# Finite element methods for flow problems with moving boundaries and interfaces

- 1.3k Downloads
- 249 Citations

## Summary

This paper is an overview of the finite element methods developed by the Team for Advanced Flow Simulation and Modeling (T*AFSM) [http://www.mems.rice.edu/TAFSM/] for computation of flow problems with moving boundaries and interfaces. This class of problems include those with free surfaces, two-fluid interfaces, fluid-object and fluid-structure interactions, and moving mechanical components. The methods developed can be classified into two main categories. The interface-tracking methods are based on the Deforming-Spatial-Domain/Stabilized Space-Time (DSD/SST) formulation, where the mesh moves to track the interface, with special attention paid to reducing the frequency of remeshing. The interface-capturing methods, typically used for free-surface and two-fluid flows, are based on the stabilized formulation, over non-moving meshes, of both the flow equations and the advection equation governing the time-evolution of an interface function marking the location of the interface. In this category, when it becomes neccessary to increase the accuracy in representing the interface beyond the accuracy provided by the existing mesh resolution around the interface, the Enhanced-Discretization Interface-Capturing Technique (EDICT) can be used to to accomplish that goal. In development of these two classes of methods, we had to keep in mind the requirement that the methods need to be applicable to 3D problems with complex geometries and that the associated large-scale computations need to be carried out on parallel computing platforms. Therefore our parallel implementations of these methods are based on unstructured grids and on both the distributed and shared memory parallel computing approaches. In addition to these two main classes of methods, a number of other ideas and methods have been developed to increase the scope and accuracy of these two classes of methods. The review of all these methods in our presentation here is supplemented by a number numerical examples from parallel computation of complex, 3D flow problems.

## Keywords

Shallow Water Equation Interface Function Particle Finite Element Method Interface Edge Propeller Surface## References

- 1.T.E. Tezduyar (1999), “CFD methods for 3D computation of complex flow problems”,
*Journal of Wind Engineering and Industrial Aerodynamics*,**81**, 97–116.CrossRefGoogle Scholar - 2.T.E. Tezduyar and Y. Osawa (1999), “Methods for parallel computation of complex flow problems”,
*Parallel Computing*,**25**, 2039–2066.CrossRefMathSciNetGoogle Scholar - 3.T.E. Tezduyar (1991), “Stabilized finite element formulations for incompressible flow computations”,
*Advances in Applied Mechanics*,**28**, 1–44.MathSciNetCrossRefGoogle Scholar - 4.T.E. Tezduyar, M. Behr and J. Liou (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 tests”,
*Computer Methods in Applied Mechanics and Engineering*,**94**, 339–351.CrossRefMathSciNetzbMATHGoogle Scholar - 5.T.E. Tezduyar, M. Behr, S. Mittal and J. Liou (1992), “A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/spacetime procedure: II. Computation of free-surface flows, two-liquid flows, and flows with drifting cylinders”,
*Computer Methods in Applied Mechanics and Engineering*,**94**, 353–371.CrossRefMathSciNetzbMATHGoogle Scholar - 6.T.J.R. Hughes and G.M. Hulbert (1988), “Space-time finite element methods for elastodynamics: formulations and error estimates”,
*Computer Methods in Applied Mechanics and Engineering*,**66**, 339–363.CrossRefMathSciNetzbMATHGoogle Scholar - 7.T.J.R. Hughes and A.N. Brooks (1979), “A multi-dimensional upwind scheme with no crosswind diffusion”, in T.J.R. Hughes (Ed.),
*Finite Element Methods for Convection Dominated Flows*, AMD-Vol. 34, 19–35, ASME, New York.Google Scholar - 8.A.N. Brooks and T.J.R. Hughes (1982), “Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations”,
*Computer Methods in Applied Mechanics and Engineering*,**32**, 199–259.CrossRefMathSciNetzbMATHGoogle Scholar - 9.T.E. Tezduyar and T.J.R. Hughes (1983), “Finite element formulations for convection dominated flows with particular emphasis on the compressible Fuler equations”, in
