Journal of Mechanical Science and Technology

The Korean Society of Mechanical Engineers (대한기계학회)
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Free Vibration and Dynamic Response Analysis by Petrov-Galerkin Natural Element Method

  • Cho, Jin-Rae (School of Mechanical Engineering, Pusan National University) ;
  • Lee, Hong-Woo (School of Mechanical Engineering, Pusan National University)
  • Published : 2006.11.01
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Abstract

In this paper, a Petrov-Galerkin natural element method (PG-NEM) based upon the natural neighbor concept is presented for the free vibration and dynamic response analyses of two-dimensional linear elastic structures. A problem domain is discretized with a finite number of nodes and the trial basis functions are defined with the help of the Voronoi diagram. Meanwhile, the test basis functions are supported by Delaunay triangles for the accurate and easy numerical integration with the conventional Gauss quadrature rule. The numerical accuracy and stability of the proposed method are verified through illustrative numerical tests.

Keywords

References

  1. Bathe, K. J., 1996, Finite Element Procedures, Prentice- Hall, Singapore
  2. Belikov, V. V., Ivanov, V. D., Kontorovich, K. K., Korytnik, S. A. and Semenov, A. Su, 1997, 'The Non-Sibsonian Interpolation; a New Method of Interpolation of the Values of a Function on an Arbitrary Set of Points,' Computational Mathematics and Mathematical Physics, Vol. 37, pp.9-15
  3. Belytschko, T., Lu, Y. Y. and Gu, L., 1994, 'Element-free Galerkin Methods,' Int. J. Numer. Methods Engng., Vol. 37, pp. 229-256 https://doi.org/10.1002/nme.1620370205
  4. Braun, J. and Sam bridge, M., 1995, 'A Numerical Method for Solving Differential Equations on Highly Irregular Evolving Grids,' Nature, Vol. 376, pp. 655-660 https://doi.org/10.1038/376655a0
  5. Cho, J. R. and Lee, H. W., 2006, 'A Petrov Galerkin Natural Element Method Securing the Numerical Integration Accuracy,' J. Mech. Sci. Technol., Vol. 20, No.1, pp. 94-109 https://doi.org/10.1007/BF02916204
  6. Cho, J. R. and Lee, H. W., 2006, '2-D Large Deformation Analysis of Nearly Incompressible Body by Natural Element Method,' Computers & Structures, Vol. 84, pp. 293-304 https://doi.org/10.1016/j.compstruc.2005.09.019
  7. Duarte, C. A. and Oden, J. T., 1996, 'An h-p Adaptive Method using Clouds,' Comput. Methods in Appl. Mech. Engrg., Vol. 139, pp.237-262 https://doi.org/10.1016/S0045-7825(96)01085-7
  8. Farin, G., 1990, 'Surface over Dirichlet Tessellations,' Computer Aided Geometric Design, Vol. 7, No. 1-4, pp.281-292 https://doi.org/10.1016/0167-8396(90)90036-Q
  9. Liu, W. K., Jun, S. and Zhang, Y. F., 1995, 'Reproducing Kernel Particle Methods,' Int. J. Numer. Methods Fluids, Vol. 20, pp. 1081-1106 https://doi.org/10.1002/fld.1650200824
  10. Melenk, J. M. and Babuska, I., 1996, 'The Partition of Unity Finite Element Method: Basic Theory and Applications,' Comput. Methods Appl. Mech. Engrg., Vol. 139, pp.289-314 https://doi.org/10.1016/S0045-7825(96)01087-0
  11. Nayroles, B., 1992, 'Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements,' Computational Mechanics, Vol. 10, pp. 307-318 https://doi.org/10.1007/BF00364252
  12. Petyt, M., 1990, Introduction to Finite Element Vibration Analysis, Cambridge University Press, New York
  13. Sibson, R., 1980, 'A Vector Identity for Dirichlet Tessellation,' Mathematical Proceedings of Cambridge Philosophical Society, Vol. 80, pp. 151-155
  14. Sukumar, N., Moran, A. and Belytschko, T., 1998, 'The Natural Element Method in Solid Mechanics,' Int. J. Numer. Methods Engng., Vol. 43, pp. 839-887 https://doi.org/10.1002/(SICI)1097-0207(19981115)43:5<839::AID-NME423>3.0.CO;2-R
  15. Zu, T. and Atluri, S. N., 1998, 'A Modified Collocation Method and a Penalty Formulation for Enforcing the Essential Boundary Condition in the Element Galerkin Method,' Computational Mechanics, Vo. 21, pp. 211- 222 https://doi.org/10.1007/s004660050296