CompA FSI Fem

Vibroacoustic Fluid-Structure Interaction with FEM

1. Governing Equations for 3-D Structure-Fluid Interaction

2. FEM for free vibrations (symmetry considerations)

3. FEM for forced vibrations (modal superposition)

Review: 1-D Structure-Fluid Interaction 3-D

HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction

x

l

Structural (normal) velocities

Fluid pressure

• Velocity-pressure equilibrium in normal direction

1

Levels of Structure-Fluid coupling*

• Weak Coupling

• No CouplingEquivalent Radiated Power = averaged normal velocities

21ERP=

2l

f n ll

c v Aρ ∑

• Normal velocities =

boundary conditions

for fluid ( FEM, BEM)


• Weak Coupling

* after Claus Pedersen

• Acoustic Transfer

Vectors (computed,

measured; direct or

inverse approach)

• Strong CouplingCoupled FEM/BEM,

FEM/FEM, …

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1. Governing Equations for 3-D Structure-Fluid Inte raction

2t

3t

P

x

y

z

τxzτ

yσ

yzτ

zσ

zyτzxτ

τ

Stress vectors Stress Tensor

Review: stress tensor, equilibrium equations, boundary tractions


1tx

xσ xyτ yyxτ

1

2

3

x x xy y xz z

yx x y y yz z

zx x zy y z z

σ τ ττ σ ττ τ σ

= + +

= + +

= + +

t e e e

t e e e

t e e e

x xy xz

xy y yz

xz yz z


=

σ

= components of stress vectors in cartesian coordinate system.

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Review: Static equilibrium

dxdy

yσ

yxyx dy

y

ττ

∂+

∂

yxτ

yy dy

y

σσ

∂+

∂

xσxyτ x

x x

σσ ∂+∂

xyyx dx

x

ττ

∂+

∂

2-D: Plane stress

0

0

yxx

xy y

x y

x y

τσ

τ σ

∂∂ + =∂ ∂

∂ ∂+ =

∂ ∂

, 0ij iσ =Symmetry of stress tensor

, 0ij jσ =


0

0

0

yxx zxx

xy y zyy

yzxz zz

fx y z

fx y z

fx y z

τσ τ

τ σ τ

ττ σ

∂∂ ∂+ + + =∂ ∂ ∂

∂ ∂ ∂+ + + =

∂ ∂ ∂∂∂ ∂+ + + =

∂ ∂ ∂xσ xyτxzτ

yσ

yzτ

zσ

zyτzxτ

yxτ

3-D:if −

In continuum mechanics, one distinguishes between tractions (acting at the faces of control volumes) and volume forces (acting on all interior particles of a volume). Examples of volume forces are gravitational forces or forces of inertia.

Components of a volume force.

, 0ij j ifσ + =4

Dynamic equilibrium equations

0

0

0

yxx zxx

xy y zyy

yzxz zz

fx y z

fx y z

f

τσ τ

τ σ τ

ττ σ

∂∂ ∂+ + + =∂ ∂ ∂

∂ ∂ ∂+ + + =

∂ ∂ ∂∂∂ ∂+ + + =

∂ ∂ ∂

xx

yy

zz

f u

f u

f u

ρ

ρ

ρ

= −

= −

= −

ii

ii

ii

Inertia forces

, 0iij j uσ ρ− =ii

Equilibrium equations of solid dynamics

Solid volume:


0zfx y z

+ + + =∂ ∂ ∂

zzf uρ= −

ext⋅ =σ n t

Solid boundary:

5

Review: Boundary tractions

tThe stress vector at a boundary point is called boundary traction. It is computed from the general relation

= ⋅t σ n

x x yx y zx z xn n n tσ τ τ+ + =


x x yx y zx z x

xy x y y zy z y

xz x yz y z z z

n n n t

n n n t

τ σ ττ τ σ

+ + =

+ + =

The boundary traction must be in equilibrium with the external load vector (or the reaction forces) at every point of the boundary.

