SLIDE 1
Respondent Mathieu Gaborit Supervisors Peter Gransson (KTH, se) - - PowerPoint PPT Presentation
Respondent Mathieu Gaborit Supervisors Peter Gransson (KTH, se) - - PowerPoint PPT Presentation
Respondent Mathieu Gaborit Supervisors Peter Gransson (KTH, se) Olivier Dazel (Le Mans Universit, fr) Faculty Opponent Lucie Rouleau (CNAM, fr) Evaluation Committees Elke Deckers (KU Leuven, be) meline Sadoulet (FEMTO ST, fr) Annie
SLIDE 2
SLIDE 3
Context
- Co-tutelle agreement between
Le Mans Université and KTH
- LAUM UMR CNRS 6613 (Olivier Dazel)
- Marcus Wallenberg Lab. (Peter Göransson)
- 6-months periods
- Funding:
- Main funder: Le Mans Acoustique
- 4th year: AERIALIST
- Support from MWL
- Support from DENORMS CA15125
- Acoustics HUB Grant
- ÖMSE grant
- Free software:
- github.com/OlivierDAZEL/PLANES
- github.com/cpplanes/pymls
1
SLIDE 4
Efgects of noise
Efgects of noise1: stress/anxiety, heart diseases, sleep troubles Focus:
- Better performance (especially at low frequency)
- Multi-objectives: broadband, context adaptation, tunability
- Multi-functional
Of course: always thinner, lighter, concealed,…
1 Babisch (2002). Noise Health, pmid: 12537836
World Health Organization (2009). OCLC: ocn475454508 World Health Organisation (2011). OCLC: 779684347 Beutel et al. (2016). PLOS ONE,. 10.1371/journal.pone.0155357 Hume et al. (2012). Noise and Health, 10.4103/1463-1741.104897. 2
SLIDE 5
Classical design
Panels with multiple layers of foam/fjbrous media Allard and Atalla ( 2009) Pros: cheap to produce, robust, broadband Cons: ineffjcient at low frequency, hard to tune
3
SLIDE 6
Classical design
Panels with multiple layers of foam/fjbrous media Allard and Atalla ( 2009) Pros: cheap to produce, robust, broadband Cons: ineffjcient at low frequency, hard to tune
3
SLIDE 7
Classical design
Panels with multiple layers of foam/fjbrous media Allard and Atalla ( 2009) Pros: cheap to produce, robust, broadband Cons: ineffjcient at low frequency, hard to tune
3
SLIDE 8
Classical design
Panels with multiple layers of foam/fjbrous media Allard and Atalla ( 2009) Pros: cheap to produce, robust, broadband Cons: ineffjcient at low frequency, hard to tune
3
SLIDE 9
Classical design
Panels with multiple layers of foam/fjbrous media Allard and Atalla ( 2009) Pros: cheap to produce, robust, broadband Cons: ineffjcient at low frequency, hard to tune
3
SLIDE 10
Classical design
Fibrous (1.5mm) – Carpet (3mm) – Film (0.5mm)
2000 4000 6000 8000 10000 Frequency (Hz) 0.0 0.2 0.4 0.6 0.8 1.0 Absorption coefficient
= 0 deg = 45 deg
Panels with multiple layers of foam/fjbrous media Allard and Atalla ( 2009) Pros: cheap to produce, robust, broadband Cons: ineffjcient at low frequency, hard to tune
3
SLIDE 11
Classical design
Fibrous (1.5mm) – Carpet (3mm) – Film (0.5mm)
2000 4000 6000 8000 10000 Frequency (Hz) 0.0 0.2 0.4 0.6 0.8 1.0 Absorption coefficient
= 0 deg = 45 deg
Panels with multiple layers of foam/fjbrous media Allard and Atalla ( 2009) Pros: cheap to produce, robust, broadband Cons: ineffjcient at low frequency, hard to tune
3
SLIDE 12
Classical design
Fibrous (1.5mm) – Carpet (3mm) – Film (0.5mm)
2000 4000 6000 8000 10000 Frequency (Hz) 0.0 0.2 0.4 0.6 0.8 1.0 Absorption coefficient
= 0 deg = 45 deg
Panels with multiple layers of foam/fjbrous media Allard and Atalla ( 2009) Pros: cheap to produce, robust, broadband Cons: ineffjcient at low frequency, hard to tune
3
SLIDE 13
Meta(-poroelastic) materials
Idea: use networks of inclusions in a poroelastic matrix to improve performance
- Elastic/PEM cylinders: Groby et al. (JASA 2009)
- Elastic Shells: Weisser et al. (JASA 2016)
- Resonators: Boutin (JASA 2013) & Groby et al. (JASA
2015)
- Split rings: Lagarrigue et al. (JASA 2013)
Deep-subwavelength with space coiling (for ex.)