*Proceedings of AIAA 21st Aerospace Sciences Meeting*, AIAA Paper 83-0125, Reno, Nevada.Google Scholar - 10.G.J. Le Beau and T.E. Tezduyar (1991), “Finite element computation of compressible flows with the SUPG formulation”, in M.N. Dhaubhadel, M.S. Engelman, and J.N. Reddy, editors,
*Advances in Finite Element Analysis in Fluid Dynamics*, FED-Vol. 123, ASME, New York, 21–27.Google Scholar - 11.G.J. Le Beau, S.E. Ray, S.K. Aliabadi and T.E. Tezduyar (1993), “SUPG finite element computation of compressible flows with the entropy and conservation variables formulations”,
*Computer Methods in Applied Mechanics and Engineering*,**104**, 397–422.CrossRefzbMATHGoogle Scholar - 12.T.E. Tezduyar, S. Mittal, S.E. Ray and R. Shih (1992), “Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements”,
*Computer Methods in Applied Mechanics and Engineering*,**95**, 221–242.CrossRefzbMATHGoogle Scholar - 13.T.J.R. Hughes, L.P. Franca and G.M. Hulbert (1989), “A new finite element formulation for computational fluid dynamics: VIII. the Galerkin/least-squares method for advective-diffusive equations”,
*Computer Methods in Applied Mechanics and Engineering*,**73**, 173–189.CrossRefMathSciNetzbMATHGoogle Scholar - 14.P. Hansbo and A. Szepessy (1990), “A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations”,
*Computer Methods in Applied Mechanics and Engineering***84**, 175–192.CrossRefMathSciNetzbMATHGoogle Scholar - 15.J. Donea (1984), “A Taylor-Galerkin method for convective transport problems,”
*International Journal for Numerical Methods in Engineering*,**20**, 101–120.CrossRefzbMATHGoogle Scholar - 16.C. Johnson, U. Navert and J. Pitkäranta (1984), “Finite element methods for linear hyperbolic problems”,
*Computer Methods in Applied Mechanics and Engineering*,**45**, 285–312.CrossRefMathSciNetzbMATHGoogle Scholar - 17.C. Johnson and J. Saranen (1986), “Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations”,
*Mathematics of Computation*,**47**, 1–18.CrossRefMathSciNetzbMATHGoogle Scholar - 18.T.J.R. Hughes, L.P. Franca and M. Mallet (1987), “A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multi-dimensional advective-diffusive systems”,
*Computer Methods in Applied Mechanics and Engineering***63**, 97–112.CrossRefMathSciNetzbMATHGoogle Scholar - 19.S. Aliabadi, S.E. Ray and T.E. Tezduyar (1993), “SUPG finite element computation of compressible flows with the entropy and conservation variables formulations”,
*Computational Mechanics*,**11**, 300–312.CrossRefzbMATHGoogle Scholar - 20.T.J.R. Hughes, L.P. Franca and M. Balestra (1986), “A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations”,
*Computer Methods in Applied Mechanics and Engineering*,**59**, 85–99.CrossRefMathSciNetzbMATHGoogle Scholar - 21.C. Farhat, M. Lesoinne and N. Maman (1995), “Mixed explicit/implicit time integration of coupled aeroelastic problems: Three-field formulation, geometric conservation and distributed solution”,
*International Journal for Numerical Methods in Fluids*,**21**, 807–835.CrossRefMathSciNetzbMATHGoogle Scholar - 22.M. Lesoinne and C. Farhat (1996), “Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations”,
*Computer Methods in Applied Mechanics and Engineering*,**134**, 71–90.CrossRefzbMATHGoogle Scholar - 23.R. Lohner, C. Yang and J.D. Baum (1996), “Rigid and flexible store separation simulations using dynamic adaptive unstructured grid technologies”, in
*Proceedings of the First AFOSR Conference on Dynamic Motion CFD*, New Brunswick, New Jersey.Google Scholar - 24.K. Stein, R. Benney, V. Kalro, T. Tezduyar and J. Leonard and M. Accorsi (1998), “Parachute fluid-structure interactions: 3-D computation”, in
*Proceedings of the 4th Japan-US Symposium on Finite Element Methods in Computational Fluid Dynamics*, Tokyo, Japan.Google Scholar - 25.J. Garcia, E. Onate, H. Sierra and S. Idelsohn (1998), “A stabilized numerical method for analysis of ship hydrodynamics”, in
*Proceedings of ECCOMAS 1998*. J. Wiley.Google Scholar - 26.E. Onate, S. Idelsohn, C. Sacco and J. Garcia (1998), “Stabilization of the numerical solution for the free surface wave equation in fluid dynamics”, in
*Proceedings of ECCOMAS 1998*. J. Wiley.Google Scholar - 27.E. Onate and J. Garcia (1999), “A methodology for analysis of fluid-structure interaction accounting for free surface waves,” in
