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A multiscale stabilized ALE formulation for incompressible flows with moving boundaries

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Abstract

This paper presents a variational multiscale stabilized finite element method for the incompressible Navier–Stokes equations. The formulation is written in an Arbitrary Lagrangian–Eulerian (ALE) frame to model problems with moving boundaries. The structure of the stabilization parameter is derived via the solution of the fine-scale problem that is furnished by the variational multiscale framework. The projection of the fine-scale solution onto the coarse-scale space leads to the new stabilized method. The formulation is integrated with a mesh moving scheme that adapts the computational grid to the evolving fluid boundaries and fluid-solid interfaces. Several test problems are presented to show the accuracy and stability of the new formulation.

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References

  1. Brooks AN, Hughes TJR (1982) Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 32: 199–259

    Article  MATH  MathSciNet  Google Scholar 

  2. Hughes TJR, Tezduyar TE (1984) Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations. Comput Methods Appl Mech Eng 45: 217–284

    Article  MATH  MathSciNet  Google Scholar 

  3. Hughes TJR, Franca LP, Balestra M (1986) A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput Methods Appl Mech Eng 59: 85–99

    Article  MATH  MathSciNet  Google Scholar 

  4. Hughes TJR, Franca LP, Hulbert GM (1989) A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective diffusive equations. Comput Methods Appl Mech Eng 73(2): 173–189

    Article  MATH  MathSciNet  Google Scholar 

  5. Hughes TJR (1995) Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput Methods Appl Mech Eng 127: 387–401

    Article  MATH  Google Scholar 

  6. Hughes TJR, Feijoo GR, Mazzei Quincy JB (1998) The variational multiscale method—a paradigm for computational mechanics. Comput Methods Appl Mech Eng 166(1–2): 3–24

    Article  MATH  Google Scholar 

  7. Franca LP, Hauke G, Masud A (2006) Revisiting stabilized finite element methods for the advective-diffusive equation. Comp Methods Appl Mech Eng 195: 1560–1572

    Article  MATH  MathSciNet  Google Scholar 

  8. Masud A, Hughes TJR (2002) A stabilized mixed finite element method for Darcy flow. Comput Methods Appl Mech Eng 191: 4341–4370

    Article  MATH  MathSciNet  Google Scholar 

  9. Ayub M, Masud A (2003) A new stabilized formulation for convective-diffusive heat transfer. Numer Heat Transf Part B 44: 1–23

    Article  Google Scholar 

  10. Masud A, Khurram R (2004) A multiscale/stabilized finite element method for the advection-diffusion equation. Comput Methods Appl Mech Eng 193: 1997–2018

    Article  MATH  Google Scholar 

  11. Masud A, Xia K (2005) A stabilized mixed finite element method for nearly incompressible elasticity. J Appl Mech 72: 711–720

    Article  MATH  MathSciNet  Google Scholar 

  12. Masud A, Khurram R (2006) A multiscale finite element method for the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 195: 1750–1777

    Article  MATH  MathSciNet  Google Scholar 

  13. Tezduyar TE, Aliabadi S, Behr M, Johnson A, Mittal S (1993) Parallel finite element computation of 3D flows. Computer 26: 27–36

    Article  Google Scholar 

  14. Johnson AA, Tezduyar TE (1994) Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces. Comput Methods Appl Mech Eng 119: 73–94

    Article  MATH  Google Scholar 

  15. Johnson AA, Tezduyar TE (1999) Advanced mesh generation and update methods for 3D flow simulations. Comput Mech 23: 130–143

    Article  MATH  Google Scholar 

  16. Masud A, Hughes TJR (1997) A space–time Galerkin/least-squares finite element formulation of the Navier-Stokes equations for moving domain problems. Comput Methods Appl Mech Eng 146: 91–126

    Article  MATH  MathSciNet  Google Scholar 

  17. Masud A (2006) Effects of mesh motion on the stability and convergence of ALE based formulations for moving boundary flows. Comput Mech 38: 430–439

    Article  MATH  MathSciNet  Google Scholar 

  18. Hughes TJR, Liu WK, Zimmermann TK (1981) Lagrangian–Eulerian finite element formulation for incompressible viscous flows. Comput Methods Appl Mech Eng 29: 329–349

    Article  MATH  MathSciNet  Google Scholar 

  19. Belytschko T, Kennedy JM, Schoeberie DF (1980) Quasi-Eulerian finite element formulation for fluid–structure interaction. ASME J Pressure Vessel Technol 102: 62–69

