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Analysis of Stokes flows by Carrera unified formulation

  • Varello, Alberto (Department of Mechanical and Aerospace Engineering, Politecnico di Torino) ;
  • Pagani, Alfonso (Department of Mechanical and Aerospace Engineering, Politecnico di Torino) ;
  • Guarnera, Daniele (Department of Mechanical and Aerospace Engineering, Politecnico di Torino) ;
  • Carrera, Erasmo (Department of Mechanical and Aerospace Engineering, Politecnico di Torino)
  • Received : 2017.09.08
  • Accepted : 2017.11.08
  • Published : 2018.05.25
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Abstract

One-dimensional (1D) models of incompressible flows, can be of interest for many applications in which fast resolution times are demanded, such as fluid-structure interaction of flows in compliant pipes and hemodynamics. This work proposes a higher-order 1D theory for the flow-field analysis of incompressible, laminar, and viscous fluids in rigid pipes. This methodology is developed in the domain of the Carrera Unified Formulation (CUF), which was first employed in structural mechanics. In the framework of 1D modelling, CUF allows to express the primary variables (i.e., velocity and pressure fields in the case of incompressible flows) as arbitrary expansions of the generalized unknowns, which are functions of the 1D computational domain coordinate. As a consequence, the governing equations can be expressed in terms of fundamental nuclei, which are invariant of the theory approximation order. Several numerical examples are considered for validating this novel methodology, including simple Poiseuille flows in circular pipes and more complex velocity/pressure profiles of Stokes fluids into non-conventional computational domains. The attention is mainly focused on the use of hierarchical McLaurin polynomials as well as piece-wise nonlocal Lagrange expansions of the generalized unknowns across the pipe section. The preliminary results show the great advantages in terms of computational costs of the proposed method. Furthermore, they provide enough confidence for future extensions to more complex fluid-dynamics problems and fluid-structure interaction analysis.

