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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

Cristian Barbarosie 1, and  Anca-Maria Toader 1,

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.
Keywords: functionally graded materials, Shape optimization, periodicity optimization, topology optimization, alternate directions algorithm, cellular problem, locally periodic homogenization, parallel computation..
Mathematics Subject Classification: Primary: 74N15, 49N45; Secondary: 49Q10, 74Q0.
Citation: Cristian Barbarosie, Anca-Maria Toader. Optimization of bodies with locally periodic microstructure by varying the periodicity pattern. Networks & Heterogeneous Media, 2014, 9 (3) : 433-451. doi: 10.3934/nhm.2014.9.433
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show all references

References:
[1]

Springer, Applied Mathematical Sciences 146, 2002. doi: 10.1007/978-1-4684-9286-6.  Google Scholar

[2]

Journal of Computational Physics, 194 (2004), 363-393. doi: 10.1016/j.jcp.2003.09.032.  Google Scholar

[3]

Computational Mechanics, 30 (2003), 235-246. doi: 10.1007/s00466-002-0382-3.  Google Scholar

[4]

Comptes Rendus Mécanique, 339 (2011), 641-648. doi: 10.1016/j.crme.2011.07.002.  Google Scholar

[5]

Structural and Multidisciplinary Optimization, 40 (2009), 381-391. doi: 10.1007/s00158-009-0378-0.  Google Scholar

[6]

Structural and Multidisciplinary Optimization, 40 (2009), 393-408. doi: 10.1007/s00158-009-0377-1.  Google Scholar

[7]

Mechanics of advanced materials and structures, 19 (2012), 290-301. doi: 10.1080/15376494.2011.642939.  Google Scholar

[8]

North-Holland, Studies in Mathematics and its Applications, 5, 1978.  Google Scholar

[9]

J. Math. Pures et Appl., 73 (1994), 47-66.  Google Scholar

[10]

Springer Verlag, 2000. doi: 10.1007/978-1-4612-1188-4.  Google Scholar

[11]

Journal of Mathematical Analysis and Applications, 71 (1979), 590-607. doi: 10.1016/0022-247X(79)90211-7.  Google Scholar

[12]

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  Google Scholar

[13]

SIAM J. Control Optim., 47 (2008), 1380-1398. doi: 10.1137/070688900.  Google Scholar

[14]

Structural and Multidisciplinary Optimization, 24 (2002), 1-10. doi: 10.1007/s00158-002-0209-z.  Google Scholar

[15]

SIAM Journal on Scientific Computing, 34 (2012), B711-B733. doi: 10.1137/110850335.  Google Scholar

[16]

SIAM Journal on Control and Optimization, 49 (2011), 195-200. doi: 10.1137/100782772.  Google Scholar

[17]

, http://webpages.fc.ul.pt/~cabarbarosie/en/anim-2011/, (web page), ().   Google Scholar

[18]

, http://webpages.fc.ul.pt/~cabarbarosie/en/examples-2013.html, (web page), ().   Google Scholar

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