Zhou et al. (APL 2019) 4
SLIDE 14
Meta(-poroelastic) materials
Idea: use networks of inclusions in a poroelastic matrix to improve performance
- Elastic/PEM cylinders: Groby et al. (JASA 2009)
- Elastic Shells: Weisser et al. (JASA 2016)
- Resonators: Boutin (JASA 2013) & Groby et al. (JASA
2015)
- Split rings: Lagarrigue et al. (JASA 2013)
Deep-subwavelength with space coiling (for ex.)
Zhou et al. (APL 2019)
. . . . . . . . . . . . . . . . . . . . . . . .
PEM Film Elastic/PEM inclusion
4
SLIDE 15
Meta(-poroelastic) materials
Idea: use networks of inclusions in a poroelastic matrix to improve performance
- Elastic/PEM cylinders: Groby et al. (JASA 2009)
- Elastic Shells: Weisser et al. (JASA 2016)
- Resonators: Boutin (JASA 2013) & Groby et al. (JASA
2015)
- Split rings: Lagarrigue et al. (JASA 2013)
Deep-subwavelength with space coiling (for ex.)
Zhou et al. (APL 2019)
. . . . . . . . . . . . . . . . . . . . . . . .
PEM Film Elastic/PEM inclusion
4
SLIDE 16
Computing the response
Transfer Matrix Method z s′ = Ts
… …
Semi-analytical approaches (eq. Multiple Scattering Theory)
- r... Numerical methods (FEM)
Ax b Need to mesh
5
SLIDE 17
Computing the response
Transfer Matrix Method z s′ = Ts
… …
Semi-analytical approaches (eq. Multiple Scattering Theory)
- r... Numerical methods (FEM)
Ax = b Need to mesh
5
SLIDE 18
Thin layers, strong efgects
On the surface...
- Wear & tear
- Dust
- Porous
Acoustic Films ... but also between layers
- bonding area
- cluttered pores
- stifg glue layer
Interfaces Zones Strong infmuence on the response Chevillotte (JASA 2012)
500 1000 1500 2000 2500 3000 3500 4000
Frequency (Hz)
0.0 0.2 0.4 0.6 0.8 1.0
Absorption coefficient
PEM PEM + fjlm
6
SLIDE 19
Thin layers, strong efgects
On the surface...
- Wear & tear
- Dust
- Porous
Acoustic Films ... but also between layers
- bonding area
- cluttered pores
- stifg glue layer
Interfaces Zones Strong infmuence on the response Chevillotte (JASA 2012)
500 1000 1500 2000 2500 3000 3500 4000
Frequency (Hz)
0.0 0.2 0.4 0.6 0.8 1.0
Absorption coefficient
PEM PEM + fjlm
6
SLIDE 20
Thin layers, strong efgects
On the surface...
- Wear & tear
- Dust
- Porous
Acoustic Films ... but also between layers
- bonding area
- cluttered pores
- stifg glue layer
Interfaces Zones Strong infmuence on the response Chevillotte (JASA 2012)
500 1000 1500 2000 2500 3000 3500 4000
Frequency (Hz)
0.0 0.2 0.4 0.6 0.8 1.0
Absorption coefficient
PEM PEM + fjlm
6
SLIDE 21
Thin layers, strong efgects
On the surface...
- Wear & tear
- Dust
- Porous
Acoustic Films ... but also between layers
- bonding area
- cluttered pores
- stifg glue layer
Interfaces Zones Strong infmuence on the response Chevillotte (JASA 2012)
500 1000 1500 2000 2500 3000 3500 4000
Frequency (Hz)
0.0 0.2 0.4 0.6 0.8 1.0
Absorption coefficient
PEM PEM + fjlm
6
SLIDE 22
Thin layers, strong efgects
On the surface...
