# American Institute of Mathematical Sciences

September  2014, 9(3): 433-451. doi: 10.3934/nhm.2014.9.433

## Optimization of bodies with locally periodic microstructure by varying the periodicity pattern

 1 CMAF, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal, Portugal

Received  October 2013 Revised  May 2014 Published  October 2014

This paper describes a numerical method to optimize elastic bodies featuring a locally periodic microscopic pattern. A new idea, of optimizing the periodicity cell itself, is considered. In previously published works, the authors have found that optimizing the shape and topology of the model hole gives a limited flexibility to the microstructure for adapting to the macroscopic loads. In the present study the periodicity cell varies during the optimization process, thus allowing the microstructure to adapt freely to the given loads. Our approach makes the link between the microscopic level and the macroscopic one. Two-dimensional linearly elastic bodies are considered, however the same techniques can be applied to three-dimensional bodies. Homogenization theory is used to describe the macroscopic (effective) elastic properties of the body. Numerical examples are presented, in which a cantilever is optimized for different load cases, one of them being multi-load. The problem is numerically heavy, since the optimization of the macroscopic problem is performed by optimizing in simultaneous hundreds or even thousands of periodic structures, each one using its own finite element mesh on the periodicity cell. Parallel computation is used in order to alleviate the computational burden.
Citation: Cristian Barbarosie, Anca-Maria Toader. Optimization of bodies with locally periodic microstructure by varying the periodicity pattern. Networks and Heterogeneous Media, 2014, 9 (3) : 433-451. doi: 10.3934/nhm.2014.9.433
##### References:
 [1] G. Allaire, Shape Optimization by the Homogenization Method, Springer, Applied Mathematical Sciences 146, 2002. doi: 10.1007/978-1-4684-9286-6. [2] G. Allaire, F. Jouve and A.-M. Toader, Structural optimization using sensitivity analysis and a level set method, Journal of Computational Physics, 194 (2004), 363-393. doi: 10.1016/j.jcp.2003.09.032. [3] C. Barbarosie, Shape optimization of periodic structures, Computational Mechanics, 30 (2003), 235-246. doi: 10.1007/s00466-002-0382-3. [4] C. Barbarosie and S. Lopes, A generalized notion of compliance, Comptes Rendus Mécanique, 339 (2011), 641-648. doi: 10.1016/j.crme.2011.07.002. [5] C. Barbarosie and A.-M. Toader, Shape and Topology Optimization for periodic problems, Part I, The shape and the topological derivative, Structural and Multidisciplinary Optimization, 40 (2009), 381-391. doi: 10.1007/s00158-009-0378-0. [6] C. Barbarosie and A.-M. Toader, Shape and Topology Optimization for periodic problems, Part II, Optimization algorithm and numerical examples, Structural and Multidisciplinary Optimization, 40 (2009), 393-408. doi: 10.1007/s00158-009-0377-1. [7] C. Barbarosie and A.-M. Toader, Optimization of bodies with locally periodic microstructre, Mechanics of advanced materials and structures, 19 (2012), 290-301. doi: 10.1080/15376494.2011.642939. [8] A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Studies in Mathematics and its Applications, 5, 1978. [9] M. Briane, Homogenization of a nonperiodic material, J. Math. Pures et Appl., 73 (1994), 47-66. [10] A. Cherkaev, Variational Methods for Structural Optimization, Springer Verlag, 2000. doi: 10.1007/978-1-4612-1188-4. [11] D. Cioranescu and J. S. J. Paulin, Homogenization in open sets with holes, Journal of Mathematical Analysis and Applications, 71 (1979), 590-607. doi: 10.1016/0022-247X(79)90211-7. [12] F. Murat and L. Tartar, H-convergence, in Topics in the Mathematical Modelling of Composite Materials, (eds. A. Cherkaev and R. Kohn), Progress in Nonlinear Differential Equations and Their Applications, 31, Birkhäuser, (1997), 21-43 [13] O. Pantz and K. Trabelsi, A post-treatment of the homogenization method for shape optimization, SIAM J. Control Optim., 47 (2008), 1380-1398. doi: 10.1137/070688900. [14] H. Rodrigues, J. M. Guedes and M. P. Bendsøe, Hierarchical optimization of material and structure, Structural and Multidisciplinary Optimization, 24 (2002), 1-10. doi: 10.1007/s00158-002-0209-z. [15] F. Schury, M. Stingl and F. Wein, Efficient two-scale optimization of manufacturable graded structures, SIAM Journal on Scientific Computing, 34 (2012), B711-B733. doi: 10.1137/110850335. [16] A. M. Toader, The topological derivative of homogenized elastic coefficients of periodic microstructures, SIAM Journal on Control and Optimization, 49 (2011), 195-200. doi: 10.1137/100782772. [17] [18]

