Optimized mesh generation by colliding bodies optimization algorithm in finite element

Document Type : Research


1 Assistant professor, Faculty of civil engineering, Shahid Rajaee Teacher Training University, Lavizan, Tehran, Iran.

2 MSc, Department of Civil Engineering, Shahid Rajaee Teacher Training University, Lavizan, Tehran, Iran.


This article presents combination method of h-refinement and node movement in finite element method to solve elasticity problems. Colliding bodies optimization algorithm (CBO), which is a meta-heuristic algorithm, is used to move nodes and in case of inaccurate answers h-refinement could be used to increase the number of nodes in the regions which have too many mistakes. Error estimate, used in both node movement and h-refinement, is made by L2-norm which is appropriate to triangle elements and another use of it is to build cost function that is used in CBO. The proposed method is suitable for finite element meshing procedure because it can solve problems in areas with high stress concentration. Two benchmark example results in linear elasticity problems with respect to other techniques, show the efficiency and acceptable accuracy of the proposed method


1. Arzani, H., Afshar, M.H., "Solving Poisson's equations by the discrete least square meshless method", WIT Transactions on Modeling and Simulation, 42, 2006, pp. 23-31. [DOI:10.2495/BE06003]
2. Arzani, H., Kaveh, A., Dehghana, M., "Adaptive node moving refinement in discrete least squares meshless method using charged system search", ScientiaIranica. Transaction A, Civil Engineering, 21, 2014, p. 1529.
3. Bank, Randolph E., "PLTMG: A Software Package for Solving Elliptic Partial Differential Equations, User's Guide 6.0", Society for Industrial and Applied Mathematics, Philadelphia, PA, 1990.
4. Johnson, C., "Numerical solution of partial differential equations by the finite element method", Studentlitteratur, Lund, Sweden, 1987
5. Johnson, C., Eriksson, K., "Adaptive finite element methods for parabolic Problems I: A Linear Model Problem", SIAM J, 28, 1991, p. 43-77. [DOI:10.1137/0728003]
6. Kaveh, A., Mahdavi, V. R. "Colliding Bodies Optimization : Extensions and Applications." Springer International Publishing, Switzerland. 2015. [DOI:10.1007/978-3-319-19659-6]
7. Kaveh, A., Mahdavi, V. R. "Colliding bodies optimization: A novel metaheuristic method." Computers & Structures. 139, pp. 18-27. 2014. [DOI:10.1016/j.compstruc.2014.04.005]
8. Nguyen-Thanh, N., Rabczuk, T., Nguyen-Xuan, H., Bordas, S. P. A. "An alternative alpha finite element method (AFEM) for free and forced structural vibration using triangular meshes", in Journal of Computational and Applied Mathematics, 233, 2010, pp. 2112-2135. [DOI:10.1016/j.cam.2009.08.117]
9. Ozyon, S., Temurta., H.,Durmu., B., Kuvat, G., "Charged system search algorithm for emission constrained economic power dispatch problem", Energy, 46, 2012, pp. 420-430. [DOI:10.1016/j.energy.2012.08.008]
10. Plaza, A., Padrón, M. A., Suárez, J. P., "Non-degeneracy study of the 8-tetrahedra longest-edge partition", Applied Numerical Mathematics, 55, 2005, pp. 458-472. [DOI:10.1016/j.apnum.2004.12.003]
11. Rosenberg, I. G., Stenger, F., "A lower bound on the angles of triangles constructed by bisecting the longest side", Math. Comp, 29, 1975, p. 390-395. [DOI:10.2307/2005558]
12. Timoshenko, S., Goodier, J.N., "Theory of elasticity, 3th ed", McGraw- Hill book, Inc., New York, USA, 1970 [DOI:10.1115/1.3408648]
13. Yershov, D. S,. Frazzoli, E., "Asymptotically optimal feedback planning using a numerical Hamilton-Jacobi-Bellman solver and an adaptive mesh refinement", Int. J. Rob. Res, 35, 2016, p. 565-584. [DOI:10.1177/0278364915602958]
14. Zeng, W., Liu, G.R., Li, D., Dong, X.W., "A smoothing technique based beta finite element method (βFEM) for crystal plasticity modeling", Computers & Structures, 162, 2016, pp. 48-67. [DOI:10.1016/j.compstruc.2015.09.007]
15. Zienkiewicz, O.C., "Achievements and some unsolved problems of the finite element method", International Journal for Numerical Methods in Engineering, 47, 2000, pp. 9-28. https://doi.org/10.1002/(SICI)1097-0207(20000110/30)47:1/3<9::AID-NME793>3.0.CO;2-P [DOI:10.1002/(SICI)1097-0207(20000110/30)47:1/33.0.CO;2-P]
16. Zienkiewicz, O.C., "The background of error estimation and adaptivity in finite element computations", Computer Methods in Applied Mechanics and Engineering, 195, 2006, pp. 207-213. [DOI:10.1016/j.cma.2004.07.053]
17. Zienkiewicz, O.C., Zhu, J.Z., "A simple error estimator and adaptive procedure for practical engineering analysis", International Journal for Numerical Methods in Engineering, 24, 1987, pp. 337-357. [DOI:10.1002/nme.1620240206]