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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)
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
• Weak Coupling
* after Claus Pedersen
• Acoustic Transfer
Vectors (computed,
measured; direct or
inverse approach)
• Strong CouplingCoupled FEM/BEM,
FEM/FEM, …
2
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
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
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 τ ττ σ ττ τ σ
=
σ
= components of stress vectors in cartesian coordinate system.
3
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 =
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
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:
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
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+ + =
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
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
6
3-D Fluid-structure coupling in linear acoustics
Pressure = normal projection of stress tensor
P− n ⋅σ n
Condition 1: Equilibrium (equal normal forces)
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
Pressure = normal projection of stress tensor
F ⋅v n
Condition 2: Kinematic compatibility (equal normal displacements)
S ⋅v n
7
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.
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
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
σ τ ττ σ ττ τ σ
+ + = −
+ + = −
+ + = −
Dynamic coupling condition
The normal is pointing out of the solid.
8
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
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
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
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
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
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
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 =
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
( ) ( )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
= + + +
= + + + −
− + + +
= −
∫ ∫ ∫
∫ ∫
∫ ∫
∫ ∫
12
Symmetry considerations
FE-partition of structure and fluid
Coupled equations (weak form)
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
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)
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
• 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
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
• 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
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
• 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
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
,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
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction 19
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
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
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
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction
• 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
HAW-M+P-CompA-Ihlenburg Acoustic Fluid-Structure Interaction 22