Shape sensitivity analysis of time-dependent flows of incompressible non-Newtonian fluids

• Autorzy: Sokołowski J. , Stebel J.
• Języki publikacji: EN

Abstrakty

EN

We study the shape differentiability of a cost function for the flow of an incompressible viscous fluid of power-law type. The fluid is confined to a bounded planar domain surrounding an obstacle. For smooth perturbations of the shape of the obstacle we express the shape gradient of the cost function which can be subsequently used to improve the initial design.

Słowa kluczowe

EN

shape optimization; shape gradient; incompressible viscous fluid; Navier-Stokes equations

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2011
• Tom: Vol. 40, no 4
• Strony: 1077--1097
• Opis fizyczny: Bibliogr. 29 poz.

Twórcy

autor

Sokołowski J.

autor

Stebel J.

• 1Institut Élie Cartan, UMR 7502 Nancy-Université-CNRS-INRIA Laboratoire de Mathématiques, Université Henri Poincaré Nancy 1, B.P. 239, 54506 Vandoeuvre Lcs Nancy Cedex, France,

Bibliografia

• Abraham,F., Behr. M and Heinkenschloss,M. (2005) Shape optimization in steady blood flow: A numerical study of non-newtonian effects. Computer Methods in Biomechanics and Biomedical Engineering, 8(2), 127-137.
• Beirão da Veiga,H., Kaplický,P. and Růžička,M. (2011) Boundary regularity of shear thickening flows. Journal of Mathematical Fluid Mechanics, 13(3), 387-404.
• Bothe,D. and Prüss, J. (2007) Lp-theory for a class of non-Newtonian fluids. SIAM J. Math. Anal., 39(2), 379-421.
• Ciarlet,P. (1994) Mathematical Elasticity: Three-dimensional Elasticity. Studies in mathematics and its applications. North-Holland.
• Bulíček, M., Haslinger, J., Málek, J. and Stebel, J. (2009) Shape optimization for Navier-Stokes equations with algebraic turbulence model: existence analysis. Applied Mathematics and Optimization, 60(2), 185-212.
• Consiglieri, L., Nečasová, Š. and Sokolowski, J. (2010) New approach to the incompressible Maxwell-Boussinesq approximation: existence, uniquness and shape sensitivity. J. Differential Equations, 249(12), 3052-3080.
• Diening, L., Råužička, M. and Wolf, J. (2010) Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Annali della Scuola Normale Superiore di Pisa. Classe di scienze, 9(1), 1-46.
• Freshe, J., Málek, J. and Steinhauer, M. (2000) On existence results for fluids with shear dependent viscosity-unsteady flows. Partial Differential Equations, Theory and Numerical Solution, 406, 121-129.
• Freshe, J., Málek, J. and Steinhauer,M. (2003) On analysis of steady floks of fluids with shear-dependent viscosity based on the Lipschitz truncation method. SIAM Journal on Mathematical Analysis, 34-1063.
• Haslinger, J. and Stebel, J. (2011) Shape optimization for Navier-Stokes equations with algebraic turbulence model: Numerical analysis and computation. Applied Mathematics and Optimization, 63(2), 277-308.
• Kaplický, P. (2005) Regularity of flows of a non-Newtonian fluid subject to Dirichlet boundary conditions. Journal for Analysis and its Applications, 24(3), 467-486.
• Ladyzhenskaya, O.A. (1967) New equations for the description of the motions of viscous incompressible fluids, and global solvability for their Bondary value problems. Trudy Mat. Inst. Steklov., 102, 85-104.
• Ladyzhenskaya, O.A. (1969) The Mathematical Theory of Viscous Incompressible Flow. Second English edition, revised and enlarged. Mathematics and its Applications, Vol. 2. Gordon and Breach Science Publishers, New York.
• Lions, J.L. (1969) Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod.
• Málek, J. and Rajagopal, K.R. (2005) Mathematical issues concerning the Navier-Stokes equations and some of its generalizations. In: C. Dafermos and E. Feireisl, eds., Evolutionary Equations. Handbook of Differential Equations., Elsevier/North-Holland, Amsterdam, 371-459.
• Málek, J., Nečas, J. and Růžička, M. (2001) On weak solutions to a class of non-Newtonian incompressible fluids in bounded Tyree-dimensional domains: the case p≥2. Advances in Differential Equations, 6(3), 257-3≥02.
• Moubachir, M. and Zolésio, J.P. (2006) Moving Shape Analysis and Control. Pure and Applied Mathematics, 277. Chapman & Hall/CRC, Boca Raton, FL, 2006.
• Plotnikov, P.I. and Sokolowski, J. (2010) Shape derivative of drag functional. SIAM J. Control Optim., 48(7), 4680-4706.
• Plotnikov, P.I. and Sokolowski, J. (2012) Compressible Navier-Stokes Equations. Theory and Shape Optimization. Mathematical Monographs, 73. Brikhäuser, Basel.
• Rajagopal, K. (1993) Mechanics of non-newtonian fluids. In: G.R. Galdi, J. Necas, eds., Recent Developments in Theoretical Fluid Mechanics. Pitman Research Notes in Mathematics Series, 291. Longman, Essex, 129-162.
• Schmidt, S. and Schulz, V. (2009) Impulse response approximations of discrete shape Hessians with application in CFD. SIAM J. Control Optim., 48(4), 2562-2580.
• Schmidt, S. and Schulz, V. (2010) Shape derivatives for general objective functions and the incompressible Navier-Stokes equations. Control Cybernet., 39(3), 677-713.
• Schowalter, W. (1978) Mechanics of Non-Newtonian Fluids. Pergamon Press.
• Slawig, T. (2005) Distributed control for a class of non-newtonian fluids. Journal of Differential Equations, 219(1), 116-143.
• Sokołowski, J. and Zolésio, J.P. (1992) Introduction to Shape optimization. Shape Sensitivity Analysis, Springer Series in Computational Mathematics, 16. Springer-Verlag, Berlin.
• Solonnikov, V.A. (2001) Lp-estimates for solutions to the initial boundaryvalue problem for the generalized Stokes system in a bounded domain. J. Math. Sci., 105(5), 2448-2484.
• Tröltzsch, F. (2010) Optimal Control of Partial Differential Equations, Graduate Studies in Mathematics, 112. American Mathematical Society, Providence, RI.
• Truesdell, C., Noll, W. and Antman, S. (2004) The Non-linear field Theories of Mechanics. Springer Verlag.
• Wachsmuth, D. and Roubíček, T. (2010) Optimal control of planar flow of incompressible non-Newtonian fluids. Z. Anal. Anwend., 29(3), 351-376.