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On the Use of Anisotropic Triangles with Mixed Finite Elements: Application to an “Immersed” Approach for Incompressible Flow Problems

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Advanced Finite Element Technologies

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 566))

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Abstract

In this chapter, we discuss the use of some common mixed finite elements in the context of a locally anisotropic remeshing strategy, close in philosophy to “immersed” approaches for interface problems. A characteristic of the present method is the presence of highly flat triangles. Such a distinctive feature may imply stability issues for mixed elements with incompressible flow problems. First, we present a review of the literature dealing with interface problems and we illustrate these results with a simple 1D framework alongside of numerical tests. Second, we present the locally anisotropic remeshing approach for interface problems in 2D with a focus on the incompressible Stokes problem. We then present numerical tests to show stability issues of common mixed elements, as well as possible stable ones. We also deal with conditioning issues. Finally, we illustrate the results with two applications, including the fluid–structure interaction of a rotational rigid bar.

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References

  • Acosta, G., & Durán, R. G. (1999). The maximum angle condition for mixed and nonconforming elements: Application to the stokes equations. SIAM Journal on Numerical Analysis, 37, 18–36.

    Article  MathSciNet  MATH  Google Scholar 

  • Ainsworth, M., Barrenechea, G. R., & Wachtel, A. (2014). Stabilisation of high aspect ratio mixed finite elements for incompressible flow. Brown: Technical report.

    MATH  Google Scholar 

  • Amdouni, S., Moakher, M., & Renard, Y. (2014). A local projection stabilization of fictitious domain method for elliptic boundary value problems. Applied Numerical Mathematics, 76, 60–75.

    Article  MathSciNet  MATH  Google Scholar 

  • Apel, T. (1999). Anisotropic finite elements: Local estimates and applications. Teubner.

    Google Scholar 

  • Apel, T., & Nicaise, S. (2004). The inf-sup condition for low order elements on anisotropic meshes. Calcolo, 41, 89–113.

    Article  MathSciNet  MATH  Google Scholar 

  • Apel, T., & Randrianarivony, H. M. (2003). Stability of discretizations of the stokes problem on anisotropic meshes. Journal Mathematics and Computers in Simulation, 61, 437–447.

    Article  MathSciNet  MATH  Google Scholar 

  • Auricchio, F., Brezzi, F., & Lovadina, C. (2004). Mixed finite element methods. Encyclopedia of computational mechanics, Chapter 9. New York: Wiley.

    Google Scholar 

  • Auricchio, F., Boffi, D., Gastaldi, L., Lefieux, A., & Reali, A. (2014). A study on unfitted 1d finite element methods. Computers and Mathematics with Applications, 68, 2080–2102.

    Article  MathSciNet  MATH  Google Scholar 

  • Auricchio, F., Brezzi, F., Lefieux, A., & Reali, A. (2015a). An “immersed” finite element method based on a locally anisotropic remeshing for the incompressible stokes problem. Computer Methods In Applied Mechanics and Engineering, 294, 428–448.

    Article  MathSciNet  Google Scholar 

  • Auricchio, F., Lefieux, A., Reali, A., Veneziani, A. (2015b). A locally anisotropic fluid-structure interaction remeshing strategy for thin structures with application to a hinged rigid leaflet. International Journal For Numerical Methods In Engineering. doi:10.1002/nme.5159.

    Google Scholar 

  • Babuška, I. (1970). The finite element method for elliptic equations with discontinuous coefficients. Computing, 5, 207–213.

    Article  MathSciNet  MATH  Google Scholar 

  • Babuška, I., & Aziz, A. K. (1976). On the angle condition in the finite element method. SIAM Journal on Numerical Analysis, 13, 214–226.

    Article  MathSciNet  MATH  Google Scholar 

  • Babuška, I., & Strouboulis, T. (2001) The finite element method and its reliability. Oxford: Oxford University Press.

    Google Scholar 

  • Baiges, J., Codina, R., Henke, F., Shahmiri, S., & Wall, W. A. (2012). A symmetric method for weakly imposing dirichlet boundary conditions in embedded finite element meshes. International Journal for Numerical Methods in Engineering, 90, 636–658.