See, e.g., Gross, Hauger, Schnell, Technische Mechanik 2, 4

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3-D Fluid-structure coupling in linear acoustics

Pressure = normal projection of stress tensor

P− n ⋅σ n

Condition 1: Equilibrium (equal normal forces)


Pressure = normal projection of stress tensor

F ⋅v n

Condition 2: Kinematic compatibility (equal normal displacements)

S ⋅v n

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Dynamic coupling contition

Consider a solid boundary in contact with an ideal fluid.Define a local coordinate system with normal and two tangential direction vectors. Then the external stress vector in local coordinates is

, , 0,0,ext n st t t P Pξ η= = − = −t n

since friction is neglected.


x x xy y xz z x

yx x y y yz z y

zx x zy y z z z

n n n Pn

n n n Pn

n n n Pn


+ + = −

+ + = −

+ + = −

Dynamic coupling condition

The normal is pointing out of the solid.

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Kinematic coupling condition

snfns f⋅ = ⋅U n U n

The normal displacements of solid and (ideal) fluid are equal. Since friction is neglected, no condition can be imposed on the tangential projections of the displacement vectors.

The fluid displacement vector can be replaced by the pressure value using the Euler


The fluid displacement vector can be replaced by the pressure value using the Euler equation in the fluid,

yielding

0 fP ρ∇ = − Uɺɺ

0 s Pρ ⋅ = −∇ ⋅U n nɺɺ

9

Equations of vibroacoustic solid-fluid interaction

The equations for a vibrating solid-fluid continuum consist of:

• dynamic equilibrium equations for solid

• acoustic equation for fluid (wave equation)

• equilibrium condition at solid-fluid interface

• continuity condition at solid-fluid interface (where fluid displacements are replaced by pressure using Euler‘s equation)

Figure: Solid-fluid continuum

sΩ

fΩwΓ


snfn

Figure: normals at interface

10

Multiply component-wise with „test displacement“ -W

Multiply with „test pressure“ -Q

… and integrate by parts …Apply first coupling condition

2. Finite Element equations for free vibrations


Apply second coupling condition

11

Illustration of Gauss Integral Theorem on rectangular domain

( ) ( )a b

σ σ τ τ σ = + + +∫ ∫ ∫

Ω

Γ

x

y

1,0n = 1,0n = −

0, 1n = −

0,1n =


( ) ( )0 0

0 00 0

0 0

, , , , ,

, , , ,

,

ij j i x x xy y x yx x y y y

b aa b

xy x y y x x yx y

y x

a b

x x x xy x y yx y x y y y

ij j i ij j i

W dA W W dxdy

W W dx W W dx

W W W W dxdy

n W ds W dA

σ σ τ τ σ

τ σ σ τ

σ τ τ σ

σ σ

Ω

= =

Γ Ω

= + + +

= + + + −

− + + +

= −

∫ ∫ ∫

∫ ∫

∫ ∫

∫ ∫

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Symmetry considerations

FE-partition of structure and fluid

Coupled equations (weak form)


Observation: the coupling matrices of the finite element model are obtained from element integrals of the product „fluid shape function * solid shape function“. Since these products are commutative, the governing equations can be scaled ( = multiplied by a constant factor) in such a way that the overall coupled system of equations is symmetric .

13

Coupled FE equations for free vibrations

• Coupled differential equations (weak form)

Introduce

• structural material law (stress strain),

• kinematic equations (strain displacements)


• kinematic equations (strain displacements)

• finite element partition of structure and fluid, finite element shape functions

s sf s

f fs f

− =

K -A M 0U U0

0 K A MPP

ii

ii

• Coupled system of FE equations

14

Coupling matrix (contains relations between structural and fluid nodes on the coupling surfaces).

