Abstract
Functionally graded materials (FGMs) are inhomogeneous composites which are characterized by gradual variation in their physical properties. This study proposes a computational approach based on the bi-directional evolutionary structural optimization (BESO) for topologically designing microstructures of such materials with multi-functional properties, e.g. bulk modulus and thermal conductivity. It is assumed that the base cells are composed of two constituents. The smooth transition between adjacent base cells is realized by considering three base cells at each stage of the optimization. Effectiveness and efficiency of the proposed approach has been demonstrated by several numerical examples.
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References
Bever, M.B., Duwez, P.E.: Gradients in composite materials. Mater. Sci. Eng. 10, 1–8 (1972)
Shen, M., Bever, M.B.: Gradients in polymeric materials. J. Mater. Sci. 7, 741–746 (1972)
Koizumi, M.: FGM activities in Japan. Compos. Part B 28(1–2), 1–4 (1997). doi:10.1016/s1359-8368(96)00016-9
Hirai, T., Chen, L.: Recent and prospective development of functionally graded materials in Japan. Mater. Sci. Forum 308–311, 509–514 (1999). doi:10.4028/www.scientific.net/MSF.308-311.509
Birman, V., Byrd, L.W.: Modeling and analysis of functionally graded materials and structures. Appl. Mech. Rev. 60(5), 195–216 (2007). doi:10.1115/1.2777164
Lin, C.-Y., Hsiao, C.-C., Chen, P.-Q., Hollister, S.J.: Interbody fusion cage design using integrated global layout and local microstructure topology optimization. Spine 29(16), 1747–1754 (2004)
Chen, K.-Z., Feng, X.-A.: CAD modeling for the components made of multi heterogeneous materials and smart materials. Comput. Aided Des. 36(1), 51–63 (2004). doi:10.1016/s0010-4485(03)00077-0
Sigmund, O.: Materials with prescribed constitutive parameters: an inverse homogenization problem. Int. J. Solids Struct. 31(17), 2313–2329 (1994). doi:10.1016/0020-7683(94)90154-6
Challis, V.J., Roberts, A.P., Wilkins, A.H.: Design of three dimensional isotropic microstructures for maximized stiffness and conductivity. Int. J. Solids Struct. 45(14–15), 4130–4146 (2008). http://dx.doi.org/10.1016/j.ijsolstr.2008.02.025
Torquato, S., Hyun, S., Donev, A.: Optimal design of manufacturable three-dimensional composites with multifunctional characteristics. J. Appl. Phys. 94(9), 5748–5755 (2003). doi:10.1063/1.1611631
Guest, J.K., Prévost, J.H.: Optimizing multifunctional materials: design of microstructures for maximized stiffness and fluid permeability. Int. J. Solids Struct. 43(22–23), 7028–7047 (2006). http://dx.doi.org/10.1016/j.ijsolstr.2006.03.001
Cadman, J., Zhou, S., Chen, Y., Li, Q.: On design of multi-functional microstructural materials. J. Mater. Sci. 48(1), 51–66 (2012). doi:10.1007/s10853-012-6643-4
de Kruijf, N., Zhou, S., Li, Q., Mai, Y.-W.: Topological design of structures and composite materials with multiobjectives. Int. J. Solids Struct. 44(22–23), 7092–7109 (2007). http://dx.doi.org/10.1016/j.ijsolstr.2007.03.028
Huang, X., Xie, Y.M.: Evolutionary Topology Optimization of Continuum Structures: Methods and Applications. John Wiley & Sons, Chichester (2010)
Huang, X., Radman, A., Xie, Y.M.: Topological design of microstructures of cellular materials for maximum bulk or shear modulus. Comput. Mater. Sci. 50(6), 1861–1870 (2011). doi:10.1016/j.commatsci.2011.01.030
Rozvany, G.I.N., Zhou, M., Birker, T.: Generalized shape optimization without homogenization. Struct. Multidiscip. Optim. 4(3), 250–252 (1992). doi:10.1007/bf01742754
Bendsøe, M.P., Sigmund, O.: Topology Optimization: Theory, Methods and Application. Springer, Berlin (2003)
Huang, X., Xie, Y., Jia, B., Li, Q., Zhou, S.: Evolutionary topology optimization of periodic composites for extremal magnetic permeability and electrical permittivity. Struct. Multidiscip. Optim. 46(3), 385–398 (2012). doi:10.1007/s00158-012-0766-8
Hassani, B., Hinton, E.: A review of homogenization and topology optimization I—homogenization theory for media with periodic structure. Comput. Struct. 69(6), 707–717 (1998). doi:10.1016/s0045-7949(98)00131-x
Hassani, B., Hinton, E.: A review of homogenization and topology opimization II—analytical and numerical solution of homogenization equations. Comput. Struct. 69(6), 719–738 (1998). doi:10.1016/s0045-7949(98)00132-1
Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71(2), 197–224 (1988). http://dx.doi.org/10.1016/0045-7825(88)90086-2
Haug, E.J., Choi, K.K., Komkov, V.: Design Sensitivity Analysis of Structural Systems. Academic, Orlando (1986)
Zhou, S., Li, Q.: Computational design of multi-phase microstructural materials for extremal conductivity. Comput. Mater. Sci. 43(3), 549–564 (2008). http://dx.doi.org/10.1016/j.commatsci.2007.12.021
Huang, X., Xie, Y.M.: Evolutionary topology optimization of continuum structures with an additional displacement constraint. Struct. Multidiscip. Optim. 40(1–6), 409–416 (2010). doi:10.1007/s00158-009-0382-4
Radman, A., Huang, X., Xie, Y.M.: Topology optimization of functionally graded cellular materials. J. Mater. Sci. 48(4), 1503–1510 (2013). doi:10.1007/s10853-012-6905-1
Radman, A., Huang, X., Xie, Y.M.: Topological optimization for the design of microstructures of isotropic cellular materials. Eng. Optim. 45(11), 1331–1348 (2013). doi:10.1080/0305215x.2012.737781
Zhou, S., Li, Q.: Microstructural design of connective base cells for functionally graded materials. Mater. Lett. 62, 4022–4024 (2008). doi:10.1016/j.matlet.2008.05.058
Huang, X., Xie, Y.M.: Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elem. Anal. Des. 43(14), 1039–1049 (2007). doi:10.1016/j.finel.2007.06.006
Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11(2), 127–140 (1963). doi:10.1016/0022-5096(63)90060-7
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Radman, A., Huang, X., Xie, Y.M. (2015). Topology Optimization of Microstructures for Multi-Functional Graded Composites. In: Gao, D., Ruan, N., Xing, W. (eds) Advances in Global Optimization. Springer Proceedings in Mathematics & Statistics, vol 95. Springer, Cham. https://doi.org/10.1007/978-3-319-08377-3_27
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DOI: https://doi.org/10.1007/978-3-319-08377-3_27
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