Skip to main content
SpringerLink
Log in

Second-order cone programming formulation for consolidation analysis of saturated porous media

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

In this paper, the incremental problem for consolidation analysis of elastoplastic saturated porous media is formulated and solved using second-order cone programming. This is achieved by the application of the Hellinger-Reissner variational theorem, which casts the governing equations of Biot’s consolidation theory as a min–max optimisation problem. The min–max problem is then discretised using the finite element method and converted into a standard second-order cone programming problem that can be solved efficiently using modern optimisation algorithms (such as the primal-dual interior-point method). The proposed computational formulation is verified against a number of benchmark examples and also applied to simulate the construction of a road embankment on soft clay.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Simon JC, Hughes TJR (1998) Computational inelasticity. Springer, New York

    MATH  Google Scholar 

  2. Sheng D, Sloan SW, Abbo AJ (2002) An automatic Newton-Raphson scheme. Int J Geomech 2(4):471–502

    Article  Google Scholar 

  3. Sloan SW (1987) Substepping schemes for the numerical integration of elastoplastic stress-strain relations. Int J Num Methods Eng 24(5):893–911

    Article  MathSciNet  MATH  Google Scholar 

  4. Sloan SW, Abbo AJ, Sheng D (2001) Refined explicit integration of elastoplastic models with automatic error control. Eng Comput 18(1/2):121–194

    Article  MATH  Google Scholar 

  5. Abbo AJ, Sloan SW (1996) An automatic load stepping algorithm with error control. Int J Num Methods Eng 39(10):1737–1759

    Article  MathSciNet  MATH  Google Scholar 

  6. Ortiz M, Popov EP (1985) Accuracy and stability of integration algorithms for elastoplastic constitutive relations. Int J Numer Methods Eng 21(9):1561–1576

    Article  MathSciNet  MATH  Google Scholar 

  7. Simo JC, Taylor RL (1985) Consistent tangent operators for rate-independent elastoplasticity. Comput Methods Appl Mech Eng 48(1):101–118

    Article  MathSciNet  MATH  Google Scholar 

  8. Borja RI, Sama KM, Sanz PF (2003) On the numerical integration of three-invariant elastoplastic constitutive models. Comput Methods Appl Mech Eng 192(9–10):1227–1258

    Article  MATH  Google Scholar 

  9. Maier G (1968) Quadratic programming and theory of elastic-perfectly plastic structures. Meccanica 3(4):265–273

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhang HW, Li JY, Pan SH (2011) New second-order cone linear complementarity formulation and semi-smooth Newton algorithm for finite element analysis of 3D frictional contact problem. Comput Methods Appl Mech Eng 200(1–4):77–88

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhang LL, Li JY, Zhang HW, Pan SH (2013) A second order cone complementarity approach for the numerical solution of elastoplasticity problems. Comput Mech 51(1):1–18

    Article  MathSciNet  MATH  Google Scholar 

  12. Lotfian Z, Sivaselvan MV (2014) A projected Newton algorithm for the dual convex program of elastoplasticity. Int J Numer Methods Eng 97(12):903–936

    Article  MathSciNet  Google Scholar 

  13. Contrafatto L, Ventura G (2004) Numerical analysis of augmented Lagrangian algorithms in complementary elastoplasticity. Int J Numer Methods Eng 60(14):2263–2287

    Article  MATH  Google Scholar 

  14. Cuomo M, Contrafatto L (2000) Stress rate formulation for elastoplastic models with internal variables based on augmented Lagrangian regularisation. Int J Solids Str 37(29):3935–3964

    Article  MATH  Google Scholar 

  15. Liew KM, Wu YC, Zou GP, Ng TY (2002) Elasto-plasticity revisited: numerical analysis via reproducing kernel particle method and parametric quadratic programming. Comput Methods Appl Mech Eng 55(6):669–683

    MATH  Google Scholar 

  16. Zhang HW, He SY, Li XS, Wriggers P (2004) A new algorithm for numerical solution of 3D elastoplastic contact problems with orthotropic friction law. Comput Mech 34(1):1–14

    Article  MATH  Google Scholar 

  17. Zhong W, Sun S (1988) A finite element method for elasto-plastic structures and contact problems by parametric quadratic programming. Int J Numer Methods Eng 26(12):2723–2738

