Application of the TLM method to the sound propagation ... · Thesis objective: sound propagation...
Transcript of Application of the TLM method to the sound propagation ... · Thesis objective: sound propagation...
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APPLICATION OF THE TLM METHOD TO THE SOUNDPROPAGATION MODELLING IN URBAN AREA
Gwenaël GUILLAUME
Laboratoire Central des Ponts et Chaussées (LCPC)
Thesis director: Judicaël PICAUT (LCPC-Nantes)Thesis co-director: Christophe AYRAULT (LAUM-Le Mans)Steering Committee: Isabelle SCHMICH (CSTB-Grenoble)
Guillaume DUTILLEUX (LRPC-Strasbourg)
October 13th, 2009
GWENAËL GUILLAUME 1
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 2
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 2
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 2
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 2
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 2
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
Global issue: noise annoyances prevention and abatement
health and societal impact of noiselegislative and regulation framework
LCPC research topic: predicting the noise level in urban environment
sound propagation modelling in urban area
GWENAËL GUILLAUME 3
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
Global issue: noise annoyances prevention and abatement
health and societal impact of noise
legislative and regulation framework
LCPC research topic: predicting the noise level in urban environment
sound propagation modelling in urban area
GWENAËL GUILLAUME 3
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
The urban noise: a complex issue
street ≡ « opened » waveguide⇒ steady-state phenomena, acoustic « leaks » by the open-tops
frontages morphology⇒ diffuse reflections, edge diffraction, absorption
long distance propagation⇒ atmospheric effects, ground effects, « unusual » micrometeorological conditions
temporal variations⇒ moving/time varying noise sources[1], micrometeorological conditions fluctutations[2]
Thesis objective: sound propagation modelling in urban area
• development of a specific time-domain numerical model
⇒ TLM method (Transmission Line Modelling)
[1] A. Can. Représentation du trafic et caractérisation dynamique du bruit en milieu urbain. PhD Thesis, Lyon, 2008.
[2] F. Junker et al.. Meteorological classification for environmental acoustics - Practical implications due to experimental accuracy and
uncertainty. ICA, Madrid (Espagne), 2007.
GWENAËL GUILLAUME 4
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
The urban noise: a complex issue
• street ≡ « opened » waveguide⇒ steady-state phenomena, acoustic « leaks » by the open-tops
• frontages morphology⇒ diffuse reflections, edge diffraction, absorption
• long distance propagation⇒ atmospheric effects, ground effects, « unusual » micrometeorological conditions
• temporal variations⇒ moving/time varying noise sources[1], micrometeorological conditions fluctutations[2]
Thesis objective: sound propagation modelling in urban area
development of a specific time-domain numerical model
⇒ TLM method (Transmission Line Modelling)
[1] A. Can. Représentation du trafic et caractérisation dynamique du bruit en milieu urbain. PhD Thesis, Lyon, 2008.
[2] F. Junker et al.. Meteorological classification for environmental acoustics - Practical implications due to experimental accuracy and
uncertainty. ICA, Madrid (Espagne), 2007.
GWENAËL GUILLAUME 4
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 5
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
HUYGENS principle (1690)
A wavefront can be broken down intoa set of secondary sources thatradiate spherical wavelets of identicalfrequency, amplitude and phase.
Numerical adaptation in electromagnetism[1]
The secondary sources are assimilated to nodes.
The « diffusion » of the field between nodes is performed by means of transmission linesin term of pulses.
[1] P.B. Johns and R.L. Beurle. Numerical solution of two dimensional scattering problems using a transmission line matrix. Proc. IEE, 118(9),
1971.
GWENAËL GUILLAUME 6
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 7
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Simple case in 2D
S R
R
R
RR
R
R
R
SA
Incident pulse
R S
R
R
RR
R
R
R
SR×A
T×A
T×A
T×A
Scattered pulses from the node
Nodal reflection and transmission coefficients:
R=ZT − ZL
ZT + ZL, R < 0
T = 1 +R =2ZT
ZT + ZL
ZT : impedance of the terminationZL : impedance of the incident transmission line(
here, ZL=Z and ZT=Z/3, soR=− 12 and T= 1
2
)
General case in 2D
.1 2
3
4
t I 1
t I 2
t I 3
t I 4
.1 2
3
4
t S 1
t S 2
t S 3
t S 4
Matrix relation: tS = D × tI
where tI =[
tI 1, tI 2, tI 3, tI 4]T,
tS =[
tS 1, tS 2, tS 3, tS 4]T,
and D =
[R T T TT R T TT T R TT T T R
].
GWENAËL GUILLAUME 8
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Simple case in 2D
S R
R
R
RR
R
R
R
SA
Incident pulse
R S
R
R
RR
R
R
R
SR×A
T×A
T×A
T×A
Scattered pulses from the node
Nodal reflection and transmission coefficients:
R=ZT − ZL
ZT + ZL, R < 0
T = 1 +R =2ZT
ZT + ZL
ZT : impedance of the terminationZL : impedance of the incident transmission line(
here, ZL=Z and ZT=Z/3, soR=− 12 and T= 1
2
)
General case in 2D
.1 2
3
4
t I 1
t I 2
t I 3
t I 4
.1 2
3
4
t S 1
t S 2
t S 3
t S 4
Matrix relation: tS = D × tI
where tI =[
tI 1, tI 2, tI 3, tI 4]T,
tS =[
tS 1, tS 2, tS 3, tS 4]T,
and D =
[R T T TT R T TT T R TT T T R
].
