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1. Continuum Mechanics
Continuum
Solid mechanics
Elasticity
materials return to their rest shape after
stresses are removed.
Plasticity
Describes materialsthat permanently
deform after a
sufficient applied
stress.RheologyThe study of materials
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 2
with both solid andfluid characteristics.
Fluid mechanics
Non-Newtonian fluids
do not undergo strain
rates proportional to
the applied shear
stress.
Newtonian fluids undergo strain rates
proportional to the applied shear stress.
http://en.wikipedia.org/wiki/Continuum_mechanics
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Solid and Structural Mechanics
Material TensorE,, ..
Solids Structures
+A, I
+ d
Rods, Beams
Plates, Shells
Dimensional Reduction
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 3
http://www.maschinenbau.fh-pforzheim.de/werkstoffkunde/bilder/
Material Science J. Rsler
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FE Models
Solid (3-D) Structure (2D+1D)
Volume elements Full stress and strain tensors
Line, Area elements Reduced stress and strain tensors
B. Bchner, R.Mllenhoff,Master Project, HAWT. Rudolph, CourseProject, HAW
Shell elements
Beam elements
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 4
+ Acoustic Fluid (3-D):
Y. Qiao, Master Thesis,
HAW
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Dynamic equilibrium of solid
0
0
0
yxx zxx
xy y zyy
yzxz zz
fx y z
fx y z
fx y z
+ + + =
+ + + =
+ + + =
xx
yy
zz
f u
f u
f u
=
=
=
ii
ii
ii , 0iij j u =
ii
Statical Equilibrium:
0 = u
components:
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 5
= n t
tBoundary:
The boundary force resultants must be in equilibrium with the external load vector (or thereaction forces) at every point of the boundary.
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Stress tensor in ideal acoustic fluid
Inviscid fluids are shear-free, so stress tensor reduces to compression forces
x xy xz
xy y yz
xz yz z
0 0
0 0
0 0
P
P
P
Structural resultant at FSI coupling surface cannot contain shear components.
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 6
P= t n
Since there are no contact tangential contact forces kinematic compatibilityholds in normal direction only.
{ {1 2, , 0, 0,ext n st t t P P= = = t n
In local coor.
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P n n
Condition 1: Normal Equilibrium (kinetic coupling)
2. FSI coupling in Linear Acoustics
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 7
=
F
v n
Condition 2: Normal Compatibility (kinematic coupling)
S
v n
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snfns f
= U n U n
Replacing fluid displacement by pressure, using the Euler equation,0 f P = U
Kinematic Coupling in terms of pressure
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 8
0 s
PP
n
= =
U n n
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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 displacementsare replaced by pressure using Eulers equation)
Figure: Solid-fluid continuum
s
f w
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 9
snf
n
Figure: normals at interface
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Multiply component-wise withtest displacement -W
Multiply with test pressure -Q
and integrate by parts Apply first coupling condition
3. FE equations for free vibrations
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 10
Apply second coupling condition
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Details of integration (rectangular domain)
a b
x
y
{ }1,0n =
{ }1,0
n=
{0, 1n =
{ }0,1n =
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 11
( ) ( )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 dy
W W W W dxdy
n W ds W dA
= =
= + + +
= + + +
+ + +
=
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FE-partition of structure and fluid
Coupled equations (weak form)
Symmetry considerations
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 12
Observation: the coupling matrices of the finite element model are obtained from element integrals ofthe product fluid shape function * solid shape function. Since these products are commutative, thegoverning equations can be scaled( = multiplied by a constant factor) in such a way that the overallcoupled system of equations is symmetric.
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Coupled differential equations (weak form)
Introduce
structural material law (stress strain),
Coupled FE equations for free vibrations
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 13
kinematic equations (strain displacements)
finite element partition of structure and fluid, finite element shape functions
s s
Tf f
+ =
K A M 0U U
00 K A MPP
ii
ii
Coupled system of FE equations
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Coupling matrix contains kinematic relations (MPC) between structural and fluidnodes on the coupling surfaces
Coupling Matrix
s s
T
f f
+ =
K A M 0U U0
0 K A MP
P
ii
ii
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The mesh size in fluid and solid/structure can differ The boundaries do not need to have exactly the same contours (search algorithm fornon-matching boundaries) The node numbers in fluid and solid must be different
FE models for FSI
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 15
Parameters for the coupling procedure: ACMODL parameter in the bulk data section. Recommended coupling procedure (BW=default)
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Eigenvalue Problem for undamped vibrations
2s s
T
f f
=
K A M 0 u0
0 K A M p
Wandinger [94]: If the stiffness matrices are p.d. then the Eigenvalues are real andpositive.