Linear Dynamics of Structure-Fluid: algebraic equat ions (FEM)

HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction 15

3. Finite Element Model for forced vibrations

• Coupled FE model of forced response

Trick (Everstine et al.):

• derive structural equation in time

• substitute

-i0

ts s s

Tf f f

e ω − − − = − − −

K A C 0 M 0U U U F0 K 0 C A MP 0P P

i ii

i ii

.

V = U


• substitute

• Coupled symmetric system

V = U

-i1

ts s s

Tf f f

e ω − − = − − −

K 0 C A M 0V V V F0 K A C 0 MP 0P P

i ii

i ii

16

Time-harmonic solution

• Coupled FE model of forced response („time domain“)

-i -i, ,t te eω ω= =V v P p

-i1

ts s s

Tf f f

e ω − − = − − −

K 0 C A M 0V V V F0 K A C 0 MP 0P P

i ii

i ii


• Coupled time-harmonic system („frequency domain“)

, ,e e= =V v P p

2+ is s sT

f f f

Ω + Ω = − − −

K 0 C A M 0 v f0 K A C 0 M p 0

17

-i0

ts s s

Tf f f

e ω − − = − − −

K A C 0 M 0U U U F0 K 0 C A MP 0P P

i ii

i ii

,≈ Φ ≈ Φv y p y• Solution of eigenvalue problems …

Modal Superposition


,s s f f≈ Φ ≈ Φv y p y

2+ iT T T T Ts s s s s s s f s s s s

T T T T Tf f f f s f f f f f f

ω ω Φ Φ Φ Φ Φ Φ Φ Φ Φ

+ = −Φ Φ Φ Φ −Φ Φ −Φ Φ

K 0 C A M 0 v f0 K A C 0 M p 0

• Modal superposition …

separately for structure and fluid

! Modal reduction in coupled system using „uncoupled modes “ !

18

Fluid-Struktur Interaktion: Modale Kopplung


Nastran Solution Deck for Coupled Analysis SOL 111

CENDTITLE = BOX coupledLABEL = Pressure at "Ear Positions"ECHO = NONE

$ Messpunkte FluidSET 123=10224,10235,10243,10251DISP(SORT1,PUNCH) = 123

$METHOD(FLUID) = 50METHOD(STRUCTURE) = 51FREQ = 20DLOAD = 1

BEGIN BULKEIGRL,50,,100.EIGRL,51,,100.

Separate Eigenvalue Analysis for Fluid, Structure

Forcing Frequencies

Dynamic Load

Data output to „punch file“ box.pch


EIGRL,51,,100.FREQ1,20,26.0,1.0,75RLOAD1, 1, 101, 0.0, 0.0, 1111 DAREA, 101, 1, 1, 1.0 TABLED1 1111 ++ 0. 1. 10000. 1. ENDT $INCLUDE 'box_fluid.nas'INCLUDE 'box.nas'$PARAM G 0.06PARAM GFL 0.06PARAM, GRDPNT, 0PARAM, AUTOSPC, YESPARAM, PRGPST, NOPARAM, SNORM, 45.ENDDATA

Grid and element data for fluid, structure

Damping loss factors for fluid, structure

20

Modeling Considerations

• The mesh size in fluid and solid can differ• The boundaries do not need to have exactly the same contours (search algorithm for non-matching boundaries)• The node numbers in fluid and solid must be different


• Parameters for the coupling procedure: ACMODL parameter in the bulk data section.

21

15:46:35 Application of Loads and Boundary Conditions to the finite element model started.15:46:35 Application of Loads and Boundary Conditions to the finite element model successfully completed.15:46:35 Solution of the system equations for normal modes started.15:46:39 Solution of the system equations for normal modes successfully completed.15:46:39 Solution of the system equations for normal modes started.15:46:39 Solution of the system equations for normal modes successfully completed.15:46:39 Solution of the system equations for frequency response started.15:46:40 Solution of the system equations for frequency response successfully completed.15:46:40 Frequency response analysis completed.

.log file


CompA FSI Fem

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