Abstract
The optimization of manufacturable extremal elastic materials can be carried out via topology optimization using the homogenization method. We combine here a standard density-based inverse homogenization technique with an anisotropic mesh adaptation procedure in the context of a finite element discretization. In this way, the optimized layouts are intrinsically smooth and ready to be manufactured.
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References
Andreassen, E., Andreasen, C.S.: How to determine composite material properties using numerical homogenization. Comput. Mater. Sci. 83, 488–495 (2014)
Bendsøe, M.P., Sigmund, O.: Topology Optimization – Theory, Methods and Applications. Springer, Berlin (2003)
Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. AMS Chelsea Publishing, Providence (2011)
Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Computer Science and Applied Mathematics. Academic, New York (1982)
Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)
Chung, P.W., Tamma, K.K., Namburu, R.R.: Homogenization of temperature-dependent thermal conductivity in composite materials. J. Thermophys. Heat Transf. 15(1), 10–17 (2001)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Curtis, F.E., Schenk, O., Wächter, A.: An interior-point algorithm for large-scale nonlinear optimization with inexact step computations. SIAM J. Sci. Comput. 32(6), 3447–3475 (2010)
Farrell, P.E., Micheletti, S., Perotto, S.: An anisotropic Zienkiewicz-Zhu-type error estimator for 3D applications. Int. J. Numer. Methods Eng. 85(6), 671–692 (2011)
Frey, P.J., George, P.L.: Mesh Generation: Application to Finite Elements. Wiley, Hoboken (2008)
Gould, P.L.: Introduction to Linear Elasticity. Springer, Paris (1994)
Hashin, Z., Shtrikman, S.: A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys. 33(10), 3125–3131 (1962)
Hecht, F.: New development in freefem++. J. Numer. Math. 20(3–4), 251–266 (2012)
Larsen, U.D., Signund, O., Bouwsta, S.: Design and fabrication of compliant micromechanisms and structures with negative Poisson’s ratio. J. Microelectron. Syst. 6(2), 99–106 (1997)
Lazarov, B.S., Sigmund, O.: Filters in topology optimization based on Helmholtz-type differential equations. Int. J. Numer. Methods Eng. 86(6), 765–781 (2011)
Micheletti, S., Perotto, S.: Anisotropic adaptation via a Zienkiewicz–Zhu error estimator for 2D elliptic problems. In: Kreiss, G., Lötstedt, P., Målqvist, A., Neytcheva, M. (eds.), Numerical Mathematics and Advanced Applications, pp. 645–653. Springer, Berlin (2010)
Micheletti, S., Perotto, S., Farrell, P.E.: A recovery-based error estimator for anisotropic mesh adaptation in CFD. Bol. Soc. Esp. Mat. Apl. SeMA 50, 115–137 (2010)
Micheletti, S., Perotto, S., Soli, L.: Ottimizzazione topologica adattativa per la fabbricazione stratificata additiva (2017). Italian patent application No. 102016000118131, filed on November 22, 2016 (extended as Adaptive topology optimization for additive layer manufacturing, International patent application PCT No. PCT/IB2017/057323)
Neves, M.M., Rodrigues, H., Guedes, J.M.: Optimal design of periodic linear elastic microstructures. Comput. Struct. 76(1–3), 421–429 (2000)
Sánchez-Palencia, E.: Homogenization Method for the Study of Composite Media. In: Asymptotic Analysis, II. Lecture Notes in Mathematics, vol. 985, pp. 192–214. Springer, Berlin (1983)
Sigmund, O.: Design of Material Structures Using Topology Optimization. Technical University of Denmark, Lyngby (1994)
Sigmund, O.: Materials with prescribed constitutive parameters: an inverse homogenization problem. Int. J. Solids Struct. 31(17), 2313–2329 (1994)
Sigmund, O., Petersson, J.: Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct. Multidiscip. Optim. 16(1), 68–75 (1998)
Yin, L., Ananthasuresh, G.: Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme. Struct. Multidiscip. Optim. 23(1), 49–62 (2001)
Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Methods Eng. 24(2), 337–357 (1987)
Zuo, W., Saitou, K.: Multi-material topology optimization using ordered SIMP interpolation. Struct. Multidiscip. Optim. 55(2), 477–491 (2017)
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Ferro, N., Micheletti, S., Perotto, S. (2020). Density-Based Inverse Homogenization with Anisotropically Adapted Elements. In: van Brummelen, H., Corsini, A., Perotto, S., Rozza, G. (eds) Numerical Methods for Flows. Lecture Notes in Computational Science and Engineering, vol 132. Springer, Cham. https://doi.org/10.1007/978-3-030-30705-9_19
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