GWENAËL GUILLAUME 8
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Diffusion in the transmission lines network⇒ connexion laws
t+∆tI 1(i,j) = tS 2
(i−1,j)
t+∆tI 2(i,j) = tS 1
(i+1,j)
t+∆tI 3(i,j) = tS 4
(i,j−1)
t+∆tI 4(i,j) = tS 3
(i,j +1)
Nodal pressure definition
t p(i,j) =12
4∑n=1
t In(i,j)
GWENAËL GUILLAUME 9
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Diffusion in the transmission lines network⇒ connexion laws
t+∆tI 1(i,j) = tS 2
(i−1,j)
t+∆tI 2(i,j) = tS 1
(i+1,j)
t+∆tI 3(i,j) = tS 4
(i,j−1)
t+∆tI 4(i,j) = tS 3
(i,j +1)
R S
R
R
RR
R
R
R
t S 2(i−1,j)
(i − 1, j) (i, j)
Nodal pressure definition
t p(i,j) =12
4∑n=1
t In(i,j)
GWENAËL GUILLAUME 9
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Diffusion in the transmission lines network⇒ connexion laws
t+∆tI 1(i,j) = tS 2
(i−1,j)
t+∆tI 2(i,j) = tS 1
(i+1,j)
t+∆tI 3(i,j) = tS 4
(i,j−1)
t+∆tI 4(i,j) = tS 3
(i,j +1)
R S
R
R
RR
R
R
R
t+∆t I 1(i,j)
(i − 1, j) (i, j)
Nodal pressure definition
t p(i,j) =12
4∑n=1
t In(i,j)
GWENAËL GUILLAUME 9
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Diffusion in the transmission lines network⇒ connexion laws
t+∆tI 1(i,j) = tS 2
(i−1,j)
t+∆tI 2(i,j) = tS 1
(i+1,j)
t+∆tI 3(i,j) = tS 4
(i,j−1)
t+∆tI 4(i,j) = tS 3
(i,j +1)
R S
R
R
RR
R
R
R
t S 1(i+1,j)
t+∆t I 2(i,j)
(i, j) (i + 1, j)
Nodal pressure definition
t p(i,j) =12
4∑n=1
t In(i,j)
GWENAËL GUILLAUME 9
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Diffusion in the transmission lines network⇒ connexion laws
t+∆tI 1(i,j) = tS 2
(i−1,j)
t+∆tI 2(i,j) = tS 1
(i+1,j)
t+∆tI 3(i,j) = tS 4
(i,j−1)
t+∆tI 4(i,j) = tS 3
(i,j +1)
R S
R
R
RR
R
R
R
t S 4(i,j−1)t+∆t I 3
(i,j)
(i, j − 1)
(i, j)
Nodal pressure definition
t p(i,j) =12
4∑n=1
t In(i,j)
GWENAËL GUILLAUME 9
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Diffusion in the transmission lines network⇒ connexion laws
t+∆tI 1(i,j) = tS 2
(i−1,j)
t+∆tI 2(i,j) = tS 1
(i+1,j)
t+∆tI 3(i,j) = tS 4
(i,j−1)
t+∆tI 4(i,j) = tS 3
(i,j +1)
R S
R
R
RR
R
R
Rt S 3
(i,j+1)t+∆t I 4(i,j)
(i, j + 1)
(i, j)
Nodal pressure definition
t p(i,j) =12
4∑n=1
t In(i,j)
GWENAËL GUILLAUME 9
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Diffusion in the transmission lines network⇒ connexion laws
t+∆tI 1(i,j) = tS 2
(i−1,j)
t+∆tI 2(i,j) = tS 1
(i+1,j)
t+∆tI 3(i,j) = tS 4
(i,j−1)
t+∆tI 4(i,j) = tS 3
(i,j +1)
R S
R
R
RR
R
R
R
t I 1(i,j)
t I 2(i,j)t I 3
(i,j)
t I 4(i,j)
Nodal pressure definition
t p(i,j) =12
4∑n=1
t In(i,j)
GWENAËL GUILLAUME 9
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 10
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Heterogeneous propagation medium modelling (micrometeorological conditions)
addition of an open-circuited branch, of impedance Z/η,to the nodal original configuration to introducerefraction and turbulence where the parameter η iscalculated by[1]:
tη(i,j) = 4
( c0
tceff(i,j)
)2
− 1
,
where tceff(i,j) =√γ R tT(i,j) + tW(i,j) .tu(i,j) .
1(Z)
2(Z)
3
(Z)
4
(Z)
5
(Z5= Z
η
)
[1] G. Dutilleux. Applicability of TLM to wind turbine noise prediction, 2nd Int. Meeting on Wind Turbine Noise, Lyon (France), 2007.
GWENAËL GUILLAUME 11
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Dissipative propagation medium modelling (atmospheric attenuation)
addition of an anechoic terminated branch, ofimpedance Z/ζ, to the original nodal configurationwhere the attenuation factor ζ is defined by[1]:
tζ(i,j) = −α√
tη(i,j) + 4 ∆lln (10)
20,
with α = f (T,P0,H) the atmospheric absorptioncoefficient (expressed in dB.m-1) and ∆l the spatial step(in m).
1(Z)
2(Z)
3
(Z)
4
(Z)
6
(Z6= Z
ζ
)
[1] J. Hofmann and K. Heutschi. Numerical simulation of sound wave propagation with sound absorption in time domain. Appl. Acoust., 68(2),
2007.
GWENAËL GUILLAUME 12
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Heterogeneous and dissipative propagation medium modelling
Matrix relation: t S(i,j) = t D(i,j) × t I(i,j),
where t I(i,j) =[
t I1 ; t I
2 ; tI3 ; tI
4 ; tI5]T,
t S(i,j) =[
tS1 ; tS
2 ; tS3 ; tS
4 ; tS5]T,
and t D(i,j) =2
tη(i,j) + tζ(i,j) + 4
t
a 1 1 1 η1 a 1 1 η1 1 a 1 η1 1 1 a η1 1 1 1 b
(i,j)
,
with ta(i,j) = −(
tη(i,j)
2+
tζ(i,j)
2+ 1
)and t b(i,j) =
tη(i,j)
2−(
tζ(i,j)
2+ 2
).
