On the Six Node Hexagon Elements for Continuum Topology Optimization of Plates Carrying in Plane Loading and Shell Structures Carrying out of Plane Loading

Document Type: Research Paper


1 CVR College of Engineering, Hyderabad, Telangana, India

2 Civil Engineering Department, Osmania University, Hyderabad, Telangana, India


The need of polygonal elements to represent the domain is gaining interest among structural engineers. The objective is to perform static analysis and topology optimization of a given continuum domain using the rational fraction type shape functions of six node hexagonal elements. In this paper, the main focus is to perform the topology optimization of two-dimensional plate structures using Evolutionary Swarm Intelligence Firefly Algorithms (ESIFA) and three-dimensional shell structures using optimality criteria. The optimization of plates carrying in plane loading is performed with minimum weight as objective. Two different types of shell structures are optimized using maximum strain energy as criteria. The optimal distribution of the material in the design domain obtained using six node hexagon elements is compared with the optimal distribution of material obtained using quadrilateral elements. A few problems from the literature have been solved and this study has proved that hexagon element gives better results over traditional quadrilateral elements.


Main Subjects

[1] S. Singh, Optimal Integration Schemes for Polygonal Finite Element Method with Schwarz-Christoffel Conformal Mapping, Master’s Thesis, Indian Institute of Science, Bangalore, 2010.

[2] M. Ishiguro, K. Higuchi, Application of hexagonal Element Scheme in Finite Element Method to Three-Dimensional Diffusion Problem of Fast Reactors, Journal of Nuclear Science and Technology, 20(11), 1983, 951-960.

[3] M. Mehrenberger, L.S. Mendoza, C. Prouveur, E. Sonnendrucker, Solving the Guiding-Center model on a regular hexagonal mesh, ESAIM: Proc., 53, 2016, 149-176.

[4] C. Talischi, G.H. Papulino, C.H. Le, Wachspress Elements for Topology optimization, Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Proceedings of the 6th International Conference on Computation of Shell and Spatial Structures, IASS-IACM, Cornell University, Ithaca, NY, USA, 2008.

[5] O. Verners, Shape optimization of a Superelement of hexagonal Lattice Structure, Riga Technical University, Modris Dobelis, Riga Technical University, 2010.

[6] T. Huang, Y. Gong, S. Zhao, Effective In-Plane Elastic Modulus of a Periodic Regular hexagonal Honeycomb Core with Thick Walls, Journal of Engineering Mechanics, 2018, 144(2), 06017019.

[7] R. Levy, W.R. Spillers, Optimal design for axisymmetric cylindrical shell buckling, Journal of Engineering Mechanics, 115(8), 1989, 1683.

[8] S.A. Falco, S.M.B. Alfonso, L.E. Vaz, Analysis and Optimal design of plates and shells under dynamic loads - II: optimization, Structural and Multidisciplinary Optimization, 27, 2004, 197-209.

[9] D. Khoza, Topology optimization of plate-like structures, Master of Engineering Thesis, University of Pretoria, 2005.

[10] I. Mekjavić, Buckling analysis of concrete spherical shells,Tehnički Vjesnik, 18(4), 2011, 633-639.

[11] T. Yin, H.F. Lam, Dynamic Analysis of Finite Length Circular Cylinder shells with a circumferential surface crack, Journal of Engineering Mechanics, 139(10), 2013, 1419-1434.

[12] A.I. Harb, K.C. Fu, Analysis and optimal design of spherical shells under axisymmetric loads, Journal of Engineering Mechanics, 116(2), 1990, 324-342.

[13] A. Aswini, G. Ramakrishna, Computer aided analysis of cylindrical shell roof structure using Matlab, International Journal for Trends in Engineering & Technology, 23(1), 2017, 6-10.

[14] S. Ahmad, B.M. Irons, O.C. Zienkiewicz, Analysis of Thick and Thin Shell Structures by Curved Finite Elements, International Journal for Numerical Methods in Engineering, 2, 1970, 419-451.

[15] K. Kant, R. Singh, Shell Dynamics with three dimensional degenerate finite elements, Computers & Structures, 50(1), 1994, 135-144.

[16] M.L. Bucalem, K.J. Bathe, Finite Element Analysis of Shell Structures, Archives of Computational Methods in Engineering, 4(1), 1997, 3-61.

[17] D. Visy, S. Adany, Local Elastic and Geometric Stiffness Matrix for the Shell element Applied in cFEM, Periodica Polytechnica Civil Engineering, 61(3), 2017, 569-580.

[18] V. den Boom, Topology optimization Including Buckling Analysis, Master of Science Thesis, Delft University of Technology, 2014.

[19] B. Archana, K.N.V. Chandrasekhar, T.M. Rao, A study on parameters of Firefly algorithm to perform Topology optimization of Continuum structures - II, i-manager’s Journal on Structural Engineering, 6(1), 2017, 16-27.

[20] X.S. Yang, Firefly algorithms for multimodal optimization, Lecture Notes in Computer Sciences, 5792, 2009, 169-178.

[21] X.S. Yang, Firefly algorithm, Levy flights and global optimization, in Research and Development in Intelligent Systems XXVI (Eds M. Bramer, R. Ellis, M. Petridis), Springer London, 2010, 209-218.

[22] A. Baghlani, M.H. Makiabadi, M. Sarcheshmehpour, Discrete Optimum Design of Truss Structures by an Improved Firefly Algorithm, Advances in Structural Engineering, 17(10), 2014, 1517-1530.

[23] Y.W. Siu, B.S.L. Lai, F.W. Wang, Z.H. Zhou, S.L. Chan, Optimization of Structures by the Optimality Criteria Method, HKIE Transactions, 10(3), 2003, 48-53.

[24] K.J. Bathe, E.N. Dvorkin, A formulation of General Shell Elements - The Use of Mixed Interpolation of Tensorial Components, International Journal for Numerical Methods in Engineering, 22, 1986, 697-722.

[25] W. Altman, F. Iguti, A Thin Cylindrical Shell Finite Element Based on a Mixed Formulation, Computers & Structures, 6, 1976, 149-155.

[26] G. Yi, Y. Sui, TIMP method for topology optimization of plate structures with displacement constraints under multiple loading cases, Structural and Multidisciplinary Optimization, 53(6), 2016, 1185-1196.

[27] H.C. Gea, Y. Fu, 3-D Topology optimization using a Design Domain Method, SAE Transactions: Journal of Passenger Cars, 104(6):1995, 1983-1989.