    Article  MathSciNet  MATH  Google Scholar 

  • Barrenechea, G. R., & Chouly, F. (2012). A local projection stabilized method for fictitious domains. Applied Mathematics Letters, 25(12), 2071–2076.

    Article  MathSciNet  MATH  Google Scholar 

  • Barrett, J. W., & Elliott, C. M. (1987). Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces. IMA Journal of Numerical Analysis, 7, 283–300.

    Article  MathSciNet  MATH  Google Scholar 

  • Basting, S., & Weismann, M. (2013). A hybrid level set-front tracking finite element approach for fluid-structure interaction and two-phase flow applications. Journal of Computational Physics, 255, 228–244.

    Article  MathSciNet  MATH  Google Scholar 

  • Bazilevs, Y., & Hughes, T. J. R. (2007). Weak imposition of dirichlet boundary conditions in fluid mechanics. Comput. Methods Appl. Mech. Engrg., 36, 12–26.

    MathSciNet  MATH  Google Scholar 

  • Béchet, É., Moës, N., & Wohlmuth, B. (2009). A stable Lagrange multiplier space for stiff interface conditions within the extended finite element method. International Journal for Numerical Methods in Engineering, 78, 931–954.

    Article  MathSciNet  MATH  Google Scholar 

  • Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L. D., & Russo, A. (2013). Basic principles of virtual element methods. Mathematical Models and Methods in Applied Sciences, 23, 199–214.

    Article  MathSciNet  MATH  Google Scholar 

  • Beirão da Veiga, L., Brezzi, F., Marini, L. D., & Russo, A. (2014). The hitchhiker’s guide to the virtual element method. Mathematical Models and Methods in Applied Sciences, 24, 1541–1573.

    Article  MathSciNet  MATH  Google Scholar 

  • Bern, M., Eppstein, D. (1992) Mesh generation and optimal triangulation. In Computing in Euclidian geometry, Lecture Notes Series on Computing, (vol. 1, pp. 23–90). Singapore, World Scientific.

    Google Scholar 

  • Bhalla, A. P. S., Bale, R., Griffith, B. E., & Patankar, N. A. (2013). A unified mathematical framework and an adaptive numerical method for fluid-structure interaction with rigid, deforming, and elastic bodies. Journal of Computational Physics, 250, 446–476.

    Article  MathSciNet  Google Scholar 

  • Boffi, D., Gastaldi, L., Heltai, L., & Peskin, C. S. (2008). On the hyper-elastic formulation of the immersed boundary method. Computer Methods in Applied Mechanics and Engineering, 197, 2210–2231.

    Article  MathSciNet  MATH  Google Scholar 

  • Boffi, D., Cavallini, N., & Gastaldi, L. (2011). Finite element approach to immersed boundary method with different fluid and solid densities. Mathematical Models and Methods in Applied Sciences, 21, 2523–2550.

    Article  MathSciNet  MATH  Google Scholar 

  • Boffi, D., Brezzi, F., & Fortin, M. (2013). Mixed finite element methods. Heidelberg: Springer.

    Google Scholar 

  • Burman, E., & Hansbo, P. (2010). Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method. Computer Methods in Applied Mechanics and Engineering, 199, 2680–2686.

    Article  MathSciNet  MATH  Google Scholar 

  • Burman, E., & Hansbo, P. (2011a). Fictitious domain finite element methods using cut elements: II a stabilized Nitsche method. Applied Numerical Mathematics, 62(4), 328–341.

    Google Scholar 

  • Burman, E., Hansbo, P. (2011b). Fictitious domain methods using cut elements: III. a stabilized nitsche method for Stokes’ problem. Technical report, Jönköping University.

    Google Scholar 

  • Carey, G. F. (1982). Derivative calculation from finite element solutions. Computer Methods in Applied Mechanics and Engineering, 35, 1–14.

    Article  MathSciNet  MATH  Google Scholar 

  • Carey, G. F., Chow, S. S., & Seager, M. K. (1985). Approximate boundary-flux calculations. Computer Methods in Applied Mechanics and Engineering, 50, 107–120.

