EuroRegio_Peplow9

7/29/2019 EuroRegio_Peplow9

1/98

Advances in Finite Element methods in Acoustics

Andrew Peplow1

1

Hoare Lea Acoustics, Bristol, UKwww.hoarelea.com [email protected]

www.ntnu.no Andrew Peplow, Computational analysis in acoustics


2/98

2

Outline

Hoare Lea Acoustics, UKSimple acoustic duct problem

Separation of variables solutionsMuffler problem : structural acoustic waves

Threedimensional problems : waves in cylindrical waveguide

3D: A question of mesh?Propagating waves for cylinder

Composite layered materials

Exterior ProblemsWind Turbine Noise

Radiation conditions : Higher Order boundary conditionsExterior problems

Conclusions



3/98

2

Outline

Hoare Lea Acoustics, UKSimple acoustic duct problem

Separation of variables solutionsMuffler problem : structural acoustic waves






Conclusions



4/98

2

OutlineHoare Lea Acoustics, UK

Simple acoustic duct problemSeparation of variables solutionsMuffler problem : structural acoustic waves






Conclusions



5/98

2








Conclusions



6/98

2








Conclusions



7/98

2








Conclusions



8/98

2








Conclusions



9/98

2








Conclusions



10/98

Hoare Lea AcousticConsultants

HLA founded 41 years ago(1969, 3 staff) in Building

Acoustics

Today 30 staff in total

8 Graduate engineers, 12senior engineers, 10

associates

3 EU projects, turnover 5million Euros

One of the largest centrefor sound and vibration in

UK

Welcome around 3-5placement students and

graduate projects each

year



11/98



Acoustics



associates



UK



year


H L A i


12/98



Acoustics



associates



UK



year


H L A ti


13/98



Acoustics



associates



UK



year


H L A ti


14/98



Acoustics



associates



UK



year


Hoare Lea Aco stic


15/98



Acoustics



associates



UK



year



16/98

4

Ducts Applications

Fan noisebuildingservices,intake/exhaustsystems

transportmufflers andpiplines.


4


17/98

4

Ducts Applications

Fan noisebuildingservices,intake/exhaustsystems

transportmufflers andpiplines.



18/98

Separation of variables

p(x, z) = X(x)Z(z)

p(x, z) = exp(ix) cos(kmz)

Gives rise to simple eigenvalueproblem

2 k2 + k2m = 0

OR =

k2 k2m

Positive values are propagating(energy transporting) waves

Spectral finite element

method p(x, z) = X(x)Z(z)

p(x, z) = exp(ix)N(z)

Eigenvalue problem. Name

spectral derived. 2 K2 k

2 K0 + K3 X = 0www.ntnu.no Andrew Peplow, Computational analysis in acoustics


19/98


p(x, z) = X(x)Z(z)



2 k2 + k2m = 0

OR =

k2 k2m









20/98


p(x, z) = X(x)Z(z)



2 k2 + k2m = 0

OR =

k2 k2m









21/98


p(x, z) = X(x)Z(z)



2 k2 + k2m = 0

OR =

k2 k2m





Eigenvalue problem. Namespectral derived.

2 K2 k2 K0 + K3 X = 0



22/98


p(x, z) = X(x)Z(z)



2 k2 + k2m = 0

OR =

k2 k2m






2 K2 k2 K0 + K3 X = 0



23/98


p(x, z) = X(x)Z(z)



2 k2 + k2m = 0

OR =

k2 k2m






2 K2 k2 K0 + K3 X = 0



24/98


p(x, z) = X(x)Z(z)



2 k2 + k2m = 0

OR =

k2 k2m






2 K2 k2 K0 + K3 X = 0



25/98


p(x, z) = X(x)Z(z)



2 k2 + k2m = 0

OR =

k2 k2m






2 K2 k2 K0 + K3 X = 0



26/98


p(x, z) = X(x)Z(z)



2 k2 + k2m = 0

OR =

k2 k2m






2 K2 k2 K0 + K3 X = 0



27/98


p(x, z) = X(x)Z(z)