1(Z)
2(Z)
3
(Z)
4
(Z)
5
(Z5= Z
η
)
6
(Z6= Z
ζ
)
Connexion laws: t+∆tI 5(i,j) = tS 5
(i,j)
Nodal pressure: t p(i,j) = 2tη(i,j)+tζ(i,j)+4
(4∑
n=1t I
n(i,j) + tη(i,j) t I
5(i,j)
)
GWENAËL GUILLAUME 13
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Heterogeneous and dissipative propagation medium modelling
Matrix relation: t S(i,j) = t D(i,j) × t I(i,j),
where t I(i,j) =[
t I1 ; t I
2 ; tI3 ; tI
4 ; tI5]T,
t S(i,j) =[
tS1 ; tS
2 ; tS3 ; tS
4 ; tS5]T,
and t D(i,j) =2
tη(i,j) + tζ(i,j) + 4
t
a 1 1 1 η1 a 1 1 η1 1 a 1 η1 1 1 a η1 1 1 1 b
(i,j)
,
with ta(i,j) = −(
tη(i,j)
2+
tζ(i,j)
2+ 1
)and t b(i,j) =
tη(i,j)
2−(
tζ(i,j)
2+ 2
).
1(Z)
2(Z)
3
(Z)
4
(Z)
5
(Z5= Z
η
)
6
(Z6= Z
ζ
)
Connexion laws: t+∆tI 5(i,j) = tS 5
(i,j)
Nodal pressure: t p(i,j) = 2tη(i,j)+tζ(i,j)+4
(4∑
n=1t I
n(i,j) + tη(i,j) t I
5(i,j)
)
GWENAËL GUILLAUME 13
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Heterogeneous and dissipative propagation medium modelling
Matrix relation: t S(i,j) = t D(i,j) × t I(i,j),
where t I(i,j) =[
t I1 ; t I
2 ; tI3 ; tI
4 ; tI5]T,
t S(i,j) =[
tS1 ; tS
2 ; tS3 ; tS
4 ; tS5]T,
and t D(i,j) =2
tη(i,j) + tζ(i,j) + 4
t
a 1 1 1 η1 a 1 1 η1 1 a 1 η1 1 1 a η1 1 1 1 b
(i,j)
,
with ta(i,j) = −(
tη(i,j)
2+
tζ(i,j)
2+ 1
)and t b(i,j) =
tη(i,j)
2−(
tζ(i,j)
2+ 2
).
1(Z)
2(Z)
3
(Z)
4
(Z)
5
(Z5= Z
η
)
6
(Z6= Z
ζ
)
Connexion laws: t+∆tI 5(i,j) = tS 5
(i,j)
Nodal pressure: t p(i,j) = 2tη(i,j)+tζ(i,j)+4
(4∑
n=1t I
n(i,j) + tη(i,j) t I
5(i,j)
)
GWENAËL GUILLAUME 13
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 14
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Wall characterized by a pressure reflection coefficient
Example: node (i, j) located at the vicinity of a west side wall defined by a pressurereflection coefficient R1
t+∆tI 1(i,j) = R1 × tS 1
(i,j)
t+∆tI 2(i,j) = tS 1
(i+1,j)
t+∆tI 3(i,j) = tS 4
(i,j−1)
t+∆tI 4(i,j) = tS 3
(i,j+1)
12
3
4
5
6
t S 1(i,j)
t+∆t I 1(i,j)
(i, j)
∆l/2
Relation between the pressure reflection coefficient R1 and the absorption coefficient inenergy α1:
α1 = 1− |R1|2
GWENAËL GUILLAUME 15
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 16
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Analogy with the wave equation
Combination of the matrix relation, the connexion laws and the nodal pressure definition:
tη(i,j) + 4
2∆t2
∆l2
∂2ttp(i,j)︷ ︸︸ ︷
t+∆t p (i,j) − 2 t p (i,j) + t−∆t p (i,j)
∆t2+tζ(i,j)
∆t∆l2
∂tp(i,j)︷ ︸︸ ︷t+∆t p (i,j) − t−∆t p (i,j)
2 ∆t=
t p (i+1,j) − 2 t p (i,j) + t p (i−1,j)
∆l2︸ ︷︷ ︸∂2
xxp(i,j)
+t p (i,j+1) − 2 t p (i,j) + t p (i,j−1)
∆l2︸ ︷︷ ︸∂2
yyp(i,j)
Helmholtz equation in a heterogeneous and dissipative medium:[∆ +
(ω2
c2TLM
− jω ζ(i,j)
c ∆l
)]P(i,j) = 0, c =
∆l∆t
Celerity correction:
cTLM =
√2
tη(i,j) + 4c ⇒ c =
√tη(i,j) + 4
2c0
GWENAËL GUILLAUME 17
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Analogy with the wave equation
Combination of the matrix relation, the connexion laws and the nodal pressure definition:
tη(i,j) + 4
2∆t2
∆l2
∂2ttp(i,j)︷ ︸︸ ︷
t+∆t p (i,j) − 2 t p (i,j) + t−∆t p (i,j)
∆t2+tζ(i,j)
∆t∆l2
∂tp(i,j)︷ ︸︸ ︷t+∆t p (i,j) − t−∆t p (i,j)
2 ∆t=
t p (i+1,j) − 2 t p (i,j) + t p (i−1,j)
∆l2︸ ︷︷ ︸∂2
xxp(i,j)
+t p (i,j+1) − 2 t p (i,j) + t p (i,j−1)
∆l2︸ ︷︷ ︸∂2
yyp(i,j)
Helmholtz equation in a heterogeneous and dissipative medium:[∆ +
(ω2
c2TLM
− jω ζ(i,j)
c ∆l
)]P(i,j) = 0, c =
∆l∆t
Celerity correction:
cTLM =
√2
tη(i,j) + 4c ⇒ c =
√tη(i,j) + 4
2c0
GWENAËL GUILLAUME 17
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Analogy with the wave equation
Combination of the matrix relation, the connexion laws and the nodal pressure definition:
tη(i,j) + 4
2∆t2
∆l2
∂2ttp(i,j)︷ ︸︸ ︷
t+∆t p (i,j) − 2 t p (i,j) + t−∆t p (i,j)
∆t2+tζ(i,j)
∆t∆l2
∂tp(i,j)︷ ︸︸ ︷t+∆t p (i,j) − t−∆t p (i,j)
2 ∆t=
t p (i+1,j) − 2 t p (i,j) + t p (i−1,j)
∆l2︸ ︷︷ ︸∂2
xxp(i,j)
+t p (i,j+1) − 2 t p (i,j) + t p (i,j−1)
∆l2︸ ︷︷ ︸∂2
yyp(i,j)
Helmholtz equation in a heterogeneous and dissipative medium:[∆ +
(ω2
c2TLM
− jω ζ(i,j)
c ∆l
)]P(i,j) = 0, c =
∆l∆t
Celerity correction:
cTLM =
√2
tη(i,j) + 4c ⇒ c =
√tη(i,j) + 4
2c0
GWENAËL GUILLAUME 17
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Analogy with the wave equation
Combination of the matrix relation, the connexion laws and the nodal pressure definition:
tη(i,j) + 4
2∆t2
∆l2
∂2ttp(i,j)︷ ︸︸ ︷
t+∆t p (i,j) − 2 t p (i,j) + t−∆t p (i,j)
∆t2+tζ(i,j)
∆t∆l2
∂tp(i,j)︷ ︸︸ ︷t+∆t p (i,j) − t−∆t p (i,j)
2 ∆t=
t p (i+1,j) − 2 t p (i,j) + t p (i−1,j)
