High-order time-domain simulation of acoustic impedance models using diffusive representation

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High-order time-domain simulation of acoustic impedance models using diffusive representation

Florian Monteghetti, Denis Matignon, Estelle Piot, Lucas Pascal

To cite this version:

Florian Monteghetti, Denis Matignon, Estelle Piot, Lucas Pascal. High-order time-domain simulation of acoustic impedance models using diffusive representation. 17th Spanish-French School Jacques- Louis Lions about Numerical Simulation in Physics and Engineering, Jun 2017, Gijón, Spain. �hal- 01618070�

High-order time-domain simulation of acoustic

impedance models using diffusive representation

Florian MONTEGHETTI ^1* , Denis MATIGNON ² , Estelle Piot ¹ and Lucas PASCAL ¹

1

Dpt. for Aerodynamics and Energetics Modeling, ONERA – The French Aerospace Lab, 31055 Toulouse (FRANCE)

2

Dpt. of Applied Mathematics, ISAE-SUPAERO, 31055 Toulouse (FRANCE)

*

florian.monteghetti@onera.fr

Context: impedance boundary condition

I Euler equation for isentropic acoustical perturbations p ,u

∀x ∈ Ω,

∂

_t

p + ∇ · u + ∇ · pu = 0

∂

_t

u + ∂

_t

pu + ∇ · u

u + ∇ · pu

u = −∇p (1) . Non-linear terms neglected in the spatial discretisation

n

Ω Γ

_z

Ω

_z

I Time-domain impedance boundary condition (TDIBC)

∀x ∈ Γ

_z

, p (t , x ) = [z (·, x ) ? u · n (·, x )] (t ) (2) . Causal and tempered distribution z ∈ D

₊⁰

( R ) ∩ S

⁰

( R )

. Integro-differential equation between p and u · n

Intuition | z ˆ | 1 (high impedance) ⇒ u · n = 0 ∂

_n

p = 0 (rigid wall) Application linear modelling of a locally-reactive propagation medium Ω

_z

Acoustical impedance models

I Ω

_z

is an acoustic liner

Γ

_z

perforated plate (p) cavity (c)

I Impedance models

¹

ˆ

z (s ) = ˆ z

_p

(s ) + ˆ z

_c

(s ) (3) ˆ

z

_p

(s ) = a

₀

+ a

¹_/2

√

s + a

₁

s (4) ˆ

z

_c

(s ) = coth b

₀

+ b

¹_/2

√

s + b

₁

s

(5)

Objectives

Ê Design a numerical TDIBC for time-domain simulation from acoustical models

(6= state of the art)

Ë Investigate well-posedness and stability

Challenges of (2):

I Generally does not reduce to Dirichlet, Neumann or Robin I Acoustical models for z only

known in the frequency domain (i.e. ˆ z (s ) only)

Analysis of acoustical impedance models: oscillatory-diffusive representation

I Recast of models (4,5) leads to the definition of ˆ h

_p

and ˆ h

_c

ˆ

z

_p

(s ) = a

₀

+ a

¹_/2

h ˆ

_p

(s ) s + a

₁

s

z

_p

? u

_n

= a

₀

u

_n

+ a

¹_/2

h

_p

? u ˙

_n

+ a

₁

u ˙

_n

ˆ

z

_c

(s ) = 1 + e

^{−∆t s}

h ˆ

_c

(s )

z

_c

? u

_n

= u

_n

+ h

_c

? u

_n

(· − ∆t ) (6) I Analysis in the complex plane reveals both poles s

_n

and cut C

0

=(s)

<(s) cut C

arg(ˆh_p)

0

=(s)

<(s) cut C

arg(ˆh_c) b₀ = 4, b₁ = 4,b_1/2 = 5b₁

0

=(s)