    Article  Google Scholar 

  20. Tezduyar TE, Behr M, Liou J (1992) A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space–time procedure. I. The concept and the preliminary numerical tests. Comput Methods Appl Mech Eng 94: 339–351

    Article  MATH  MathSciNet  Google Scholar 

  21. Tezduyar TE, Behr M, Mittal S, Liou J (1992) A new strategy for finite element computations involving moving boundaries and interfaces – the deforming-spatial-domain/space-time procedure: II. Computation of free-surface flows, two-liquid flows, and flows with drifting cylinders. Comput Methods Appl Mech Eng 94: 353–371

    Article  MATH  MathSciNet  Google Scholar 

  22. Lesoinne M, Farhat C (1996) Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations. Comput Methods Appl Mech Eng 134(1–2): 71–90

    Article  MATH  Google Scholar 

  23. Oñate E, Garcia J, Idelsohn SR, Del Pin F (2004) Finite calculus formulations for finite element analysis of incompressible flows. Eulerian, ALE and Lagrangian approaches. Comput Methods Appl Mech Eng 195(23–24): 3001–3037

    Google Scholar 

  24. Farhat C, Geuzaine P (2004) Design and analysis of robust ALE time-integrators for the solution of unsteady flow problems on moving grids. Comput Methods Appl Mech Eng 193(39–41): 4073–4095

    Article  MATH  MathSciNet  Google Scholar 

  25. Khurram R, Masud A (2006) A multiscale/stabilized formulation of the incompressible Navier-Stokes equations for moving boundary flows and fluid–structure interaction. Comput Mech 38(4–5): 403–416

    Article  MATH  Google Scholar 

  26. Masud A, Calderer R (2009) A variational multiscale stabilized formulation for the incompressible Navier–Stokes equations. Comput Mech 44: 145–160

    Article  MATH  MathSciNet  Google Scholar 

  27. Masud A, Bhanabhagvanwala M, Khurram R (2007) An adaptive mesh rezoning scheme for moving boundary flows and fluid– structure interaction. Comput Fluids 36: 77–91

    Article  MATH  MathSciNet  Google Scholar 

  28. Kanchi H, Masud A (2007) A 3D adaptive mesh moving scheme. Int J Numer Methods Fluids 54: 923–944

    Article  MATH  MathSciNet  Google Scholar 

  29. Masud A, Franca LP (2008) A hierarchical multiscale framework for problems with multiscale source terms. Comput Methods Appl Mech Eng 197: 2692–2700

    Article  MathSciNet  Google Scholar 

  30. Masud A, Kwack J (2008) A stabilized mixed finite element method for the first-order form of convection-diffusion equation. Int J Numer Methods Fluids 57: 1321–1348

    Article  MATH  MathSciNet  Google Scholar 

  31. Bazilevs Y, Calo VM, Cottrell JA, Hughes TJR, Reali A, Scovazzi G (2007) Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput Methods Appl Mech Eng 197: 173–201

    Article  MATH  MathSciNet  Google Scholar 

  32. Behr M, Hastreiter D, Mittal S, Tezduyar TE (1995) Incompressible flow past a circular cylinder: dependence of the computed flow field on the location of the lateral boundaries. Methods Appl Mech Eng 123: 309–316

    Article  Google Scholar 

  33. Hauke G, Hughes TJR (1998) A comparative study of different sets of variables for solving compressible and incompressible flows. Comput Methods Appl Mech Eng 153: 1–44

    Article  MATH  MathSciNet  Google Scholar 

  34. Mittal S, Tezduyar TE (1992) A finite element study of incompressible flows past oscillating cylinders and airfoils. Int J Numer Methods Fluids 15: 1073–1118

    Article  Google Scholar 

  35. Wall WA, Ramm E (1998) Fluid–Structure interaction based upon a stabilized (ALE) finite element method. In: Idelsohn S, Oñate E, Dvorkin E (eds) Computational mechanics—new trends and applications (Proceedings of WCCM IV), CIMNE, Barcelona

  36. Kalro V, Tezduyar TE (1997) Parallel 3D computation of unsteady flows around circular cylinders. Parallel Comput. 23: 1235–1248

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Ramon Calderer & Arif Masud

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  1. Ramon Calderer
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  2. Arif Masud
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Correspondence to Arif Masud.

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Calderer, R., Masud, A. A multiscale stabilized ALE formulation for incompressible flows with moving boundaries. Comput Mech 46, 185–197 (2010). https://doi.org/10.1007/s00466-010-0487-z

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  • DOI: https://doi.org/10.1007/s00466-010-0487-z

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