- Wear & tear
- Dust
- Porous
Acoustic Films ... but also between layers
- bonding area
- cluttered pores
- stifg glue layer
Interfaces Zones Strong infmuence on the response Chevillotte (JASA 2012)
500 1000 1500 2000 2500 3000 3500 4000
Frequency (Hz)
0.0 0.2 0.4 0.6 0.8 1.0
Absorption coefficient
PEM PEM + fjlm
6
SLIDE 23
Thin layers, strong efgects
On the surface...
- Wear & tear
- Dust
- Porous
Acoustic Films ... but also between layers
- bonding area
- cluttered pores
- stifg glue layer
Interfaces Zones Strong infmuence on the response Chevillotte (JASA 2012)
500 1000 1500 2000 2500 3000 3500 4000
Frequency (Hz)
0.0 0.2 0.4 0.6 0.8 1.0
Absorption coefficient
PEM PEM + fjlm
6
SLIDE 24
Thin layers, strong efgects
On the surface...
- Wear & tear
- Dust
- Porous
Acoustic Films ... but also between layers
- bonding area
- cluttered pores
- stifg glue layer
Interfaces Zones Strong infmuence on the response Chevillotte (JASA 2012)
500 1000 1500 2000 2500 3000 3500 4000
Frequency (Hz)
0.0 0.2 0.4 0.6 0.8 1.0
Absorption coefficient
PEM PEM + fjlm
6
SLIDE 25
Acoustic fjlms
Paper B
- Woven or non-woven
- From fmexible to stifg
- high fmow resistivity
(105 N s m
4 and above)
- very thin (less than 1 mm)
7
SLIDE 26
Acoustic fjlms
Paper B
- Woven or non-woven
- From fmexible to stifg
- high fmow resistivity
(105 N · s · m−4 and above)
- very thin (less than 1 mm)
7
SLIDE 27
Models: Equivalent fmuid — Rigid frame
Determine a ρeq and a Keq For instance Johnson-Champoux-Allard (JCA) model: σ (Flow resistivity) φ, Λ, α∞
Johnson et al. (JFM 1987)
Λ′
Champoux and Allard (JAP 1991)
α0, α′
0, k′ Lafarge et al. (JASA 1997), Lafarge ( 1993), Pride et al. (PRB 1993) 8
SLIDE 28
Models: Biot’s theory — Elastic frame
Coupled motion & constitutive equations
Biot (JASA 1956), Biot and Willis (JAMech 1957), Allard and Atalla ( 2009), Dazel et al. (JASA 2007)
- fmuid/skeleton interactions
- 3 waves (2 compressional, 1 shear)
- Fluid phase properties: JCA model
ρ1 Mass density E, ν Young’s modulus, Poisson’s ratio η Loss factor
9
SLIDE 29
Characterisation uncertainties
Paper B 40 samples per fjlm, 2 fjlms (based on ISO 9053-1:2018)
1000 2000 3000 4000 4520
Frequency (Hz)
1 2 3 4 5 6 7 8
Static air flow resistivity 1e6
Woven
1000 2000 3000 4000 4520
Frequency (Hz)
0.0 0.2 0.4 0.6 0.8 1.0
Static air flow resistivity 1e6
Non-woven Samples Mean
10
SLIDE 30
Characterisation uncertainties
Paper B 40 samples per fjlm, 2 fjlms (based on ISO 9053-1:2018)
′ 1
d Characterized Parameters 40 30 20 10 10 20 30 40 Deviation w.r.t. to mean (%) Non-woven film Woven film
10
SLIDE 31
C a n w e g e t a s i m p l e r m
- d
e l f
- r
fj l m s ? H
- w
t
- a
c c
- u
n t f
- r
u n c e r t a i n t i e s ? H
- w
t
- d
e a l w i t h e x p e n s i v e m
- d
e l s ?
SLIDE 32
C a n w e g e t a s i m p l e r m
- d
e l f
- r
fj l m s ? H
- w
t
- a
c c
- u
n t f
- r
u n c e r t a i n t i e s ? H
- w
t
- d
e a l w i t h e x p e n s i v e m
- d
e l s ?
