Publicerad: 2004-12-28
ISBN:
ISSN: 1650-3686 (tryckt), 1650-3740 (online)
We consider the problem of optimal design of flow domains for Navier–Stokes flows in order to minimize a given performance functional. We attack the problem using topology optimization techniques; or control in coefficients; which are widely known in structural optimization of solid structures for their flexibility; generality; and yet ease of use and integration with existing FEM software. Topology optimization rapidly finds its way into other areas of optimal design; yet until recently it has not been applied to problems in fluid mechanics. The success of topology optimization methods for the minimal drag design of domains for Stokes fluids has lead to attempts to use the same optimization model for designing domains for incompressible Navier–Stokes flows. We show that the optimal control problem obtained as a result of such a straightforward generalization is ill-posed; at least if attacked by the direct method of calculus of variations. We illustrate the two key difficulties with simple numerical examples and propose changes in the optimization model that allow us to overcome these difficulties. Namely; to deal with impenetrable inner walls that may appear in the flow domain we slightly relax the incompressibility constraint as typically done in penalty methods for solving the incompressible Navier–Stokes equations. In addition; to prevent discontinuous changes in the flow due to very small impenetrable parts of the domain that may disappear; we consider so-called filtered designs; that has become a “classic” tool in the topology optimization toolbox. Technically; however; our use of filters differs significantly from their use in the structural optimization problems in solid mechanics; owing to the very unlike design parametrizations in the two models. We rigorously establish the well-posedness of the proposed model and then discuss related computational issues.
1Gunzburger; M. D.; Perspectives in flow control and optimization; Advances in Design and Control; Society for Industrial and Applied Mathematics (SIAM); Philadelphia; PA; 2003.
2Mohammadi; B. and Pironneau; O.; Applied shape optimization for fluids; Numerical Mathematics and Scientific Computation; Oxford University Press; New York; 2001.
3Bendsøe; M. P. and Sigmund; O.; Topology Optimization: Theory; Methods; and Applications; Springer-Verlag; Berlin; 2003.
4Feireisl; E.; “Shape optimization in viscous compressible fluids;” Appl. Math. Optim.; Vol. 47; No. 1; 2003; pp. 59–78.
5Ton; B. A.; “Optimal shape control problem for the Navier-Stokes equations;” SIAM J. Control Optim.; Vol. 41; No. 6; 2003; pp. 1733–1747.
6Gunzburger; M. D.; Kim; H.; and Manservisi; S.; “On a shape control problem for the stationary Navier-Stokes equations;” M2AN Math. Model. Numer. Anal.; Vol. 34; No. 6; 2000; pp. 1233–1258.
7Gunzburger; M. D. and Kim; H.; “Existence of an optimal solution of a shape control problem for the stationary Navier-Stokes equations;” SIAM J. Control Optim.; Vol. 36; No. 3; 1998; pp. 895–909.
8Borrvall; T. and Petersson; J.; “Topology optimization of fluids in Stokes flow;” Internat. J. Numer. Methods Fluids; Vol. 41; No. 1; 2003; pp. 77–107.
9Evgrafov; A.; “On the limits of porous materials in the topology optimization of Stokes flows;” Bl°a serien; Department of Mathematics; Chalmers University of Technology; Gothenburg; Sweden; 2003; Submitted for publication.
10Klarbring; A.; Petersson; J.; Torstenfelt; B.; and Karlsson; M.; “Topology optimization of flow networks;” Comput. Methods Appl. Mech. Engrg.; Vol. 192; No. 35-36; 2003; pp. 3909–3932.
11Darrigol; O.; “Between hydrodynamics and elasticity theory: the first five births of the Navier- Stokes equation;” Arch. Hist. Exact Sci.; Vol. 56; No. 2; 2002; pp. 95–150.
12Gersborg-Hansen; A.; Topology optimization of incompressible Newtonian flows at moderate Reinolds numbers; Master’s thesis; Department of Mechanical Engineering; Technical University of Den- mark; December 2003.
13Allaire; G.; “Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework; a volume distribution of holes;” Arch. Rational Mech. Anal.; Vol. 113; No. 3; 1990; pp. 209–259.
14Sigmund; O.; “On the design of compliant mechanisms using topology optimization;” Mech. Struct. Mach.; Vol. 25; No. 4; 1997; pp. 493–524.
15Sigmund; O. and Petersson; J.; “Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards; mesh-dependencies and local minima;” Struct. Multidisc. Optim.; Vol. 16; No. 1; 1998; pp. 68–75.
16Bourdin; B.; “Filters in topology optimization;” Internat. J. Numer. Methods Engrg.; Vol. 50; No. 9; 2001; pp. 2143–2158.
17Bruns; T. E. and Tortorelli; D. A.; “Topology optimization of non-linear elastic structures and compliant mechanisms;” Comput. Methods Appl. Mech. Engrg.; Vol. 190; No. 26–27; 2001; pp. 3443– 3459.
18Allaire; G.; “Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. II. Noncritical sizes of the holes for a volume distribution and a surface distribution of holes;” Arch. Rational Mech. Anal.; Vol. 113; No. 3; 1990; pp. 261–298.
19Borrvall; T. and Petersson; J.; “Topology optimization using regularized intermediate density control;” Comput. Methods Appl. Mech. Engrg.; Vol. 190; No. 37–38; 2001; pp. 4911–4928.
20Hornung; U.; editor; Homogenization and porous media; Vol. 6 of Interdisciplinary Applied Math- ematics; Springer-Verlag; New York; 1997.
21Soko lowski; J. and Zol´esio; J.-P.; Introduction to shape optimization; Vol. 16 of Springer Series in Computational Mathematics; Springer-Verlag; Berlin; 1992.
22Gunzburger; M. D.; Finite element methods for viscous incompressible flows; Computer Science and Scientific Computing; Academic Press Inc.; Boston; MA; 1989.
23Temam; R.; Navier-Stokes equations; AMS Chelsea Publishing; Providence; RI; 2001; Reprint of the 1984 edition.
24Heinrich; J. C. and Vionnet; C. A.; “The penalty method for the Navier-Stokes equations;” Arch. Comput. Methods Engrg.; Vol. 2; No. 2; 1995; pp. 51–65.
25Fuchs; M. and Seregin; G.; Variational methods for problems from plasticity theory and for gener- alized Newtonian fluids; Vol. 1749 of Lecture Notes in Mathematics; Springer-Verlag; Berlin; 2000.
26Carey; G. F. and Krishnan; R.; “Penalty finite element method for the Navier-Stokes equations;” Comput. Methods Appl. Mech. Engrg.; Vol. 42; No. 2; 1984; pp. 183–224.
27Lin; S. Y.; Chin; Y. S.; and Wu; T. M.; “A modified penalty method for Stokes equations and its applications to Navier-Stokes equations;” SIAM J. Sci. Comput.; Vol. 16; No. 1; 1995; pp. 1–19.
28Petersson; J. and Sigmund; O.; “Slope constrained topology optimization;” Internat. J. Numer. Methods Engrg.; Vol. 41; No. 8; 1998; pp. 1417–1434.
29Bellido; J. C.; “Existence of classical solutions for a one-dimensional optimal design problem in wave propagation;” 2003; Preprint;Mathematical Institute; University of Oxford; Oxford; UK. Submitted for publication.
