Automatic Adaptive Finite Element Enrichement using Interpolation Cover Functions
Authors
Abstract
In this paper, a method is proposed to improve the results of the standard finite element method. L2 norm is used to determine the nodal error. In the next step, the appropriate order of the interpolation cover is seclected to be proportional to the nodal error and the results are corrected. The error computation procedure and the use of covering enrichment functions will continue until the error reaches the specified value. Cover enrichment interpolation functions will consider the effects of the adjacent elements of each node, in addition to the values obtained from the standard interpolation for each element. Computation rules are programmed in the matlab program and considered for the same examples. Comparison of the results of the proposed method with the exact solutions and the results of the methods proposed by the other researchers in the field of linear elasticity indicates the efficiency and accuracy of the proposed method.
Keywords
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13. Zienkiewicz, O. C., and Zhu, J. Z., “A Simple Error Estimator and Adaptive Procedure for Practical Engineerng Analysis”, International Journal for Numerical Methods in Engineering, Vol. 24, pp. 337-357, 1987.
14. Timoshenko, S., and Goodier, J. N., Theory of Elasticity, 3th ed, New York: McGraw- Hill book, 1970.
15. Ebrahimnejad, M., Fallah, N., and Khoei, A. R., “Adaptive Refinement in the Meshless Finite Volume Method for Elasticity Problems”, Computers & Mathematics with Applications, Vol. 69, pp. 1420-1443, 2015.
2. Johnson, C., and Eriksson, K., “Adaptive Finite Element Methods for Parabolic Problems I: A Linear Model Problem”, SIAM Journal, Vol. 28, pp. 43-77, 1991.
3. Yang, R., and Yuan, G., “h-Refinement for Simple Corner Balance Scheme of SN Transport Equation on Distorted Meshes”, Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 184, pp. 241-253, 2016.
4. Zander, N., Bog, T., Elhaddad, M., Frischmann, F., Kollmannsberger, S., and Rank, E., “The Multi-level -Method for Three-dimensional Problems: Dynamically Changing High-order Mesh Refinement with Arbitrary Hanging Nodes”, Computer Methods in Applied Mechanics and Engineering, Vol. 310, pp. 252-277, 2016.
5. Yang, J., Zhang, B., Liang, C., and Rong, Y., “A High-order Flux Reconstruction Method with Adaptive Mesh Refinement and Artificial Diffusivity on Unstructured Moving/deforming Mesh for Shock Capturing”, Computers & Fluids, Vol. 139, pp. 17-35, 2016.
6. Shi, G. H., Block System Modeling by Discontinuous Deformation Analysis, Southampton, UK: Computational Mechanics Publications, 1993.
7. Ghasemzadeh, H., Ramezanpour, M. A., and Bodaghpour, S., “Dynamic High Order Numerical Manifold Method Based on Weighted Residual Method”, International Journal for Numerical Methods in Engineering, Published online in Wiley Online Library, Vol. 100, No. 8, pp. 596-619, 2014.
8. Kim, J,. and Bathe, K. J., “The Finite Element Method Enriched by Interpolation Covers”, Computers & Structures, Vol. 116, pp. 35-49, 2013.
9. Kim, J,. and Bathe, K. J., “Towards a Procedure to Automatically Improve Finite Element Solutions by Interpolation Covers”, Computers & Structures, Vol. 131, pp. 81-97, 2014.
10. Arzani, H., Kaveh, A., and Dehghana, M., “Adaptive Node Moving Refinement in Discrete Least Squares Meshless Method using Charged System Searc”, International Journal of Science & Technology, Transaction A, Civil Engineering, Vol. 21, pp. 1529-1538, 2014.
11. Zeng, W., Liu, G. R., Li, D., and Dong, X. W., “A Smoothing Technique Based Beta Finite Element Method (βFEM) for Crystal Plasticity Modeling”, Computers & Structures, Vol. 162, pp. 48-67, 2016.
12. Arzani, H., Kaveh, A., and Taheri Taromsari, M., “Optimum Two-dimensional Crack Modeling in Discrete Least Square Meshless Method by Charged System Search Algorithm”, International Journal of Science & Technology. Transaction A, Civil Engineering, Vol. 24, pp. 143-152, 2017.
13. Zienkiewicz, O. C., and Zhu, J. Z., “A Simple Error Estimator and Adaptive Procedure for Practical Engineerng Analysis”, International Journal for Numerical Methods in Engineering, Vol. 24, pp. 337-357, 1987.
14. Timoshenko, S., and Goodier, J. N., Theory of Elasticity, 3th ed, New York: McGraw- Hill book, 1970.
15. Ebrahimnejad, M., Fallah, N., and Khoei, A. R., “Adaptive Refinement in the Meshless Finite Volume Method for Elasticity Problems”, Computers & Mathematics with Applications, Vol. 69, pp. 1420-1443, 2015.
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