Acoustics Group – Department of Information Technology – Ghent University/iMinds

CEAS, X-Noise : Atm. and ground effects on aircraft noise – Sevilla, 2013

Timothy Van Renterghem

Efficient outdoor sound propagation modelling in

time-domain

Sound propagation is essentially a time-domain process

Response over broad frequency range possible with a single run

Including of non-linear effectsModelling realistic sources (moving,

transient) Fluid-flow acoustics coupling can be

treated more easily

Why time-domain models?

Finite-difference time-domain (FDTD) method Solving LEE Numerical discretisation strongly influences

modelling efficiency computational cost numerical accuracy numerical stability

In absence/presence of flow Finite absorbers

Long distance sound propagation moving frame approach hybrid modelling

Outline

Linear continuous sound propagation equation in still air Navier-Stokes equations reduced to

Momentum equation (velocity equation) Continuity equation (pressure equation) + (linear)

pressure-density relation Assumptions

Linearization in acoustical quantities Non-moving propagation medium No thermal, viscous effects and molecular relaxation No gravity

Sound propagation equations

0

1 0pt

v

20 0p c

t

v2

22 0p c p

t

, , , ,

ldt lidx jdy kdz i j kp p

Lowest possible spatial stencil Two options for central differences

Collocated-in-place (CIP) Staggered-in-place (SIP)

FDTD spatial aspects

2 2 3 3

1, , , , 2 3, , , , , ,

2 2 3 3

1, , , , 2 3, , , , , ,

1, , 1, ,

,

2 3

3, ,,

...1! 2! 3!

...1! 2! 3!

3...

!2

i j k i j ki j k i j k i j k

i j k i j ki j k i j k i j k

i j k i j k

i jj k i k

dx p dx p dx pp px x x

dx p dx p dx pp px x x

p dxppx

pxdx

2 2 3 3

1, , 0.5, , 2 2 3 30.5, , 0.5, , 0.5, ,

2 2 3 3

, , 0.5, , 2 2 3 30.5, , 0.5, , 0.5, ,

1, , , ,

0.5, ,

2.1! 2 .2! 2 .3!

2.1! 2 .2! 2 .3!

i j k i j ki j k i j k i j k

i j k i j ki j k i j k i j k

i j k i j

i j k

dx p dx p dx pp px x x

dx p dx p dx pp px x x

p ppx

2 3

30.5, ,

.3!

..4. i j k

k dx pd xx

i i+1i-1 i+0.5 i+1i

dx

Extended spatial stencil Take more neighbouring cells to better approach

the gradient E.g. Involve 6 neighbouring cells (7-point stencil,

CIP, central differences)

Dispersion-relation preserving (DRP) schemes– Numerically optimize values of ai to further

decrease phase error

Drawbacks Point source representation difficult Complicated boundary treatment Reduced time steps for numerical stability

FDTD spatial aspects

)(60

)3()2(9)(45)(45)2(9)3( 6000000

0

dxOdx

dxxpdxxpdxxpdxxpdxxpdxxpxp

x

n

nii

x

idxxpadxx

p )(10

0

ai Taylor DRP

a-3=-a3 -0.0167 -0.0208

a-2=-a2 0.1500 0.1667

a-1=-a1 -0.7500 -0.7709

a0 0 0

Lowest-order schemes Two options for explicit schemes

Collocated-in-time (CIT) Staggered-in-time (SIT)

SIT is advantageous Higher numerical accuracy with lowest order central

difference scheme (see spatial discretisation) Halves memory use compared to CIT (in-place

computation possible) Doubles time step compared to CIT (stability)

FDTD temporal aspects

l l+1l-1

dt

pv

pv

pv

l l+1l-1

dt

p p pv vv v

l-0.5 l+0.5 l+1.5l-1.5

2 2 102l l lp p dtc v 1 2 l-0.5

0l lp p dtc v

High-order schemes By Taylor expansion

In general : improves accuracy Strongly increases memory cost

Advanced schemes Runge-Kutta Crank-Nickolson

– Implicit scheme– Stability guaranteed

FDTD temporal aspects

Numerical stability Time-delay system

Update equations can be written as a discrete time-delay system (z-transform of SIT scheme)

Poles of system should have a modulus smaller than or equal to one (or abs of eigenvalues of A should be smaller than 1)

FDTD stability

0.5

ll

li

PX

V

1 1 1 1 11 , 1.5

1 1, 1.5

i i i i i i

i i i i

M M M R N S M R N NA

N S N N

mm AXX 1

1 0.51

0.5 1.5 1, 1.5

l l li i

il l l

i i i i i

MP M P RV

N V N V S P

Numerical stability Time-delay system : pole plots (SIT,SIP)

FDTD stability

CN=1

CN=1.1

CN=0.5

2 2

1 1 1CN cdtdx dy

Numerical accuracy Two aspects

Phase error Amplitude error

SIP/SIT p-v FDTD is amplitude-error free At all Courant Numbers

FDTD results in phase errors Phase error decreases with finer spatial

discretisation Phase error vanishes when

– CN=1– Propagation along the diagonal of square cells

FDTD accuracy

2

2

2

2

2

2 )2/(sin)2/(sin)2/(sinarcsin2dz

dzkdy

dykdx

dxkdtc zyx

Numerical accuracy

FDTD accuracy

Including meteorological effects Concept of “background flow”

Most relevant interactions between wind and acoustics in outdoor applications near the ground included

– Convection in uniform flows– Refraction in non-uniform flows– Scattering of sound