∆l2︸ ︷︷ ︸∂2
xxp(i,j)
+t p (i,j+1) − 2 t p (i,j) + t p (i,j−1)
∆l2︸ ︷︷ ︸∂2
yyp(i,j)
Helmholtz equation in a heterogeneous and dissipative medium:[∆ +
(ω2
c2TLM
− jω ζ(i,j)
c ∆l
)]P(i,j) = 0, c =
∆l∆t
Celerity correction:
cTLM =
√2
tη(i,j) + 4c ⇒ c =
√tη(i,j) + 4
2c0
GWENAËL GUILLAUME 17
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 18
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Room acoustics applications
Figure: Eigenmode (2, 1) of a 2D room with perfectlyreflecting walls• dimensions: (6.75 m × 5.23 m)• discretization: ∆l = 16 cm and ∆t = 0.3 ms• sinusoidal source frequency: 60.5 Hz
Figure: Reverberation time of a 3D room• dimensions: (5 m × 4 m × 3 m)
• discretization: ∆l = 5 cm and ∆t = 8× 10−5 s• gaussian pulse source frequency: 500 Hz
GWENAËL GUILLAUME 19
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
« Open-space » application
SR
HSHR
xR
HS = 1 m, HR = 2 m and xR = 20 m
perfectly reflective ground
discretization: ∆l = 2 cm and∆t = 4.1× 10−5 s
gaussian pulse source frequency: 1500 Hz
GWENAËL GUILLAUME 20
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Literature study
few developments
few validations sometimes limited or even arguable
Thesis contributions
analytical formulation of a TLM model combining most of the propagative phenomena
achievement of a generic 2D/3D formulation and numerical implementation
rigorous validation of the model for academic cases
Main limitations of the model
no relevant virtual boundary condition formulation in TLM for acoustic modelling
no realistic boundaries conditions
GWENAËL GUILLAUME 21
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Literature study
few developments
few validations sometimes limited or even arguable
Thesis contributions
analytical formulation of a TLM model combining most of the propagative phenomena
achievement of a generic 2D/3D formulation and numerical implementation
rigorous validation of the model for academic cases
Main limitations of the model
no relevant virtual boundary condition formulation in TLM for acoustic modelling
no realistic boundaries conditions
GWENAËL GUILLAUME 21
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM METHOD PRINCIPLEHOMOGENEOUS AND NON-DISSIPATIVE ATMOSPHERE MODELLINGHETEROGENEOUS AND DISSIPATIVE ATMOSPHERE MODELLINGBOUNDARY CONDITION: PRESSURE REFLECTION COEFFICIENTANALOGY WITH THE WAVE EQUATIONNUMERICAL VERIFICATIONS AND CONCLUSIONS
Literature study
few developments
few validations sometimes limited or even arguable
Thesis contributions
analytical formulation of a TLM model combining most of the propagative phenomena
achievement of a generic 2D/3D formulation and numerical implementation
rigorous validation of the model for academic cases
Main limitations of the model
no relevant virtual boundary condition formulation in TLM for acoustic modelling
no realistic boundaries conditions
GWENAËL GUILLAUME 21
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CLASSICAL FORMULATIONIMPEDANCE REPRESENTATIONTLM IMPEDANCE BOUNDARY CONDITION FORMULATIONNUMERICAL VALIDATION
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 22
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CLASSICAL FORMULATIONIMPEDANCE REPRESENTATIONTLM IMPEDANCE BOUNDARY CONDITION FORMULATIONNUMERICAL VALIDATION
Classical impedance boundary condition formulation:
in the frequency domain:
P(b) (ω) = Z (ω)× Vn(b) (ω)
in the time domain:
p(b) (t) = z (t) ∗ vn(b) (t) =
∫ +∞
−∞z(t′)× vn(b)
(t − t′
)dt′
where z (t) = F−1 [Z (ω)]
Necessary conditions to transpose Z (ω) in the time domain:[1]
causality
passivity
reality
[1] S.W. Rienstra. Impedance models in time domain including the extended Helmholtz resonator model. 12th AIAA/CEAS Conf., Cambridge,
Massachusetts (USA), 2006.
GWENAËL GUILLAUME 23
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CLASSICAL FORMULATIONIMPEDANCE REPRESENTATIONTLM IMPEDANCE BOUNDARY CONDITION FORMULATIONNUMERICAL VALIDATION
Classical impedance boundary condition formulation:
in the frequency domain:
P(b) (ω) = Z (ω)× Vn(b) (ω)
in the time domain:
p(b) (t) = z (t) ∗ vn(b) (t) =
∫ +∞
−∞z(t′)× vn(b)
(t − t′
)dt′
where z (t) = F−1 [Z (ω)]
Necessary conditions to transpose Z (ω) in the time domain:[1]
causality
passivity
reality
[1] S.W. Rienstra. Impedance models in time domain including the extended Helmholtz resonator model. 12th AIAA/CEAS Conf., Cambridge,
Massachusetts (USA), 2006.
