Optimal design in small amplitude homogenization

Grégoire Allaire; Sergio Gutiérrez

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 3, page 543-574
  • ISSN: 0764-583X

Abstract

top
This paper is concerned with optimal design problems with a special assumption on the coefficients of the state equation. Namely we assume that the variations of these coefficients have a small amplitude. Then, making an asymptotic expansion up to second order with respect to the aspect ratio of the coefficients allows us to greatly simplify the optimal design problem. By using the notion of H-measures we are able to prove general existence theorems for small amplitude optimal design and to provide simple and efficient numerical algorithms for their computation. A key feature of this type of problems is that the optimal microstructures are always simple laminates.

How to cite

top

Allaire, Grégoire, and Gutiérrez, Sergio. "Optimal design in small amplitude homogenization." ESAIM: Mathematical Modelling and Numerical Analysis 41.3 (2007): 543-574. <http://eudml.org/doc/250029>.

@article{Allaire2007,
abstract = { This paper is concerned with optimal design problems with a special assumption on the coefficients of the state equation. Namely we assume that the variations of these coefficients have a small amplitude. Then, making an asymptotic expansion up to second order with respect to the aspect ratio of the coefficients allows us to greatly simplify the optimal design problem. By using the notion of H-measures we are able to prove general existence theorems for small amplitude optimal design and to provide simple and efficient numerical algorithms for their computation. A key feature of this type of problems is that the optimal microstructures are always simple laminates. },
author = {Allaire, Grégoire, Gutiérrez, Sergio},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Optimal design; H-measures; homogenization.; small amplitude homogenization; -measures; shape optimization},
language = {eng},
month = {8},
number = {3},
pages = {543-574},
publisher = {EDP Sciences},
title = {Optimal design in small amplitude homogenization},
url = {http://eudml.org/doc/250029},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Allaire, Grégoire
AU - Gutiérrez, Sergio
TI - Optimal design in small amplitude homogenization
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/8//
PB - EDP Sciences
VL - 41
IS - 3
SP - 543
EP - 574
AB - This paper is concerned with optimal design problems with a special assumption on the coefficients of the state equation. Namely we assume that the variations of these coefficients have a small amplitude. Then, making an asymptotic expansion up to second order with respect to the aspect ratio of the coefficients allows us to greatly simplify the optimal design problem. By using the notion of H-measures we are able to prove general existence theorems for small amplitude optimal design and to provide simple and efficient numerical algorithms for their computation. A key feature of this type of problems is that the optimal microstructures are always simple laminates.
LA - eng
KW - Optimal design; H-measures; homogenization.; small amplitude homogenization; -measures; shape optimization
UR - http://eudml.org/doc/250029
ER -

References

top
  1. G. Allaire, Shape Optimization by the Homogenization Method. Springer-Verlag (2002).  
  2. G. Allaire and S. Gutiérrez, Optimal design in small amplitude homogenization (extended version). Preprint available at (2005).  URIhttp://www.cmap.polytechnique.fr/preprint/repository/576.pdf
  3. G. Allaire and F. Jouve, Optimal design of micro-mechanisms by the homogenization method. Eur. J. Finite Elements11 (2002) 405–416.  
  4. G. Allaire, F. Jouve and H. Maillot, Topology optimization for minimum stress design with the homogenization method. Struct. Multidiscip. Optim.28 (2004) 87–98.  
  5. J.C. Bellido and P. Pedregal, Explicit quasiconvexification for some cost functionals depending on derivatives of the state in optimal designing. Discr. Contin. Dyn. Syst.8 (2002) 967–982.  
  6. M.P. Bendsøe and O. Sigmund, Topology Optimization. Theory, Methods, and Applications. Springer-Verlag, New York (2003).  
  7. A. Cherkaev, Variational Methods for Structural Optimization. Springer Verlag, New York (2000).  
  8. A. Donoso and P. Pedregal, Optimal design of 2D conducting graded materials by minimizing quadratic functionals in the field. Struct. Multidiscip. Optim.30 (2005) 360–367.  
  9. P. Duysinx and M.P. Bendsøe, Topology optimization of continuum structures with local stress constraints. Int. J. Num. Meth. Engng.43 (1998) 1453–1478.  
  10. P. Gérard, Microlocal defect measures. Comm. Partial Diff. Equations16 (1991) 1761–1794.  
  11. Y. Grabovsky, Optimal design problems for two-phase conducting composites with weakly discontinuous objective functionals. Adv. Appl. Math.27 (2001) 683–704.  
  12. F. Hecht, O. Pironneau and K. Ohtsuka, FreeFem++ Manual. Downloadable at  URIhttp://www.freefem.org
  13. L. Hörmander, The analysis of linear partial differential operators III. Springer, Berlin (1985).  
  14. R.V. Kohn, Relaxation of a double-well energy. Cont. Mech. Thermodyn.3 (1991) 193–236.  
  15. R. Lipton, Relaxation through homogenization for optimal design problems with gradient constraints. J. Optim. Theory Appl.114 (2002) 27–53.  
  16. R. Lipton, Stress constrained G closure and relaxation of structural design problems. Quart. Appl. Math.62 (2004) 295–321.  
  17. R. Lipton and A. Velo, Optimal design of gradient fields with applications to electrostatics. Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. XIV, Stud. Math. Appl.31 (2002) 509–532.  
  18. G. Milton, The theory of composites. Cambridge University Press (2001).  
  19. F. Murat and L. Tartar, Calcul des Variations et Homogénéisation, Les Méthodes de l'Homogénéisation Théorie et Applications en Physique, Coll. Dir. Études et Recherches EDF, 57, Eyrolles, Paris (1985) 319–369. English translation in Topics in the mathematical modelling of composite materials, A. Cherkaev and R. Kohn Eds., Progress in Nonlinear Differential Equations and their Applications31, Birkhäuser, Boston (1997).  
  20. U. Raitums, The extension of extremal problems connected with a linear elliptic equation. Soviet Math.19 (1978) 1342–1345.  
  21. L. Tartar, H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations. Proc. Royal Soc. Edinburgh115A (1990) 93–230.  
  22. L. Tartar, Remarks on optimal design problems. Calculus of variations, homogenization and continuum mechanics (Marseille, 1993), World Sci. Publishing, River Edge, NJ, Ser. Adv. Math. Appl. Sci.18 (1994) 279–296.  
  23. L. Tartar, An introduction to the homogenization method in optimal design, in Optimal shape design (Tróia, 1998), A. Cellina and A. Ornelas Eds., Springer, Berlin, Lect. Notes Math.1740 (2000) 47–156.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.