SLIDE 33
C a n w e g e t a s i m p l e r m
- d
e l f
- r
fj l m s ? H
- w
t
- a
c c
- u
n t f
- r
u n c e r t a i n t i e s ? H
- w
t
- d
e a l w i t h e x p e n s i v e m
- d
e l s ?
SLIDE 34
C a n w e g e t a s i m p l e r m
- d
e l f
- r
fj l m s ? H
- w
t
- a
c c
- u
n t f
- r
u n c e r t a i n t i e s ? H
- w
t
- d
e a l w i t h e x p e n s i v e m
- d
e l s ?
SLIDE 35
C a n w e g e t a s i m p l e r m
- d
e l f
- r
fj l m s ?
SLIDE 36
Towards a simplifjcation
Paper A To represent the layer of fjlm: transfer matrix
Thomson (JAP 1950), Dazel et al. (JAP 2013) s(0) s(d)
d z s′ = T(d)s How to get T d ? d dzs z s z T d d Motion + constitutive eqs. d 1 T d I d
11
SLIDE 37
Towards a simplifjcation
Paper A To represent the layer of fjlm: transfer matrix
Thomson (JAP 1950), Dazel et al. (JAP 2013) s(0) s(d)
d z s′ = T(d)s How to get T(d)? d dzs(z) = αs(z) ⇒ T(d) = exp(−dα) Motion + constitutive eqs. d 1 T d I d
11
SLIDE 38
Towards a simplifjcation
Paper A To represent the layer of fjlm: transfer matrix
Thomson (JAP 1950), Dazel et al. (JAP 2013) s(0) s(d)
d z s′ = T(d)s How to get T(d)? d dzs(z) = αs(z) ⇒ T(d) = exp(−dα) Motion + constitutive eqs. d 1 T d I d
11
SLIDE 39
Towards a simplifjcation
Paper A To represent the layer of fjlm: transfer matrix
Thomson (JAP 1950), Dazel et al. (JAP 2013) s(0) s(d)
d z s′ = T(d)s How to get T(d)? d dzs(z) = αs(z) ⇒ T(d) = exp(−dα) Motion + constitutive eqs. d ≪ 1 ⇒ T(d) ≈ I − dα
11
SLIDE 40
Pruning the matrix
Paper A From Dazel et al. (JAP 2013) and Dazel et al. (JASA 2007) s = { ˆ σxz, us
z, ut z, ˆ
σzz, p, us
x
}T
xz xz
us
z
us
z
ut
z
ut
z zz zz
p p us
x
us
x
1 1 1 1 1 1 us 0 us d No coupling between us
x & ut z
No coupling between p &
xz 12
SLIDE 41
Pruning the matrix
Paper A From Dazel et al. (JAP 2013) and Dazel et al. (JASA 2007) ˆ σxz ˆ σxz′ us
z
us
z ′
ut
z
ut
z ′
ˆ σzz ˆ σzz′ p p′ us
x
us
x ′
1 1 1 1 1 1 us 0 us d No coupling between us
x & ut z
No coupling between p &
xz 12
SLIDE 42
Pruning the matrix
Paper A From Dazel et al. (JAP 2013) and Dazel et al. (JASA 2007) ˆ σxz ˆ σxz′ us
z
us
z ′
ut
z
ut
z ′
ˆ σzz ˆ σzz′ p p′ us
x
us
x ′
1 1 1 1 1 1 us 0 us d No coupling between us
x & ut z
No coupling between p &
xz 12
SLIDE 43
Pruning the matrix
Paper A From Dazel et al. (JAP 2013) and Dazel et al. (JASA 2007) ˆ σxz ˆ σxz′ us
z
us
z ′
ut
z
ut
z ′
ˆ σzz ˆ σzz′ p p′ us
x
us
x ′
1 1 1 1 1 1 us 0 us d No coupling between us
x & ut z
No coupling between p &
xz 12
SLIDE 44
Pruning the matrix
Paper A From Dazel et al. (JAP 2013) and Dazel et al. (JASA 2007) ˆ σxz ˆ σxz′ us
z
us
z ′
ut
z
ut
z ′
ˆ σzz ˆ σzz′ p p′ us
x
us
x ′
1 1 1 1 1 1 us(0) ≈ us(d) No coupling between us
x & ut z
No coupling between p &
xz 12
SLIDE 45
Pruning the matrix
Paper A From Dazel et al. (JAP 2013) and Dazel et al. (JASA 2007) ˆ σxz ˆ σxz′ us
z
us
