Optimisation de forme pour la conception des cavités de liner

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HAL Id: hal-01959233

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Submitted on 18 Dec 2018

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Optimisation de forme pour la conception des cavités de liner

Gilles Tissot, Gwénaël Gabard

To cite this version:

Gilles Tissot, Gwénaël Gabard. Optimisation de forme pour la conception des cavités de liner. CFA’18 - 14ème Congrès Français d’acoustique, Apr 2018, Le Havre, France. pp.1-30. �hal-01959233�

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Context Outline Gov. equations Shape optimisation Implementation Results Conclusion 1/17

Shape optimisation for acoustic liners design

GillesTissot, GwénaëlGabard

Laboratoire d’Acoustique de l’Université du Mans

Congrès Français d’Acoustique Le Havre, Wednesday, 25 April 2018

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Context

Acoustic liners:

Schematic representation.

Integrated in nacelles of aricrafts.

Constraints:

Low frequency absorption requires deep cavities.

Challenge to push away this limitation.

MACIAANR project: LAUM, SAFRAN.

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Context

Objective:

Optimise cavity shape of liners.

Make it resonate at lower frequencies.

Strategy:

Shape optimisation.

Based on frequency response.

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Outline

1 Governing equations

2 Shape optimisation

3 Numerical implementation

4 Results

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Governing equations

Helmholtz equation:











∆p+k²p= 0 inΩ

∇p.n= 0 onΓs (free slip).

∇p.n=δvⁱ⁻¹2 ∆Tp−δTk²^{(i−1)(γ−1)}₂ p

onΓw (walls)

∇p.n+ikp= 2ike^ikL onΓZ (Impedance B.C.).

Γs

L

Bossart et al. (2003) Berggren et al. (2018)

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Governing equations

Helmholtz equation:











∆p+k²p= 0 inΩ

∇p.n= 0 onΓs (free slip)

∇p.n=δvⁱ⁻¹2 ∆Tp−δTk²^{(i−1)(γ−1)}₂ p

onΓw (walls).

δv= q 2

ωRea ; δT = q 2

ωReaP r.

∇f=∇Tf+∂f

∂nn;

∆f= ∆Tf+∂²f

∂n² +∇T.n∂f

∂n

Γs

Γw

L

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Governing equations

Helmholtz equation:











∆p+k²p= 0 inΩ

∇p.n=δvⁱ⁻¹2 ∆Tp−δTk²^{(i−1)(γ−1)}₂ p

onΓw (walls)

Unitary amplitude of incoming plane wave.

Reflexion coefficient:

R= 1

L_Z Z

ΓZ

p(x, L) dx−e^ikL

e^ikL.

Impedance: Z= 1 +R 1−R.

ΓZ

Γs

Γw

L

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Governing equations

Helmholtz equation:











∆p+k²p= 0 inΩ

∇p.n=δvⁱ⁻¹2 ∆Tp−δTk²^{(i−1)(γ−1)}₂ p

onΓw (walls)

Modify cavity shape for improving behaviour: Target impedance.

J(Ω, p) = 1 2

Z k2

k1

|Z(p, k)−Z_T(k)|²dk

ΓZ

Γs

Γw

L

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Context Outline Gov. equations Shape optimisation Implementation Results Conclusion

Generalities Impedance matching 6/17

Shape optimisation

Generalities

Infinitesimal domain variations:

Reference domain:

Ω

Shape variation:

Ω +hδθ

h small.

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Shape optimisation

Generalities

Cost functional: unconstrainedminimisation problem

J(Ω) = Z

Ω

f(x) dx Shape derivative: ∀δθ we have

(∇_δθJ, δθ) = lim

h→0

J(Ω +hδθ)− J(Ω)

h =

Z

Γθ

δθ(x).n(x)f(x) dx.

Descent direction:

J(Ω +hδθ) =J(Ω) +h Z

Γθ

δθ(x).n(x)f(x) dx+O(h²)

⇒δθ(x) =−f(x)n(x) ensures a descent direction! Indeed: J(Ω +hδθ) =J(Ω)−h

Z

Γθ

f²(x) dx+O(h²)

Allaire (2003,2004)

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Shape optimisation

Generalities

J(Ω) = Z

Ω

h→0

h =

Z

Γθ

Descent direction:

Γθ

⇒δθ(x) =−f(x)n(x) ensures a descent direction! Indeed: J(Ω +hδθ) =J(Ω)−h

Z

Γθ

f²(x) dx+O(h²)

Allaire (2003,2004)

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Shape optimisation

Generalities

J(Ω) = Z

Ω

h→0

h =

Z

Γθ

Descent direction:

Γθ

⇒δθ(x) =−f(x)n(x) ensures a descent direction!

Indeed: J(Ω +hδθ) =J(Ω)−h Z

Γθ

f²(x) dx+O(h²)

Allaire (2003,2004)

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Shape optimisation

Impedance matching

Cost functional:

J(Ω, p) = 1 2

Z k2

k1

|Z(p, k)−ZT(k)|²dk

subject to

∆p+k²p= 0 + B.C.

Impedance: Z(p, k) = e^−ikL−e^ikL+ ¯p

e^−ikL+e^ikL−p¯ with p¯= 1 LZ

Z

Γ_Z

pdl.

Bängtsson et al. (2003)

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Shape optimisation

Impedance matching

Cost functional:

J(Ω, p) = 1 2

Z k2

k1

subject to

∆p+k²p= 0 + B.C.

