ライブラリ登録: Guest
Begell Digital Portal Begellデジタルライブラリー 電子書籍 ジャーナル 参考文献と会報 リサーチ集
International Journal for Multiscale Computational Engineering
インパクトファクター: 1.016 5年インパクトファクター: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN 印刷: 1543-1649
ISSN オンライン: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2019029111
pages 261-280

MAKING USE OF SYMMETRIES IN THE THREE-DIMENSIONAL ELASTIC INVERSE HOMOGENIZATION PROBLEM

C. Méndez
CIMEC-UNL-CONICET, Predio Conicet Dr Alberto Cassano, CP 3000 Santa Fe, Argentina
J.M. Podestá
CIMEC-UNL-CONICET, Predio Conicet Dr Alberto Cassano, CP 3000 Santa Fe, Argentina
S. Toro
CIMEC-UNL-CONICET, Predio Conicet Dr Alberto Cassano, CP 3000 Santa Fe, Argentina
Alfredo E. Huespe
CIMEC-UNL-CONICET, Predio Conicet Dr Alberto Cassano, CP 3000 Santa Fe, Argentina; Centre Internacional de Metodes Numerics en Enginyeria (CIMNE), Campus Nord UPC; 3E.T.S d'Enginyers de Camins, Canals i Ports, Technical University of Catalonia (UPC), Barcelona Tech, Campus Nord, Mòdul C-1, c/ Jordi Girona 1-3, 08034, Barcelona, Spain
J. Oliver
Centre Internacional de Metodes Numerics en Enginyeria (CIMNE), Campus Nord UPC; E.T.S d'Enginyers de Camins, Canals i Ports, Technical University of Catalonia (UPC), Barcelona Tech, Campus Nord, Mòdul C-1, c/ Jordi Girona 1-3, 08034, Barcelona, Spain

要約

The objective of this paper is the design of three-dimensional elastic metamaterials with periodic microarchitectures. The microarchitectures of these materials are attained by following an inverse design technique jointly with an homogenization-based topology optimization algorithm. In this context, we have particularly studied the connection between the symmetry of the material layout at the microscale of 3D periodic composites and the symmetry of the effective elastic properties.We have analyzed some possible Bravais lattices and space groups, which are typically associated with crystallography, to study the way in which the symmetries of these geometrical objects can be usefully used for the microarchitecture design of 3D elastic metamaterial. Following a previous work of the authors for two-dimensional problems, we suggest adopting the design domain of the topology optimization problem coincident with the Wigner-Seitz cells of specific Bravais lattices having the same point group to that of the target elasticity tensor. The numerical assessment described in this paper aims at the design of an extreme material. The solutions obtained with this procedure show that different composite microarchitectures emerge depending on the cell shape selection.

参考

  1. Amman, H., Calmon, P., and Iakovleva, E., Direct Elastic Imaging of a Small Inclusion, SIAMJ. Imaging Sci., vol. 1, no. 2, pp. 169-187, 2008.

  2. Amstutz, S., Analysis of a Level Set Method for Topology Optimization, Optim. Methods Software, vol. 26, nos. 4-5, pp. 555-573, 2011.

  3. Amstutz, S. and Andra, H., A New Algorithm for Topology Optimization Using a Level-Set Method, J. Comput. Phys., vol. 216, no. 2, pp. 573-588, 2006.

  4. Amstutz, S., Giusti, S., Novotny, A., and de Souza Neto, E., Topological Derivative for Multi-Scale Linear Elasticity Models Applied to the Synthesis of Microstructures, Int. J. Numer. Methods Eng., vol. 84, no. 6, pp. 733-756, 2010.

  5. Andreassen, E., Lazarov, B.S., and Sigmund, O., Design of Manufacturable 3D Extremal Elastic Microstructure, Mech. Mater., vol. 69, no. 1, pp. 1-10, 2014.

  6. Bendsoe, M. and Sigmund, O., Topology Optimization: Theory, Methods, and Applications, New York: Springer Science and Business Media, 2003.

  7. Cadman, J.E., Zhou, S., Chen, Y, and Li, Q., On Design of Multi-Functional Microstructural Materials, J. Mater. Sci., vol. 48, no. 1,pp. 51-66,2013.

  8. Coelho, P., Amiano, L., Guedes, J., and Rodrigues, H., Scale-Size Effects Analysis of Optimal Periodic Material Microstructures Designed by the Inverse Homogenization Method, Comput. Struct., vol. 174, pp. 21-32, 2016.

  9. Diaz, A. and Benard, A., Designing Materials with Prescribed Elastic Properties Using Polygonal Cells, Int. J. Numer. Methods Eng., vol. 57, no. 3, pp. 301-314, 2003.

  10. Feyel, F. and Chaboche, J., Fe2 Multiscale Approach for Modelling the Elastoviscoplastic Behaviour of Long Fibre SiC/Ti Composite Materials, Comput. Nethods Appl. Mech. Eng., vol. 183, nos. 3-4, pp. 309-330, 2000.

  11. Hashin, Z. and Shtrikman, S., A Variational Approach to the Theory of the Elastic Behaviour of Multiphase Materials, J. Mech. Phys. Solids, vol. 11, no. 2, pp. 127-140, 1963.

  12. Huang, X., Radman, A., and Xie, Y, Topological Design of Microstructures of Cellular Materials for Maximum Bulk or Shear Modulus, Comput. Mater. Sci., vol. 50, no. 6, pp. 1861-1870, 2011.

  13. Huang, X., Zhou, S., Xie, Y, and Li, Q., Topology Optimization of Microstructures of Cellular Materials and Composites for Macrostructures, Comput. Mater. Sci., vol. 67, pp. 397-407, 2013.