2 k2 + k2m = 0

OR =

k2 k2m






2 K2 k2 K0 + K3 X = 0


6


28/98

Comparison of spectral method vs

modal solution.1


7


29/98

Geometric Step Problem.1

1Peplow & Finnveden, JASA, 2004



30/98

Beam coupled to a fluid

Beam has stiffness Ds and

massdensity ms Coupling is through vertical

displacement, W(x)

Expect an eigenvalue problem tosolve

A dispersion relation is produced

A relation between frequency and wavenumber



31/98




displacement, W(x)






32/98




displacement, W(x)






33/98




displacement, W(x)






34/98




displacement, W(x)






35/98

[K4]

d4

dx4+ [K2]

d2

dx2+ 2[K0] [K3] + [K0]

S = 0

with (N+ 1) (N+ 1) FE matrices

[K4] =

DS 0

T

0 0

,

[K2] =

0 0T

0

, [K0] =

0 T

,

[K3] =

0 0T

0

, [K0] =

ms 0

T

0

.



36/98

[K4]

d4

dx4+ [K2]

d2

dx2+ 2[K0] [K3] + [K0]

S = 0


[K4] =

DS 0

T

0 0

,

[K2] =

0 0T

0

, [K0] =

0 T

,

[K3] =

0 0T

0

, [K0] =

ms 0

T

0

.



37/98

[K4]

d4

dx4+ [K2]

d2

dx2+ 2[K0] [K3] + [K0]

S = 0


[K4] =

DS 0

T

0 0

,

[K2] =

0 0T

0

, [K0] =

0 T

,

[K3] =

0 0T

0

, [K0] =

ms 0

T

0

.


10

Ei l bl f l d


38/98

Eigenvalue problem for coupled

structuralacoustic finite element

4[K4] +

2[K2] + [K0]

S = 0.

0 I

K0 K2

+ 4

I 00 K4

S

2S

= 0.


Quadratic eigenvalue

problem in 2

. However, rank[K4] = 1,

rank[K2] = N.

Not wellconditionedeigenvalue problem but

is "regular". Size of "regular" problem

is now N+ 2. Nacousticunknowns and twostructural variables, W,

Wx or Wxxwww.ntnu.no Andrew Peplow, Computational analysis in acoustics

10

Ei l bl f l d


39/98



4[K4] +

2[K2] + [K0]

S = 0.

0 I

K0 K2

+ 4

I 00 K4

S

2S

= 0.



problem in 2


rank[K2] = N.





10

Eigen al e problem for co pled


40/98



4[K4] +

2[K2] + [K0]

S = 0.

0 I

K0 K2

+ 4

I 00 K4

S

2S

= 0.



problem in 2


rank[K2] = N.





10



41/98



4[K4] +

2[K2] + [K0]

S = 0.

0 I

K0 K2

+ 4

I 00 K4

S

2S

= 0.



problem in 2


rank[K2] = N.





10



42/98



4[K4] +

2[K2] + [K0]

S = 0.

0 I

K0 K2

+ 4

I 00 K4

S

2S

= 0.



problem in 2


rank[K2] = N.





11


43/98

Phase velocities for fluidfilled pipe


12


44/98

Phase velocities for waterfilled pipe


13

Transmission loss results with


45/98

Transmission loss results with

aluminium panel

Transmission Loss, W = 750 mm,

H= 60 mm, S= 20 mm, for duct

and expansion chamber supporting

aluminimum plates



46/98

Figure: Typical cylindricalwaveguide and FEM mesh



47/98

16

Some propagating modes of rigid and


48/98

Some propagatingmodes of rigid and

absorbing muffler

Figure: Symmetric (FE)

and asymmetricwww.ntnu.no Andrew Peplow, Computational analysis in acoustics

17

A l b t t d


49/98

A laboratory study

Laminate plates find all wave-types

during the FE process

under plane-stress 2Dconditions 10 times

faster than COMSOL

to be continued for 3Dconditions expected 100

times quicker


17

A l b t t d


50/98

A laboratory study




faster than COMSOL


times quicker


17

A l b t t d


51/98

A laboratory study




faster than COMSOL


times quicker


18

A laborator st d


52/98

A laboratory study

Comments wave-types change

from evanescent to

propagating at a certain

frequency

difference betweenmeasurements and two

finite element codes

similarly for phase


18

A laboratory study


53/98

A laboratory study

Comments wave-types change

from evanescent to

propagating at a certain

frequency

difference betweenmeasurements and two

finite element codes

similarly for phase


19

A practical perspective


54/98


Prediction methods are critical demonstrate acceptable wind

farm noise impact at the

planning stage

achieve this acceptable noise

impact ... ... in practice balance noise

impact against generating

capacity

empirical engineering methods

(e.g. ISO 9613-2, CONCAWE) approximate semi-analytical

methods (e.g. ray tracing)

