Abstract
In the present paper we deduce formulae for the shape and topological derivatives for elliptic problems in unbounded domains subject to periodicity conditions. Note that the known formulae of shape and topological derivatives for elliptic problems in bounded domains do not apply to the periodic framework. We consider a general notion of periodicity, allowing for an arbitrary parallelepiped as periodicity cell. Our calculations are useful for optimizing periodic composite materials by gradient type methods using the topological derivative jointly with the shape derivative for periodic problems. Important particular cases of functionals to minimize/maximize are presented. A numerical algorithm for optimizing periodic composites using the topological and shape derivatives is the subject of a second paper (Barbarosie and Toader, Struct Multidiscipl Optim, 2009).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Allaire G (1992) Homogenization and two-scale convergence. SIAM J Math Anal 23(6):1482–1518
Allaire G, Jouve F, Toader A-M (2004) Structural optimization using sensitivity analysis and a level set method. J Comput Phys 194:363–393
Barbarosie C (2002) Optimização de forma aplicada a materiais compósitos. In: Goicolea JM, Mota Soares C, Pastor M, Bugeda G (eds) Métodos Numéricos en Ingenieria V, SEMNI, Sociedad Española de Métodos Numéricos en Ingenieria, p 369
Barbarosie C (2003) Shape optimization of periodic structures. Comput Mech 30:235–246
Barbarosie C, Toader A-M (2009) Shape and topology optimization for periodic problems, part II, optimization algorithm and numerical examples. Struct Multidiscipl Optim. doi:10.1007/s00158-009-0377-1
Eschenauer H, Kobelev V, Schumacher A (1994) Bubble method for topology and shape optimization of structures. Struct Optim 8:42–51
Garreau S, Guillaume P, Masmoudi M (2001) The topological asymptotic for PDE systems: the elasticity case. SIAM J Control Optim 39:1756–1778
Murat F, Simon J (1976) Etudes de problèmes d’optimal design. Lecture notes in computer science, vol 41. Springer, Berlin, pp 54–62
Sokołowski J, Żochowski A (2001) Topological derivatives of shape functionals for elasticity systems. Mech Struct Mach 29:331–349
Sokołowski A, Zolezio JP (1992) Introduction to shape optimization: shape sensitivity analysis. Springer series in computational mathematics, vol 10. Springer, Berlin
Suquet PM (1982) Une méthode duale en homogénéisation: application aux milieux élastiques. J Méc Théor Appl 79–98 (Numéro spécial)
Toader A-M (2008) Optimization of periodic microstructures using shape and topology derivatives. Communication at WCCM8, ECCOMAS 2008, Venice, July 2008
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barbarosie, C., Toader, AM. Shape and topology optimization for periodic problems. Struct Multidisc Optim 40, 381–391 (2010). https://doi.org/10.1007/s00158-009-0378-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-009-0378-0