GWENAËL GUILLAUME 23
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CLASSICAL FORMULATIONIMPEDANCE REPRESENTATIONTLM IMPEDANCE BOUNDARY CONDITION FORMULATIONNUMERICAL VALIDATION
Classical impedance boundary condition formulation:
in the frequency domain:
P(b) (ω) = Z (ω)× Vn(b) (ω)
in the time domain:
p(b) (t) = z (t) ∗ vn(b) (t) =
∫ +∞
−∞z(t′)× vn(b)
(t − t′
)dt′
where z (t) = F−1 [Z (ω)]
Necessary conditions to transpose Z (ω) in the time domain:[1]
causalitypassivity
reality
[1] S.W. Rienstra. Impedance models in time domain including the extended Helmholtz resonator model. 12th AIAA/CEAS Conf., Cambridge,
Massachusetts (USA), 2006.
GWENAËL GUILLAUME 23
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CLASSICAL FORMULATIONIMPEDANCE REPRESENTATIONTLM IMPEDANCE BOUNDARY CONDITION FORMULATIONNUMERICAL VALIDATION
Classical impedance boundary condition formulation:
in the frequency domain:
P(b) (ω) = Z (ω)× Vn(b) (ω)
in the time domain:
p(b) (t) = z (t) ∗ vn(b) (t) =
∫ +∞
−∞z(t′)× vn(b)
(t − t′
)dt′
where z (t) = F−1 [Z (ω)]
Necessary conditions to transpose Z (ω) in the time domain:[1]
causality
passivityreality
[1] S.W. Rienstra. Impedance models in time domain including the extended Helmholtz resonator model. 12th AIAA/CEAS Conf., Cambridge,
Massachusetts (USA), 2006.
GWENAËL GUILLAUME 23
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CLASSICAL FORMULATIONIMPEDANCE REPRESENTATIONTLM IMPEDANCE BOUNDARY CONDITION FORMULATIONNUMERICAL VALIDATION
Classical impedance boundary condition formulation:
in the frequency domain:
P(b) (ω) = Z (ω)× Vn(b) (ω)
in the time domain:
p(b) (t) = z (t) ∗ vn(b) (t) =
∫ +∞
−∞z(t′)× vn(b)
(t − t′
)dt′
where z (t) = F−1 [Z (ω)]
Necessary conditions to transpose Z (ω) in the time domain:[1]
causality
passivity
reality
[1] S.W. Rienstra. Impedance models in time domain including the extended Helmholtz resonator model. 12th AIAA/CEAS Conf., Cambridge,
Massachusetts (USA), 2006.
GWENAËL GUILLAUME 23
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CLASSICAL FORMULATIONIMPEDANCE REPRESENTATIONTLM IMPEDANCE BOUNDARY CONDITION FORMULATIONNUMERICAL VALIDATION
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 24
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CLASSICAL FORMULATIONIMPEDANCE REPRESENTATIONTLM IMPEDANCE BOUNDARY CONDITION FORMULATIONNUMERICAL VALIDATION
Impedance representation by a sum of 1st order linear systems[1]
in the frequency domain (response of K linear systems):
Z (ω) =
K∑k=1
Ak
λk − jω
where λk are real poles (λk > 0)
in the time domain (sum of K impulse responses):
z (t) =K∑
k=1
Ake−λk tH (t)
where H (t) is the HEAVISIDE function
[1] Y. Reymen et al.. Time-domain impedance formulation based on recursive convolution. 12th AIAA/CEAS Conf., Cambridge, Massachusetts
(USA), 2006.
GWENAËL GUILLAUME 25
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CLASSICAL FORMULATIONIMPEDANCE REPRESENTATIONTLM IMPEDANCE BOUNDARY CONDITION FORMULATIONNUMERICAL VALIDATION
ZWIKKER and KOSTEN impedance model application[1]
Model expression:
Z (ω) = Z∞
√1 + jωτ
jωτwhere τ =
ρ0q2γ
RSΩa time constant
and Z∞=ρ0c0q
Ωthe impedance at the limit ωτ →∞
Transposition in the time domain[2]:
z (t) = Z∞
[δ (t) +
1τ
f (t)]
where t = t/τ
Impulse response approximation f(t):
f (t) =e−t/2
2
[I1
(t2
)+ I0
(t2
)]H (t) =
K∑k=1
Ake−λk tH (t)
[1] C. Zwikker and C. W. Kosten. Sound absorbing materials. Elsevier Ed., New York, 1949.
[2] V. E. Ostashev et al. Padé approximation in time-domain boundary conditions of porous surfaces. JASA, 122(1), 2007.
GWENAËL GUILLAUME 26
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CLASSICAL FORMULATIONIMPEDANCE REPRESENTATIONTLM IMPEDANCE BOUNDARY CONDITION FORMULATIONNUMERICAL VALIDATION
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 27
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CLASSICAL FORMULATIONIMPEDANCE REPRESENTATIONTLM IMPEDANCE BOUNDARY CONDITION FORMULATIONNUMERICAL VALIDATION
Time domain impedance boundary condition formulation[1]
p (m∆t) = Z′[
vn (m∆t) +K∑
k=1
A′kψk (m∆t)
]
where the accumulators ψk are given by (⇒ recursive convolution method)
ψk (m∆t) = vn (m∆t)
(1− e−λk∆t′
)λk
+ e−λk∆t′ψk ((m− 1) ∆t)
ZWIKKER and KOSTEN model: Z′ = Z∞, A′k = Ak and ∆t′ = ∆t
MIKI model[2]: Z′ = Z0, A′k = AkµΓ(−bM)
and ∆t′ = ∆t.
[1] Y. Reymen et al. Time-domain impedance formulation based on recursive convolution. 12th AIAA/CEAS Conf., Cambridge, Massachusetts
(USA), 2006.
