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Accelerated proximal gradient method for elastoplastic analysis with von Mises yield criterion

  • Wataru Shimizu
  • Yoshihiro KannoEmail author
Original Paper Area 2

Abstract

It is known that, under certain conditions, the quasi-static incremental analysis problem of elastoplastic structures with the von Mises yield criterion can be formulated as a second-order cone programming (SOCP) problem, which can be solved with a primal-dual interior-point method. Alternatively, this paper proposes to solve an equivalent unconstrained nonsmooth convex optimization problem, which has a form similar to a class of regularized least-square problems, known as group LASSO. We propose an accelerated proximal gradient method with an adaptive restart scheme for solving this unconstrained optimization problem. The algorithm is easy to implement, and free from numerical solution of linear equations unlike conventional methods in computational mechanics. Numerical experiments suggest that the presented algorithm outperforms a standard solver that implements a primal-dual interior-point method for conic optimization.

Keywords

Second-order cone programming Accelerated gradient scheme Proximal gradient method Group LASSO Plasticity Incremental analysis 

Mathematics Subject Classification

90C25 65K10 90C90 

Notes

Acknowledgements

The work of the second author is partially supported by JSPS KAKENHI 26420545 and 17K06633.

References

  1. 1.
    Alfano, G., Rosati, L., Valoroso, N.: A displacement-like finite element model for \(J_{2}\) elastoplasticity: variational formulation and finite-step solution. Comput. Methods Appl. Mech. Eng. 155, 325–358 (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    Anjos, M.F., Lasserre, J.B. (eds.): Handbook on Semidefinite, Conic and Polynomial Optimization. Springer, New York (2012)zbMATHGoogle Scholar
  3. 3.
    Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183–202 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. SIAM, Philadelphia (2001)CrossRefzbMATHGoogle Scholar
  5. 5.
    Benson, H.Y., Shanno, D.F.: An exact primal-dual penalty method approach to warm-starting interior-point methods for linear programming. Comput. Optim. Appl. 38, 371–399 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bisbos, C.D., Makrodimopoulos, A., Pardalos, P.M.: Second-order cone programming approaches to static shakedown analysis in steel plasticity. Optim. Methods Softw. 20, 25–52 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bleyer, J., de Buhan, P.: A numerical approach to the yield strength of shell structures. Eur. J. Mech. A/Solids 59, 178–194 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bleyer, J., Maillard, M., de Buhan, P., Coussot, P.: Efficient numerical computations of yield stress fluid flows using second-order cone programming. Comput. Methods Appl. Mech. Eng. 283, 599–614 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Calafiore, G., El Ghaoui, L.: Optimization Models. Cambridge University Press, Cambridge (2014)zbMATHGoogle Scholar
  10. 10.
    Capurso, M., Maier, G.: Incremental elastoplastic analysis and quadratic optimization. Meccanica 5, 107–116 (1970)CrossRefzbMATHGoogle Scholar
  11. 11.
    Curnier, A.: Computational Methods in Solid Mechanics. Kluwer Academic Publishers, Dordrecht (1994)CrossRefzbMATHGoogle Scholar
  12. 12.
    de Souza Neto, E.A., Perić, D., Owen, D.R.J.: Computational Methods for Plasticity. Wiley, Chichester (2008)CrossRefGoogle Scholar
  13. 13.
    Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  14. 14.
    Engau, A., Anjos, M.F., Vannelli, A.: On interior-point warmstarts for linear and combinatorial optimization. SIAM J. Optim. 20, 1828–1861 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gondzio, J.: Warm start of the primal-dual method applied in the cutting-plane scheme. Math. Program. 83, 125–143 (1998)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Han, W., Reddy, B.D.: Plasticity, 2nd edn. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  17. 17.
    Herfelt, M.A., Poulsen, P.N., Hoang, L.C., Jensen, J.F.: Numerical limit analysis of keyed shear joints in concrete structures. Struct. Concr. 17, 481–490 (2016)CrossRefGoogle Scholar
  18. 18.
    Huang, J., Griffiths, D.V.: Observations on return mapping algorithms for piecewise linear yield criteria. Int. J. Geomech. 8, 253–265 (2008)CrossRefGoogle Scholar
  19. 19.