2
The undamped structure-fluid system has real (coupled) eigenfrequencies.
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 17
Alternatively, the system can be rewritten in the symmetric form
leading to a quadratic eigenproblem (QEP).
Nastran Sol106 (Complex Modes)
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Coupled FE model of the forced response
Trick (Everstine et al.):
derive structural equation in time
-i
0
ts s s
T
f f f
e
+ + =
K A C 0 M 0U U U F
0 K 0 C A MP 0
P P
i ii
i ii
.
4. FE equations of the forced vibrations
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 18
substitute
Coupled symmetric system
=
-i
1
ts s s
T
f f f
e
+ + =
K 0 C A M 0V V V F
0 K A C 0 MP 0P P
i ii
i ii
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time domain
-i -it te e = =V v P
-i
1
ts s s
T
f f f
e
+ + =
K 0 C A M 0V V V F
0 K A C 0 MP 0
P P
i ii
i ii
Time-harmonic solution
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 19
frequency domain
2i
s s s
T
f f f
=
K 0 C A M 0 v f
0 K A C 0 M p 0
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2i
s s s
T
f f f
=
K 0 C A M 0 v f
0 K A C 0 M p 0
direct: SOL108
modal: SOL111
Solution procedures
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 20
o a re uc on can e per orme w uncoup e rea -va ue mo es
Solve:
[ ] 0
0
s s s
f f f
=
=
K M x
K M x
{
{ }
2
1, ,
21, ,
,
,
js js jsj m
jf jf jfj n
=
=
=
=
x
x
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-i
0
ts s s
T
f f f
e
+ + =
K A C 0 M 0U U U F
0 K 0 C A MP 0P P
i ii
i ii
Solution of eigenvalue problems
Modal Superposition
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 21
,s s f f v y p y
2i
T T T T Ts s s s s s s f s s s s
T T T T T f f f f s f f f f f f
=
K 0 C A M 0 v f
0 K A C 0 M p 0
Modal superposition separately for structure and fluid
! Modal reduction in coupled system using uncoupled modes !
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Fluid-Struktur Interaktion: Modale Kopplung
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SOL 111
CEND
TITLE = BOX coupled
LABEL = Pressure at "Ear Positions"
ECHO = NONE
$ Messpunkte Fluid
SET 123=10224,10235,10243,10251
DISP(SORT1,PUNCH) = 123
$
METHOD(FLUID) = 50METHOD(STRUCTURE) = 51
FREQ = 20
DLOAD = 1
BEGIN BULK
EIGRL,50,,100.
Separate Eigenvalue Analysisfor Fluid, Structure
Forcing Frequencies
Dynamic Load
Data output to punch file box.pch
Nastran Solution Deck for FSI simulation (modal)
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 23
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 0.06
PARAM GFL 0.06
PARAM, GRDPNT, 0
PARAM, AUTOSPC, YES
PARAM, PRGPST, NO
PARAM, SNORM, 45.
ENDDATA
Grid and element data for fluid, structure
Damping loss factors for fluid, structure
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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 (extract)
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No Coupling
Structural VelocitiesEquivalentRadiated Power
Normal velocities =boundary conditionsfor fluid ( FEM, BEM)
5. Levels of Structure-Fluid coupling*
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 25
* after Claus edersen
Acoustic TransferVectors (computed,measured; direct orreciprocal approach)
Strong Coupling CoupledFEM/BEM,FEM/FEM,
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Equivalent Radiated Power (ERP)
If p and v are known on the surface of a radiating panel (e.g., from a coupledcomputation) then one can compute the power radiated from the panel as the sumof all elements.
In FE-practice, one often computes the so-called equivalent radiated power,assuming that each finite element emits a plane wave. Then the pressure can be
replaced by normal velocity, allowing to estimate the radiated power from theuncoupled (structural) simulation.
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Equivalent Radiated Power (Marburg et al.*)
HAW/M+P, Ihlenburg, CompA VA FSI with FEM 27
*S. Marburg, B. Nolte (eds.), Compuational Acoustics of Noise Propagation in Fluids, Springer Verlag 2008,pp. 429--431
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Sound power vs. ERP on surface of diesel engine
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Vibroacoustic evaluation of passenger car cabin
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