    Article  MATH  Google Scholar 

  18. Hjiaj M, Fortin J, de Saxcé G (2003) A complete stress update algorithm for the non-associated Drucker-Prager model including treatment of the apex. Int J Eng Sci 41(10):1109–1143

    Article  Google Scholar 

  19. Le CV, Nguyen-Xuan H, Nguyen-Dang H (2010) Upper and lower bound limit analysis of plates using FEM and second-order cone programming. Comput Str 88(1–2):65–73

    Article  Google Scholar 

  20. Liu F, Zhao J (2013) Upper bound limit analysis using radial point interpolation meshless method and nonlinear programming. Int J Mech Sci 70:26–38

    Article  Google Scholar 

  21. Portioli F, Casapulla C, Gilbert M, Cascini L (2014) Limit analysis of 3D masonry block structures with non-associative frictional joints using cone programming. Comput Str 143:108–121

    Article  Google Scholar 

  22. Sloan SW (1988) Lower bound limit analysis using finite elements and linear programming. Int J Numer Anal Methods Geomech 12(1):61–77

    Article  MATH  Google Scholar 

  23. Lyamin AV, Sloan SW (2002) Upper bound limit analysis using linear finite elements and non-linear programming. Int J Numer Anal Methods Geomech 26(2):181–216

    Article  MATH  Google Scholar 

  24. Krabbenhoft K, Lyamin AV, Hjiaj M, Sloan SW (2005) A new discontinuous upper bound limit analysis formulation. Int J Numer Methods Eng 63(7):1069–1088

    Article  MATH  Google Scholar 

  25. Hjiaj M, Lyamin AV, Sloan SW (2005) Numerical limit analysis solutions for the bearing capacity factor N\(\gamma \). Int J Solids Str 42(5–6):1681–1704

    Article  MATH  Google Scholar 

  26. Tang C, Toh K-C, Kok-Kwang P (2014) Axisymmetric lower-bound limit analysis using finite elements and second-order cone programming. J Eng Mech 140(2):268–278

    Article  Google Scholar 

  27. Makrodimopoulos A, Martin CM (2006) Lower bound limit analysis of cohesive-frictional materials using second-order cone programming. Int J Numer Methods Eng 66(4):604–634

    Article  MATH  Google Scholar 

  28. Li HX, Yu HS (2006) A non-linear programming approach to kinematic shakedown analysis of composite materials. Int J Numer Methods Eng 66(1):117–146

    Article  MATH  Google Scholar 

  29. König JA, Maier G (1981) Shakedown analysis of elastoplastic structures: a review of recent developments. Nucl Eng Design 66(1):81–95

    Article  Google Scholar 

  30. Belytschko T (1972) Plane stress shakedown analysis by finite elements. Int J Mech Sci 14(9):619–625

    Article  Google Scholar 

  31. Feng X-Q, Yu S-W (1995) Damage and shakedown analysis of structures with strain-hardening. Int J Plast 11(3):237–249

    Article  MathSciNet  MATH  Google Scholar 

  32. Spiliopoulos KV, Patsios TN (2010) An efficient mathematical programming method for the elastoplastic analysis of frames. Eng Str 32(5):1199–1214

    Article  Google Scholar 

  33. Moharrami H, Mahini MR, Cocchetti G (2015) Elastoplastic analysis of plane stress/strain structures via restricted basis linear programming. Comput Str 146:1–11

    Article  Google Scholar 

  34. Krabbenhøft K, Lyamin A, Sloan S (2007) Formulation and solution of some plasticity problems as conic programs. Int J Solids Str 44(5):1533–1549

    Article  MATH  Google Scholar 

  35. Krabbenhoft K, Lyamin AV (2012) Computational Cam clay plasticity using second-order cone programming. Comput Methods Appl Mech Eng 209–212:239–249

    Article  MathSciNet  MATH  Google Scholar 

  36. Krabbenhoft K, Lyamin AV, Sloan SW, Wriggers P (2007) An interior-point algorithm for elastoplasticity. Int J Numer Methods Eng 69(3):592–626

