Advertisement

Applied Mathematics and Mechanics

, Volume 37, Issue 12, pp 1707–1720 | Cite as

Strong-form framework for solving boundary value problems with geometric nonlinearity

  • J. P. YangEmail author
  • W. T. Su
Article

Abstract

In this paper, we present a strong-form framework for solving the boundary value problems with geometric nonlinearity, in which an incremental theory is developed for the problem based on the Newton-Raphson scheme. Conventionally, the finite element methods (FEMs) or weak-form based meshfree methods have often been adopted to solve geometric nonlinear problems. However, issues, such as the mesh dependency, the numerical integration, and the boundary imposition, make these approaches computationally inefficient. Recently, strong-form collocation methods have been called on to solve the boundary value problems. The feasibility of the collocation method with the nodal discretization such as the radial basis collocation method (RBCM) motivates the present study. Due to the limited application to the nonlinear analysis in a strong form, we formulate the equation of equilibrium, along with the boundary conditions, in an incremental-iterative sense using the RBCM. The efficacy of the proposed framework is numerically demonstrated with the solution of two benchmark problems involving the geometric nonlinearity. Compared with the conventional weak-form formulation, the proposed framework is advantageous as no quadrature rule is needed in constructing the governing equation, and no mesh limitation exists with the deformed geometry in the incremental-iterative process.

Key words

geometric nonlinearity incremental-iterative algorithm radial basis collocation method (RBCM) strong form 

