Nonsmooth Computational Mechanics

II. Algorithms and Examples
  • Vladimir F. Dem’yanov
  • Georgios E. Stavroulakis
  • Ludmila N. Polyakova
  • Panagiotis D. Panagiotopoulos
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 10)


In this Chapter nonsmooth computational mechanics algorithms are proposed and studied. They are based on the quasidifferentiable and codifferentiable optimization algorithms, discussed thoroughly in Chapter 6, and on some cases of them (e.g. the d.c. optimization techniques). A review of classical computational mechanics algorithms and the link with the here proposed methods are also given. The techniques are illustrated by means of numerical examples. This Chapter concerns pilot applications which can be followed for the numerical treatment of other quasidifferential models in engineering.


Potential Energy Function Frictional Stress Hemivariational Inequality Unilateral Contact Tangential Interface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Allgower E.L. and Georg K. (1990), Numerical continuation methods: an introduction, Springer Verlag, Berlin, Heidelberg.CrossRefzbMATHGoogle Scholar
  2. 2.
    Auchmuty G. (1989), Duality algorithms for nonconvex variational principles. Num. Functional Analysis and Optimization, 10, 211–264.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bathe K.-J. (1981), Finite element procedures in engineering analysis, Prentice-Hall Inc., New Jersey.Google Scholar
  4. 4.
    Cottle R.W., Pang J.S. and Stone R.E. (1992), The linear complementarity problem, Academic Press, New York.zbMATHGoogle Scholar
  5. 5.
    Crisfield M.A. (1991), Non-linear finite element analysis of solids and structures, John Wiley and Sons, Chichester Intern.Google Scholar
  6. 6.
    Curnier A. (1993), Methodes numeriques en mecanique des solides, Presses polytech-niques et universitaires romandes, Lausanne.zbMATHGoogle Scholar
  7. 7.
    Demyanov V.F. and Vasiliev L.N. (1985), Nondifferentiable Optimization, Optimization Software, New York.CrossRefzbMATHGoogle Scholar
  8. 8.
    Eaves Curtis B., Gould F.J., Peitgen H.-O. and Todd M.J. (eds.) (1983), Homotopy methods and global convergence, Plenum Press, New York.zbMATHGoogle Scholar
  9. 9.
    Fletcher R. (1990), Practical methods of optimization, J. Willey and Sons.Google Scholar
  10. 10.
    Forsgren A. and Ringertz U. (1993), On the use of a modified Newton method for nonlinear finite element analysis. Comp. Meths in Appl. Mech. and Engineering, 110, 275–283.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Glowinski R., Lions J.-L. and Tremolieres R. (1981), Numerical analysis of variational inequalities, Studies in Mathematics and its Applications Vol. 8, North-Holland, Elsevier, Amsterdam, New York.CrossRefzbMATHGoogle Scholar
  12. 12.
    Guddat J., Guerra Vasquez F. and Jongen J.Th. (1990), Parametric Optimization: singularities, path following and jumps, B.G. Teubner, Stuttgart and John Wiley, Chichester International.CrossRefGoogle Scholar
  13. 13.
    Hiriart-Urruty J.-B. and Lemarechal C. (1993), Convex analysis and minimization algorithms I, Springer-Verlag, Berlin, Heidelberg.CrossRefGoogle Scholar
  14. 14.
    Holnicki-Szulc J. and Gierlinski J.T. (1995), Structural analysis, design and control by the virtual distortion method, J. Wiley anc Sons Ltd., Chichester.Google Scholar
  15. 15.
    Horst R., Pardalos P.M. and Thoai N.V. (1995), Introduction to global optimization, Kluwer Academic, Doldrecht, Boston, London.zbMATHGoogle Scholar
  16. 16.
    Horst R. and Pardalos P.M. (eds.) (1995), Handbook of global optimization, Kluwer Academic, Doldrecht, Boston, London.zbMATHGoogle Scholar
  17. 17.
    Jongen J.Th., Jonker P. and Twilt F. (1983), Nonlinear optimization in Rn. I. Morse theory, Chebychev approximation, Peter Lang Verlag, Frankfurt a.M.Google Scholar
  18. 18.
