Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
On Constitutive Models for the Momentum Transfer to Particles in Fluid-Dominated Two-Phase Flows Read chapter Read first chapter

2018 | OriginalPaper | Chapter

Remarks on a Fluid-Structure Interaction Scheme Based on the Least-Squares Finite Element Method at Small Strains

Authors : Carina Nisters, Alexander Schwarz, Solveigh Averweg, Jörg Schröder

Published in: Advances in Mechanics of Materials and Structural Analysis

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The present contribution introduces a least-squares finite element method (LSFEM) based fluid-structure interaction (FSI) approach. The proposed method is based on the formulation of mixed finite elements in terms of stresses and velocities for both the fluid and the solid regime. The LSFEM offers the advantage of a flexibility to construct functionals with sophisticated physical quantities as e.g. stresses, velocities and displacements. The approximation of the stresses and velocities in suitable spaces, namely in the spaces \(H(\text {div})\) and \(H^1\), respectively, leads to the inherent fulfillment of the coupling conditions of a FSI method. A numerical example considering an incompressible, linear elastic material behavior at small deformations and the incompressible Navier–Stokes equations demonstrates the applicability of the LSFEM-FSI method.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now
previous chapter Application of Piezoelements in a Structured Thin-Walled Shaft Material to Active Control of Stability and Vibration
next chapter Dynamic Analysis of Gradient Poroelastic Solids and Structures
Appendix
Available only for authorised users
Literature
1.
Ali, A., Reimann, T., Sternel, D.C., Schäfer, M.: Comparison of advanced turbulence modeling approaches for fluid-structure interaction. In: Coupled Problems in Science and Engineering VI, CIMNE (2015)
2.
Balzani, D., Deparis, S., Fausten, S., Forti, D., Heinlein, A., Klawonn, A., Quarteroni, A., Rheinbach, Q., Schröder, J.: Numerical modeling of fluid-structure interaction in arteries with anisotropic polyconvex hyperelastic and anisotropic viscoelastic material models at finite strains. Int. J. Numer. Method Biomed. Eng. (2015)
3.
Bazilevs, Y., Hsu, M., Kiendl, J., Wüchner, R., Bletzinger, K.: 3D Simulation of wind turbine rotors at full scale. Part II: Fluid - structure interaction modeling with composite blades. Int. J. Numer. Methods Fluids 65, 236–253 (2011)
4.
Bochev, P.B., Gunzburger, M.D.: Least-squares methods for the velocity-pressure-stress formulation of the Stokes equations. Comput. Methods Appl. Mech. Eng. 126(3–4), 267–287 (1995)MathSciNetCrossRefMATH
5.
Bochev, P.B., Gunzburger, M.D.: Least-Squares Finite Element Methods, 1st edn. Springer, New York (2009)MATH
6.
Bodnár, T., Galdi, G., Nečasová, S.: Fluid-Structure Interaction and Biomedical Applications. Springer, Berlin (2014)CrossRefMATH
7.
Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Berlin (2013)CrossRefMATH
8.
9.
Brooks, A.N., Hughes, T.J.R.: 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 (1982)MathSciNetCrossRefMATH
10.
Bungartz, H.-J., Mehl, M., Schäfer, M.: Fluid Structure Interaction II: Modelling, Simulation, Optimization. Springer, Berlin (2010)CrossRefMATH
11.
Cai, Z., Lee, B., Wang, P.: Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems. SIAM J. Numer. Anal. 42(2), 843–859 (2004)MathSciNetCrossRefMATH
12.
Cori, J.-F., Etienne, S., Garon, A., Pelletier, D.: High-order implicit Runge–Kutta time integrators for fluid-structure interactions. Int. J. Numer. Methods Fluids 78, 385–412 (2015)MathSciNetCrossRef
13.
Deang, J.M., Gunzburger, M.D.: Issues related to least-squares finite element methods for the Stokes equations. SIAM J. Sci. Comput. 20(3), 878–906 (1998)MathSciNetCrossRefMATH
14.
Donea, J., Huerta, A.: Finite Element Methods for Flow Problems. Wiley, New York (2003)CrossRef
15.
Förster, Ch., Wall, W.A., Ramm, E.: Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows. Comput. Methods Appl. Mech. Eng. 196(7), 1278–1293 (2007)MathSciNetCrossRefMATH
16.
Glück, M., Breuer, M., Durst, F., Halfmann, A., Rank, E.: Computation of fluid-structure interaction on lightweight structures. J. Wind Eng. Ind. Aerodyn. 89(14), 1351–1368 (2001)CrossRef
17.
Hron, J., Turek, S.: A monolithic FEM solver for an ALE formulation of fluid-structure interaction with configuration for numerical benchmarking. In: European Conference on Computational Fluid Dynamics ECCOMAS CFD (2006)
18.
Hughes, T.J.R., Brooks, A.N.: Multi-dimensional upwind scheme with no crosswind diffusion. ASME, Appl. Mech. Div. 34, 19–35 (1979)MATH
19.
Hughes, T.J.R., Franca, L.P., Hulbert, G.M.: 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, 173–189 (1989)MathSciNetCrossRefMATH
20.
Jiang, B.-N.: The Least-Squares Finite Element Method, Scientific Computation. Springer, Berlin (1998)CrossRef
21.
Kayser-Herold, O., Matthies, H.G.: A unified least-squares formulation for fluid-structure interaction problems. Comput. Struct. 85, 998–1011 (2007)MathSciNetCrossRef