*Proceedings of European Conference on Computational Mechanics*, Munich, Germany.Google Scholar - 28.M. Cruchaga and E. Onate (1999), “A generalized streamline finite element approach for the analysis of incompressible flow problems including moving surfaces”,
*Computer Methods in Applied Mechanics and Engineering*,**173**, 241–255.CrossRefzbMATHGoogle Scholar - 29.S. Aliabadi and T.E. Tezduyar (1993), “Space-time finite element computation of compressible flows involving moving boundaries and interfaces”,
*Computer Methods in Applied Mechanics and Engineering***107**, 209–224.CrossRefzbMATHGoogle Scholar - 30.M. Behr and T.E. Tezduyar (1994), “Finite element solution strategies for large-scale flow simulations”,
*Computer Methods in Applied Mechanics and Engineering*,**112**, 3–24.CrossRefMathSciNetzbMATHGoogle Scholar - 31.A.A. Johnson and T.E. Tezduyar (1997), “Parallel computation of incompressible flows with complex geometries”,
*International Journal for Numerical Methods in Fluids*,**24**, 1321–1340.CrossRefzbMATHGoogle Scholar - 32.A.A. Johnson and T.E. Tezduyar (1996), “Simulation of multiple spheres falling in a liquid-filled tube”,
*Computer Methods in Applied Mechanics and Engineering*,**134**, 351–373.CrossRefMathSciNetzbMATHGoogle Scholar - 33.T.E. Tezduyar, S. Aliabadi, M. Behr, A. Johnson, V. Kalro and M. Litke (1996), “Flow simulation and high performance computing”,
*Computational Mechanics*,**18**, 397–412.CrossRefzbMATHGoogle Scholar - 34.A.A. Johnson and T.E. Tezduyar (2001), “Methods for 3D computation of fluid-object interactions in spatially-periodic flows”,
*Computer Methods in Applied Mechanics and Engineering*,**190**, 3201–3221.CrossRefzbMATHGoogle Scholar - 35.S. Mittal and T.E. Tezduyar (1994), “Massively parallel finite element computation of incompressible flows involving fluid-body interactions”,
*Computer Methods in Applied Mechanics and Engineering*,**112**, 253–282.CrossRefMathSciNetzbMATHGoogle Scholar - 36.T. Tezduyar, S. Aliabadi, M. Behr, A. Johnson and S. Mittal (1993), “Parallel finite-element computation of 3D flows”,
*IEEE Computer*,**26**, 27–36.Google Scholar - 37.T.E. Tezduyar, M. Behr, S. Mittal and A.A. Johnson (1992), “Computation of unsteady incompressible flows with the finite element methods—space-time formulations, iterative strategies and massively parallel implementations”, in P. Smolinski, W.K. Liu, G. Hulbert, and K. Tamma, editors,
*New Methods in Transient Analysis*, AMD-Vol.143, ASME, New York, 7–24.Google Scholar - 38.D.R. Lynch (1982), “Wakes in liquid-liquid systems”,
*Journal of Computational Physics*,**47**, 387–411.CrossRefMathSciNetzbMATHGoogle Scholar - 39.J.T. Batina (1989), “Unsteady Euler airfoil solutions using unstructured dynamics meshes”, in
*Proceedings of AIAA 27th Aerospace Sciences Meeting*, AIAA Paper 89-0115, Reno, Nevada.Google Scholar - 40.J.A. Benek, P.G. Buning and J.L. Steger (1985), “A 3-D Chimera grid embedding technique”, Paper 85-1523,
*AIAA*.Google Scholar - 41.M. Behr and T.E. Tezduyar (1999), “Shear-Slip Mesh Update Method”,
*Computer Methods in Applied Mechanics and Engineering*,**174**, 261–274.CrossRefzbMATHGoogle Scholar - 42.M. Behr and T.E. Tezduyar (2001), “Shear-Slip Mesh update in 3D computation of complex flow problems with rotating mechanical components”,
*Computer Methods in Applied Mechanics and Engineering*,**190**, 3189–3200.CrossRefzbMATHGoogle Scholar - 43.T.E. Tezduyar, S. Aliabadi and M. Behr (1998), “Enhanced-Discretization Interface-Capturing T echnique (EDICT) for computation of unsteady flows with interfaces”,
*Computer Methods in Applied Mechanics and Engineering*,**155**, 235–248.CrossRefzbMATHGoogle Scholar - 44.T.E. Tezduyar, S. Aliabadi and M. Behr (1997), “Enhanced-Discretization Interface-Capturing Technique”, in Y. Matsumoto and A. Prosperetti, editors,
*Proceedings of the ISAC '97 High Performance Computing on Multiphase Flows*, 1–6, Japan Society of Mechanical Engineers.Google Scholar - 45.C.W. Hirt and B.D. Nichols (1981), “Volume of fluid (VOF) method for the dynamics of free boundaries”,
*Journal of Computational Physics*,**39**, 201–225.CrossRefzbMATHGoogle Scholar - 46.D. L. Youngs (1984), “Time-dependent multimaterial flow with large fluid distortion”, in K. W. Morton and M. J. Baines, editors,
