zbMATH — the first resource for mathematics

Topological optimization method for a geometric control problem in Stokes flow. (English) Zbl 1165.76011
Summary: We consider a geometric control problem in Stokes flow. We propose a simple and fast algorithm using topological optimization techniques. Our approach consists in studying the variation of a cost function with respect to the insertion of a small obstacle in the domain. Theoretical results are derived in two- and three-dimensional case for large class of cost functions. Some numerical experiments are presented in 2D and 3D, showing the efficiency of our approach.

76D55 Flow control and optimization for incompressible viscous fluids
76D07 Stokes and related (Oseen, etc.) flows
Full Text: DOI
[1] Allaire, G.; Jouve, F.; Toader, A.-M., Structural optimization using sensitivity analysis and a level-set method, J. comput. phys., 194, 1, 363-393, (2004) · Zbl 1136.74368
[2] Arnold, D.; Brezzi, F.; Fortin, M., A stable finite element for the Stokes equations, Calcolo, 21, 4, 337-344, (1984) · Zbl 0593.76039
[3] M. Bendsoe, Optimal topology design of continuum structure: an introduction, Technical report, Departement of mathematics, Technical University of Denmark, DK2800 Lyngby, Denmark, 1996
[4] Borrvall, T.; Petersson, J., Topological optimization of fluids in Stokes flow, Inter. J. numer. methods fluids, 41, 1, 77-107, (2003) · Zbl 1025.76007
[5] Buttazzo, G.; Dal Maso, G., Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions, Appl. math. optim., 23, 17-49, (1991) · Zbl 0762.49017
[6] Cabuk, H.; Modi, V., Optimum plane diffusers in laminar flow, J. fluid mech., 237, 373-393, (1992) · Zbl 0825.76176
[7] Cliff, E.M.; Heinkenschloss, M.; Shenoy, A., Airfoil design by an all-at-once method, Inter. J. compu. fluid mech., 11, 3-25, (1998) · Zbl 0940.76084
[8] Dautray, R.; Lions, J., Analyse mathémathique et calcul numérique pour LES sciences et LES techniques, Collection CEA, (1987), Masson
[9] Ghattas, O.; Bark, J.-H., Optimal control of two and three dimensional incompressible navier – stokes flows, J. comput. phys., 136, 231-244, (1997) · Zbl 0893.76067
[10] Glowinski, R.; Pironneau, O., On the numerical computation of the minimum drag profile in laminar flow, J. fluid mech., 72, 385-389, (1975) · Zbl 0323.76024
[11] Guest, J.K.; Prévost, J.H., Topology optimization of creeping fluid flows using a darcy – stokes finite element, Inter. J. numer. methods engrg., 66, 461-484, (2006) · Zbl 1110.76310
[12] Guillaume, Ph.; Sid Idris, K., Topological sensitivity and shape optimization for the Stokes equations, SIAM J. control optim., 43, 1, 1-31, (2004) · Zbl 1093.49029
[13] Gunzburger, M.D., Perspectives in flow control and optimization, Advances in design and control, (2003), SIAM Philadelphia, PA · Zbl 1088.93001
[14] Gunzburger, M.; Kim, H.; Manservisi, S., On a shape control problem for the stationary navier – stokes equations, M2AN math. model. numer. anal., 34, 6, 1233-1258, (2000) · Zbl 0981.76027
[15] Hassine, M.; Jan, S.; Masmoudi, M., From differential calculus to 0-1 topological optimization, SIAM J. control. optim., 45, 6, 1965-1987, (2007) · Zbl 1139.49039
[16] Hassine, M.; Masmoudi, M., The topological asymptotic expansion for the quasi-Stokes problem, Esaim cocv j., 10, 4, 478-504, (2004) · Zbl 1072.49027
[17] Kim, D.W.; Kim, M.U., Minimum drag shape in two-dimensional viscous flow, Inter. J. numer. methods fluids, 21, 93-111, (1995) · Zbl 0840.76079
[18] Mohammadi, B.; Pironneau, O., Applied shape optimization for fluids, Numerical mathematics and scientific computation, (2001), Oxford University Press New York · Zbl 0970.76003
[19] Pironneau, O., On optimum design in fluid mechanics, J. fluid mech., 64, 97-110, (1974) · Zbl 0281.76020
[20] Pironneau, O., Optimal shape design for elliptic systems, (1984), Springer Berlin · Zbl 0496.93029
[21] Sokolowski, J.; Zochowski, A., On the topological derivative in shape optimization, SIAM J. control optim., 37, 4, 1251-1272, (1999) · Zbl 0940.49026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.