Shape optimization in the Navier-Stokes flow with thermal effects. (English) Zbl 1308.76075
Summary: We consider the shape optimization problem of a body immersed in the incompressible fluid governed by Navier-Stokes equations coupling with a thermal model. Based on the continuous adjoint method, we formulate and analyze the shape optimization problem. Then we derive the structure of shape gradient for the cost functional by using the differentiability of a minimax formulation involving a Lagrange functional with the function space parametrization technique. Moreover, we present a gradient-type algorithm to the shape optimization problem. Finally, numerical examples demonstrate the feasibility and effectiveness of the proposed algorithm.
MSC:
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76D55 | Flow control and optimization for incompressible viscous fluids |
| 76R50 | Diffusion |
| 65K10 | Numerical optimization and variational techniques |
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 74P10 | Optimization of other properties in solid mechanics |
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
Keywords:
shape optimization; computational fluid dynamics; shape gradient; Navier-Stokes equations; minimax principle; convective heat transferReferences:
| [1] | O.Pironneau, Optimal shape design for elliptic systems, Springer, Berlin, 1984. · Zbl 0534.49001 |
| [2] | B.Mohammadi and O.Pironneau, Applied shape optimization for fluids, Clardendon Press, Oxford, 2001. · Zbl 0970.76003 |
| [3] | J.Simon, Domain variation for drag in Stokes flow, Proceedings of IFIP Conference in Shanghai, Lecture Notes in Control and Information Science, Springer, New York, 1990. |
| [4] | J.Bello, E.Fernndez‐Cara, and J.Simon, The variation of the drag with respect to the domain in Navier‐Stokes flow, optimization, optimal control, partial differential equations, Int Ser Numer Math107 (1992), 287-296. · Zbl 0763.76066 |
| [5] | J.Bello, E.Fernndez‐Cara, J.Lemoine, and J.Simon, The differentiability of the drag with respect to the variations of a lipschitz domain in a Navier‐Stokes flow, SIAM J Control Optim35 (1997), 626-640. · Zbl 0873.76019 |
| [6] | M. C.Delfour and J.‐P.Zolésio, Shapes and geometries: analysis, differential calculus, and optimization, SIAM, Philadelphia, 2002. |
| [7] | J.Sokolowski and J.‐P.Zolésio, Introduction to shape optimization: shape sensitivity analysis, Springer‐Verlag, Berlin, 1992. · Zbl 0761.73003 |
| [8] | W. J.Yan and Y. C.Ma, Shape reconstruction of an inverse Stokes problem, J Comput Appl Math216 (2008), 554-562. · Zbl 1138.76031 |
| [9] | W. J.Yan and Y. C.Ma, The application of domain derivative of the nonhomogeneous Navier‐Stokes equations in shape reconstruction, Comput Fluids38 (2009), 1101-1107. · Zbl 1242.76037 |
| [10] | D.Chenais, J.Monnier, and J. P.Vila, A shape optimal design problem with convective radiative thermal transfer, J Optim Theory Appl110 (2001), 75-117. · Zbl 1001.49041 |
| [11] | R.Adams and J.Fournier, Sobolev spaces, Academic Press, New York, 2003. · Zbl 1098.46001 |
| [12] | O.Pironneau, Optimal shape design by local boundary variations, Springer‐Verlag, Berlin, 1988. |
| [13] | J.Hadamard, Mémoire sur le problème d’analyse relatif à l’équilibre des plaques élastiques encastrées, Mémoire des savants étrangers, 33 (1908), 515-629. · JFM 39.1022.01 |
| [14] | J.Simon, Differentiation with respect to the domain in boundary value problems, Numer Funct Anal Optim2 (1980), 649-687. · Zbl 0471.35077 |
| [15] | G.Dogan, P.Morin, R. H.Nochetto, and M.Verani, Discrete gradient flows for shape optimization and applications, Comput Methods Appl Mech Eng196 (2007), 3898-3914. · Zbl 1173.49307 |
| [16] | R.Temam, Navier Stokes equations, theory and numerical analysis, AMS Chelsea, Rhode Island, 2001. · Zbl 0981.35001 |
| [17] | V.Girault and P. A.Raviart, Finite element methods for Navier‐Stokes Equations, Springer‐Verlag, Berlin, 1986. · Zbl 0585.65077 |
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