    John, E., Yıldırım, E.A.: Implementation of warm-start strategies in interior-point methods for linear programming in fixed dimension. Comput. Optim. Appl. 41, 151–183 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kaneko, I.: Piecewise linear elastic-plastic analysis. Int. J. Numer. Methods Eng. 14, 757–767 (1979)CrossRefzbMATHGoogle Scholar
  21. 21.
    Kanno, Y.: Nonsmooth Mechanics and Convex Optimization. CRC Press, Boca Raton (2011)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kanno, Y.: A fast first-order optimization approach to elastoplastic analysis of skeletal structures. Optim. Eng. 17, 861–896 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Krabbenhøft, K., Lyamin, A.V.: Computational Cam clay plasticity using second-order cone programming. Comput. Methods Appl. Mech. Eng. 209–212, 239–249 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Krabbenhøft, K., Lyamin, A.V., Sloan, S.W.: Formulation and solution of some plasticity problems as conic programs. Int. J. Solids Struct. 44, 1533–1549 (2007)CrossRefzbMATHGoogle Scholar
  25. 25.
    Krabbenhøft, K., Lyamin, A.V., Sloan, S.W., Wriggers, P.: An interior-point algorithm for elastoplasticity. Int. J. Numer. Methods Eng. 69, 592–626 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Krichene, W., Bayen, A.M., Bartlett, P.L.: Accelerated mirror descent in continuous and discrete time. In: Cortes, C., Lawrence, N.D., Lee, D.D., Sugiyama, M., Garnett, R. (eds.) Advances in Neural Information Processing Systems 28 (NIPS 2015), pp. 2845–2853. Curran Associates, Red Hook (2015)Google Scholar
  27. 27.
    Krieg, R.D., Krieg, D.B.: Accuracies of numerical solution methods for the elastic-perfectly plastic model. J. Press. Vessel Technol. (ASCE) 99, 510–515 (1977)CrossRefGoogle Scholar
  28. 28.
    Maier, G.: A quadratic programming approach for certain classes of non-linear structural problems. Meccanica 3, 121–130 (1968)CrossRefzbMATHGoogle Scholar
  29. 29.
    Maier, G., Munro, J.: Mathematical programming applications to engineering plastic analysis. Appl. Mech. Rev. (ASME) 35, 1631–1643 (1982)Google Scholar
  30. 30.
    Makrodimopoulos, A.: Computational formulation of shakedown analysis as a conic quadratic optimization problem. Mech. Res. Commun. 33, 72–83 (2006)CrossRefzbMATHGoogle Scholar
  31. 31.
    Makrodimopoulos, A., Martin, C.M.: Lower bound limit analysis of cohesive-frictional materials using second-order cone programming. Int. J. Numer. Methods Eng. 66, 604–634 (2006)CrossRefzbMATHGoogle Scholar
  32. 32.
    Makrodimopoulos, A., Martin, C.M.: Upper bound limit analysis using simplex strain elements and second-order cone programming. Int. J. Numer. Anal. Methods Geomech. 31, 835–865 (2007)CrossRefzbMATHGoogle Scholar
  33. 33.
    Mitchell, J.E.: Restarting after branching in the SDP approach to MAX-CUT and similar combinatorial optimization problem. J. Comb. Optim. 5, 151–166 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Mosci, S., Rosasco, L., Santoro, M., Verri, A., Villa, S.: Solving structured sparsity regularization with proximal methods. In: Balcázar, J.L., Bonchi, F., Gionis, A., Sebag, M. (eds.) Machine Learning and Knowledge Discovery in Databases, pp. 418–433. Springer, Berlin (2010)CrossRefGoogle Scholar
  35. 35.
    Nesterov, Y.: A method of solving a convex programming problem with convergence rate \(O(1/k^{2})\). Sov. Math. Dokl. 27, 372–376 (1983)zbMATHGoogle Scholar
  36. 36.
    Nesterov, Y.: Introductory Lectures on Convex Optimization: A Basic Course. Kluwer Academic Publishers, Dordrecht (2004)CrossRefzbMATHGoogle Scholar
  37. 37.
    O’Donoghue, B., Candès, E.: Adaptive restart for accelerated gradient schemes. Found. Comput. Math. 15, 715–732 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Oñate, E.: Structural Analysis with the Finite Element Method. Linear Statics, vol. 1. Springer, Berlin (2009)Google Scholar
  39. 39.
    Ortiz, M., Popov, E.P.: Accuracy and stability of integration algorithms for elastoplastic constitutive relations. Int. J. Numer. Methods Eng. 21, 1561–1576 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Parikh, N., Boyd, S.: Proximal algorithms. Found. Trends Optim. 1, 127–239 (2014)CrossRefGoogle Scholar
  41. 41.
    Pastor, J., Thoré, Ph, Pastor, F.: Limit analysis and numerical modeling of spherically porous solids with Coulomb and Drucker–Prager matrices. J. Comput. Appl. Math. 234, 2162–2174 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Pérez-Foguet, A., Rodríguez-Ferran, A., Huerta, A.: Consistent tangent matrices for substepping schemes. Comput. Methods Appl. Mech. Eng. 190, 4627–4647 (2001)CrossRefzbMATHGoogle Scholar