exact numerical methods(e.g.parabolic equation, fast field

program)www.ntnu.no Andrew Peplow, Computational analysis in acoustics

19



55/98


Prediction methods are critical demonstrate acceptable wind

farm noise impact at the

planning stage




capacity






19



56/98


Prediction methods are critical

demonstrate acceptable windfarm noise impact at the

planning stage




capacity






19



57/98




planning stage




capacity






19



58/98




planning stage




capacity






19



59/98




planning stage




capacity







60/98

UPWIND Sound shadow region results

under temperature lapse

and-or upwind conditions

Result is large decreases over

neutral of typically -10dB(A) to-15dB(A) coupled with highly

variable noise level

DOWNWIND

Sound enhancement resultsdue to multiple paths under

temperature inversion and/or

downwind conditions

Result is small increases overneutral of typically +1dB(A) to

+3dB(A) and much more stable

noise level



61/98







DOWNWIND



downwind conditions



noise level



62/98







DOWNWIND



downwind conditions



noise level



63/98







DOWNWIND



downwind conditions



noise level



64/98

UPWIND

(PML) Sound shadow region

results under temperature lapseand-or upwind conditions

(ABC) Result is large decreasesover neutral of typically -10dB(A)

to -15dB(A) coupled with highly


DOWNWIND

Sound enhancement results due to

multiple paths under temperatureinversion and/or downwind

conditions


+3dB(A) and much more stablenoise level



65/98

UPWIND






DOWNWIND



conditions





66/98

UPWIND






DOWNWIND



conditions





67/98

UPWIND






DOWNWIND



conditions




22

Case studies


68/98

Wind Turbine with Tones.wav


23

Minimize reflection coefficient


69/98

Figure: Direction of Incident

Plane Wave on fictitiousboundary illustrating reflected

wave.

Consider the total acousticfield due to Incident PlaneWave

I = A exp(ik x) exp(it)

(x, z) =

I(x, z) + RM() R(x, z).

For the transmitting boundarycondition

n

= ik

R0() = cos 1

cos +1 , = 1


23



70/98



wave.



(x, z) =

I(x, z) + RM() R(x, z).


n

= ik

R0() = cos 1

cos +1 , = 1


23



71/98



wave.



(x, z) =

I(x, z) + RM() R(x, z).


n

= ik

R0() = cos 1

cos +1 , = 1



72/98

23



73/98



wave.



(x, z) =

I(x, z) + RM() R(x, z).


n

= ik

R0() = cos 1

cos +1 , = 1


Expression for reflection coefficient

R1() =cos 1+a1 sin

2

cos +1+a1 sin2


74/98

cos +1+a1 sin

Model A. Take a series expansion around = 0 yields a1 =12

to minimise

R1()2

.

Continuing withexpansion inincident angle

a2 =1

8

a3 =1

16

Figure: Absolute values of reflection

coefficient.www.ntnu.no Andrew Peplow, Computational analysis in acoustics


R1() =cos 1+a1 sin

2

cos +1+a1 sin2


75/98

cos +1+a1 sin


to minimise

R1()2

.


a2 =1

8

a3 =1

16




R1() =cos 1+a1 sin

2

cos +1+a1 sin2


76/98

+ + 1


to minimise

R1()2

.


a2 =1

8

a3 =1

16




R1() =cos 1+a1 sin

2

cos +1+a1 sin2


77/98

1


to minimise

R1()2

.


a2 =1

8

a3 =1

16




R1() =cos 1+a1 sin

2

cos +1+a1 sin2


78/98


to minimise

R1()2

.


a2 =1

8

a3 =1

16



25

Minimize close to grazing incidence or

averaged sound power ?


79/98


Minimise RM() around different incident angles other than = 0 above.

RM() =cos 1+

Mm=1 amsin

2m

cos +1+

Mm=1

amsin2m

the corresponding higher order boundary conditions :

n

= a0(ik) +m=M

m=1am

(ik)2m2ms2m

.



80/98

25




81/98



RM() =cos 1+

Mm=1 amsin

2m

cos +1+

Mm=1

amsin2m


n

= a0(ik) +m=M

m=1am

(ik)2m2ms2m

.


25




82/98



RM() =cos 1+

Mm=1 amsin

2m

cos +1+

Mm=1

amsin2m


n

= a0(ik) +m=M

m=1am

(ik)2m2ms2m

.


26

Finite - Boundary Element Method :

Song and Wolf


83/98

Song and Wolf

, r

For exterior space.

Two elements. 0 < a< r1 < b,b< r2 <

Polynomial elements in

azimuthal direction 0 < 2

Separation of variables in(r, )

p(r, ) = R(r)N() p(r, ) = rN()

Another quadratic eigenvalueproblem. Similar to Bessel

ODE.

{( + 1)[K2] [K1] +r[K3] k

2[K0]}R = 0

Hence requires requires extraanalysis for real modelling ...

but benefits are worthwhilewww.ntnu.no Andrew Peplow, Computational analysis in acoustics

26


Song and Wolf


84/98

Song and Wolf

, r

For exterior space.





p(r, ) = R(r)N() p(r, ) = rN()


ODE.

{( + 1)[K2] [K1] +r[K3] k

2[K0]}R = 0



26


Song and Wolf


85/98

Song and Wolf

, r

For exterior space.





p(r, ) = R(r)N() p(r, ) = rN()


ODE.

{( + 1)[K2] [K1] +r[K3] k

2[K0]}R = 0



26


Song and Wolf


86/98

So g a d o

, r

For exterior space.



azimuthal direction 0<

2


p(r,

) = R(r)N(

) p(r, ) = rN()


ODE.

{( + 1)[K2] [K1] +r[K3] k

2[K0]}R = 0



26


Song and Wolf


87/98

g

, r

For exterior space.


Polynomial elements inazimuthal direction. 0 <

2


p

(r,

) =R

(r

)N

(

) p(r, ) = rN()


ODE.

{( + 1)[K2] [K1] +r[K3] k

2[K0]}R = 0



26


Song and Wolf


88/98

g

, r

For exterior space.


Polynomial elements inazimuthal direction. 0 <

2


p(r,

) =R

(r

)N

(

) p(r, ) = rN()


ODE.

{( + 1)[K2] [K1] +r[K3] k

2[K0]}R = 0



26


Song and Wolf


89/98

g

, r

For exterior space.


Polynomial elements inazimuthal direction. 0 < 2


p(r, ) = R(r)N()

p(r, ) = rN()


ODE.

{( + 1)[K2] [K1] +r[K3] k

2[K0]}R = 0



26


Song and Wolf


90/98

g

, r

For exterior space.


Polynomial elements inazimuthal direction. 0 < 2


p(r, ) = R(r)N()

p(r, ) = rN()


ODE.

{( + 1)[K2] [K1] +r[K3] k

2[K0]}R = 0



27

Summary

S t l th d i t f fi it l t d t l ti


91/98

Spectral methods are mixture of finite elements and exact solutions.

Work extremely well for pipes, ducts & waveguides. Advantage workload is considerably reduced from full Finite Element

method.

Generate dispersion relations giving group velocity etc. Use eigsfrom MATLAB.

Full solution for 3D problems. Only require 2D Mesh.

Wolf, Song and Deeks extended this formulation to exterior domains(for WT noise for example)

Otherwise use Non-reflecting or Perfectly Matched Layers for FE/PE

modelliing or BEM.


27

Summary

S t l th d i t f fi it l t d t l ti


92/98



method.





modelliing or BEM.


27

Summary

Spectral methods are mixture of finite elements and exact solutions


93/98



method.





modelliing or BEM.


27

Summary



94/98



method.





modelliing or BEM.


27

Summary



95/98



method.





modelliing or BEM.


27

Summary



96/98



method.





modelliing or BEM.


27

Summary



97/98



method.





modelliing or BEM.


27

Summary



98/98



method.





modelliing or BEM.


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