Keywords

References

  1. Brezzi, F. (1974), "On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers", Revue Francaise de Automatique, Informatique, Recherche Operationnelle. Analyse Numerique, 8,129-151. https://doi.org/10.1051/m2an/197408R201291
  2. Carrera, E., Petrolo, M. and Varello, A. (2012), "Advanced beam formulations for free-vibration analysis of conventional and joined wings", J. Aerosp. Eng., 25(2), 282-293. https://doi.org/10.1061/(ASCE)AS.1943-5525.0000130
  3. Carrera, E. and Pagani,A. (2014), "Free vibration analysis of civil engineering structures by component-wise models", J. Sound Vibr., 333(19), 4597-4620. https://doi.org/10.1016/j.jsv.2014.04.063
  4. Carrera, E. and Giunta, G. (2010), "Refined beam theories based on a unified formulation", Int. J. Appl. Mech., 2(1), 117-143. https://doi.org/10.1142/S1758825110000500
  5. Carrera, E. and Petrolo, M. (2012), "Refined beam elements with only displacement variables and plate/shell capabilities", Meccan., 47(3), 537-556. https://doi.org/10.1007/s11012-011-9466-5
  6. Carrera, E., and Cinefra, M., Petrolo, M. and Zappino, E. (2014), Finite Element Analysis of Structures Through Unified Formulation, John Wiley & Sons Ltd.
  7. Carrera, E. and Zappino, E. (2017), "One-dimensional finite element formulation with node-dependent kinematics", Comput. Struct., 192(1), 114-125. https://doi.org/10.1016/j.compstruc.2017.07.008
  8. Carrera, E., Cinefra, M., Petrolo, M. and Zappino, E. (2014), Finite Element Analysis of Structures through Unified Formulation, John Wiley & Sons Ltd.
  9. Euler, L. (1775) Principia Pro Motu Sanguinis Per Arterias Determinando.
  10. Fung, Y.C. (1997), Biomechanics: Circulation, Springer.
  11. Filippi, M. and Carrera, E. (2016), "Capabilities of 1d cuf-based models to analyse metallic/composite rotors" Adv. Aircraft Spacecraft Sci., 3(1), 1-14. https://doi.org/10.12989/aas.2016.3.1.001
  12. Formaggia, L., Gerbeau, J.F., Nobile, F. and Quarteroni, A. (2001), "On the coupling of 3d and 1d Navier-Stokes equations for flow problems in compliant vessels", Comput. Meth. Appl. Mech. Eng., 561-582.
  13. Formaggia, L., Lamponi, D. and Quarteroni, A. (2003), "One-dimensional models for blood flow in arteries", J. Eng. Math., 251-279.
  14. Giunta, G., Biscani, F., Belouettar, S., Ferreira, A.J.M. and Carrera, E. (2013), "Free vibration analysis of composite beams via refined theories", Compos. Part B: Eng., 44(1), 540-552. https://doi.org/10.1016/j.compositesb.2012.03.005
  15. Miglioretti, F. and Carrera, E. (2015), "Application of a refined multi-field beam model for the analysis of complex configurations", Mech. Adv. Mater. Struct., 22(1-2), 52-66. https://doi.org/10.1080/15376494.2014.912365
  16. OpenFOAM Fundation (2011), OpenFOAM, .
  17. Pagani, A. (2015) "Component-wise models for static,dynamic and aeroelastic analyses of metallic and composite aerospace structures", Ph.D. Dissertation, Politecnico di Torino, Italy.
  18. Pagani, A., De Miguel, A., Petrolo, M. and Carrera, E. (2016), "Analysis of laminated beams via unified formulation and legendre polynomial expansions", Compos. Struct., 156, 78-92 . https://doi.org/10.1016/j.compstruct.2016.01.095
  19. Perotto, S., Ern, A. and Veneziani, A. (2009), "Hierarchical local model reduction for elliptic problems i: A domain decomposition approach", A SIAM Interdiscipl. J., 8, 1102-1127.
  20. Perotto, S., Reali, A., Rusconi, P. and Veneziani, A. (2017), " Higamod: A hierarchical isogeometric approach for {MODel} reduction in curved pipes", Comput. Flu., 142, 21-29. https://doi.org/10.1016/j.compfluid.2016.04.014
  21. Quarteroni, A., Formaggia, L. and Veneziani, A. (2009), Cardiovascular Mathematics, 1.
  22. Quarteroni, A. and Rozza, G. (2006), "Numerical solution of parametrized navier-stokes equations by reduced basis methods", Numer. Meth. Part. Differ. Eq., 923 -948.
  23. Quarteroni, A. (2009), Numerical Models for Differential Problems, Springer.
  24. Ravindran, S.S. (2000), "Reduced-order approach for optimal control of fluids using proper orthogonal decomposition", Int. J. Numer. Meth. Flu.
  25. Stera, S. and Skalak, R. (1993), The History of Poiseuille's Law.
  26. Salmoiraghi, F., Ballarin, F., Heltai, L. and Rozza, G. (2016), "Isogeometric analysis-based reduced order modelling for incompressible linear viscous flows in parametrized shapes", Adv. Model. Simul. Eng. Sci., 3-21 .
  27. Smith, N.P., Pullan, A.J. and Hunter, P.J. (2002), "An anatomically based model of transient coronary blood flow in the heart", SIAM J. Appl. Math., 990-1018.
  28. Sherewin, S.J., Franke, V. and Peiro, J. (2003), "One-dimensional modelling of a vascular network in spacetime variables", J. Eng. Math., 217-250 .
  29. Vreugdenhil, C.B. (1998), Numerical Methods for Shallow/Water Flows.
  30. Varello, A. (2013), "Advanced higher-order one-dimensional models for fluid-structure interaction analysis", Ph.D. Dissertation, Politecnico di Torino, Italy.
  31. Zienkiewicz, O.C. and Taylor, R.L. (1977), The Finite Element Method, McGraw-Hill, London, U.K.

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