Applied Mathematics and Mechanics

, Volume 25, Issue 1, pp 47–58 | Cite as

Non-interior smoothing algorithm for frictional contact problems

  • Zhang Hong-wu
  • He Su-yan
  • Li Xing-si


A new algorithm for solving the three-dimensional elastic contact problem with friction is presented. The algorithm is a non-interior smoothing algorithm based on an NCP-function. The parametric variational principle and parametric quadratic programming method were applied to the analysis of three-dimensional frictional contact problem. The solution of the contact problem was finally reduced to a linear complementarity problem, which was reformulated as a system of nonsmooth equations via an NCP-function. A smoothing approximation to the nonsmooth equations was given by the aggregate function. A Newton method was used to solve the resulting smoothing nonlinear equations. The algorithm presented is easy to understand and implement. The reliability and efficiency of this algorithm are demonstrated both by the numerical experiments of LCP in mathematical way and the examples of contact problems in mechanics.

Key words

three-dimensional frictional contact problem parametic quadratic programming method linear complementarity problem NCP-function aggregate function non-interior smoothing algorithm 

Chinese Library Classification

O221 O242.21 

2000 Mathematics Subject Classification

65K05 90C90 74M10 74M15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Demyanov V F, Stavroulakis G E, Polyakova L N.Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics [M]. Dordrecht: Kluwer Academic Publishers, 1996.Google Scholar
  2. [2]
    Kikuchi N, Oden J T.Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods[M].Philadelphia:SIAM, 1988.zbMATHGoogle Scholar
  3. [3]
    ZHONG Wan-xie, ZHANG Hong-wu, WU Cheng-wei.Parametric Variational Principles and Applications in Engineering[M]. Beijing: Science Press, 1997. (in Chinese)Google Scholar
  4. [4]
    CHEN Guo-qing, CEN Wan-ji, FENG En-min. Nonlinear complementarity principle for three-dimension contact problem and solution method[J].Science in China, Ser A, 1995, 25(11): 1181–1190. (in Chinese).Google Scholar
  5. [5]
    LI Xue-wen, CHEN Wan-ji. Nonsmooth method for solving three-dimensional frictional contact problems[J].Chinese Journal of Computational Methanics, 2000,17(1):43–49. (in Chinese)Google Scholar
  6. [6]
    Christensen P W, Klarbring A, Pang J S,et al Formulation and comparison of algorithms for frictional contact problems[J].International Journal for Numerical Methods in Engineering, 1998,42: 145–173.zbMATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    ZHANG Hong-wu, ZHANG Wan-xie, GU Yuan-xian. A combined parametric quadratic programming and iteration method for 3D elastic-plastic frictional contact problem analysis[J].Comput. Meths. Appl. Mech. 1998,155: 307–324.zbMATHCrossRefGoogle Scholar
  8. [8]
    ZHANG Hong-wu. Parametric variational principle for elastic-plastic consolidation analysis of saturated porous media[J].Int. J. Numer. Anal. Meth. Geomechanics, 1995,19: 851–867.zbMATHCrossRefGoogle Scholar
  9. [9]
    ZHANG Hong-wu. Schrefler B A. Gradient-dependent plasticity model and dynamic strain localisation analysis of saturated and partially saturated porous media: one dimensional model[J].European Journal of Solid Mechanics, A/ Solids, 2000,19(3): 503–524.CrossRefGoogle Scholar
  10. [10]
    ZHANG Hong-wu, Galvanetto U, Schrefler BA. Local analysis and global nonlinear behaviour of periodic assemblies of bodies in elastic contact[J].Computational Mechanics, 1999,24(4): 217–229.zbMATHCrossRefGoogle Scholar
  11. [11]
    ZHANG Hong-wu, GU Yuan-xian, ZHONG Wan-xie. The finite element analysis for coupled problems between heat transfer and contact processes[J].Acta Solida Mechanica, 2000,21(3): 217–224. (in Chinese)Google Scholar
  12. [12]
    Billups S C, Murty K G. Complementarity problems[J].Journal of Computational and Applied Mathematics, 2000,124: 303–318.zbMATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    Wright S J,Primal-Dual Interior-Point Methods[M]. Philadelphia: SIAM Publications, 1997.zbMATHGoogle Scholar
  14. [14]
    XIU Nai-hua, Gao Zi-you. The new advances in methods for complementarity problems[J].Advances in Mathematics, 1999,28(3): 193–210. (in Chinese)zbMATHMathSciNetGoogle Scholar
  15. [15]
    Ferris M C, Kanzow C. Complementarity and related problems: A survey[A]. In: Pardalos P M, Resende M G C Eds:Handbook on Applied Optimization[C]. New York: Oxford University Press, 2002,514–530.Google Scholar
  16. [16]
    CHEN Chun-hui, Mangasarian O L. Smoothing methods for convex inequalities and linear complementarity problems[J].Mathematical Programming, 1995,71: 51–69.zbMATHMathSciNetCrossRefGoogle Scholar
  17. [17]
    CHEN Bing-tong, XIU Nai-hua. A Global linear and local quadratic non-interior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions [J].SIAM Journal on Optimization, 1999,9, 605–623.zbMATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    Burke J V, XU Song: The global linear convergence of a non-interior path following algorithm for linear complementarity problems[J].Mathematics of Operations Research, 1998,23: 719–734.zbMATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    LI Xing-si. An efficient method for non-differentiable optimization problem[J].Science in China, Ser A, 1994, 24(4): 371–377. (in Chinese)Google Scholar
  20. [20]
    Kanzow C. Some noninterior continuation methods for linear complementarity problems[J].SIAM J Matrix Anal Appl, 1996,17(4): 851–868.zbMATHMathSciNetCrossRefGoogle Scholar
  21. [21]
    Buczkowski R, Kleiber M. Elasto-plastic interface model for 3D-frictional orthotropic contact problems[J].International Journal for Numerical Methods in Engineering, 1997,40: 599–619.zbMATHCrossRefGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics and Mechanics 2004

Authors and Affiliations

  • Zhang Hong-wu
    • 1
  • He Su-yan
    • 1
  • Li Xing-si
    • 1
  1. 1.State Key Laboratory of Structural Analysis of Industrial EquipmentDalian University of TechnologyDalianP. R. China

Personalised recommendations