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Research Papers: Fluid-Structure Interaction

Linear Stability Analysis and Application of a New Solution Method of the Elastodynamic Equations Suitable for a Unified Fluid-Structure-Interaction Approach

[+] Author and Article Information
C. G. Giannopapa1

Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlandsc.g.giannopapa@tue.nl

G. Papadakis

Experimental and Computational Laboratory for the Analysis of Turbulence, Department of Mechanical Engineering, King’s College London, Strand WC2R 2LS, UKgeorge.papadakis@kcl.ac.uk

1

Corresponding author.

J. Pressure Vessel Technol 130(3), 031303 (Jul 11, 2008) (8 pages) doi:10.1115/1.2937764 History: Received January 29, 2006; Revised January 03, 2007; Published July 11, 2008

In the conventional approach for fluid-structure-interaction problems, the fluid and solid components are treated separately and information is exchanged across their interface. According to the conventional terminology, the current numerical methods can be grouped in two major categories: partitioned methods and monolithic methods. Both methods use separate sets of equations for fluid and solid that have different unknown variables. A unified solution method has been presented in the previous work of Giannopapa and Papadakis (2004, “A New Formulation for Solids Suitable for a Unified Solution Method for Fluid-Structure Interaction Problems  ,” ASME PVP 2004, San Diego, CA, July, PVP Vol. 491–1, pp. 111–117), which is different from these methods. The new approach treats both fluid and solid as a single continuum; thus, the whole computational domain is treated as one entity discretized on a single grid. Its behavior is described by a single set of equations, which are solved fully implicitly. In this paper, the elastodynamic equations are reformulated so that they contain the same unknowns as the Navier–Stokes equations, namely, velocities and pressure. Two time marching and one spatial discretization scheme, widely used for fluid equations, are applied for the solution of the reformulated equations for solids. Using linear stability analysis, the accuracy and dissipation characteristics of the resulting difference equations are examined. The aforementioned schemes are applied to a transient structural problem (beam bending) and the results compare favorably with available analytic solutions and are consistent with the conclusions of the stability analysis. A parametric investigation using different meshes, time steps, and beam dimensions is also presented. For all cases examined, the numerical solution was stable and robust and therefore is suitable for the next stage of application to full fluid-structure-interaction problems.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

Classification of FSI methods

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Figure 2

Integration of velocity

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Figure 3

(a) Stencil for the 1D displacement formulation and (b) stencil of the 1D velocity formulation

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Figure 4

Comparison of amplitude portraits for the 1D velocity and displacement formulations

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Figure 5

Two-dimensional beam-bending case

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Figure 6

Comparison of end displacement obtained with the standard stress analysis and the velocity-pressure formulation (Co=0.33)

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Figure 7

Comparison of Euler implicit and backward differencing scheme (envelope of end displacement)

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Figure 8

Effect of time step on the envelope of end displacement (Euler implicit scheme with time steps 10−4s, 10−5s, and 10−6s)

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Figure 9

Comparison of envelope of end displacement obtained with Euler implicit (time step 10−5s) and backward differencing (time step 10−4s)

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Figure 10

Effect of mesh resolution on the envelope of end displacement (meshes 40×10, 60×20, and 200×50)

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Figure 11

End displacement for a beam with dimensions 10×5m2

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Figure 12

End displacement for a beam with dimensions 40×5m2

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Figure 13

End displacement for a beam with dimensions 20×5m2 and applied end shear τ=5×105Pa

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