  14. Lazarov, B.S. and Sigmund, O., Filters in Topology Optimization based on Helmholtz-Type Differential Equations, Int. J. Numer. Methods Eng., vol. 86, pp. 765-781,2011.

  15. Li, H., Luo, Z., Gao, L., and Walker, P., Topology Optimization for Functionally Graded Cellular Composites with Metamaterials by Level Sets, Comput. Methods Appl. Mech. Eng., vol. 328, pp. 340-364,2018.

  16. Lopes, C., Santos, R.D., and Novotny, A., Topological Derivative-Based Topology Optimization of Structures Subject to Multiple Load-Cases, Latin Am. J. Solids Struct., vol. 12, no. 5, pp. 834-860, 2015.

  17. Meille, S. and Garboczi, E., Linear Elastic Properties of 2D and 3D Models of Porous Materials Made from Elongated Objects, Modell. Simul. Mater. Sci. Eng., vol. 9, no. 5, pp. 371-390, 2001.

  18. Mendez, C., Podesta, J., Lloberas-Valls, O., Toro, S., Huespe, A., and Oliver, J., Computational Material Design for Acoustic Cloaking, Int. J. Numer. Methods Eng., vol. 112, no. 10, pp. 1353-1380, 2017.

  19. Michel, J., Moulinec, H., and Suquet, P., Effective Properties of Composite Materials with Periodic Microstructure: A Computational Approach, Comput. Methods Appl. Mech. Eng., vol. 172, nos. 1-4, pp. 109-143, 1999.

  20. Neves, M., Rodrigues, H., and Guedes, J.M., Optimal Design of Periodic Linear Elastic Microstructures, Comput. Struct., vol. 76, nos. 1-3, pp. 421-429,2000.

  21. Novotny, A. and Sokolowski, J., Topological Derivatives in Shape Optimization, Springer Science and Business Media, 2012.

  22. Nye, J., Physical Properties of Crystals: Their Representation by Tensors and Matrices, vol. 146, Oxford: Clarendon Press, 2006.

  23. Oliver, J., Ferrer, A., Cante, J., Giusti, S., and Lloberas-Valls, O., On Multi-Scale Computational Design of Structural Materials Using the Topological Derivative, Adv. Comput. Plasticity, vol. 46, pp. 289-308, 2018.

  24. Osanov, M. and Guest, J., Topology Optimization for Architected Materials Design, Annu. Rev. Mater. Sci., vol. 46, pp. 211-233, 2016.

  25. Podesta, J., Mendez, C., Toro, S., Huespe, A., and Oliver, J., Material Design of Elastic Structures Using Voronoi Cells, Int. J. Numer. Methods Eng., vol. 115, no. 3, pp. 269-292, 2018.

  26. Podesta, J., Mendez, C., Toro, S., and Huespe, A., Symmetry Considerations for Topology Design in the Elastic Inverse Homogenization Problem, J. Mech. Phys. Solids, vol. 128, pp. 54-78,2019.

  27. Sigmund, O., Materials with Prescribed Constitutive Parameters: An Inverse Homogenization Problem, Int. J. Solids Struct., vol. 31, no. 17, pp. 2313-2329, 1994.

  28. Sigmund, O., Tailoring Materials with Prescribed Elastic Properties, Mech. Mater., vol. 20, no. 4, pp. 351-368, 1995.

  29. Sigmund, O., A New Class of Extremal Composites, J. Mech. Phys. Solids, vol. 48, no. 2, pp. 397-428, 2000.

  30. Solyom, J., Fundamentals of the Physics of Solids, vol. 1, New York: Springer Science and Business Media, 2007.

  31. Souvignier, B., A General Introduction to Space Groups, Int. Tables for Crystallog., vol. A, pp. 22-41, 2016.

  32. Ting, T., Anisotropic Elasticity: Theory and Applications, No. 45, Oxford University Press, 1996.

  33. Wang, F., Lazarov, B. S., and Sigmund, O., On Projection Methods, Convergence and Robust Formulations in Topology Optimization, Struct. Multidisc. Optim., vol. 43, no. 6,pp. 767-784,2011.

  34. Wang, Y, Luo, Z., Zhang, N., and Kang, Z., Topological Shape Optimization of Microstructural Metamaterials Using a Level Set Method, Comput. Mater. Sci., vol. 87, pp. 178-186,2014.


Articles with similar content:

On the Principles of Calculating Radiating Devices with Impedance Elements
Telecommunications and Radio Engineering, Vol.60, 2003, issue 5&6
V. V. Ovsyanikov
The CSANN Neural Model Modification for Production Scheduling
Journal of Automation and Information Sciences, Vol.36, 2004, issue 10
Tadeush Witkowski, Arkadiush Antchak, Pavel Antchak
A NUMERICAL ALGORITHM OF SOLVING PROBLEMS OF ELECTROMAGNETIC WAVE DIFFRACTION BY A PLANE LAYER WITH THE KERR NONLINEARITY
Telecommunications and Radio Engineering, Vol.76, 2017, issue 16
A. V. Brovenko, O. S. Troshchylo, A. Ye. Poyedinchuk, P. N. Melezhik
PSEUDO-MULTI-SCALE FUNCTIONS FOR THE STABILIZATION OF CONVECTION-DIFFUSION EQUATIONS ON RECTANGULAR GRIDS
International Journal for Multiscale Computational Engineering, Vol.11, 2013, issue 4
Ali I. Nesliturk, Onur Baysal
Vector Optimization of Hierarchical Structures
Journal of Automation and Information Sciences, Vol.36, 2004, issue 11
Albert N. Voronin