Numerical simulation and optimal shape for viscous flow by a fictitious domain method. (English) Zbl 0837.76068

Summary: We discuss the fictitious domain solution of the Navier-Stokes equations modelling unsteady incompressible viscous flow. The method is based on a Lagrange multiplier treatment of the boundary conditions to be satisfied and is particularly well suited to the treatment of no-slip boundary conditions. This approach allows the use of structured meshes and fast specialized solvers for problems on complicated geometries. Another interesting feature of the fictitious domain approach is that it allows the solution of optimal shape problems without regriding. The resulting methodology is applied to the solution of flow problems including external incompressible viscous flow modelled by the Navier-Stokes equations and then to an optimal shape problem for Stokes and Navier- Stokes flow.


76M30 Variational methods applied to problems in fluid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
76D07 Stokes and related (Oseen, etc.) flows
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[1] Young, J. Comp. Phys. 92 pp 1– (1991)
[2] , , and , ’EM-TRANAIR: Steps toward solution of general 3D Maxwell’s equations’, in Computer Methods in Applied Sciences and Engineering, (ed.), Nova Science, Commack, NY, 1991, pp. 49-72.
[3] Buzbee, SIAM J. Numer. Anal. 8 pp 722– (1971)
[4] Glowinski, Comput. Methods Appl. Mech. Eng. 111 pp 283– (1994)
[5] Glowinski, Comput. Methods Appl. Mech. Eng. 112 pp 133– (1994)
[6] Glowinski, Japan J. Ind. Appl. Math.
[7] Borgers, Numer. Math. 57 pp 435– (1990)
[8] Numerical Methods for Nonlinear Variational Problems, Springer, New York, 1984.
[9] , , and , ’Numerical methods for incompressible and compressible Navier-Stokes problems’, in , , and (eds.), Finite Elements in Fluids, Vol. 6, Wiley, Chichester, 1985, pp. 1-40.
[10] ’Viscous flow simulation by finite element methods and related numerical techniques’, in and (eds.), Progress and Supercomputing in Computational Fluid Dynamics, Birkhauser, Boston, 1985, pp. 173-210.
[11] Bristeau, Comp. Phys. Rep. 6 pp 73– (1987)
[12] ’Finite element methods for the numerical simulation of incompressible viscous flow. Introduction to the control of the Navier-Stokes equations’, Lectures in Applied Mathematics, 28, AMS, Providence, R. I., 1991, pp. 219-301. · Zbl 0751.76046
[13] and , ’A one shot domain decomposition/fictitious domain method for the solution of elliptic equations’, to appear.
[14] Finite Element Methods for Fluids, Wiley, Chichester, 1989. · Zbl 0712.76001
[15] Shape Optimization, Springer, New York, 1984.
[16] Begis, Appl. Math. Optim. 2 pp 130– (1975)
[17] Haslinger, East-West J. Numer. Math. 1 pp 111– (1992)
[18] ’Optimization methods for partial differential equations from science and engineering’, Ph.D. Dissertation. Department of Computational and Applied Mathematics, Rice University, Houston, TX, 1995.
[19] Optimisation: Thèorie et Algorithmes, Dunod, Paris, 1971.
[20] Dennis, SIAM J. Optim. 1 pp 448– (1991)
[21] Torczon, TOMS, Trans. Meth. Software
[22] and , ’On the use of parallel direct search methods for nonlinear programming problems’, Technical Report # 93-33, Department of Computational & Applied Mathematics. Rice University, houston. TX, 1993.
[23] and , ’A fictious domain method for the incompressible Navier-Stokes equations’, in , (eds.), The Finite element Method in the 90’s, Springer, Berlin, 1991, pp. 440-417.
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