Abstract
This work deals on the optimization and computational material design using the topological derivative concept. The necessary details to obtain the anisotropic topological derivative are first presented. In the context of multi-scale topology optimization, it is crucial since the homogenization of the constitutive tensor of a micro-structure confers in general an anisotropic response. In addition, this work addresses the multi-scale material design problem in which the goal is then to minimize the structural (macro-scale) compliance by appropriately designing the material distribution (micro-structure) at a lower scale (micro-scale). To overcome the exorbitant computational cost, a consultation during the iterative process of a discrete material catalog (computed off-line) of micro-scale optimized topologies (Computational Vademecum) is proposed in this work. This results into a large diminution of the resulting computational costs, which make affordable the proposed methodology for multi-scale computational material design. Some representative examples assess the performance of the considered approach.
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References
G. Allaire, Shape Optimization by the Homogenization Method, vol. 146 (Springer Science & Business Media, 2012)
H. Ammari, H. Kang, Polarization and moment tensors with applications to inverse problems and effective medium theory. Applied Mathematical Sciences, vol. 162 (Springer, New York, 2007)
S. Amstutz, S.M. Giusti, A.A. Novotny, E.A. De Souza Neto, Topological derivative for multi-scale linear elasticity models applied to the synthesis of microstructures. Int. J. Numer. Methods Eng. 84(6), 733–756 (2010)
S. Amstutz, H. Andrä, A new algorithm for topology optimization using a level-set method. J. Comput. Phys. 216(2), 573–588 (2006)
M.P. Bendsoe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71(2), 197–224 (1988)
G. Cardone, S.A. Nazarov, J. Sokolowski, Asymptotic analysis, polarization matrices, and topological derivatives for piezoelectric materials with small voids. SIAM J. Control Optim. 48(6), 3925–3961 (2010)
F. Chinesta, A. Leygue, F. Bordeu, J.V. Aguado, E. Cueto, D. Gonzalez, I. Alfaro, A. Ammar, A. Huerta, PGD-Based computational vademecum for efficient design, optimization and control. Arch. Comput. Methods Eng. 20(1), 31–59 (2013)
E.A. De Souza Neto, R.A. Feijóo, Variational foundations of large strain multiscale solid constitutive models: kinematical formulation. Advanced Computational Materials Modeling: From Classical to Multi-Scale Techniques-Scale Techniques, pp. 341–378, 2010
D. Esteves Campeão, S. Miguel Giusti, A. Antonio Novotny, Topology design of plates considering different volume control methods. Eng. Comput. 31(5), 826–842 (2014)
A. Ferrer, Multi-scale topological design of structural materials: an integrated approach. Ph.D. thesis, Universitat Politècnica de Catalunya, 2017
S.M. Giusti, L.A.M. Mello, E.C.N. Silva, Piezoresistive device optimization using topological derivative concepts. Struct. Multidiscip. Optim. 50(3), 453–464 (2014)
S.M. Giusti, A. Ferrer, J. Oliver, Topological sensitivity analysis in heterogeneous anisotropic elasticity problem. Theoretical and computational aspects. Comput. Methods Appl. Mech. Eng. 311, 134–150 (2016)
J.A. Hernandez, J. Oliver, A.E. Huespe, M.A. Caicedo, J.C. Cante, High-performance model reduction techniques in computational multiscale homogenization. Comput. Methods Appl. Mech. Eng. 276, 149–189 (2014)
J. Kato, D. Yachi, K. Terada, T. Kyoya, Topology optimization of micro-structure for composites applying a decoupling multi-scale analysis. Struct. Multidiscip. Optim. 49(4), 595–608 (2014)
A.A. Novotny, J. Sokolowski, Topological derivatives in shape optimization. Interaction of Mechanics and Mathematics (Springer, Berlin, 2013)
J. Sokolowski, A. Zochowski, The topological derivative method in shape optimization. SIAM J. Control Optim. 37(4), 1251–1272 (1999)
L. Xia, P. Breitkopf, Concurrent topology optimization design of material and structure within FE2 nonlinear multiscale analysis framework. Comput. Methods Appl. Mech. Eng. 278, 524–542 (2014)
Acknowledgements
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Program (FP/2007–2013)/ERC Grant Agreement n. 320815, Advanced Grant Project COMP-DES-MAT.
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Oliver, J., Ferrer, A., Cante, J.C., Giusti, S.M., Lloberas-Valls, O. (2018). On Multi-scale Computational Design of Structural Materials Using the Topological Derivative. In: Oñate, E., Peric, D., de Souza Neto, E., Chiumenti, M. (eds) Advances in Computational Plasticity. Computational Methods in Applied Sciences, vol 46. Springer, Cham. https://doi.org/10.1007/978-3-319-60885-3_14
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DOI: https://doi.org/10.1007/978-3-319-60885-3_14
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