    Article  MathSciNet  MATH  Google Scholar 

  • Chin, E. B., Lasserre, J. B., & Sukumar, N. (2015). Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra. Computational Mechanics.

    Google Scholar 

  • Court, S., Fournier, M., & Lozinski, A. (2014). A fictitious domain approach for the Stokes problem based on the extended finite element method. International Journal for Numerical Methods in Fluids, 74, 73–99.

    Google Scholar 

  • Demkowicz, L. (1991). A note on symmetry boundary conditions in finite element methods. Applied Mathematics Letters, 4(5), 27–30.

    Article  MathSciNet  MATH  Google Scholar 

  • Diniz dos Santos, N., Gerbeau, J.-F., & Bourgat, J.-F. (2008). A partitioned fluid-structure algorithm for elastic thin valves with contact. Computer Methods in Applied Mechanics and Engineering, 197, 1750–1761.

    Article  MathSciNet  MATH  Google Scholar 

  • Elman, D., Silvester, H., & Wathen, A. (2005). Finite elements and fast iterative solvers with applications in incompressible fluid dynamics. Oxford: Oxford Press.

    MATH  Google Scholar 

  • Ern, A., & Guermond, J.-L. (2004). Theory and practice of finite elements. Heidelberg: Springer.

    Book  MATH  Google Scholar 

  • Fabrèges, B. (2012). Une méthode de prolongement régulier pour la simulation d’écoulements fluide/particules. PhD thesis, Université Paris-Sud.

    Google Scholar 

  • Frei, S., & Richter, R. (2014). A locally modified parametric finite element method for interface problems. SIAM Journal on Numerical Analysis, 52(5), 2315–2334.

    Article  MathSciNet  MATH  Google Scholar 

  • Fries, T.-P., & Belytschko, T. (2010). The extended/generalized finite element method: An overview of the method and its applications. International Journal for Numerical Methods in Engineering, 84, 253–304.

    MathSciNet  MATH  Google Scholar 

  • Gerstenberger, A., & Wall, W. A. (2008). An eXtended Finite Element Method/Lagrange multiplier based approach for fluid-structure interaction. Computer Methods in Applied Mechanics and Engineering, 197, 1699–1714.

    Article  MathSciNet  MATH  Google Scholar 

  • Girault, V., & Glowinski, R. (1995). Error analysis of fictitious domain method applied to a dirichlet problem. Japan Journal of Industrial and Applied Mathematics, 12, 487–514.

    Article  MathSciNet  MATH  Google Scholar 

  • Glowinski, R. (2003). Handbook of numerical analysis: Numerical methods for fluids (Part 3) (vol. 9), chapter VIII. North-Holland.

    Google Scholar 

  • Glowinski, R., Pan, T., & Périeux, J. (1994). A fictitious domain method for external incompressible viscous flow modeled by navier-stokes equations. Computer Methods In Applied Mechanics and Engineering, 112, 133–148.

    Article  MathSciNet  MATH  Google Scholar 

  • Hachem, E., Feghali, S., Codina, R., & Coupez, T. (2013). Immersed stress method for fluid-structure interaction using anisotropic mesh adaptation. International Journal for Numerical Methods in Engineering, 94(9), 805–825.

    Article  MathSciNet  Google Scholar 

  • Hannukainen, A., Korotov, S., & Křížek, M. (2012). The maximum angle condition is not necessary for convergence of the finite element method. Numerical Mathematics, 120, 79–88.

    Article  MathSciNet  MATH  Google Scholar 

  • Hansbo, A., & Hansbo, P. (2002). An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Computer Methods in Applied Mechanics and Engineering, 191, 5537–5552.

    Article  MathSciNet  MATH  Google Scholar 

  • Hansbo, P., Larson, M. G., & Zahedi, S. (2013). A cut finite element method for a Stokes interface problem. Applied Numerical Mathematics, 85, 90–114.

    Google Scholar 

  • Haslinger, J., & Renard, Y. (2009). A new fictitious domain approach inspired by the extended finite element method. SIAM Journal on Numerical Analysis, 47, 1474–1499.

