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

Numerical Simulation

Noise and Vibration Short Course

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

3 Dept. of Mech. Engineering University of Kentucky

Numerical Simulation

Noise and Vibration Short Course

4 Dept. of Mech. Engineering University of Kentucky

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

  Pressure   Impedance or Admittance

Boundary Conditions

nvpZ =

nvp

ZA ==

1

Numerical Simulation

Noise and Vibration Short Course

5 Dept. of Mech. Engineering University of Kentucky

Finite Element Method

022 =+∇ pkp

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

Absorption Velocity BC’s

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

⎠

⎞

⎜⎜

⎝

⎛−+∇⋅∇

21

22

2

11

Sn

VSV

dSvNipdVNNc

dSNNz

idVNN ρωω

ρω

In Depth

Numerical Simulation

Noise and Vibration Short Course

6 Dept. of Mech. Engineering University of Kentucky

Boundary Element Method

022 =+∇ pkp

( ) ( ) ( ) ( )∫ ⎟

⎠

⎞⎜⎝

⎛∂

∂+−=

Ssn dS

nrGprGviPpPC ρω

Boundary Integral Equation

Boundary Condtions

rerGikr

π4)(

−

=

Numerical Simulation

Noise and Vibration Short Course

7 Dept. of Mech. Engineering University of Kentucky

tjnev

ω

tjep ω1

( ) ( ) ( ) ( )∫ ⎟

⎠

⎞⎜⎝

⎛∂

∂+−=

Ssn dS

nrGprGviPpPC ρω

Boundary Integral Equation

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.

Numerical Simulation

Noise and Vibration Short Course

8 Dept. of Mech. Engineering University of Kentucky

tjep ω1

( ) ( ) ( ) ( )∫ ⎟

⎠

⎞⎜⎝

⎛∂

∂+−=

Ssn dS

nrGprGviPpPC ρω

Boundary Integral Equation

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.

Numerical Simulation

Noise and Vibration Short Course

9 Dept. of Mech. Engineering University of Kentucky

BEM Versus FEM

BEM FEM

Numerical Simulation

Noise and Vibration Short Course

10 Dept. of Mech. Engineering University of Kentucky

Overview

  Introduction to Numerical Methods

  BEM Overview

  Simulation Example - Engine Cover

Numerical Simulation

Noise and Vibration Short Course

11 Dept. of Mech. Engineering University of Kentucky

Exterior (radiation)

vn or ps or Zin everywhere Boundary mesh

(2-D surface mesh)

Interior (cavity)

vn Zin ps

Direct BEM

Numerical Simulation

Noise and Vibration Short Course

12 Dept. of Mech. Engineering University of Kentucky

  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

Numerical Simulation

Noise and Vibration Short Course

13 Dept. of Mech. Engineering University of Kentucky

Indirect BEM

Noise

source

Openings

Zin vn

vn

ps

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

Numerical Simulation

Noise and Vibration Short Course

14 Dept. of Mech. Engineering University of Kentucky

Example

ps

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

n̂+ Side

- Side

Numerical Simulation

Noise and Vibration Short Course

15 Dept. of Mech. Engineering University of Kentucky

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

Numerical Simulation

Noise and Vibration Short Course

16 Dept. of Mech. Engineering University of Kentucky

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

ü Solution ü Postprocessing

Steps in a BEM Analysis

Numerical Simulation

Noise and Vibration Short Course

17 Dept. of Mech. Engineering University of Kentucky

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

Numerical Simulation

Noise and Vibration Short Course

18 Dept. of Mech. Engineering University of Kentucky

What about Triangles?

Linear Quadrilateral Linear Triangle

Linear triangles are nearly as accurate as linear quadrilaterals

=

Numerical Simulation

Noise and Vibration Short Course

19 Dept. of Mech. Engineering University of Kentucky

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

Numerical Simulation

Noise and Vibration Short Course

20 Dept. of Mech. Engineering University of Kentucky

Implementation in Virtual Lab

Elements have varying sizes

Numerical Simulation

Noise and Vibration Short Course

21 Dept. of Mech. Engineering University of Kentucky

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

Numerical Simulation

Noise and Vibration Short Course

22 Dept. of Mech. Engineering University of Kentucky

Steps in a BEM Analysis

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

ü Solution ü Postprocessing

Numerical Simulation

Noise and Vibration Short Course

23 Dept. of Mech. Engineering University of Kentucky

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

Numerical Simulation

Noise and Vibration Short Course

24 Dept. of Mech. Engineering University of Kentucky

Steps in a BEM Analysis

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

ü Solution ü Postprocessing

Numerical Simulation

Noise and Vibration Short Course

25 Dept. of Mech. Engineering University of Kentucky

Point Sources

Input as ü  Sound Power ü  Pressure Amplitude/Phase

r

p(r)

rAerp

ikr−

=)(

Numerical Simulation

Noise and Vibration Short Course

26 Dept. of Mech. Engineering University of Kentucky

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

Numerical Simulation

Noise and Vibration Short Course

27 Dept. of Mech. Engineering University of Kentucky

Steps in a BEM Analysis

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

ü Solution ü Postprocessing

Numerical Simulation

Noise and Vibration Short Course

28 Dept. of Mech. Engineering University of Kentucky

Automatic Junction Zero Jump Condition

Automatic in Virtual.Lab

Numerical Simulation

Noise and Vibration Short Course

29 Dept. of Mech. Engineering University of Kentucky

Steps in a BEM Analysis

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

ü Solution ü Postprocessing

Numerical Simulation

Noise and Vibration Short Course

30 Dept. of Mech. Engineering University of Kentucky

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

Quadrature and Field Point Processing Side

Numerical Simulation

Noise and Vibration Short Course

31 Dept. of Mech. Engineering University of Kentucky

Indirect BEM

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

Absorbing Panels

Numerical Simulation

Noise and Vibration Short Course

32 Dept. of Mech. Engineering University of Kentucky

Steps in a BEM Analysis

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

ü Solution ü Postprocessing

Numerical Simulation

Noise and Vibration Short Course

33 Dept. of Mech. Engineering University of Kentucky

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

Numerical Simulation

Noise and Vibration Short Course

34 Dept. of Mech. Engineering University of Kentucky

Steps in a BEM Analysis

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

ü Solution ü Postprocessing