z ′
ut
z
ut
z ′
ˆ σzz ˆ σzz′ p p′ us
x
us
x ′
1 1 1 1 1 1 us(0) ≈ us(d) No coupling between us
x & ut z
No coupling between p &
xz 12
SLIDE 46
Pruning the matrix
Paper A From Dazel et al. (JAP 2013) and Dazel et al. (JASA 2007) ˆ σxz ˆ σxz′ us
z
us
z ′
ut
z
ut
z ′
ˆ σzz ˆ σzz′ p p′ us
x
us
x ′
1 1 1 1 1 1 us(0) ≈ us(d) No coupling between us
x & ut z
No coupling between p & ˆ σxz
12
SLIDE 47
Pruning the matrix
Paper A From Dazel et al. (JAP 2013) and Dazel et al. (JASA 2007) ˆ σxz ˆ σxz′ us
z
us
z ′
ut
z
ut
z ′
ˆ σzz ˆ σzz′ p p′ us
x
us
x ′
1 1 1 1 1 1 us(0) ≈ us(d) No coupling between us
x & ut z
No coupling between p & ˆ σxz
12
SLIDE 48
Results
Paper A
2000 4000 6000 8000 10000 Frequency (Hz) 0.0 0.2 0.4 0.6 0.8 1.0 Absorption coefficient
Biot-JCA 13
SLIDE 49
Results
Paper A
2000 4000 6000 8000 10000 Frequency (Hz) 0.0 0.2 0.4 0.6 0.8 1.0 Absorption coefficient
Biot-JCA JCA 13
SLIDE 50
Results
Paper A
2000 4000 6000 8000 10000 Frequency (Hz) 0.0 0.2 0.4 0.6 0.8 1.0 Absorption coefficient
Biot-JCA JCA Proposed approach 13
SLIDE 51
Did we reach the goal?
Paper A
- Close to an equivalent fmuid model
- Solid displacement on both sides equal
- Importance of the shear term
- Additional simplifjcation of the equivalent density and compressibility
14
SLIDE 52
C a n w e g e t a s i m p l e r m
- d
e l f
- r
fj l m s ? H
- w
t
- a
c c
- u
n t f
- r
u n c e r t a i n t i e s ?
SLIDE 53
C a n w e g e t a s i m p l e r m
- d
e l f
- r
fj l m s ? H
- w
t
- a
c c
- u
n t f
- r
u n c e r t a i n t i e s ?
SLIDE 54
A (not so) naive approach
Paper B
ZB ZS
d z θ Film on backing ZB known
- Equiv. Fluid
Impedance translation Linearisation Zs ZB j d
f
Z2
BK 1 f
Zs Zs Zs Zs f Zs f ZB
15
SLIDE 55
A (not so) naive approach
Paper B
ZB ZS
d z θ Film on backing ZB known
- Equiv. Fluid
Impedance translation Linearisation Zs ZB j d
f
Z2
BK 1 f
Zs Zs Zs Zs f Zs f ZB
15
SLIDE 56
A (not so) naive approach
Paper B
ZB ZS
d z θ Film on backing ZB known
- Equiv. Fluid
Impedance translation Linearisation Zs ≈ ZB + jωd cos θ(ρf − Z2
BK−1 f
) Zs Zs Zs Zs f Zs f ZB
15
SLIDE 57
A (not so) naive approach
Paper B
ZB ZS
d z θ Film on backing ZB known
- Equiv. Fluid
Impedance translation Linearisation Zs ≈ ZB + jωd cos θ(ρf − Z2
BK−1 f
) ξ = ¯ ξ + ∆ξ Zs Zs Zs Zs f Zs f ZB
15
SLIDE 58
A (not so) naive approach
Paper B
ZB ZS
d z θ Film on backing ZB known
- Equiv. Fluid
Impedance translation Linearisation Zs ≈ ZB + jωd cos θ(ρf − Z2
BK−1 f
) ξ = ¯ ξ + ∆ξ Zs = ¯ Zs + ∆Zs Zs f Zs f ZB
15
SLIDE 59
A (not so) naive approach
Paper B
ZB ZS
d z θ Film on backing ZB known
- Equiv. Fluid
Impedance translation Linearisation Zs ≈ ZB + jωd cos θ(ρf − Z2
BK−1 f
) ξ = ¯ ξ + ∆ξ Zs = ¯ Zs + ∆Zs ∆Zs = f(∆ξ) Zs f ZB
15
SLIDE 60
A (not so) naive approach
Paper B
ZB ZS
d z θ Film on backing ZB known
- Equiv. Fluid