    Jongen J.Th., Jonker P. and Twilt F. (1986), Nonlinear optimization in Rn. II. Transversality, flows, parametric aspects, Peter Lang Verlag, Frankfurt a.M.Google Scholar
  19. 19.
    Jongen J.Th., Jonker P. and Twilt F. (1986), Critical sets in parametric optimization. Mathematical Programming, 34, 333–353.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jongen J.Th. and Pallaschke D. (1988), On linearization and continuous selection of functions. Optimization, 19 (3), 343–353.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Koltsakis E.K., Mistakidis E.S. and Tzaferopoulos M.A. (1995), On the numerical treatment of nonconvex energy problems of mechanics. J. of Global Optimization, 6 (4), 427–448.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kuntz L. and Scholtes S. (1995), Qualitative aspects of the local approximation of a piecewise differentiate function. Nonlinear Analysis. Theory, Methods and Applications, 25 (2), 197–215.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kurutz M. (1993), Stability of structures with nonsmooth nonconvex energy functional. The one dimensional case. Eur. J. of Mechanics, A/Solids, 12 (3), 347–385.MathSciNetzbMATHGoogle Scholar
  24. 24.
    Matthies H. and Strang G. (1979), The solution of nonlinear finite element equations. Intern. J. of Numerical Methods in Engineering, 14, 1613–1626.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Miettinen M. (1993), Approximation of hemivariational inequalities and optimal control problems PhD Thesis, Univ. of Jyvaskyla, Dept of Mathematics, Rep. 59.Google Scholar
  26. 26.
    Miettinen M. (1995), On constrained hemivariational inequalities and their approximation, Applicable Analysis, 56, 303–326.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Miettinen M., Makela M.M., and Haslinger J. (1995), On numerical solution of hemivariational inequalities by nonsmooth optimization methods. Journal of Global Optimization, 8 (4), 401–425.MathSciNetCrossRefGoogle Scholar
  28. 28.
    Mistakidis E.S. and Panagiotopoulos P.D. (1994), On the approximation of nonmono-tone multivalued problems by monotone subproblems. Computer Methods in Applied Mechanics and Engineering, 114, 55–76.MathSciNetCrossRefGoogle Scholar
  29. 29.
    Murty K.G. (1988), Linear complementarity, linear and nonlinear programming, Heldermann Verlag, Berlin.zbMATHGoogle Scholar
  30. 30.
    Panagiotopoulos P.D. (1985), Inequality problems in mechanics and applications. Convex and nonconvex energy functions, Birkhauser Verlag, Basel - Boston - Stuttgart, (russian translation, MIR Publ., Moscow 1988 ).Google Scholar
  31. 31.
    Panagiotopoulos P.D. (1993), Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer Verlag, Berlin — Heidelberg — New York.CrossRefzbMATHGoogle Scholar
  32. 32.
    Panagiotopoulos P.D. and Tzaferopoulos M.A. (1995), On the numerical treatment of nonconvex energy problems. Multilevel decomposition techniques for hemivariational inequalities. Comp. Meths in Appl. Mech. Engineering, 123 (1–4), 81–94.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Pardalos P.M. and Rosen J.B. (1987), Constrained global optimization: algorithms and applications, Springer Lecture Notes in Computer Science, Vol. 268.Google Scholar
  34. 34.
    Rohde A. and Stavroulakis G.E. (1995), Path following energy optimization in unilateral contact problems. Journal of Global Optimization, 6 (4), 347–365.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Rohde A. and Stavroulakis G.E. (1995), Genericity results for path-following in dis-cretized unilateral contact mechanics, in: Parametric Optimization and Related Topics IV, Eds. J. Guddat, H. Th. Jongen, F. Nozicka, G. Still and F. Twilt, Peter Lang Verlag, Frankfurt am Main (submitted)Google Scholar
  36. 36.
    Shor N.Z. (1985), Minimization methods for nondifferentiable functions, Springer Verlag, Berlin.CrossRefGoogle Scholar
  37. 37.
