Optimization of structural topology in the high-porosity regime

Journal of the Mechanics and Physics of Solids

Volumn 56, Issue 3, 2008, Pages 1043-1064

  Bourdin, Blaise ^a   Kohn, Robert V ^b  

^a LOUISIANA STATE UNIVERSITY   (United States)

^b NEW YORK UNIVERSITY   (United States)

Author keywords

Effective properties; Extremal microstructures; High porosity limit; Homogenization; Optimal design

Indexed keywords

HOMOGENIZATION METHOD; MICROSTRUCTURE; NUMERICAL METHODS; OPTIMIZATION; QUADRATIC PROGRAMMING; TOPOLOGY;

EFFECTIVE PROPERTIES; EXTREMAL MICROSTRUCTURES;

POROSITY;

EID: 39049146499     PISSN: 00225096     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jmps.2007.06.002     Document Type: Article

References (31)

1. Allaire G. Shape Optimization by the Homogenization Method (2002), Springer, New York
2. Allaire G., and Aubry S. On optimal microstructures for a plane shape optimization problem. Struct. Opt. 17 2 (1999) 86-94
3. Allaire G., and Francfort G. Existence of minimizers for non-quasiconvex functionals arising in optimal design. Ann. Inst. H. Poincaré Anal. Nonlinéaire 15 3 (1998) 301-339
4. Allaire G., and Kohn R. Optimal bounds on the effective behavior of a mixture of two well-ordered elastic materials. Q. Appl. Math. 51 4 (1993) 643-674
5. Allaire G., and Kohn R. Optimal design for minimum weight and compliance using extremal microstructures. Eur J. Mech. A/Solids 12 (1993) 839-878
6. Avellaneda M. Optimal bounds and microgeometries for elastic two-phase composites. SIAM J. Appl. Math. 47 6 (1987) 1216-1228
7. Ben-Tal A., and Nemirovski A. Robust truss topology design via semidefinite programming. SIAM J. Optim. 7 4 (1997) 991-1016
8. Ben-Tal A., Kocvara M., Nemirovski A., and Zowe J. Free material design via semidefinite programming: the multiload case with contact conditions. SIAM J. Opt. 9 4 (1999) 813-832
9. Bendsøe M. Optimization of Structural Topology, Shape, and Material (1995), Springer, Berlin, Heidelberg
10. Bendsøe M., and Haber R. The Michell layout problem as a low volume fraction limit of the perforated plate optimization problem: an asymptotic study. Struct. Optim. 6 (1993) 263-267
11. Bendsøe M., Guedes J., Haber R., Pedersen P., and Taylor J. An analytical model to predict optimal material properties in the context of optimal structural design. J. Appl. Mech. 61 4 (1994) 930-937
12. Bouchitté G. Optimization of light structures: the vanishing mass conjecture in homogenization. GAKUTO Internat. Ser. Math. Sci. Appl. 18 (2003) 131-145
13. Bouchitté G., and Buttazzo G. Characterization of optimal shapes and masses through the Monge-Kantorovitch equation. J. Eur. Math. Soc. 3 (2001) 139-168
14. Bouchitté G., Buttazzo G., and Fragalà I. Bounds for the effective coefficients and homogenized low-dimensional structures. J. Math. Pures Appl. (9) 81 5 (2002) 453-469
15. Cherkaev A. Variational Methods for Structural Optimization (2000), Springer, New York
16. Cherkaev, A., Kohn, R. (Eds.), 1997. Topics in the Mathematical Modelling of Composite Materials. Birkhäuser Boston, Boston, MA.
17. Christensen R. Mechanics of cellular and other low-density materials. Int. J. Solids Struct. 37 (2000) 93-104
18. Christensen R.M. Mechanics of low-density materials. J. Mech. Phys. Solids 34 6 (1986) 563-578
19. Cioranescu D., and Saint Jean Paulin J. Homogenization of Reticulated Structures (1999), Springer, New York
20. Francfort G., and Murat F. Homogenization and optimal bounds in linear elasticity. Arch. Ration. Mech. Anal. 94 4 (1986) 307-334
21. 2 and applications. Arch. Ration. Mech. Anal. 131 4 (1995) 305-333
22. Gibson, L., Ashby, M., 1997. Cellular Solids, Structure and Properties, second ed. Cambridge Solid State Science Series. Cambridge University Press, Cambridge, UK.
23. Grabovsky Y., and Kohn R. Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. II. The Vigdergauz construction. J. Mech. Phys. Solids 43 6 (1995) 949-972
24. Hyun S., and Torquato S. Optimal and manufacturable two-dimensional, kagoma-like cellular solids. J. Mater. Res. 17 (2002) 137-144
25. Michell A. The limits of economy of material in frame-structures. Philos. Mag. 8 (1904) 589-597
26. Sigmund O. Materials with prescribed constitutive parameters: an inverse homogenization problem. Int. J. Solids Struct. 31 17 (1994) 2313-2329
27. Sigmund O. On the design of compliant mechanisms using topology optimization. Mech. Struct. Mach. 25 4 (1997) 495-526
28. Tartar, L., 1985. Estimations fines des coefficients homogénéisés. In: Ennio De Giorgi's Colloquium (Paris, 1983). Pitman, Boston, MA, pp. 168-187.
29. Vandenberghe L., and Boyd S. Semidefinite programming. SIAM Rev. 38 1 (1996) 49-95
30. Vigdergauz S. Two-dimensional grained composites of extremal rigidity. J. Appl. Mech. 66 2 (1994) 390-394
31. Zhikov V.V. Averaging of problems in the theory of elasticity on singular structures. Izv. Math. 66 2 (2002) 299-365