  43. 43.
    Romano, G., Barretta, R., Diaco, M.: Algorithmic tangent stiffness in elastoplasticity and elastoviscoplasticity: a geometric insight. Mech. Res. Commun. 37, 289–292 (2010)CrossRefzbMATHGoogle Scholar
  44. 44.
    de Saxcé, G., Oueslati, A., Charkaluk, E., Tritsch, J.-B. (eds.): Limit State of Materials and Structures: Direct Methods 2. Springer, Dordrecht (2013)Google Scholar
  45. 45.
    Simo, J.C., Hughes, T.J.R.: Computational Inelasticity. Springer, New York (1998)zbMATHGoogle Scholar
  46. 46.
    Simo, J.C., Taylor, R.L.: A return mapping algorithm for plane stress elastoplasticity. Int. J. Numer. Methods Eng. 22, 649–670 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Simon, J.-W., Höwer, D., Weichert, D.: A starting-point strategy for interior-point algorithms for shakedown analysis of engineering structures. Eng. Optim. 46, 648–668 (2014)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Skajaa, A., Andersen, E.D., Ye, Y.: Warmstarting the homogeneous and self-dual interior point method for linear and conic quadratic problems. Math. Program. Comput. 5, 1–25 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Su, W., Boyd, S., Candès, E.J.: A differential equation for modeling Nesterov’s accelerated gradient method: theory and insights. In: Ghahramani, Z., Welling, M., Cortes, C., Lawrence, N.D., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems 27 (NIPS 2014), pp. 2510–2518. Curran Associates, Red Hook (2014)Google Scholar
  50. 50.
    Tang, C., Toh, K.-C., Phoon, K.-K.: Axisymmetric lower-bound limit analysis using finite elements and second-order cone programming. J. Eng. Mech. (ASCE) 140, 268–278 (2014)CrossRefGoogle Scholar
  51. 51.
    Tangaramvong, S., Tin-Loi, F.: A complementarity approach for elastoplastic analysis of strain softening frames under combined bending and axial force. Eng. Struct. 29, 742–753 (2007)CrossRefGoogle Scholar
  52. 52.
    Tangaramvong, S., Tin-Loi, F.: Simultaneous ultimate load and deformation analysis of strain softening frames under combined stresses. Eng. Struct. 30, 664–674 (2008)CrossRefGoogle Scholar
  53. 53.
    Tangaramvong, S., Tin-Loi, F., Song, C.: A direct complementarity approach for the elastoplastic analysis of plane stress and plane strain structures. Int. J. Numer. Methods Eng. 90, 838–866 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B (Methodological) 58, 267–288 (1996)Google Scholar
  55. 55.
    Tibshirani, R.: Regression shrinkage and selection via the lasso: a retrospective. J. R. Stat. Soc. Ser. B (Statistical Methodology) 73, 273–282 (2011)Google Scholar
  56. 56.
    Tin-Loi, F., Xia, S.H.: Nonholonomic elastoplastic analysis involving unilateral frictionless contact as a mixed complementarity problem. Comput. Methods Appl. Mech. Eng. 190, 4551–4568 (2001)CrossRefzbMATHGoogle Scholar
  57. 57.
    Trillat, M., Pastor, J.: Limit analysis and Gurson’s model. Eur. J. Mech. A/Solids 24, 800–819 (2005)CrossRefzbMATHGoogle Scholar
  58. 58.
    Tütüncü, R.H., Toh, K.C., Todd, M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program. B95, 189–217 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Yamaguchi, T., Kanno, Y.: Ellipsoidal load-domain shakedown analysis with von Mises yield criterion: a robust optimization approach. Int. J. Numer. Methods Eng. 107, 1136–1144 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Yonekura, K., Kanno, Y.: Second-order cone programming with warm start for elastoplastic analysis with von Mises yield criterion. Optim. Eng. 13, 181–218 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Yuan, M., Lin, Y.: Model selection and estimation in regression with grouped variables. J. R. Stat. Soc. Ser. B (Statistical Methodology) 68, 49–67 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Zhang, X., Sheng, D., Sloan, S.W., Krabbenhoft, K.: Second-order cone programming formulation for consolidation analysis of saturated porous media. Comput. Mech. 58, 29–43 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK 2017

Authors and Affiliations

  1. 1.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyThe University of TokyoTokyoJapan
  2. 2.IHI CorporationTokyoJapan
  3. 3.Mathematics and Informatics CenterThe University of TokyoTokyoJapan

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