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

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 16, Issue: 3, page 523-544
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topAmstutz, Samuel. "A penalty method for topology optimization subject to a pointwise state constraint." ESAIM: Control, Optimisation and Calculus of Variations 16.3 (2010): 523-544. <http://eudml.org/doc/250726>.

@article{Amstutz2010,

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.
},

author = {Amstutz, Samuel},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Topology optimization; topological derivative; penalty methods; pointwise state constraints; topology optimization},

language = {eng},

month = {7},

number = {3},

pages = {523-544},

publisher = {EDP Sciences},

title = {A penalty method for topology optimization subject to a pointwise state constraint},

url = {http://eudml.org/doc/250726},

volume = {16},

year = {2010},

}

TY - JOUR

AU - Amstutz, Samuel

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/7//

PB - EDP Sciences

VL - 16

IS - 3

SP - 523

EP - 544

AB -
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.

LA - eng

KW - Topology optimization; topological derivative; penalty methods; pointwise state constraints; topology optimization

UR - http://eudml.org/doc/250726

ER -

## References

top- R.A. Adams, Sobolev spaces, Pure and Applied Mathematics65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975).
- G. Allaire, Shape optimization by the homogenization method, Applied Mathematical Sciences146. Springer-Verlag, New York (2002). Zbl0990.35001
- G. Allaire, F. Jouve and H. Maillot, Topology optimization for minimum stress design with the homogenization method. Struct. Multidiscip. Optim.28 (2004) 87–98. Zbl1243.74148
- G. Allaire, F. Jouve and A.-M. Toader, Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys.194 (2004) 363–393. Zbl1136.74368
- G. Allaire, F. de Gournay, F. Jouve and A.-M. Toader, Structural optimization using topological and shape sensitivity via a level set method. Control Cybern.34 (2005) 59–80. Zbl1167.49324
- S. Amstutz, Sensitivity analysis with respect to a local perturbation of the material property. Asymptot. Anal.49 (2006) 87–108. Zbl1187.49036
- S. Amstutz and H. Andrä, A new algorithm for topology optimization using a level-set method. J. Comput. Phys.216 (2006) 573–588. Zbl1097.65070
- J. Appell and P.P. Zabrejko, Nonlinear superposition operators, Cambridge Tracts in Mathematics95. Cambridge University Press, Cambridge (1990).
- M.P. Bendsøe and N. Kikuchi, Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Engrg.71 (1988) 197–224. Zbl0671.73065
- 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).
- M. Burger and R. Stainko, Phase-field relaxation of topology optimization with local stress constraints. SIAM J. Control Optim.45 (2006) 1447–1466 (electronic). Zbl1116.74053
- M. Burger, B. Hackl and W. Ring, Incorporating topological derivatives into level set methods. J. Comput. Phys.194 (2004) 344–362. Zbl1044.65053
- P. Duysinx and M.P. Bendsøe, Topology optimization of continuum structures with local stress constraints. Internat. J. Numer. Methods Engrg.43 (1998) 1453–1478. Zbl0924.73158
- H. Eschenauer, V.V. Kobolev and A. Schumacher, Bubble method for topology and shape optimization of structures. Struct. Optimization8 (1994) 42–51.
- S. Garreau, P. Guillaume and M. Masmoudi, The topological asymptotic for PDE systems: the elasticity case. SIAM J. Control Optim.39 (2001) 1756–1778 (electronic). Zbl0990.49028
- 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. Zbl1042.35002
- P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics24. Pitman (Advanced Publishing Program), Boston, USA (1985). Zbl0695.35060
- A. Henrot and M. Pierre, Variation et optimisation de formes, Mathématiques et applications48. Springer-Verlag, Heidelberg (2005). Zbl1098.49001
- M. Hintermüller and K. Kunisch, Stationary optimal control problems with pointwise state constraints (to appear).
- M. Hintermüller and W. Ring, A level set approach for the solution of a state-constrained optimal control problem. Numer. Math.98 (2004) 135–166. Zbl1059.65057
- K. Ito and K. Kunisch, Semi-smooth Newton methods for state-constrained optimal control problems. Systems Control Lett.50 (2003) 221–228. Zbl1157.49311
- C. Meyer, A. Rösch and F. Tröltzsch, Optimal control of PDEs with regularized pointwise state constraints. Comput. Optim. Appl.33 (2006) 209–228.
- F. Murat and J. Simon, Étude de problèmes d'optimal design, in Lecture Notes in Computer Sciences41, Springer-Verlag, Berlin (1976) 54–62.
- S.A. Nazarov and J. Sokołowski, Asymptotic analysis of shape functionals. J. Math. Pures Appl.82 (2003) 125–196. Zbl1031.35020
- J.A. Norato, M.P. Bendsøe, R.B. Haber and D.A. Tortorelli, A topological derivative method for topology optimization. Struct. Multidiscip. Optim.33 (2007) 375–386. Zbl1245.74074
- M. Petzoldt, Regularity results for Laplace interface problems in two dimensions. Z. Anal. Anwendungen20 (2001) 431–455. Zbl1165.35333
- J.-J. Rückmann and J.A. Gómez, On generalized semi-infinite programming. Top14 (2006) 1–59.
- J.J. Rückmann and A. Shapiro, First-order optimality conditions in generalized semi-infinite programming. J. Optim. Theory Appl.101 (1999) 677–691. Zbl0956.90055
- G. Savaré, Regularity results for elliptic equations in Lipschitz domains. J. Funct. Anal.152 (1998) 176–201.
- J. Simon, Differentiation with respect to the domain in boundary value problems. Numer. Funct. Anal. Optim.2 (1980) 649–687. Zbl0471.35077
- J. Sokołowski and A. Żochowski, On the topological derivative in shape optimization. SIAM J. Control Optim.37 (1999) 1251–1272 (electronic).
- J. Sokołowski and J.-P. Zolésio, Introduction to shape optimization – Shape sensitivity analysis, Springer Series in Computational Mathematics16. Springer-Verlag, Berlin (1992).
- G. Still, Generalized semi-infinite programming: numerical aspects. Optimization49 (2001) 223–242. Zbl1039.90083
- M.Y. Wang, X. Wang and D. Guo, A level set method for structural topology optimization. Comput. Methods Appl. Mech. Engrg.192 (2003) 227–246. Zbl1083.74573

## NotesEmbed ?

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