    Article  MathSciNet  MATH  Google Scholar 

  • Hautefeuille, M., Annavrapu, C., & Dolbow, J. E. (2012). Robust imposition of dirichlet boundary conditions on embedded surfaces. International Journal for Numerical Methods in Engineering, 90, 40–64.

    Article  MathSciNet  MATH  Google Scholar 

  • Heltai, L., & Costanzo, F. (2012). Variational implementation of immersed finite element methods. Computer Methods in Applied Mechanics and Engineering, 229, 110–127.

    Article  MathSciNet  MATH  Google Scholar 

  • Hyman, M. A. (1952). Non-iterative numerical solution of boundary-value problems. Applied Scientific Research, Section B, 2, 325–351.

    Article  MathSciNet  MATH  Google Scholar 

  • Ilinca, F., & Hétu, J.-F. (2011). A finite element immersed boundary method for fluid flow around rigid objects. International Journal for Numerical Methods in Fluids, 65, 856–875.

    Article  MathSciNet  MATH  Google Scholar 

  • Jamet, P. (1976). Estimations d’erreur pour des éléments finis droits presque dégénérés. RAIRO. Analyse Numérique, 10, 43–60.

    MathSciNet  MATH  Google Scholar 

  • Kamenski, L., Huang, W., & Xu, H. (2014). Conditioning of finite element equations with arbitrary anistropic meshes. Mathematics of Computations, 83(289), 2187–2211.

    Article  MathSciNet  MATH  Google Scholar 

  • Kellogg, R. B. (1971). Singularities in interface problems. Numerical solution of partial differential equations (pp. 351–400). Cambridge: Academic Press.

    Google Scholar 

  • Křížek, M. (1991). On semiregular families of triangulations and linear interpolation. Applications of Mathematics, 36(3), 223–232.

    MathSciNet  MATH  Google Scholar 

  • Lefieux, A. (2014). On the use of anisotropic triangles in an immersed finite element approach with application to fluid-structure interaction problems. PhD thesis, Istituto Universitario degli Studi Superiori di Pavia.

    Google Scholar 

  • Lemrabet, K. (1977). Régularité de la solution d’un problème de transmission. Journal de Mathèmatiques Pures et Appliquèes, 9(56): 1–38.

    Google Scholar 

  • Lew, A. J., & Buscaglia, G. C. (2008). A discontinuous-Galerkin-based immersed boundary method. International Journal for Numerical Methods in Engineering, 76, 427–454.

    Google Scholar 

  • Li, J., Melenk, J. M., Wohlmuth, B., & Zou, J. (2010). Optimal a priori estimates for higher order finite elements for elliptic interface problems. Applied Numerical Mathematics, 60, 19–37.

    Article  MathSciNet  MATH  Google Scholar 

  • Li, Z., & Ito, K. (2006). The immersed interface method. SIAM.

    Google Scholar 

  • Liao, Q., & Silvester, D. (2013). Robust stabilized stokes approximation methods for highly stretched grids. IMA Journal of Numerical Analysis, 33, 413–431.

    Article  MathSciNet  MATH  Google Scholar 

  • Massing, A., Larson, M. G., Logg, A., & Rognes, M. E. (2012). A stabilized Nitsche fictitious domain method for the stokes problem. Journal of Scientific Computing, 61(3), 604–628.

    Google Scholar 

  • Maury, B. (2001). A fat boundary method for the poisson problem in a domain with holes. Journal of Scientific Computing, 16, 319–339.

    Article  MathSciNet  MATH  Google Scholar 

  • Maury, B. (2009). Numerical analysis of a finite element/volume penalty method. SIAM Journal on Numerical Analysis, 47(2), 1126–1148.

    Article  MathSciNet  MATH  Google Scholar 

  • Melenk, J. M., & Babuška, I. (1996). The partition of unity finite element method: basic theory and applications. Computer Methods in Applied Mechanics and Engineering, 139, 289–314.

    Article  MathSciNet  MATH  Google Scholar 

  • Micheletti, S., Perotto, S., & Picasso, M. (2004). Stabilized finite elements on anisotropic meshes: A priori error estimates for the advection-diffusion and the stokes problems. SIAM Journal on Numerical Analysis, 41(3), 1131–1162.

