Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T13:16:39.549Z Has data issue: false hasContentIssue false
  • English
  • Français

A penalty method for topology optimization subject to a pointwise state constraint

Published online by Cambridge University Press:  18 June 2009

Samuel Amstutz
Samuel Amstutz*
Affiliation:
Laboratoire d'Analyse Non Linéaire et Géométrie, Faculté des Sciences, 33 rue Louis Pasteur, 84000 Avignon, France. samuel.amstutz@univ-avignon.fr
Get access

Abstract

This paper deals with topology optimization of domains subject to a pointwise constraint on the gradient of the state. To realize this constraint, a class of penalty functionals is introduced and the expression of the corresponding topological derivative is obtained for the Laplace equation in two space dimensions. An algorithm based on these concepts is proposed. It is illustrated by some numerical applications.

Keywords

Topology optimizationtopological derivativepenalty methodspointwise state constraints
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 3 , July 2010 , pp. 523 - 544
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.A. Adams, Sobolev spaces, Pure and Applied Mathematics 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975).
G. Allaire, Shape optimization by the homogenization method, Applied Mathematical Sciences 146. Springer-Verlag, New York (2002).
Allaire, G., Jouve, F. and Maillot, H., Topology optimization for minimum stress design with the homogenization method. Struct. Multidiscip. Optim. 28 (2004) 87–98. CrossRef
Allaire, G., Jouve, F. and Toader, A.-M., Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194 (2004) 363–393. CrossRef
Allaire, G., de Gournay, F., Jouve, F. and Toader, A.-M., Structural optimization using topological and shape sensitivity via a level set method. Control Cybern. 34 (2005) 59–80.
Amstutz, S., Sensitivity analysis with respect to a local perturbation of the material property. Asymptot. Anal. 49 (2006) 87–108.
Amstutz, S. and Andrä, H., A new algorithm for topology optimization using a level-set method. J. Comput. Phys. 216 (2006) 573–588. CrossRef
J. Appell and P.P. Zabrejko, Nonlinear superposition operators, Cambridge Tracts in Mathematics 95. Cambridge University Press, Cambridge (1990).
Bendsøe, M.P. and Kikuchi, N., Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Engrg. 71 (1988) 197–224. CrossRef
M.P. Bendsøe and O. Sigmund, Topology optimization, Theory, methods and applications. Springer-Verlag, Berlin (2003).
J.F. Bonnans, J.C. Gilbert, C. Lemaréchal and C.A. Sagastizábal, Numerical optimization, Theoretical and practical aspects. Universitext, Springer-Verlag, Berlin, Second Edition (2006).
Burger, M. and Stainko, R., Phase-field relaxation of topology optimization with local stress constraints. SIAM J. Control Optim. 45 (2006) 1447–1466 (electronic). CrossRef
Burger, M., Hackl, B. and Ring, W., Incorporating topological derivatives into level set methods. J. Comput. Phys. 194 (2004) 344–362. CrossRef
Duysinx, P. and Bendsøe, M.P., Topology optimization of continuum structures with local stress constraints. Internat. J. Numer. Methods Engrg. 43 (1998) 1453–1478. 3.0.CO;2-2>CrossRef
Eschenauer, H., Kobolev, V.V. and Schumacher, A., Bubble method for topology and shape optimization of structures. Struct. Optimization 8 (1994) 42–51. CrossRef
Garreau, S., Guillaume, P. and Masmoudi, M., The topological asymptotic for PDE systems: the elasticity case. SIAM J. Control Optim. 39 (2001) 1756–1778 (electronic). CrossRef
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition.
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman (Advanced Publishing Program), Boston, USA (1985).
A. Henrot and M. Pierre, Variation et optimisation de formes, Mathématiques et applications 48. Springer-Verlag, Heidelberg (2005).
M. Hintermüller and K. Kunisch, Stationary optimal control problems with pointwise state constraints (to appear).
Hintermüller, M. and Ring, W., A level set approach for the solution of a state-constrained optimal control problem. Numer. Math. 98 (2004) 135–166. CrossRef
Ito, K. and Kunisch, K., Semi-smooth Newton methods for state-constrained optimal control problems. Systems Control Lett. 50 (2003) 221–228. CrossRef
Meyer, C., Rösch, A. and Tröltzsch, F., Optimal control of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33 (2006) 209–228. CrossRef
F. Murat and J. Simon, Étude de problèmes d'optimal design, in Lecture Notes in Computer Sciences 41, Springer-Verlag, Berlin (1976) 54–62.
Nazarov, S.A. and Sokołowski, J., Asymptotic analysis of shape functionals. J. Math. Pures Appl. 82 (2003) 125–196. CrossRef
Norato, J.A., Bendsøe, M.P., Haber, R.B. and Tortorelli, D.A., A topological derivative method for topology optimization. Struct. Multidiscip. Optim. 33 (2007) 375–386. CrossRef
Petzoldt, M., Regularity results for Laplace interface problems in two dimensions. Z. Anal. Anwendungen 20 (2001) 431–455. CrossRef
Rückmann, J.-J. and Gómez, J.A., On generalized semi-infinite programming. Top 14 (2006) 1–59. CrossRef
Rückmann, J.J. and Shapiro, A., First-order optimality conditions in generalized semi-infinite programming. J. Optim. Theory Appl. 101 (1999) 677–691. CrossRef
Savaré, G., Regularity results for elliptic equations in Lipschitz domains. J. Funct. Anal. 152 (1998) 176–201. CrossRef
Simon, J., Differentiation with respect to the domain in boundary value problems. Numer. Funct. Anal. Optim. 2 (1980) 649–687. CrossRef
Sokołowski, J. and A. Żochowski, On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251–1272 (electronic). CrossRef
J. Sokołowski and J.-P. Zolésio, Introduction to shape optimization – Shape sensitivity analysis, Springer Series in Computational Mathematics 16. Springer-Verlag, Berlin (1992).
Still, G., Generalized semi-infinite programming: numerical aspects. Optimization 49 (2001) 223–242. CrossRef
Wang, M.Y., Wang, X. and Guo, D., A level set method for structural topology optimization. Comput. Methods Appl. Mech. Engrg. 192 (2003) 227–246. CrossRef