Impedance translation Linearisation Zs ≈ ZB + jωd cos θ(ρf − Z2
BK−1 f
) ξ = ¯ ξ + ∆ξ Zs = ¯ Zs + ∆Zs ∆Zs = f(∆ξ) ∆Zs = f(∆ξ, ZB)
15
SLIDE 61
Melamine backing — Woven Film
Paper B
1000 2000 3000 4000 4520 Frequency (Hz) 0.0 0.2 0.4 0.6 0.8 1.0 Absorption coefficient Naive approach - Woven facing 1000 2000 3000 4000 4520 Frequency (Hz) 0.0 0.2 0.4 0.6 0.8 1.0 Absorption coefficient Proposed approach - Woven facing
- Exp. Responses
Standard dev. envelopes Computed envelopes 16
SLIDE 62
Melamine backing — Woven Film
Paper B
1000 2000 3000 4000 4520 Frequency (Hz) 0.0 0.2 0.4 0.6 0.8 1.0 Absorption coefficient Naive approach - Woven facing 1000 2000 3000 4000 4520 Frequency (Hz) 0.0 0.2 0.4 0.6 0.8 1.0 Absorption coefficient Proposed approach - Woven facing
- Exp. Responses
Standard dev. envelopes Computed envelopes 16
SLIDE 63
Highlights
Paper B
- Fast computation: 2 traces only
- Merge characterisation data with system response
- Cope with resonances
- Reuse characterisation data
- Sensible results
17
SLIDE 64
C a n w e g e t a s i m p l e r m
- d
e l f
- r
fj l m s ? H
- w
t
- a
c c
- u
n t f
- r
u n c e r t a i n t i e s ? H
- w
t
- d
e a l w i t h e x p e n s i v e m
- d
e l s ?
SLIDE 65
C a n w e g e t a s i m p l e r m
- d
e l f
- r
fj l m s ? H
- w
t
- a
c c
- u
n t f
- r
u n c e r t a i n t i e s ? H
- w
t
- d
e a l w i t h e x p e n s i v e m
- d
e l s ?
SLIDE 66
Numerical methods and meta-poroelastic materials
Paper C Core
- Porous matrix
- Embedded structures
- Added layers
… …
Want to design complex cores? Compute with FEM!
18
SLIDE 67
Numerical methods and meta-poroelastic materials
Paper C } Core
- Porous matrix
- Embedded structures
- Added layers
… …
Want to design complex cores? Compute with FEM!
18
SLIDE 68
Numerical methods and meta-poroelastic materials
Paper C } Core
- Porous matrix
- Embedded structures
- Added layers
… …
Want to design complex cores? Compute with FEM!
18
SLIDE 69
Numerical methods and meta-poroelastic materials
Paper C } Core
- Porous matrix
- Embedded structures
- Added layers
… …
? Want to design complex cores? Compute with FEM!
18
SLIDE 70
Thin fjlms and FEM
Paper C FEM works best on meshes without too much distortion The problem is the same for tiny embedded elements... Previous work: coupling FEM and PW DGM Gaborit et al. (Int. J. NME 2018)
19
SLIDE 71
Thin fjlms and FEM
Paper C FEM works best on meshes without too much distortion The problem is the same for tiny embedded elements... Previous work: coupling FEM and PW DGM Gaborit et al. (Int. J. NME 2018)
19
SLIDE 72
Numerical methods and meta-poroelastic materials
Paper C
- Plane wave incidence
Periodicity Bloch waves
- Expansion of incident and coating fjelds
- Propagation of loads and unknowns
- Assemble the FEM domain
- Balance the system using orthogonality
… …
?