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhang H-W, Wang H (2006) Parametric variational principle based elastic-plastic analysis of heterogeneous materials with Voronoi finite element method. Appl Math Mech 27(8):1037–1047

    Article  MATH  Google Scholar 

  38. Yonekura K, Kanno Y (2012) Second-order cone programming with warm start for elastoplastic analysis with von Mises yield criterion. Optim Eng 13(2):181–218

    Article  MathSciNet  MATH  Google Scholar 

  39. Bemporad A, Paggi M (2015) Optimization algorithms for the solution of the frictionless normal contact between rough surfaces. Int J Solids Str 69–70:94–105

    Article  Google Scholar 

  40. Klarbring A (1986) A mathematical programming approach to three-dimensional contact problems with friction. Comput Methods Appl Mech Eng 58(2):175–200

    Article  MathSciNet  MATH  Google Scholar 

  41. Klarbring A, Björkman G (1988) A mathematical programming approach to contact problems with friction and varying contact surface. Comput Str 30(5):1185–1198

    Article  MATH  Google Scholar 

  42. Bleyer J, Maillard M, de Buhan P, Coussot P (2015) Efficient numerical computations of yield stress fluid flows using second-order cone programming. Comput Methods Appl Mech Eng 283:599–614

    Article  MathSciNet  Google Scholar 

  43. Krabbenhoft K, Huang J, da Silva MV, Lyamin AV (2012) Granular contact dynamics with particle elasticity. Granul Matter 14(5):607–619

    Article  Google Scholar 

  44. Krabbenhoft K, Lyamin AV, Huang J, da Silva M Vicente (2012) Granular contact dynamics using mathematical programming methods. Comput Geotech 43:165–176

    Article  Google Scholar 

  45. Lim K-W, Krabbenhoft K, Andrade J (2014) On the contact treatment of non-convex particles in the granular element method. Comput Particle Mech 1(3):257–275

    Article  Google Scholar 

  46. Huang J, da Silva MV, Krabbenhoft K (2013) Three-dimensional granular contact dynamics with rolling resistance. Comput Geotech 49:289–298

    Article  Google Scholar 

  47. Lim K-W, Krabbenhoft K, Andrade JE (2014) A contact dynamics approach to the granular element method. Comput Methods Appl Mech Eng 268:557–573

    Article  MathSciNet  MATH  Google Scholar 

  48. Zhang H (1995) Parametric variational principle for elastic-plastic consolidation analysis of saturated porous media. Int J Numer Anal Methods Geomech 19(12):851–867

    Article  MATH  Google Scholar 

  49. Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12(2):155–164

    Article  MATH  Google Scholar 

  50. Biot MA (1941) Consolidation settlement under a rectangular load distribution. J Appl Phys 12(5):426–430

    Article  MATH  Google Scholar 

  51. Reissner E (1950) On a variational theorem in elasticity. J Math Phys 24:90–95

    Article  MathSciNet  MATH  Google Scholar 

  52. Zhang X, Krabbenhoft K, Pedroso D, Lyamin A, Sheng D, Da Silva MV, Wang D (2013) Particle finite element analysis of large deformation and granular flow problems. Comput Geotech 54:133–142

    Article  Google Scholar 

  53. Andersen ED, Roos C, Terlaky T (2003) On implementing a primal-dual interior-point method for conic quadratic optimization. Math Progr 95(2):249–277

    Article  MathSciNet  MATH  Google Scholar 

  54. Sivaselvan MV (2011) Complementarity framework for non-linear dynamic analysis of skeletal structures with softening plastic hinges. Int J Numer Methods Eng 86(2):182–223

    Article  MathSciNet  MATH  Google Scholar 

  55. Bolzon G, Maier G, Tin-Loi F (1997) On multiplicity of solutions in quasi-brittle fracture computations. Computat Mech 19(6):511–516

    Article  MATH  Google Scholar 

  56. Tin-Loi F, Tseng P (2003) Efficient computation of multiple solutions in quasibrittle fracture analysis. Comput Methods Appl Mech Eng 192(11–12):1377–1388

    Article  MATH  Google Scholar 

  57. Zhang X (2014) Particle finite element method in geomechanics (PhD thesis). Faculty of Engineering & Built Environment, University of Newcastle, Newcastle