    Schramm H. and Zowe J. (1992), A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results. SIAM J. Optimization, 2, 121–152.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Stavroulakis G.E. (1991), Analysis of structures with interfaces. Formulation and study of variational-hemivariational inequality problems, Ph.D. Dissertation, Aristotle University, Thessaloniki.Google Scholar
  39. 39.
    Stavroulakis G.E., Panagiotopoulos P.D. and Al-Fahed A.M. (1991), On the rigid body displacements and rotations in unilateral contact problems and applications. Computers and Structures, 40, 599–614.CrossRefzbMATHGoogle Scholar
  40. 40.
    Stavroulakis G.E. (1993), Convex decomposition for nonconvex energy problems in elastostatics and applications, European Journal of Mechanics A/Solids 12 (1), 1–20.MathSciNetzbMATHGoogle Scholar
  41. 41.
    Stavroulakis G.E. and Panagiotopoulos P.D. (1993), Convex multilevel decomposition algorithms for non-monotone problems, Int. J. Num. Meth. Engng. 36, 1945–1966.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Stavroulakis G.E. and Mistakidis E.S. (1995), Numerical treatment of hemivariational inequalities. Computational Mechanics, 16, 406–416.MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Stavroulakis G.E. and Panagiotopoulos P.D. (1994), A new class of multilevel decomposition algorithms for nonmonotone problems based on the quasidifferentiability concept, Comp. Meth. Appl. Mech. Engng. 117, 289–307.MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Stavroulakis G.E., Demyanov V.F. and Polyakova L.N. (1995), Quasidifferentiability in Mechanics, Journal of Global Optimization, 6 (4), 327–345.MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Stein E., Wagner W. and Wriggers P. (1989), Grundlagen nichtlinearer Berech-nungsverfahren in der Strukturmechanik, In: Nichtlineare Berechnungen im Kon-struktiven Ingenieurbau, Ed. Stein E., Wagner W. and Wriggers P, 1–53, Springer Verlag.Google Scholar
  46. 46.
    Strodiot J.J. and Nguyen V.H. (1988), On the numerical treatment of the inclusion 0 ∈ ∂f(x), In: Topics in Nonsmooth Mechanics, Eds. Strodiot J.J. and Nguyen V.H, 267–294, Birkhauser Verlag.Google Scholar
  47. 47.
    Tzaferopoulos M.Ap. (1993), On an efficient method for the frictional contact problem of structures with convex energy density. Computers and Structures, 48 (1), 87–106.MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Tzaferopoulos M.Ap., Mistakidis E.S., Bisbos C.D. and Panagiotopoulos P.D. (1995), Comparison of two methods for the solution of a class of nonconvex energy problems using convex minimization algorithms. Computers and Structures, 57 (6), 959–971.MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Womersley R.S. and Flecher R. (1986), An algorithm for composite nonsmooth optimization problems, Journal Optimization Theory and Applications, 48, 493–523.CrossRefzbMATHGoogle Scholar
  50. 50.
    Wriggers P. (1988), Konsistente Linearisierungen in der Kontinuumsmechanik und ihre Anwendungen auf die Finite-Element-Methode, Habilitationsschrift, Univer-sitat Hannover, Bericht F88/4.Google Scholar
  51. 51.
    Wriggers P. (1993), Continuum mechanics, nonlinear finite element techniques and computational stability. In: Progrress in computational analysis of inelastic structures, CISM Lect. Notes No. 321, Ed. Wriggers P, 245–287, Springer Verlag.Google Scholar
  52. 52.
    Zienkiewicz O.C. and Taylor R.L. (1991), The finite element method. Vol. II: Solid and fluid mechanics, dynamics and non-linearity, Fourth Ed., McGraw Hill.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • Vladimir F. Dem’yanov
    • 1
  • Georgios E. Stavroulakis
    • 2
  • Ludmila N. Polyakova
    • 1
  • Panagiotis D. Panagiotopoulos
    • 3
    • 4
  1. 1.Department of MathematicsSt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Lehr- und Forschungsgebiet für Mechanik; Lehrstuhl C für MathematikRWTHAachenGermany
  3. 3.Department of Civil EngineeringAristotle UniversityThessalonikiGreece
  4. 4.Faculty of Mathematics and PhysicsRWTHAachenGermany

Personalised recommendations