DOI: https://doi.org/10.9744/ced.17.3.152-157

### Generalization of FEM Using Node-Based Shape Functions

Kanok-Nukulchai Worsak, Wong F.T., Sommanawat W.

#### Abstract

In standard FEM, the stiffness of an element is exclusively influenced by nodes associated with the element via its element-based shape functions. In this paper, the authors present a method that can be viewed as a generalization of FEM for which the influence of a node is not limited by a hat function around the node. Shape functions over an element can be interpolated over a predefined set of nodes around the element. These node-based shape functions employ Kriging Interpolations commonly found in geostatistical technique. In this study, a set of influencing nodes are covered by surrounding layers of elements defined as its domain of influence (DOI). Thus, the element stiffness is influenced by not only the element nodes, but also satellite nodes outside the element. In a special case with zero satellite nodes, the method is specialized to the conventional FEM. This method is referred to as Node-Based Kriging FEM or K-FEM. The K-FEM has been tested on 2D elastostatic, Reissner-Mindlin’s plate and shell problems. In all cases, exceptionally accurate displacement and stress fields can be achieved with relatively coarse meshes. In addition, the same set of Kringing shape functions can be used to interpolate the mesh geometry. This property is very useful for representing the curved geometry of shells. The distinctive advantage of the K-FEM is its inheritance of the computational procedure of FEM. Any existing FE code can be easily extended to K-FEM; thus, it has a higher chance to be accepted in practice.

#### Keywords

Finite element; kriging interpolation; node-based shape function; satellite nodes.

PDF

#### References

1. Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., and Krysl, P., Meshless Methods: An Overview and Recent Developments, Computer Methods in Applied Mechanics and Engineering, 139, 1996, pp. 3-47.
2. Fries, T.P. and Matthies, H.G., Classification and Overview of Meshfree Methods, Institute of Scientific Computing, Technical University Braunschweig, Brunswick, Germany, 2004.
3. Gu, Y.T., Meshfree Methods and Their Comparisons, International Journal of Computa-tional Methods, 2, 2005. pp. 477-515.
4. Belytschko, T., Lu, Y.Y., and Gu, L., Element-free Galerkin Methods, International Journal for Numerical Methods in Engineering, 37, 1994, pp. 229-256.
5. Nayroles, B., Touzot, G., and Villon, P., Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements, Computational Mechanics, 10, 1992, pp. 307-318.
6. Gu, L., Moving Kriging Interpolation and Element-free Galerkin Method, International Journal for Numerical Methods in Engineering, 56, 2003, pp. 1-11.
7. Tongsuk, P. and Kanok-Nukulchai, W., Further Investigation of Element-Free Galerkin Method using Moving Kriging Interpolation, International Journal of Computational Methods, 1, 2004, pp. 345-365.
8. Sayakoummane, V. and Kanok-Nukulchai, W., A Meshless Analysis of Shells Based on Moving Kriging Interpolation, International Journal of Computational Methods, 4, 2007, pp. 543-565.
9. Plengkhom, K. and Kanok-Nukulchai, W., An Enhancement of Finite Element Methods with Moving Kriging Shape Functions, International Journal of Computational Methods, 2, 2005, pp. 451-475.
10. Olea, R.A., Geostatistics for Engineers and Earth Scientists, Boston, Kluwer Academic Publishers, 1999.
11. Wackernagel, H., Multivariate Geostatistics, the 2nd, completely revised edition. Berlin, Springer, 1998.
12. Wong, F.T. and Kanok-Nukulchai, W., Kriging-based Finite Element Method for Analyses of Reissner-Mindlin Plates, Emerging Trends: Keynote Lectures and Symposia, Proc. 10th East Asia-Pacific Conference Structure Engineering Construction (EASEC-10), Bangkok, Thailand, August 3-5, 2006, Asian Institute of Technology, 2006, pp. 509-514.
13. Dai, K.Y., Liu, G.R., Lim, K.M., and Gu, Y.T., Comparison between the Radial Point Interpolation and the Kriging Interpolation Used in Meshfree Methods, Computational Mechanics, 32, 2003, pp. 60-70.
14. Wong, F.T. and Kanok-Nukulchai, W., On the Convergence of the Kriging-based Finite Element Method, Proc. 3rd Asia-Pacific Congress on Computational Mechanics (APCOM’07) in conjunction with 11th International Conference Enhancement and Promotion of Computational Methods in Engineering and Science (EPMESC XI), Kyoto, Japan, December 3-6, 2007, Asian-Pacific Association for Computational Mechanics, Paper No. MS38-2-3, 2007.
15. Wong, F.T. and Kanok-Nukulchai, W., On the Convergence of the Kriging-based Finite Element Method, International Journal of Computational Methods, 2008.
16. Masood, Z. and Kanok-Nukulchai, W., An Adaptive Mesh Generation for Kriging Element-free Galerkin Method Based on Delaunay Triangulation, Emerging Trends: Keynote Lectures and Symposia, Proc. 10th East Asia-Pacific Conf. Struct. Eng. Constr. (EASEC-10), Bangkok, Thailand, August 3-5, 2006, Asian Institute of Technology, 2006, pp. 499-508.
17. Sommanawat, W. and Kanok-Nukulchai, W., The Enrichment of Material Discontinuity in Moving Kriging Methods, Emerging Trends: Keynote Lectures and Symposia, Proc. 10th East Asia-Pacific Conference Structure Engineering Construction (EASEC-10), Bangkok, Thailand, August 3-5, 2006, Asian Institute of Technology, 2006, pp. 525-530.
18. Wicaksana, C. and Kanok-Nukulchai, W., Dynamic Analysis of Timoshenko Beam and Mindlin Plate by Kriging-Based Finite Element Methods, Emerging Trends: Keynote Lectures and Symposia, Proc. 10th East Asia-Pacific Conf. Struct. Eng. Constr. (EASEC-10), Bangkok, Thailand, August 3-5, 2006, Asian Institute of Technology, 2006, pp. 515-524.
19. Wong, F.T. and Kanok-Nukulchai, W., A Kriging-based Finite Element Method for Analyses of Shell Structures, Proceedings of the 8th World Congress on Computational Mechanics (WCCM8) and the 5th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2008), Venice, Italy, June 30-July 5, 2008, International Association for Computational Mechanics, Paper No. 1247, 2008.

DOI: https://doi.org/10.9744/ced.17.3.152-157

CED is published by The Institute of Research & Community Outreach - Petra Christian University, Surabaya, Indonesia

©All right reserved 2016.Civil Engineering Dimension, ISSN: 1410-9530, e-ISSN: 1979-570X

View My Stats