No generation of sound The acoustics will not influence the macro-fluid flow

Inhomogeneous atmosphere No additional cost Highest sound speed determines stability criterion

time-domain in OSP

0

1 0pt

0 0v

v v v v20 0c pp

t

0v + v

Numerical discretisation Spatial discretisation : SIP still interesting Temporal discretisation needs care

SIT is moderately unstable CIT is fully stable but computationally costly PSIT weakly unstable

time-domain in OSP

Numerical discretisation PSIT scheme

– Second order terms in the flow speed are neglected during discretisation as wind speeds are typically low

– Explicit, efficient scheme still possible– Numerical error

» Small amplitude-errors appear» Phase error is not affected

– see Van Renterghem et al.(Appl. Acoust., 2007)– Can be efficiently implemented

time-domain in OSP

1 2 0.50

l l l-0.5 lnoflowp p dtc dt p 0 v v

1l+0.5 l-0.5 l l lnoflow noflow

0

dt p dt dtρ

0 0v v v v v v

Numerical discretisation

time-domain in OSP

Impedance boundary condition Locally reacting surfaces Models reflection at surfaces only

Including a second medium in the simulation domain Extended reaction (non-local reaction) Models both reflection at surfaces, absorption

inside, and transmission through materials Spatially heterogeneous materials

Finite absorbers

Direct convolution Frequency domain impedance definition:

Each frequency domain signal or function has a time domain analogy

In time domain, we need a convolution which is a computationally costly operation

Impedance boundary cond.

P Z V

*p t Z t v t

1Z t Z

0

t tp t Z t v d Z t v d

Using exp. decaying time-domain functions Efficient direct convolution Recursive approach

Series in j Easy time-domain equivalent Mass-spring-damper system

Pade approximants Examples of application

Attenborough 4-parameter model modified Zwikker and Kosten model

Digital filters Efficient IIR filters Highly flexibility to approach any w-Z curve

Impedance boundary cond.

01

aZj t

1 0 11Z a a a jj

1 0 1

t dv tp t a v t dt a v t a

dt

0

0

ttaZ t e

t

0

0

( )

ni

iim

ii

i

a zZ z

b z

Poro-rigid frame model : Zwikker and Kosten Only the air in between the material matrix is

allowed to vibrate Reasonable when density of the frame and the

stiffness is significantly larger than those of air 3-parameter model

(flow resistivity) (porosity) ks (structure factor)

Including porous medium

0 skpt

0v v

20 0 0cp

t

v

2

0 0

sjZ

kZ

Poro-elastic models : M.A. Biot Coupled movement of frame and air inside the

porous medium included slightly adapted version Parameters

Tortuosity: mt

Porosity: a , f= 1-a

Flow resistivity: Bulk modulus of frame: Kf

Frame density: f

Frame damping coefficient: Rf

Including porous medium

0a aa

a a ffp K Pt

K

v v

2 1aa a a f a

t

aa f

mpt t

v v v v v

f af

ff

a

p pt t

K

v

2 1ff f

tf f a f a a f

a

pt

Rt

m

vv v v v v

Long distance propagation Volume discretisation techniques not well suited

for long-distance propagation Solutions

Moving-frame FDTD– Use of short, broadband pulses– Mainly software challenge

» Allocate and de-allocate memory in an efficient way

– Mainly efficient if propagation is essentially in one-direction

time-domain in OSP

Region a Region b Region c

Long distance propagation Solutions

Hybrid modelling– Many techniques highly efficient in particular

cases– Coupling in an attempt to combine “best of both

worlds”– Coupling in same domain

» BEM-PE– Cross-domain coupling

» Raytracing-analytical formulae (e.g. diffraction)

» BEM-raytracing» FDTD-PE

time-domain in OSP

Long distance propagation Green’s Function Parabolic Equation method

PE-type model : one-way sound propagation, effective sound speed approach, range-dependent impedance and profiles, inclusion of terrain possible (CMM,GTPE,rGFPE), diffraction over thin screen, etc.

Works with vertical array of acoustic pressures Extrapollation towards next array based on Green’s

function Discretisation

– in vertical direction : strong discretisation– in forward direction : stepping at several

wavelengths without loss in accuracy Efficiently uses FFT

time-domain in OSP

Long distance propagation FDTD-GFPE

Typical road traffic noise application Complex source region, less complex receiver zone

Hybrid modelling

Long distance propagation FDTD-PE

One-way coupling Interface

– vertical array of receivers in FDTD– Time signal at each receiver recorded– Fourier transform gives starting function for PE for

all frequencies of interest– Should be well chosen

1FDTD, multiple PE calulations

Hybrid modelling

Long distance propagation FDTD-PE

Example : evaluation of T-noise barriers in wind

Hybrid modelling

Long distance propagation

Hybrid modelling

Long distance propagation FDTD-PE

Starting fields

Hybrid modelling

0 10 20 30 40 500

0.5

1

1.5

2

height z (m)

mag

nitu

de (

Pa) 100 Hz

0 10 20 30 40 500

0.1

0.2

0.3

0.4

height z (m)

mag

nitu

de (

Pa) 300 Hz

0 10 20 30 40 500

0.1

0.2

0.3

0.4

height z (m)

mag

nitu

de (

Pa) 500 Hz

Hybrid modelling

FDTD 2 mFDTD 4 mFDTD+PE 2 mFDTD+PE 4 m

locationstarting field

locationdownwindnoise barrier

Time-domain modelling in outdoor sound propagation has become mature

Low-order schemes are well suited in outdoor sound propagation when carefully choosing the numerical discretisation scheme

Hybrid modelling for long-distance sound propagation

Current trends in time-domain modelling Parallellisation (by using GPU) Pseudo-spectral time-domain technique (PSTD)

Conclusions