[2] B. Cotté Propagation acoustique en milieu extérieur complexe: problèmes spécifiques au ferroviaire dans le contexte des trains à grande
vitesse. PhD Thesis, LMFA, École Centrale de Lyon (France), 2008.
GWENAËL GUILLAUME 28
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CLASSICAL FORMULATIONIMPEDANCE REPRESENTATIONTLM IMPEDANCE BOUNDARY CONDITION FORMULATIONNUMERICAL VALIDATION
Boundaries modelling in a TLM model: case of the ground
Introduction of a virtual node
Boundary pressure definition:
p(
t +∆t2
)= tS 3
(i,j) + tS 4(i,j−1)
Normal particle velocity:
vn
(t +
∆t2
)=
tS 3(i,j) − tS 4
(i,j−1)
ρ0 c
1 2
3
4
5
6
(i, j)
1 2
3
4
5
6
(i, j − 1)
t S 3(i,j)
t S 4(i,j−1) ∆l/2
∆l/2
GWENAËL GUILLAUME 29
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CLASSICAL FORMULATIONIMPEDANCE REPRESENTATIONTLM IMPEDANCE BOUNDARY CONDITION FORMULATIONNUMERICAL VALIDATION
TLM model matching
Scattered pulse from the virtual node:
tm S 4(i,j−1) = tm S 3
(i,j)
[−1 + Λk
1 + Λk
]+
Z′
1 + Λk
K∑k=1
A′ke−λk∆t′tm−∆tψk
where Λk =Z′
ρ0 c
(1 +
K∑k=1
A′k1− e−λk∆t′
λk
) 1 2
3
4
5
6
(i, j)
1 2
3
4
5
6
(i, j − 1)
t S 3(i,j)
t S 4(i,j−1)
Accumulators:
tm−∆tψk =
(tm−∆tS3 (i, j)− tm−∆tS4 (i, j− 1)
ρ0 c
)(1− e−λk∆t′
λk
)+ e−λk∆t′
tm−2∆tψk
GWENAËL GUILLAUME 30
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CLASSICAL FORMULATIONIMPEDANCE REPRESENTATIONTLM IMPEDANCE BOUNDARY CONDITION FORMULATIONNUMERICAL VALIDATION
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 31
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CLASSICAL FORMULATIONIMPEDANCE REPRESENTATIONTLM IMPEDANCE BOUNDARY CONDITION FORMULATIONNUMERICAL VALIDATION
Heterogeneous plane ground: ZWIKKER and KOSTEN impedance model
SR
HSHR
xR
xD
(RS1, q1, Ω1) (RS2
, q2, Ω2)
HS = 1 m, HR = 2 m and xR = 20 m
discontinuity at xD = 10 m from the source
RS1 = 10 kN.s.m-4, q1 =√
3.5 and Ω1 = 0.2
RS2 = 100 kN.s.m-4, q2 =√
10 and Ω2 = 0.5
GWENAËL GUILLAUME 32
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CLASSICAL FORMULATIONIMPEDANCE REPRESENTATIONTLM IMPEDANCE BOUNDARY CONDITION FORMULATIONNUMERICAL VALIDATION
Heterogeneous plane ground: MIKI impedance model
SR
HSHR
xR
xD
(RS1) (RS2
)
HS = 1 m, HR = 2 m and xR = 20 m
discontinuity at xD = 10 m from the source
RS1 = 10 kN.s.m-4
RS2 = 1000 kN.s.m-4
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM ABSORBING CONDITIONS REVIEWPROPOSED TLM ABSORBING LAYERS FORMULATIONNUMERICAL VALIDATION
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 34
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM ABSORBING CONDITIONS REVIEWPROPOSED TLM ABSORBING LAYERS FORMULATIONNUMERICAL VALIDATION
Absorbing boundaries: application of a « non-reflective »termination
definition of the pressure field on the limit by a TAYLORseries expansion[1]
application of a purely real impedance condition[2]
Absorbing layers: introduction of an anisotropic absorbing region
Perfectly Matched Layers (PML)?
modification of the whole connexion laws for the nodeslocated inside the layer[3]
[1] S. El-Masri et al.. Vocal tract acoustics using the transmission line matrix (TLM). ICSLP, Philadelphia (USA), 1996.
[2] J. Hofmann and K. Heutschi. Numerical simulation of sound wave propagation with sound absorption in time domain. Appl. Acoust., 2007.
[3] D. De Cogan et al.. Transmission Line Matrix in Computational Mechanics. CRC Press, 2005.
GWENAËL GUILLAUME 35
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM ABSORBING CONDITIONS REVIEWPROPOSED TLM ABSORBING LAYERS FORMULATIONNUMERICAL VALIDATION
Absorbing boundaries: application of a « non-reflective »termination
definition of the pressure field on the limit by a TAYLORseries expansion[1]
application of a purely real impedance condition[2]
Absorbing layers: introduction of an anisotropic absorbing region
Perfectly Matched Layers (PML)?
modification of the whole connexion laws for the nodeslocated inside the layer[3]
[1] S. El-Masri et al.. Vocal tract acoustics using the transmission line matrix (TLM). ICSLP, Philadelphia (USA), 1996.
[2] J. Hofmann and K. Heutschi. Numerical simulation of sound wave propagation with sound absorption in time domain. Appl. Acoust., 2007.
[3] D. De Cogan et al.. Transmission Line Matrix in Computational Mechanics. CRC Press, 2005.
GWENAËL GUILLAUME 35
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM ABSORBING CONDITIONS REVIEWPROPOSED TLM ABSORBING LAYERS FORMULATIONNUMERICAL VALIDATION
Absorbing boundaries: application of a « non-reflective »termination
definition of the pressure field on the limit by a TAYLORseries expansion[1]
application of a purely real impedance condition[2]
Absorbing layers: introduction of an anisotropic absorbing region
Perfectly Matched Layers (PML)?
modification of the whole connexion laws for the nodeslocated inside the layer[3]
[1] S. El-Masri et al.. Vocal tract acoustics using the transmission line matrix (TLM). ICSLP, Philadelphia (USA), 1996.