  58. Makrodimopoulos A (2010) Remarks on some properties of conic yield restrictions in limit analysis. Int J Numer Methods Biomed Eng 26(11):1449–1461

    Article  MATH  Google Scholar 

  59. Sturm JF (2002) Implementation of interior point methods for mixed semidefinite and second order cone optimization problems. Optim Methods Softw 17(6):1105–1154

    Article  MathSciNet  MATH  Google Scholar 

  60. Makrodimopoulos A, Martin CM (2007) Upper bound limit analysis using simplex strain elements and second-order cone programming. Int J Numer Anal Methods Geomech 31(6):835–865

    Article  MATH  Google Scholar 

  61. Zhang X, Krabbenhoft K, Sheng D (2014) Particle finite element analysis of the granular column collapse problem. Granul Matter 16(4):609–619

    Article  Google Scholar 

  62. Zhang X, Krabbenhoft K, Sheng D, Li W (2015) Numerical simulation of a flow-like landslide using the particle finite element method. Comput Mech 55(1):167–177

    Article  MathSciNet  MATH  Google Scholar 

  63. Zhang X, Sheng D, Kouretzis GP, Krabbenhoft K, Sloan SW (2015) Numerical investigation of the cylinder movement in granular matter. Phys Rev E 91(2):022204

  64. Li X, Zhang X, Han X, Sheng DC (2010) An iterative pressure-stabilized fractional step algorithm in saturated soil dynamics. Int J Numer Anal Methods Geomech 34(7):733–753

    MATH  Google Scholar 

  65. Washizu K (1982) Variational methods in elasticity and plasticity, 3rd edn. Pergamon Press, New York

    MATH  Google Scholar 

  66. Krabbenhoft K, Karim MR, Lyamin AV, Sloan SW (2012) Associated computational plasticity schemes for nonassociated frictional materials. Int J Numer Methods Eng 90(9):1089–1117

    Article  MathSciNet  MATH  Google Scholar 

  67. Martin CM, Makrodimopoulos A (2008) Finite-element limit analysis of Mohr-Coulomb materials in 3D using semidefinite programming. J Eng Mech 134(4):339–347

    Article  Google Scholar 

  68. Yu S, Zhang X, Sloan SW (2015) A 3D upper bound limit analysis using radial point interpolation meshless method and second-order cone programming. Int J Numer Methods Eng (under review)

  69. Kardani O, Lyamin A, Krabbenhøft K (2015) Application of a GPU-accelerated hybrid preconditioned conjugate gradient approach for large 3D problems in computational geomechanics. Comput Math Appl 69(10):1114–1131

    Article  MathSciNet  Google Scholar 

  70. Terzaghi K (1943) Theoretical soil mechanics. Wiley, New York

    Book  Google Scholar 

  71. Abbo A (1997) Finite element algorithmns for elastoplasticity and consolidation (PhD thesis). Faculty of Engineering & Built Environment, University of Newcastle, Newcastle

  72. Mandel J (1953) Condolidation des Sols (Étude mathématique). Géotechnique 3(7):287–299

    Article  Google Scholar 

  73. Coussy O (2004) Poromechanics. Wiley, Chichester

    MATH  Google Scholar 

  74. Cryer CW (1963) A comparison of the three-dimensional consolidation theories of biot and terzaghi. Quart J Mech Appl Math 16(4):401–412

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors wish to acknowledge the support of the Australian Research Council Centre of Excellence for Geotechnical Science and Engineering and Australian Research Council’s Discovery Projects funding scheme (Project Number DP150104257).

Author information

Authors and Affiliations

  1. ARC Centre of Excellence for Geotechnical Science and Engineering, The University of Newcastle, Newcastle, Australia

    Xue Zhang, Daichao Sheng, Scott W. Sloan & Kristian Krabbenhoft

Authors
  1. Xue Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daichao Sheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Scott W. Sloan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kristian Krabbenhoft
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xue Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, X., Sheng, D., Sloan, S.W. et al. Second-order cone programming formulation for consolidation analysis of saturated porous media. Comput Mech 58, 29–43 (2016). https://doi.org/10.1007/s00466-016-1280-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-016-1280-4

Keywords

Search

Navigation