# Presentation: 226783

## Finite Element Solution of Fluid-Structure Interaction Problems

- Gordon C. Everstine
- Naval Surface Warfare Center, Carderock Div.
- Bethesda, Maryland 20817
- 18 May 2000
- EverstineGC@nswccd.navy.mil

## Fluid-Structure Interaction

- Exterior Problems: Vibrations	, Radiation and Scattering,
- 		       Shock Response
- Interior Problems:  Acoustic Cavities, Piping Systems

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## Structural Acoustics

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## Large-Scale Fluid-Structure Modeling Approaches

- Structure
- Finite elements
- Fluid
- Boundary elements
- Finite elements with absorbing boundary
- Infinite elements
- ρc impedance
- Doubly asymptotic approximations (shock)
- Retarded potential integral equation (transient)

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## Exterior Fluid Mesh

- 383,000 Structural DOF

- 248,000 Fluid DOF

- 631,000 Total DOF

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## Structural-Acoustic Analogy

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## Fluid-Structure Interaction Equations

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## Fluid Finite Elements

- Pressure Formulation
- Ee = 1020Ge, e=Ge/c2, Ge arbitrary
- Direct input of areas in K and M matrices
- Symmetric Potential Formulation
- uz represents velocity potential
- New unknown: q= p dt  (velocity potential)
- Ge=-1/, Ee=-1020/, e =-1/(c2)
- Direct input of areas in B (damping) matrix

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## Finite Element Formulations of FSI

- + 3-variable formulations

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## Displacement Formulation

- Fundamental unknown: fluid displacement (3 DOF/point)
- Model fluid domain with elastic F.E. (e.g., elastic solids in 3-D, membranes in 2-D)
- Any coordinate systems; constrain rotations (DOF 456)
- Material properties (3-D): Ge0    Ee=(6)c2, e=1⁄2-, e= , where  =10-4
- Boundary conditions:
- Free surface: natural B.C.
- Rigid wall: un=0 (SPC or MPC)
- Accelerating boundary: un continuous (MPC), slip
- Real and complex modes, frequency and transient response
- 3 DOF/point, spurious modes

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## Displacement Method Mode Shapes

- 0 Hz
- Spurious

- 1506 Hz
- Good

- 1931 Hz
- Spurious

- 1971 Hz
- Good

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## Helmholtz Integral Equations

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## Matrix Formulation of Fluid-Structure Problem

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## Spherical Shell With Sector Drive

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## Added Mass by Boundary Elements

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## Frequencies of Submerged Cylindrical Shell

- N=circumferential, M=longitudinal, L=radial (end)

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## Low Frequency F.E. Piping Model

- Beam model for pipe
- 1-D acoustic fluid model for fluid (rods)
- Two sets of coincident grid points
- Pipe and fluid have same transverse motion
- Elbow flexibility factors are used
- Adjusted fluid bulk modulus for fluid in elastic pipes E=B/[1+BD/Est)]
- Arbitrary geometry, inputs, outputs
- Applicable below first lobar mode

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## Planar Piping System: Free End Response 

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## Needs

- Link between CAD model and FE model
- Infinite elements
- Meshing (e.g., between hull and outer fluid FE surface
- Modeling difficulties (e.g., joints, damping, materials, mounts)
- Error estimation and adaptive meshing

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## References

- G.C. Everstine, "Structural Analogies for Scalar Field Problems," *Int. J. Num. Meth. in Engrg*., Vol. 17, No. 3, pp. 471-476 (March 1981).
- G.C. Everstine, "A Symmetric Potential Formulation for Fluid-Structure Interaction," *J. Sound and Vibration*, Vol. 79, No. 1, pp. 157-160 (Nov. 8, 1981).
- G.C. Everstine, "Dynamic Analysis of Fluid-Filled Piping Systems Using Finite Element Techniques," *J. Pressure Vessel Technology*, Vol. 108, No. 1, pp. 57-61 (Feb. 1986).
- G.C. Everstine and F.M. Henderson, "Coupled Finite Element/Boundary Element Approach for Fluid-Structure Interaction," *J. Acoust. Soc. Amer*., Vol. 87, No. 5, pp. 1938-1947 (May 1990).
- G.C. Everstine, "Prediction of Low Frequency Vibrational Frequencies of Submerged Structures," *J. Vibration and Acoustics*, Vol. 13, No. 2, pp. 187-191 (April 1991).
- G.C. Everstine, "Finite Element Formulations of Structural Acoustics Problems," *Computers and Structures*, Vol. 65, No. 3, pp. 307-321 (1997).

- http://ocean.dt.navy.mil/pubs/_pubs.htm

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