[2] J. Hofmann and K. Heutschi. Numerical simulation of sound wave propagation with sound absorption in time domain. Appl. Acoust., 2007.
[3] D. De Cogan et al.. Transmission Line Matrix in Computational Mechanics. CRC Press, 2005.
GWENAËL GUILLAUME 35
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM ABSORBING CONDITIONS REVIEWPROPOSED TLM ABSORBING LAYERS FORMULATIONNUMERICAL VALIDATION
Absorbing boundaries: application of a « non-reflective »termination
definition of the pressure field on the limit by a TAYLORseries expansion[1]
application of a purely real impedance condition[2]
Absorbing layers: introduction of an anisotropic absorbing region
Perfectly Matched Layers (PML)?
modification of the whole connexion laws for the nodeslocated inside the layer[3]
[1] S. El-Masri et al.. Vocal tract acoustics using the transmission line matrix (TLM). ICSLP, Philadelphia (USA), 1996.
[2] J. Hofmann and K. Heutschi. Numerical simulation of sound wave propagation with sound absorption in time domain. Appl. Acoust., 2007.
[3] D. De Cogan et al.. Transmission Line Matrix in Computational Mechanics. CRC Press, 2005.
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM ABSORBING CONDITIONS REVIEWPROPOSED TLM ABSORBING LAYERS FORMULATIONNUMERICAL VALIDATION
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM ABSORBING CONDITIONS REVIEWPROPOSED TLM ABSORBING LAYERS FORMULATIONNUMERICAL VALIDATION
Proposed absorbing layers formulation
modification of the connexion law only forthe incident pulse propagating in thedirection of the computational domainlimit
DE COGAN et al. formulation
t+∆tI 1(i,j) = F(i,j) × tS2
(i−1,j)
t+∆tI 2(i,j) = F(i,j) × tS1
(i+1,j)
t+∆tI 3(i,j) = F(i,j) × tS4
(i,j−1)
t+∆tI 4(i,j) = F(i,j) × tS3
(i,j+1)
⇒
Proposed formulation
t+∆tI 1(i,j) = F(i,j) × tS2
(i−1,j)
t+∆tI 2(i,j) = tS1
(i+1,j)
t+∆tI 3(i,j) = tS4
(i,j−1)
t+∆tI 4(i,j) = tS3
(i,j+1)
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM ABSORBING CONDITIONS REVIEWPROPOSED TLM ABSORBING LAYERS FORMULATIONNUMERICAL VALIDATION
Proposed absorbing layers formulation
modification of the connexion law only forthe incident pulse propagating in thedirection of the computational domainlimit
DE COGAN et al. formulation
t+∆tI 1(i,j) = F(i,j) × tS2
(i−1,j)
t+∆tI 2(i,j) = F(i,j) × tS1
(i+1,j)
t+∆tI 3(i,j) = F(i,j) × tS4
(i,j−1)
t+∆tI 4(i,j) = F(i,j) × tS3
(i,j+1)
⇒
Proposed formulation
t+∆tI 1(i,j) = F(i,j) × tS2
(i−1,j)
t+∆tI 2(i,j) = tS1
(i+1,j)
t+∆tI 3(i,j) = tS4
(i,j−1)
t+∆tI 4(i,j) = tS3
(i,j+1)
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM ABSORBING CONDITIONS REVIEWPROPOSED TLM ABSORBING LAYERS FORMULATIONNUMERICAL VALIDATION
Attenuation factor for an absorbing layer of thickness eAL
Looking for a function such as:
F(d(iN ,jN) = 0
)= 1 at the interface
F(d(i1,j1) = eAL
)= ε on the limit, ε ∈ ]0, 1]
F(d(i,j)
)= (1 + ε)− exp
[−(d(i,j) − eAL
)2
B
]
with eAL =λNλAL
∆l and B = −e 2
ALln ε
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM ABSORBING CONDITIONS REVIEWPROPOSED TLM ABSORBING LAYERS FORMULATIONNUMERICAL VALIDATION
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM ABSORBING CONDITIONS REVIEWPROPOSED TLM ABSORBING LAYERS FORMULATIONNUMERICAL VALIDATION
Comparison of the virtual boundary conditions efficiency
error (x, y) = 10 log10
T∑t=0|pff (x, y, t)− p (x, y, t)|2
T∑t=0|pff (x, y, t)|2
Figure: Computational domain
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM ABSORBING CONDITIONS REVIEWPROPOSED TLM ABSORBING LAYERS FORMULATIONNUMERICAL VALIDATION
Comparison of the virtual boundary conditions efficiency
error (x, y) = 10 log10
T∑t=0|pff (x, y, t)− p (x, y, t)|2
T∑t=0|pff (x, y, t)|2
Figure: Free-field computation
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM ABSORBING CONDITIONS REVIEWPROPOSED TLM ABSORBING LAYERS FORMULATIONNUMERICAL VALIDATION
Comparison of the virtual boundary conditions efficiency
error (x, y) = 10 log10
T∑t=0|pff (x, y, t)− p (x, y, t)|2
T∑t=0|pff (x, y, t)|2
Figure: Virtual free-field computation
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM ABSORBING CONDITIONS REVIEWPROPOSED TLM ABSORBING LAYERS FORMULATIONNUMERICAL VALIDATION
Comparison of the virtual boundary conditions efficiency
error (x, y) = 10 log10
T∑t=0|pff (x, y, t)− p (x, y, t)|2
T∑t=0|pff (x, y, t)|2
Figure: Virtual free-field computation
Figure: Virtual boundary conditions efficiency(AL: NλAL = 5 and ε = 10−5)
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
TLM ABSORBING CONDITIONS REVIEWPROPOSED TLM ABSORBING LAYERS FORMULATIONNUMERICAL VALIDATION
Urban application
Figure: Street section Figure: Sound levels along the receivers axis
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CONCLUSIONURBAN APPLICATION EXAMPLESOUTLOOK
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CONCLUSIONURBAN APPLICATION EXAMPLESOUTLOOK
Work done
analytical formulation and numerical implementation of a 2D/3D TLM model integratingmost of the propagative phenomenaimprovement of the method
matched impedance boundary condition formulationnew formulation of absorbing layers
validation of the model by comparison with analytical and numerical solutions inacademic cases (room acoustics, outdoor sound propagation)
