Vibroacoustic Fluid-Structure Interaction With FEM

8/9/2019 Vibroacoustic Fluid-Structure Interaction With FEM

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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


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


+ 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:


= 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.


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


=

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


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


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


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 =


( ) ( )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


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


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


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.


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


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


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


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


,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)


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*


* 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.*)


*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


Vibroacoustic Fluid-Structure Interaction With FEM

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