Optimal shape design in three-dimensional Brinkman flow using asymptotic analysis techniques
Author:
Houcine Meftahi
Journal:
Quart. Appl. Math. 75 (2017), 525-537
MSC (2010):
Primary 65M32, 76B75, 49Q10, 74S30
DOI:
https://doi.org/10.1090/qam/1464
Published electronically:
March 16, 2017
MathSciNet review:
3636167
Full-text PDF
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Additional Information
Abstract: The aim of this paper is to reconstruct an obstacle $\omega$ immersed in a fluid governed by the Brinkman equation in a three-dimensional bounded domain $\Omega$ from internal data. We reformulate the inverse problem in an optimization one by using a least square functional. We prove the existence of an optimal solution for the optimization problem. We perform the asymptotic expansion of the cost function using a straightforward way based on a penalization technique. An important advantage of this method is that it avoids the truncation method used in the literature. Finally, we make some numerical results, exploring the efficiency of the method.
References
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- Giuseppe Buttazzo and Gianni Dal Maso, Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions, Appl. Math. Optim. 23 (1991), no. 1, 17–49. MR 1076053, DOI https://doi.org/10.1007/BF01442391
- Jean Céa, Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût, RAIRO-Modélisation mathématique et analyse numérique 20 (1986), no. 3, 371–402.
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- Anders Logg, Kent-Andre Mardal, and Garth N. Wells (eds.), Automated solution of differential equations by the finite element method, Lecture Notes in Computational Science and Engineering, vol. 84, Springer, Heidelberg, 2012. The FEniCS book. MR 3075806
- Mohamed Masmoudi, The topological asypmtotic expansion, in Computational Methods for Control Applications, GAKUTO Internt. Ser. Math. Sci. Appl, H. Kawarada and J. Periaux, eds., Gakhtosho, Tokyo 16 (2001), 53–76.
- L. F. N. Sá, R. C. R. Amigo, A. A. Novotny, and E. C. N. Silva, Topological derivatives applied to fluid flow channel design optimization problems, Struct. Multidiscip. Optim. 54 (2016), no. 2, 249–264. MR 3518568, DOI https://doi.org/10.1007/s00158-016-1399-0
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References
- M. Abdelwahed, M. Hassine, and M. Masmoudi, Optimal shape design for fluid flow using topological perturbation technique, J. Math. Anal. Appl. 356 (2009), no. 2, 548–563. MR 2524289, DOI https://doi.org/10.1016/j.jmaa.2009.02.045
- Martin S Alnaes, Anders Logg, Kristian B Olgaard, Marie E Rognes, and Garth N Wells, Unified form language: A domain-specific language for weak formulations of partial differential equations, ACM Transactions on Mathematical Software (TOMS) 40 (2014), no. 2, 9.
- Martin Sandve Alnæs, Anders Logg, Kent-Andre Mardal, Ola Skavhaug, and Hans Petter Langtangen, Unified framework for finite element assembly, International Journal of Computational Science and Engineering 4 (2009), no. 4, 231–244.
- Thomas Borrvall and Joakim Petersson, Topology optimization of fluids in Stokes flow, Internat. J. Numer. Methods Fluids 41 (2003), no. 1, 77–107. MR 1949585, DOI https://doi.org/10.1002/fld.426
- Giuseppe Buttazzo and Gianni Dal Maso, Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions, Appl. Math. Optim. 23 (1991), no. 1, 17–49. MR 1076053, DOI https://doi.org/10.1007/BF01442391
- Jean Céa, Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût, RAIRO-Modélisation mathématique et analyse numérique 20 (1986), no. 3, 371–402.
- Stéphane Garreau, Philippe Guillaume, and Mohamed Masmoudi, The topological asymptotic for PDE systems: the elasticity case, SIAM J. Control Optim. 39 (2001), no. 6, 1756–1778. MR 1825864, DOI https://doi.org/10.1137/S0363012900369538
- Ph. Guillaume and K. Sid Idris, Topological sensitivity and shape optimization for the Stokes equations, SIAM J. Control Optim. 43 (2004), no. 1, 1–31. MR 2081970, DOI https://doi.org/10.1137/S0363012902411210
- Maatoug Hassine and Mohamed Masmoudi, The topological asymptotic expansion for the quasi-Stokes problem, ESAIM Control Optim. Calc. Var. 10 (2004), no. 4, 478–504. MR 2111076, DOI https://doi.org/10.1051/cocv%3A2004016
- Anders Logg, Kent-Andre Mardal, and Garth N. Wells (eds.), Automated solution of differential equations by the finite element method, Lecture Notes in Computational Science and Engineering, vol. 84, Springer, Heidelberg, 2012. The FEniCS book. MR 3075806
- Mohamed Masmoudi, The topological asypmtotic expansion, in Computational Methods for Control Applications, GAKUTO Internt. Ser. Math. Sci. Appl, H. Kawarada and J. Periaux, eds., Gakhtosho, Tokyo 16 (2001), 53–76.
- L. F. N. Sá, R. C. R. Amigo, A. A. Novotny, and E. C. N. Silva, Topological derivatives applied to fluid flow channel design optimization problems, Struct. Multidiscip. Optim. 54 (2016), no. 2, 249–264. MR 3518568, DOI https://doi.org/10.1007/s00158-016-1399-0
- Axel Schumacher, Topologieoptimierung von bauteilstrukturen unter verwendung von lochpositionierungskriterien, Ph.D. thesis, Forschungszentrum für Multidisziplinäre Analysen und Angewandte Strukturoptimierung. Institut für Mechanik und Regelungstechnik, 1996.
- J. Sokołowski and A. Żochowski, On the topological derivative in shape optimization, SIAM J. Control Optim. 37 (1999), no. 4, 1251–1272. MR 1691940, DOI https://doi.org/10.1137/S0363012997323230
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Additional Information
Houcine Meftahi
Affiliation:
Technical University of Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany
MR Author ID:
873346
Email:
meftahi@math.tu-berlin.de
Received by editor(s):
December 10, 2016
Received by editor(s) in revised form:
February 3, 2017
Published electronically:
March 16, 2017
Article copyright:
© Copyright 2017
Brown University