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##### References:
 [1] G. Allaire, Shape Optimization by the Homogenization Method, Springer, Applied Mathematical Sciences 146, 2002. doi: 10.1007/978-1-4684-9286-6. [2] G. Allaire, F. Jouve and A.-M. Toader, Structural optimization using sensitivity analysis and a level set method, Journal of Computational Physics, 194 (2004), 363-393. doi: 10.1016/j.jcp.2003.09.032. [3] C. Barbarosie, Shape optimization of periodic structures, Computational Mechanics, 30 (2003), 235-246. doi: 10.1007/s00466-002-0382-3. [4] C. Barbarosie and S. Lopes, A generalized notion of compliance, Comptes Rendus Mécanique, 339 (2011), 641-648. doi: 10.1016/j.crme.2011.07.002. [5] C. Barbarosie and A.-M. Toader, Shape and Topology Optimization for periodic problems, Part I, The shape and the topological derivative, Structural and Multidisciplinary Optimization, 40 (2009), 381-391. doi: 10.1007/s00158-009-0378-0. [6] C. Barbarosie and A.-M. Toader, Shape and Topology Optimization for periodic problems, Part II, Optimization algorithm and numerical examples, Structural and Multidisciplinary Optimization, 40 (2009), 393-408. doi: 10.1007/s00158-009-0377-1. [7] C. Barbarosie and A.-M. Toader, Optimization of bodies with locally periodic microstructre, Mechanics of advanced materials and structures, 19 (2012), 290-301. doi: 10.1080/15376494.2011.642939. [8] A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Studies in Mathematics and its Applications, 5, 1978. [9] M. Briane, Homogenization of a nonperiodic material, J. Math. Pures et Appl., 73 (1994), 47-66. [10] A. Cherkaev, Variational Methods for Structural Optimization, Springer Verlag, 2000. doi: 10.1007/978-1-4612-1188-4. [11] D. Cioranescu and J. S. J. Paulin, Homogenization in open sets with holes, Journal of Mathematical Analysis and Applications, 71 (1979), 590-607. doi: 10.1016/0022-247X(79)90211-7. [12] F. Murat and L. Tartar, H-convergence, in Topics in the Mathematical Modelling of Composite Materials, (eds. A. Cherkaev and R. Kohn), Progress in Nonlinear Differential Equations and Their Applications, 31, Birkhäuser, (1997), 21-43 [13] O. Pantz and K. Trabelsi, A post-treatment of the homogenization method for shape optimization, SIAM J. Control Optim., 47 (2008), 1380-1398. doi: 10.1137/070688900. [14] H. Rodrigues, J. M. Guedes and M. P. Bendsøe, Hierarchical optimization of material and structure, Structural and Multidisciplinary Optimization, 24 (2002), 1-10. doi: 10.1007/s00158-002-0209-z. [15] F. Schury, M. Stingl and F. Wein, Efficient two-scale optimization of manufacturable graded structures, SIAM Journal on Scientific Computing, 34 (2012), B711-B733. doi: 10.1137/110850335. [16] A. M. Toader, The topological derivative of homogenized elastic coefficients of periodic microstructures, SIAM Journal on Control and Optimization, 49 (2011), 195-200. doi: 10.1137/100782772. [17] [18]
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