<(s) cut C

arg(ˆh_c)

poles s_n

b₀ = 0.1,b₁ = 4,b_1/2 = 5b₁

−π

−π

2 0

π 2 π

I Oscillatory-diffusive representation

^2,3,4

in the time domain h (t ) = X

n∈Z

r

_n

e

^snt

+ ˆ

C

e

^−ξt

dµ (ξ ) (7) I Convolution is then recast into an observer of an

infinite-dimensional state-space representation h ? u

_n

= X

n∈Z

r

_n

ψ

_n

+ ˆ

C

ϕ

_ξ

dµ (ξ ) (8) where variables ϕ

_ξ

and ψ

_n

follow a first order dynamic

˙

ϕ

_ξ

= −ξ ϕ

_ξ

+ u

_n

ψ ˙

_n

= s

_n

ψ

_n

+ u

_n

(9)

! Time-local realisation of a hereditary operator

! Analysis applicable to a wide class of acoustical models

⁴

Numerical impedance boundary condition

I Numerical TDIBC stems from the discretisation of (8)

I DoF: N

_ψ

poles s

_n

and weights ˜ r

_n

/ N

_ϕ

poles ξ

_k

and weight ˜ µ

_k

I Optimisation of ˜ r

_n

and ˜ µ

_k

in the frequency domain

³

(10)

! Broadband approximation through elementary linear least squares

! No optimisation against experimental data

h (jω ) ' h ˜ (jω ) = X

|n|≤N_ψ

˜ r

_n

jω − s

_n

+ X

k≤N_ϕ

˜ µ

_k

jω + ξ

_k

(10)

0 1 2 3 4 5 6 7 8 9 10

N_ϕ = 2 N_ϕ = 6

ξ_min ξ_max

f (kHz)

<[ˆhp(s)s]

0 1 2 3 4 5 6 7 8 9 10

N_ψ = 1 N_ψ = 4

N_ϕ = 2 f (kHz)

<[ˆz2(s)]

0 1 2 3 4 5 6 7 8 9 10

f (kHz)

=[ˆz2(s)]

Spatial discretisation

! Discontinuous Galerkin method (high spatial order)

I Numerical flux for the TDIBC

⁵

: centred, with fictive state q

_z

n

q

_z

Γ

_z

Ω q Ω

_z

f = A

_i

n

_i

q + q

_z

(q )

2 q =

p u I Additional variables (ψ

_n

, ϕ

_k

)

_n,k

at each node of Γ

_z

I Spatial discretisation leads to the following global formulation

X ˙ (t ) = A X (t ) + C X (t − ∆t ) (11) on extended state X =

p

_i

u

_i

ψ

_n

ϕ

_k

! TDIBC integrated at the semi-discrete level

I If C 6= 0, (11) is a delay differential equation (DDE)

Time discretisation

! High-order time integration with Runge-Kutta method (a

_ij

, b

_j

) I If C 6= 0, use of a continuous extension

⁶

(a

_ij

, b

_i

(θ))

Validation

I Asymptotic stability study of (11) on a monodimensional case

! TDIBC does not impact stability limit as long as ξ

_max

≤ f

_max^DG

Well-posedness

I Admissibility conditions for z : reality, passivity, causality and stability

⁷

I Semigroup theory on extended state

⁸

p (·, t ) u (·, t ) ψ

_n

ϕ

_ξ

Acknowledgments & Bibliography

This research has initially been supported by ISAE-SUPAERO and is now supported jointly by ONERA (The French Aerospace Lab) and DGA

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3T. H´elie and D. Matignon, “Diffusive representations for the analysis and simulation of flared acoustic pipes with visco-thermal losses”, Mathematical Models and Methods in Applied Sciences 16, 503–536 (2006).

4F. Monteghetti, D. Matignon, E. Piot and L. Pascal, “Design of broadband time-domain impedance boundary conditions using the oscillatory-diffusive representation of acoustical models”, The Journal of the Acoustical Society of America, (Submitted) (2016).

5F. Monteghetti, D. Matignon, E. Piot and L. Pascal, “Simulation temporelle d’un mod`ele d’imp´edance de liner en utilisant la repr´esentation diffusive d’op´erateurs”, in 13e Congr`es Fran¸cais d’Acoustique, 000130 (11th–15th Apr. 2016), pp. 2549–2555.

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Poster session of the XVII Spanish-French School Jacques-Louis Lions about Numerical Simulation in Physics and Engineering Gij´ on, June 6-10, 2016