Applied Mathematics and Mechanics

, Volume 37, Issue 12, pp 1707–1720

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

• J. P. Yang
• 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

O343.5

74S30

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)
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)
4. [4]
Monaghan, J. J. An introduction to SPH. Computer Physics Communications, 48, 89–96 (1988)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
27. [27]
Hon, Y. C. and Schaback, R. On unsymmetric collocation by radial basis functions. Applied Mathematics and Computation, 119, 177–186 (2001)
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)
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)
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)