A C C A x q f f f0
20
SLIDE 73
Numerical methods and meta-poroelastic materials
Paper C
- Plane wave incidence
Periodicity Bloch waves
- Expansion of incident and coating fjelds
- Propagation of loads and unknowns
- Assemble the FEM domain
- Balance the system using orthogonality
… …
?
A C C A x q f f f0
20
SLIDE 74
Numerical methods and meta-poroelastic materials
Paper C
- Plane wave incidence
Periodicity Bloch waves
- Expansion of incident and coating fjelds
- Propagation of loads and unknowns
- Assemble the FEM domain
- Balance the system using orthogonality
… …
?
A C C A x q f f f0
20
SLIDE 75
Numerical methods and meta-poroelastic materials
Paper C
- Plane wave incidence
Periodicity Bloch waves
- Expansion of incident and coating fjelds
- Propagation of loads and unknowns
- Assemble the FEM domain
- Balance the system using orthogonality
… …
?
A C C A x q f f f0
20
SLIDE 76
Numerical methods and meta-poroelastic materials
Paper C
- Plane wave incidence
Periodicity Bloch waves
- Expansion of incident and coating fjelds
- Propagation of loads and unknowns
- Assemble the FEM domain
- Balance the system using orthogonality
… …
?
A C C A x q f f f0
20
SLIDE 77
Numerical methods and meta-poroelastic materials
Paper C
- Plane wave incidence
Periodicity Bloch waves
- Expansion of incident and coating fjelds
- Propagation of loads and unknowns
- Assemble the FEM domain
- Balance the system using orthogonality
… …
?
A C C A x q f f f0
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SLIDE 78
Numerical methods and meta-poroelastic materials
Paper C
- Plane wave incidence
Periodicity Bloch waves
- Expansion of incident and coating fjelds
- Propagation of loads and unknowns
- Assemble the FEM domain
- Balance the system using orthogonality
… …
?
A C C A x q f f f0
20
SLIDE 79
Numerical methods and meta-poroelastic materials
Paper C
- Plane wave incidence
Periodicity Bloch waves
- Expansion of incident and coating fjelds
- Propagation of loads and unknowns
- Assemble the FEM domain
- Balance the system using orthogonality
… …
?
[ A C C′ A′ ] { x q } = { f f′ } + { f0 }
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Results
Paper C
… …
d ≪ 1 d ≪ 1 Compare:
- Transfer matrices
- FEM everywhere (with mesh refjned for convergence at 1000Hz)
- FEM in the core (same mesh) & Proposed approach
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SLIDE 81
Results (oblique incidence)
Paper C
1000 2000 3000 4000 5000 Frequency (Hz) 0.0 0.2 0.4 0.6 0.8 1.0 Absorpion Coefficient TMM Reference Coatings in FEM Proposed Method
Pure FEM ( ) fails before the hybrid ( ). The reference is pure TMM ( ).
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Key aspects
- Simple strategy not to mesh fjlms
- Designed for periodic systems
- Extension of the FEM system
- Generalised to difgerent media
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SLIDE 83
Conclusion
SLIDE 84
Contributions
Achievements
- Better understanding of the physics of fjlms
- Statistical characterisation of fjlms
- ... and simple approach to model uncertainties
- Strategies to include surface layers at a reduced cost
- Work on thin layers & applicable to interface zones
Output
- Dataset on statistics of fjlms
- Open, reusable source code available online
- Extension and additional ideas presented in conferences
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SLIDE 85
Contributions
Achievements
- Better understanding of the physics of fjlms
- Statistical characterisation of fjlms
- ... and simple approach to model uncertainties
- Strategies to include surface layers at a reduced cost
- Work on thin layers & applicable to interface zones
Output
- Dataset on statistics of fjlms
- Open, reusable source code available online
- Extension and additional ideas presented in conferences
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SLIDE 86
What’s next?
- Still lacking a proper model of a randomly mixed boundary condition
- Refjned techniques needed to analyse the efgect of uncertainties
(constrictions)
- Ideas from Paper B could be extended beyond surface layers
- More to do towards engineering and making tailored absorbers
- Unfjnished works...
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SLIDE 87
Thanks! — Tack! — Merci !
gaborit@kth.se
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