Chinese Library Classification

O343.5 

2010 Mathematics Subject Classification

74S30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Yang, Y. B. and Shieh, M. S. Solution method for nonlinear problems with multiple critical points. AIAA Journal, 28, 2110–2116 (1990)CrossRefGoogle Scholar
  2. [2]
    Yang, Y. B. and Kuo, S. R. Theory and Analysis of Nonlinear Framed Structures, Prentice-Hall, Singapore (1994)Google Scholar
  3. [3]
    Lee, J. D. A large-strain elastic-plastic finite element analysis of rolling process. Computer Methods in Applied Mechanics and Engineering, 161, 315–347 (1998)CrossRefzbMATHGoogle Scholar
  4. [4]
    Monaghan, J. J. An introduction to SPH. Computer Physics Communications, 48, 89–96 (1988)CrossRefzbMATHGoogle Scholar
  5. [5]
    Nayroles, B., Touzot, G., and Villon, P. Generalizing the finite element method: diffuse approximation and diffuse elements. Computational Mechanics, 10, 307–318 (1992)CrossRefzbMATHGoogle Scholar
  6. [6]
    Belytschko, T., Lu, Y. Y., and Gu, L. Element free Galerkin methods. International Journal for Numerical Methods in Engineering, 37, 229–256 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Lu, Y. Y., Belytschko, T., and Gu, L. A new implementation of the element free Galerkin method. Computer Methods in Applied Mechanics and Engineering, 113, 397–414 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Zhu, T. and Atluri, S. N. A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method. Computational Mechanics, 21, 211–222 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Liu, W. K., Jun, S., and Zhang, Y. F. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids, 20, 1081–1106 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Jun, S., Liu, W. K., and Belytschko, T. Explicit reproducing kernel particle methods for large deformation problems. International Journal for Numerical Methods in Engineering, 41, 137–166 (1998)CrossRefzbMATHGoogle Scholar
  11. [11]
    Liu, W. K. and Jun, S. Multi-scale reproducing kernel particle methods for large deformation problems. International Journal for Numerical Methods in Engineering, 4, 1339–1362 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Chen, J. S., Pan, C., Wu, C. T., and Liu, W. K. Reproducing kernel particle methods for large deformation analysis of nonlinear structures. Computer Methods in Applied Mechanics and Engineering, 139, 195–227 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Chen, J. S., Pan, C., and Wu, C. T. Large deformation analysis of rubber based on a reproducing kernel particle method. Computational Mechanics, 19, 153–168 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Chen, J. S., Pan, C., and Wu, C. T. Application of reproducing kernel particle methods to large deformation and contact analysis of elastomers. Rubber Chemistry and Technology, 7, 191–213 (1998)CrossRefGoogle Scholar
  15. [15]
    Li, S., Hao, W., and Liu, W. K. Mesh-free simulations of shear banding in large deformation. International Journal of Solids and Structures, 37, 7185–7206 (2000)CrossRefzbMATHGoogle Scholar
  16. [16]
    Liew, K. M., Ng, T. Y., and Wu, Y. C. Meshfree method for large deformation analysis: a reproducing kernel particle approach. Engineering Structures, 24, 543–551 (2002)CrossRefGoogle Scholar
  17. [17]
    Duarte, C. A. and Oden, J. T. An h-p adaptive method using clouds. Computer Methods in Applied Mechanics and Engineering, 139, 237–262 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Chen, J. S., Wu, C. T., Yoon, S., and You, Y. A stabilized conforming nodal integration for Galerkin meshfree method. International Journal for Numerical Methods in Engineering, 50, 435–466 (2001)CrossRefzbMATHGoogle Scholar
  19. [19]
    Wang, D. and Sun, Y. A Galerkin meshfree formulation with stabilized conforming nodal integration for geometrically nonlinear analysis of shear deformable plates. International Journal of Computational Methods, 8, 685–703 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Chen, J. S., Hillman, M., and Ruter, M. An arbitrary order variationally consistent integration for Galerkin meshfree methods. International Journal for Numerical Methods in Engineering, 95, 387–418 (2013)MathSciNetCrossRefGoogle Scholar
  21. [21]
    Wang, D. and Peng, H. A Hermite reproducing kernel Galerkin meshfree approach for buckling analysis of thin plates. Computational Mechanics, 51, 1013–1029 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Wang, D. and Wu, J. An efficient nesting sub-domain gradient smoothing integration algorithm with quadratic exactness for Galerkin meshfree methods. Computer Methods in Applied Mechanics and Engineering, 298, 485–519 (2016)MathSciNetCrossRefGoogle Scholar
  23. [23]
    Hu, H. Y., Chen, J. S., and Hu, W. Weighted radial basis collocation method for boundary value problems. International Journal for Numerical Methods in Engineering, 69, 2736–2757 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Hu, H. Y., Chen, J. S., and Hu, W. Error analysis of collocation method based on reproducing kernel approximation. Numerical Methods for Partial Differential Equations, 27, 554–580 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Chi, S. W., Chen, J. S., Hu, H. Y., and Yang, J. P. A gradient reproducing kernel collocation method for boundary value problems. International Journal for Numerical Methods in Engineering, 93, 1381–1402 (2013)MathSciNetCrossRefGoogle Scholar
  26. [26]
    Hu, H. Y., Li, Z. C., and Cheng, A. H. D. Radial basis collocation method for elliptic equations. Computers and Mathematics with Applications, 50, 289–320 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Hon, Y. C. and Schaback, R. On unsymmetric collocation by radial basis functions. Applied Mathematics and Computation, 119, 177–186 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Hu, H. Y. and Chen, J. S. Radial basis collocation method and quasi-Newton iteration for nonlinearelliptic problems. Numerical Methods for Partial Differential Equations, 24, 991–1017 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Yang, J. P. and Su, W. T. Investigation of radial basis collocation method for incremental-iterative analysis. International Journal of Applied Mechanics, 8, 1650007 (2016)CrossRefGoogle Scholar
  30. [30]
    Liu, Y., Sun, L., Xu, F., Liu, Y., and Cen, Z. Bspline-based method for 2-D large deformation analysis. Engineering Analysis with Boundary Elements, 35, 761–767 (2011)CrossRefzbMATHGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Civil EngineeringChiao Tung UniversityTaiwanChina

Personalised recommendations