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CONCLUSIONURBAN APPLICATION EXAMPLESOUTLOOK
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 44
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CONCLUSIONURBAN APPLICATION EXAMPLESOUTLOOK
Parallel streets geometry (quiet street)
Figure: Gaussian pulse propagation
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CONCLUSIONURBAN APPLICATION EXAMPLESOUTLOOK
Urban noise barriers
Figure: Without barrier Figure: Green flat barrier
Figure: Perfectly reflective L-shaped barrier Figure: Green L-shaped barrier
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CONCLUSIONURBAN APPLICATION EXAMPLESOUTLOOK
1 ISSUE AND OBJECTIVES
2 TLM METHODTLM method principleHomogeneous and non-dissipative atmosphere modellingHeterogeneous and dissipative atmosphere modellingBoundary condition: pressure reflection coefficientAnalogy with the wave equationNumerical verifications and conclusions
3 IMPEDANCE BOUNDARY CONDITIONClassical formulationImpedance representationTLM impedance boundary condition formulationNumerical validation
4 VIRTUAL BOUNDARY CONDITIONTLM absorbing conditions reviewProposed TLM absorbing layers formulationNumerical validation
5 CONCLUSIONS AND OUTLOOKConclusionUrban application examplesOutlook
GWENAËL GUILLAUME 47
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CONCLUSIONURBAN APPLICATION EXAMPLESOUTLOOK
Outlook concerning our contribution
thickness consideration in the impedance boundary condition
⇒ coefficients identification in the frequency domain
rigorous PML formulation for TLM in acoustics
Outlook concerning the TLM model
atmospheric attenuation frequency dependency
⇒ digital filters[1]
sound transmission
⇒ transmission coefficient
⇒ wall acoustic propagation modelling
tetrahedral 3D mesh[2]
⇒ 3D simulations with 2D cartesian simulations computational burden
numerical scheme analysis
[1] T. Tsuchiya. Numerical simulation of sound wave propagation with sound absorption in time domain. 13th Int. Cong. Sound Vib., Vienne,
2006.
[2] S.J. Miklavcic and J. Ericsson Practical implementation of the 3D tetrahedral TLM method and visualization of room acoustics. ITN Resarch
Report ISSN, 2004.
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CONCLUSIONURBAN APPLICATION EXAMPLESOUTLOOK
Outlook concerning our contribution
thickness consideration in the impedance boundary condition
⇒ coefficients identification in the frequency domain
rigorous PML formulation for TLM in acoustics
Outlook concerning the TLM model
atmospheric attenuation frequency dependency
⇒ digital filters[1]
sound transmission
⇒ transmission coefficient
⇒ wall acoustic propagation modelling
tetrahedral 3D mesh[2]
⇒ 3D simulations with 2D cartesian simulations computational burden
numerical scheme analysis
[1] T. Tsuchiya. Numerical simulation of sound wave propagation with sound absorption in time domain. 13th Int. Cong. Sound Vib., Vienne,
2006.
[2] S.J. Miklavcic and J. Ericsson Practical implementation of the 3D tetrahedral TLM method and visualization of room acoustics. ITN Resarch
Report ISSN, 2004.
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CONCLUSIONURBAN APPLICATION EXAMPLESOUTLOOK
Outlook in terms of validation
micrometeorological conditions implementation
comparison with experimental results
Outlook in terms of applications
auralization (soundscape virtual modelling)
coupling with road trafic models
. . .
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
CONCLUSIONURBAN APPLICATION EXAMPLESOUTLOOK
Outlook in terms of validation
micrometeorological conditions implementation
comparison with experimental results
Outlook in terms of applications
auralization (soundscape virtual modelling)
coupling with road trafic models
. . .
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
Thank you for your attention
This thesis’ work is supported by the following organizations’ scientific drafts:
LCPC - Opération 11M061: « Prévoir le bruit en milieu urbain » (« Forecast the noise level in urban environment »)
IRSTV CNRS 2488, PRF « Environnements sonores urbains » (« Sound urban environments »)
GdR CNRS 2493, thème 2: « Propagation en espace urbain et en milieu ouvert » (« Propagation in urban area and in open-space »)
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ISSUE AND OBJECTIVESTLM METHOD
IMPEDANCE BOUNDARY CONDITIONVIRTUAL BOUNDARY CONDITION
CONCLUSIONS AND OUTLOOK
Wave propagation equations in absorbing layers
theoretical wave propagation equation in PML[1]:
1c2
0
∂2p∂t2−∂2p∂x2
= −1c2
0qx∂p∂t
+ ρ0qx∂ux
∂x+ ρ0ux
∂qx
∂x
discrete wave propagation equation obtained with the proposed method:
∆t2
∆l2t+∆tP(i) − 2 tP(i) + t−∆tP(i)
∆t2−
tP(i+1) − 2 tP(i) + tP(i−1)
∆l2=
−F(i)∆t2
∆l2tP(i) − t−∆tP(i)
∆t+ ρ0 F(i)
tu(i+1) − tu(i)
∆l+ ρ0 tu(i)
F(i+1) − F(i)
∆l+
Θ
∆l2
with Θ = −F(i+1)
[tS2
(i) − tS1(i)
]+[F(i) − 1
]t−∆tS1
(i) − 2 F(i) tS2(i−1)
−[F(i+1) − 1
]t−∆tS2
(i) + tS1(i) + tS2
(i) − t−∆tS1(i).
[1] Q. Qi and T.L. Geers. Evaluation of the perfectly matched layer for computational acoustics. J. Comput. Phys., 139(1), 1997.
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