*Numerical Methods in Fluid Dynnamics*, Notes on Numberical Fluid Mechanics, 273–285, Academic Press, New York.Google Scholar - 47.C.M. Lemos (1996), “Higher-order schemes for free-surface flows with arbitrary configurations”,
*International Journal for Numerical Methods in Fluids*,**23**, 545–566.CrossRefzbMATHGoogle Scholar - 48.T.E. Tezduyar and Y. Osawa (2000), “Finite element stabilization parameters computed from element matrices and vectors”,
*Computer Methods in Applied Mechanics and Engineering*,**190**, 411–430.CrossRefzbMATHGoogle Scholar - 49.S. Aliabadi and T. Tezduyar (2000), “Stabilized-Finite-Element/Interface-Capturing Technique for parallel computation of unsteady flows with interfaces”,
*Computer Methods in Applied Mechanics and Engineering*,**190**, 243–261.CrossRefzbMATHGoogle Scholar - 50.S. Mittal, S. Aliabadi and T.E. Tezduyar (1999), “Parallel computation of unsteady compressible flows with the EDICT”,
*Computational Mechanics*,**23**, 151–157.CrossRefzbMATHGoogle Scholar - 51.T.E. Tezduyar, Y. Osawa, K. Stein, R. Benney, V. Kumar and J. McCune (2000), “Numerical methods for computer assisted analysis of parachute mechanics”, to appear in
*Proceedings of 8th Conference on Numerical Methods in Continuum Mechanics*, Liptovsky Jan, Slovakia, CD-ROM.Google Scholar - 52.T.E. Tezduyar, Y. Osawa, K. Stein, R. Benney, V. Kumar and J. McCune (2001), “Computational methods for parachute aerodynamics”,
*Proceedings of Computational Fluid Dynamics for the 21st Century*, Kyoto, Japan.Google Scholar - 53.R.G. Dean and R.A. Dalrymple (1984),
*Water Wave Mechanics for Engineers and Scientists*. Prentice-Hall.Google Scholar - 54.Y. Saad and M. Schultz (1986), “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems”,
*SIAM Journal of Scientific and Statistical Computing*,**7**, 856–869.CrossRefMathSciNetzbMATHGoogle Scholar - 55.T. E. Tezduyar, S. Aliabadi, M. Behr, A. Johnson, V. Kalro and M. Litke (1996), “High performance computing techniques for flow simulations”, in M. Papadrakakis, editor,
*Solving Large-Scale Problems in Mechanics: Parallel Solution Methods in Computational Mechanics*, 363–398, John Wiley & Sons.Google Scholar - 56.V. Kalro and T. Tezduyar (1998), “Parallel iterative computational methods for 3D finite element flow simulations”,
*Computer Assisted Mechanics and Engineering Sciences*,**5**, 173–183.zbMATHGoogle Scholar - 57.Z. Johan, T.J.R. Hughes and F. Shakib (1991), “A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids”,
*Computer Methods in Applied Mechanics and Engineering*,**87**, 281–304.CrossRefMathSciNetzbMATHGoogle Scholar - 58.Z. Johan, K.K. Mathur, S.L. Johnsson and T.J.R. Hughes (1995), “A case study in parallel computation: Viscous flow around an Onera M6 wing”,
*International Journal for Numerical Methods in Fluids*,**21**, 877–884.CrossRefzbMATHGoogle Scholar - 59.T.E. Tezduyar, M. Behr, S.K. Aliabadi, S. Mittal and S.E. Ray (1992), “A new mixed preconditioning method for finite element computations”,
*Computer Methods in Applied Mechanics and Engineering*,**99**, 27–42.CrossRefMathSciNetzbMATHGoogle Scholar - 60.J. Smagorinsky (1963), “General circulation experiments with the primitive equations”,
*Monthly Weather Review*,**91**, 99–165.CrossRefGoogle Scholar - 61.C. Kato and M. Ikegawa (1961), “Large eddy simulation of unsteady turbulent wake of a circular cylinder using the finite element method”, in I. Celik, T. Kobayashi, K.N. Ghia and J. Kurokawa, editors,
*Advances in Numerical Simulation of Turbulent Flows*, FED-Vol. 117, ASME, New York, 49–56.Google Scholar - 62.A.A. Johnson and T.E. Tezduyar (1999), “Advanced mesh generation and update methods for 3D flow simulations”,
*Computational Mechanics*,**23**, 130–143.CrossRefzbMATHGoogle Scholar - 63.I. Güler, M. Behr and T.E. Tezduyar (1999), “Parallel finite element computation of free-surface flows”,
*Computational Mechanics*,**23**, 117–123.CrossRefzbMATHGoogle Scholar - 64.T.E. Tezduyar and S. Aliabadi (1998), “EDICT for computation of unsteady flows with interfaces”, in
*Modeling and Simulation Based Engineering (eds. S. Atluri and P. O'Donoghue). Proceedings of International Conference on Computational Engineering Science*, Atlanta, Georgia.Google Scholar