22.
Korelc, J.: Automatic generation of finite-element code by simultaneous optimization of expressions. Theor. Comput. Sci. 187(1), 231–248 (1997)CrossRefMATH
23.
Korelc, J.: Multi-language and multi-environment generation of nonlinear finite element codes. Eng. Comput. 18(4), 312–327 (2002)CrossRef
24.
Küttler, U., Gee, M., Förster, Ch., Comerford, A., Wall, W.A.: Coupling strategies for biomedical fluid - structure interaction problems. Int. J. Numer. Methods Biomed. Eng. 26, 305–321 (2010)MathSciNetCrossRefMATH
25.
Küttler, U., Wall, W.A.: Fixed-point fluid-structure interaction solvers with dynamic relaxation. Comput. Mech. 43(1), 61–72 (2008)CrossRefMATH
26.
Münzenmaier, S., Starke, G.: First-order system least squares for coupled Stokes–Darcy flow. SIAM J. Numer. Anal. 49(1), 387–404 (2011)MathSciNetCrossRefMATH
27.
Nisters, C., Schwarz, A.: Efficient stress-velocity least-squares finite element formulations for the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. (2017, in revision)
28.
Ozcelikkale, A., Sert, C.: Least-squares spectral element solution of incompressible Navier–Stokes equations with adaptive refinement. J. Comput. Phys. 231(9), 3755–3769 (2012)MathSciNetCrossRefMATH
29.
Pian, T.H.H., Sumihara, K.: A rational approach for assumed stress finite elements. Int. J. Numer. Methods Eng. 20, 1685–1695 (1984)CrossRefMATH
30.
Prabhakar, V., Pontaza, J.P., Reddy, J.N.: A collocation penalty least-squares finite element formulation for incompressible flows. Comput. Methods Appl. Mech. Eng. 197, 449–463 (2008)MathSciNetCrossRefMATH
31.
Raviart, P.-A., Thomas, J.M.: A mixed finite element method for 2-nd order elliptic problems. Mathematical Aspects of Finite Element Methods, pp. 292–315. Springer, Berlin (1977)CrossRef
32.
Reddy, J.N.: Penalty-finite-element analysis of 3-D Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 35, 87–97 (1982)CrossRefMATH
33.
Reddy, J.N., Gartling, D.K.: The Finite Element Method in Heat Transfer and Fluid Dynamics, 3rd edn. CRC Press, Boca Raton (2010)MATH
34.
Schröder, J., Schwarz, A., Steeger, K.: Least-squares finite element formulations for isotropic and anisotropic elasticity at small and large strains. Advanced Finite Element Technologies. CISM Courses and Lectures, pp. 131–175. Springer, Berlin (2016)CrossRef
35.
Schröder, W.: Summary of Flow Modulation and Fluid-Structure Interaction Findings. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 109. Springer, Berlin (2010)CrossRef
36.
Schwarz, A., Nickaeen, M., Serdas, S., Nisters, C., Ouazzi, A., Schröder, J., Turek, S.: A comparative study of mixed least-squares FEMs for the incompressible Navier–Stokes equations. Int. J. Comput. Sci. Eng. (2016)
37.
Schwarz, A., Schröder, J., Serdas, S., Turek, S., Ouazzi, A., Nickaeen, M.: Performance aspects of a mixed s-v LSFEM for the incompressible Navier–Stokes equations with improved mass conservation. Proc. Appl. Math. Mech. 13, 97–98 (2013)CrossRef
38.
Schwarz, A., Steeger, K., Schröder, J.: Weighted overconstrained least-squares mixed finite elements for static and dynamic problems in quasi-incompressible elasticity. Comput. Mech. 54(3), 603–612 (2014)MathSciNetCrossRefMATH
39.
Simo, J.C., Rifai, M.S.: A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Methods Eng. 29, 1595–1638 (1990)MathSciNetCrossRefMATH
40.
41.
Tian, F.-B., Dai, H., Luo, H., Doyle, J.F., Rousseau, B.: Fluid-structure interaction involving large deformations: 3D simulations and applications to biological systems. J. Comput. Phys. 258, 451–469 (2014)MathSciNetCrossRefMATH
42.
Turek, S., Hron, J., Razzaq, M., Wobker, H., Schäfer, M.: Numerical benchmarking of fluid-structure interaction: a comparison of different discretization and solution approaches (2010)
43.
van Zuijlen, A.H., Bijl, H.: A higher-order time integration algorithm for the simulation of nonlinear fluid-structure interaction. Nonlinear Anal. Theory Methods Appl. 63(5–7), e1597–e1605 (2005)CrossRefMATH
44.
Wall, W.A., Genkinger, S., Ramm, E.: A strong coupling partitioned approach for fluid-structure interaction with free surfaces. Comput. Fluids 36(1), 169–183 (2007)CrossRefMATH
45.
Wilson, E.L., Taylor, R.L., Doherty, W.P., Ghaboussi, J.: Incompatible displacement models. Numer. Comput. Methods Struct. Mech. 43 (1973)
46.
Wolfram Research Inc. Mathematica. Campaign: Wolfram Research, Inc. Version 10.1 edition (2015)
47.
Zienkiewicz, O.C., Taylor, R.L., Nithiarasu, P.: Fluid-structure interaction. The Finite Element Method for Fluid Dynamics, vol. 3, pp. 423–449 (2014)
Metadata
Title
Remarks on a Fluid-Structure Interaction Scheme Based on the Least-Squares Finite Element Method at Small Strains
Authors
Carina Nisters
Alexander Schwarz
Solveigh Averweg
Jörg Schröder
Copyright Year
2018
Book
Advances in Mechanics of Materials and Structural Analysis

Print ISBN: 978-3-319-70562-0

Electronic ISBN: 978-3-319-70563-7

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-70563-7_12

Premium Partners

    Image Credits