scopus
  44. Y. Ning, Z. Yang, B. Wei, and B. Gu, “Advances in two-dimensional discontinuous deformation analysis for rock-mass dynamics,” International Journal of Geomechanics, 2016. View at Publisher · View at Google Scholar
  45. J. C. Trinkle, J.-S. Pang, M. S. Sudarsky, and M. G. Lo, “On dynamic multi-rigid-body contact problems with Coulomb friction,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 77, no. 4, pp. 267–279, 1997. View at Publisher · View at Google Scholar · View at Scopus
  46. M. Anitescu and F. A. Potra, “Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems,” Nonlinear Dynamics, vol. 14, no. 3, pp. 231–247, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  47. A. M. Al-Fahed Nuseirat and G. E. Stavroulakis, “Complementarity problem formulation of the frictional grasping problem,” Computer Methods in Applied Mechanics and Engineering, vol. 190, no. 8–10, pp. 941–952, 2000. View at Publisher · View at Google Scholar · View at Scopus
  48. V. S. Arikatla and S. De, “An iterative predictor-corrector approach for modeling static and kinetic friction in interactive simulations,” Graphical Models, vol. 82, pp. 29–42, 2015. View at Publisher · View at Google Scholar · View at Scopus
  49. F. Gholami, M. Nasri, J. Kövecses, and M. Teichmann, “A linear complementarity formulation for contact problems with regularized friction,” Mechanism and Machine Theory, vol. 105, pp. 568–582, 2016. View at Publisher · View at Google Scholar
  50. P. W. Christensen, A. Klarbring, J. S. Pang, and N. Strömberg, “Formulation and comparison of algorithms for frictional contact problems,” International Journal for Numerical Methods in Engineering, vol. 42, no. 1, pp. 145–173, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus