02 Introduction to Numerical Simulationweb.engr.uky.edu/~dherrin/...Numerical_Simulation.pdf · Numerical Simulation Noise and Vibration Short Course Dept. of Mech. Engineering 7

Introduction to Numerical Acoustics

D. W. Herrin, Ph.D., P.E. University of Kentucky

Department of Mechanical Engineering

Numerical Simulation

Noise and Vibration Short Course

The Wave Equation

In 1D

∂2p∂x2

−1c2∂2p∂t2

= 0

In 3D

∇2p− 1c2∂2p∂t2

= 0

∇2p = ∂2

∂x2+∂2

∂y2+∂2

∂z2$

%&

'

() p

2 Dept. of Mech. Engineering University of Kentucky

Helmholtz Equation (steady state harmonic)

∇2p+ k2p = 0



The Speed of Sound

c = c0TT0

The speed of sound is a function of temperature

c = γRTM0

The speed of sound at one temperature is easily related to that at another





  Normal Velocity - Default B.C. is vn = 0

  Pressure   Impedance or Admittance

Boundary Conditions

nvpZ =

nvp

ZA ==

1




Finite Element Method

022 =+∇ pkp

[ ]{ } [ ]{ } [ ]{ } { }FpKpCpM =++− ωω2

Absorption Velocity BC’s

⎣ ⎦ { } ⎣ ⎦{ } ⎣ ⎦{ } { } ⎣ ⎦{ }∫∫∫∫ −=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−+∇⋅∇

21

22

2

11

Sn

VSV

dSvNipdVNNc

dSNNz

idVNN ρωω

ρω

In Depth




Boundary Element Method

022 =+∇ pkp

( ) ( ) ( ) ( )∫ ⎟

⎠

⎞⎜⎝

⎛∂

∂+−=

Ssn dS

nrGprGviPpPC ρω

Boundary Integral Equation

Boundary Condtions

rerGikr

π4)(

−

=




tjnev

ω

tjep ω1

( ) ( ) ( ) ( )∫ ⎟

⎠

⎞⎜⎝

⎛∂

∂+−=

Ssn dS

nrGprGviPpPC ρω


If the Green's function is the point monopole source, the first term on the right hand side of the boundary integral equation is the summation of the point monopole sources.




tjep ω1

( ) ( ) ( ) ( )∫ ⎟

⎠

⎞⎜⎝

⎛∂

∂+−=

Ssn dS

nrGprGviPpPC ρω


If the Green's function is the point monopole source, the scattering effect from the rigid boundary on the acoustic field from each of the volumetric point sources.




BEM Versus FEM

BEM FEM




Overview

  Introduction to Numerical Methods

  BEM Overview

  Simulation Example - Engine Cover




Exterior (radiation)

vn or ps or Zin everywhere Boundary mesh

(2-D surface mesh)

Interior (cavity)

vn Zin ps

Direct BEM




  Boundary must be closed §  Cannot model open systems such as partial

enclosures §  Directly predict radiation efficiency §  Directly predict contribution

  Matrix is non-symmetric §  Coupling to FEM not efficient §  Inefficient for large meshes

  Non-uniqueness problem (radiation only) §  Must add overdetermination points

ODP’s

Direct BEM




Indirect BEM

Noise

source

Openings

Zin vn

vn

ps

BE CAREFUL Boundary conditions must be specified on both sides of the boundary




Example

ps

Elements have a positive and negative side defined by the normal vector

n̂+ Side

- Side




Indirect BEM

  Boundary can be open §  More general §  More difficult to use §  No radiation efficiency

  Symmetric matrix §  Faster than direct BEM

  Non-existence problem (radiation only) §  Must add absorbing planes




ü Preprocessing   Mesh Definition   Fluid Properties   Sources and Symmetry   Boundary Conditions   Treatment for Radiation

ü Solution ü Postprocessing

Steps in a BEM Analysis




v  Linear Elements – at least 6 elements/wavelength v  Parabolic Elements – at least 2 elements/wavelength v  Acoustic wavelength is a function of frequency

BEM solution time: (Nodes)2 or (Nodes)3

Is the BEM Mesh Fine Enough?

fc

=λ




What about Triangles?

Linear Quadrilateral Linear Triangle

Linear triangles are nearly as accurate as linear quadrilaterals

=




Example

What element size do you need if you want to solve up to 800 Hz using linear elements?

m43.0Hz800sm343

==λ

m07.06m429.

6===

λl




Implementation in Virtual Lab

Elements have varying sizes




Normal Consistency

The element normal direction must be consistent for any boundary element analysis (direct or indirect). Seems to be handled automatically in Virtual Lab










Fluid Material Properties

§  Speed of sound §  Density §  Add Dissipation in Fluid (loss factor η)

ü  Complex wave number (complex speed of sound)

)2/ :ratio (damping .'

, 2

1' ηζωηω

==⎟⎠

⎞⎜⎝

⎛−=

kc'j

ck










Point Sources

Input as ü  Sound Power ü  Pressure Amplitude/Phase

r

p(r)

rAerp

ikr−

=)(




Symmetry

A symmetry plane can be used to model the floor of a hemi-anechoic chamber

VL Tip: Define symmetry planes before defining mesh preprocessing set










Automatic Junction Zero Jump Condition

Automatic in Virtual.Lab










Direct BEM ü  Specify overdetermination points ü  50+ is normally sufficient

Quadrature and Field Point Processing Side




Indirect BEM

ü  Specify absorbing panels ü  Specify small real admittance (0.002)

Absorbing Panels










Quadrature Selection Default is 2 2 1 We Recommend ü  3 3 2 for linear elements ü  4 4 2 for parabolic elements

Using the default quadrature will produce wrong results for parabolic elements







02 Introduction to Numerical Simulationweb.engr.uky.edu/~dherrin/...Numerical_Simulation.pdf · Numerical Simulation Noise and Vibration Short Course Dept. of Mech. Engineering 7

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