    Article  MathSciNet  MATH  Google Scholar 

  • Moës, N., Dolbow, J., & Belytschko, T. (1999). A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 46, 131–150.

    Article  MATH  Google Scholar 

  • Nicaise, S. (1993). Polygonal interface problems. Switzerland: Peter Lang.

    Google Scholar 

  • Parvizian, J., Düster, A., & Rank, E. (2007). Finite cell method. Computational Mechanics, 41, 121–133.

    Article  MathSciNet  MATH  Google Scholar 

  • Pedrizzetti, G. (2005). Kinematic characterization of valvular opening. Physical Review Letters, 95, 194502.

    Article  Google Scholar 

  • Peskin, C. (1977). Numerical analysis of blood flow in the heart. Journal of Computational Physics, 25, 220–252.

    Article  MathSciNet  MATH  Google Scholar 

  • Peskin, C. S. (2002). The immersed boundary method. Acta Numerica, 11, 1–39.

    Article  MathSciNet  MATH  Google Scholar 

  • Rand, A. (2009). Delaunay refinement algorithms for numerical methods. PhD thesis, Carnegie Mellon University.

    Google Scholar 

  • Sanders, J. D., Laursen, T. A., & Puso, M. A. (2012). A Nitsche embedded mesh method. Computational Mechanics, 49, 243–257.

    Article  MathSciNet  MATH  Google Scholar 

  • van Brummelen, E. H., van der Zee, K. G., Garg, V. V., & Prudhomme, S. (2011). Flux evaluation in primal and dual boundary-coupled problems. Journal of Applied Mechanics, 79, 010904.

    Article  Google Scholar 

  • van Loon, R., Anderson, P. D., & van de Vosse, F. N. (2006). A fluid-structure interaction method with solid-rigid contact for heart valve dynamics. Journal of Computational Physics, 217, 806–823.

    Article  MathSciNet  MATH  Google Scholar 

  • Yu, Z. (2005). A DLM/FD method for fluid/flexible-body interactions. Journal of Computational Physics, 207, 1–27.

    Article  MATH  Google Scholar 

  • Ženíšek, A. (1969). The convergence of the finite element method for boundary value problems of a system of elliptic equations (in czech). APL Materials, 14, 355–377.

    MATH  Google Scholar 

  • Zlámal, M. (1968). On the finite element method. Numerical Mathematics, 12, 394–409.

    Article  MATH  Google Scholar 

  • Zunino, P., Cattaneo, L., & Colciago, C. M. (2011). An unfitted interface penalty method for the numerical approximation of contrast problems. Applied Numerical Mathematics, 61, 1059–1076.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This work is partially funded by iCardioCloud project by Cariplo Foundation (No. 2013-1779)) and Lumbardy Region (No. 42938382; No. 46554874); ERC Starting Grant through the Project ISOBIO: Isogeometric Methods for Biomechanics (No. 259229); The authors would also like to acknowledge the support of Franco Brezzi and Alessandro Veneziani in the realization of this work.

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Authors and Affiliations

  1. Department of Civil Engineering and Architecture, University of Pavia, Pavia, Italy

    Ferdinando Auricchio & Alessandro Reali

  2. Department of Medicine, Emory University, Atlanta, GA, USA

    Adrien Lefieux

  3. Institute for Advanced Study, Technische Universitat München, Munchen, Germany

    Alessandro Reali

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  1. Ferdinando Auricchio
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  2. Adrien Lefieux
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  3. Alessandro Reali
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Correspondence to Adrien Lefieux .

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Editors and Affiliations

  1. Institute of Mechanics, University of Duisburg-Essen, Essen, Germany

    Jörg Schröder

  2. University of Hannover, Hannover, Germany

    Peter Wriggers

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Auricchio, F., Lefieux, A., Reali, A. (2016). On the Use of Anisotropic Triangles with Mixed Finite Elements: Application to an “Immersed” Approach for Incompressible Flow Problems. In: Schröder, J., Wriggers, P. (eds) Advanced Finite Element Technologies. CISM International Centre for Mechanical Sciences, vol 566. Springer, Cham. https://doi.org/10.1007/978-3-319-31925-4_8

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