We define the Lagrangian(constrained7→unconstrained)

L(Ω, p, λ) =J(Ω, p)−real Z k2

k1

λ,∆p+k²p

Ω dk

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Shape optimisation

Impedance matching

Cost functional:

J(Ω, p) = 1 2

Z k2

k1

subject to

∆p+k²p= 0 + B.C.

We define the Lagrangian(constrained7→unconstrained)

L(Ω, p, λ) =J(Ω, p)−real Z k2

k1

λ,∆p+k²p

Ω dk

Derivative with respect to each variable:

(∇_δλL, δλ) = (∇_δpL, δp) = 0 (∇_δθL, δθ) = 0⇒(∇_δθJ, δθ)

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Impedance matching

Derivative with respect toλ: ∆p+k²p= 0.

Derivative with respect top: Adjoint equation











∆λ+ (k^∗)²λ= 0 x∈Ω

∇λ.n= 0 x∈Γs

∇λ.n+δ_vi+ 1

2 ∆_Tλ−δ_T(k^∗)²(i+ 1)(γ−1)

2 λ= 0 x∈Γ_w

∇λ.n−ik^∗λ= 2 LZ

e^ikL(Z(p, k)−ZT(k))

(e^−ikL+e^ikL−p)¯²∗ x∈ΓZ.

Derivative with respect toΩ: For any displacement directionδθ, (∇δθJ, δθ) =

Z

Γθ

(δθ(x).n(x)) Z k₂

k1

Real

−∇λ^∗.∇p+k²λ^∗p dkdx.

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Impedance matching











∆λ+ (k^∗)²λ= 0 x∈Ω

∇λ.n= 0 x∈Γs

∇λ.n+δ_vi+ 1

2 ∆_Tλ−δ_T(k^∗)²(i+ 1)(γ−1)

2 λ= 0 x∈Γ_w

(e^−ikL+e^ikL−p)¯²

∗ x∈ΓZ.

Z

Γθ

k1

Real

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Impedance matching











∆λ+ (k^∗)²λ= 0 x∈Ω

∇λ.n= 0 x∈Γs

∇λ.n+δ_vi+ 1

2 ∆_Tλ−δ_T(k^∗)²(i+ 1)(γ−1)

2 λ= 0 x∈Γ_w

(e^−ikL+e^ikL−p)¯²

∗ x∈ΓZ.

Z

Γθ

k1

Real

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Impedance matching

Algorithm:

Solve iteratively:

Solve Helmholtz direct ⇒ p(k) (frequency response). Solve Helmholtz adjoint ⇒ λ(k) (frequency response). Optimality condition ⇒ ∇_δθJ.

Move shape with linesearch algorithm(Armijo backtracking). Iterate until convergence.

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Numerical implementation

Finite elements with immersed boundary:

XFEMfor integration methods on cut elements (GETFEM++).

Level set:







ψ= 0 x∈Γ_θ ψ >0 x∈Ω ψ <0 x /∈Ω.

Transport equation for moving the shape:

∂ψ

∂t +vn.∇ψ= 0.

with

v=−∇_δθJ.n. (Regularised by Sobolev Gradient) nis the outward wall normal direction. SUPG discretisation.

Moës et al. (1999) Osher et al. (2001) Protas et al. (2004)

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Inviscid problem Viscous problem 12/17

Results

No viscosity: Z_T = 0,k= [0.2 : 0.6]

Initial condition. Converged.

Coloured by∇δθJ.

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Results

No viscosity: Z_T = 0,k= [0.2 : 0.6]

−20

−10 0 10 20

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Z

k

Initial Final

ReactanceIm(Z).

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Results

No viscosity: Z_T = 0,k= [0.15 : 0.25]

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Results

No viscosity: Z_T = 0,k= [0.15 : 0.25]

−20

−10 0 10 20

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Z

k

Initial Final

ReactanceIm(Z).

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Results

Viscosity: Z_T = 1.0,k= [0.2 : 0.6]

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Results

Viscosity: Z_T = 1.0,k= [0.2 : 0.6]

−20

−10 0 10 20

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Z

k

Initial Final

0 0.2 0.4 0.6 0.8 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Z

k

Initial Final

ReactanceIm(Z). ResistanceRe(Z).

0 0.02 0.04 0.06 0.08 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

α

k

Initial Final

Absorptionα= 1− |R|².

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Results

Viscosity: Z_T = 1.0,k= [0.15 : 0.25]

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Results

Viscosity: Z_T = 1.0,k= [0.15 : 0.25]

−20

−10 0 10 20

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Z

k

Initial Final

0 0.2 0.4 0.6 0.8 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Z

k

Initial Final

ReactanceIm(Z). ResistanceRe(Z).

0 0.02 0.04 0.06 0.08 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

α

k

Initial Final

Absorptionα= 1− |R|².

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Conclusion

Summary:

Shape optimisation for impedance matching.

Viscous model: compromise between Helmholtz and full linearised Navier-Stokes.

Perspectives:

Optimise from efficient/realistic configuration.

Different cost functional (Penalty to initial guess, absorption, . . . ).

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Optimisation de forme pour la conception des cavités de liner

HAL Id: hal-01959233

https://hal.inria.fr/hal-01959233

Optimisation de forme pour la conception des cavités de liner

Gilles Tissot, Gwénaël Gabard

To cite this version:

Shape optimisation for acoustic liners design

Context

Context

Outline

Governing equations

Governing equations

Governing equations

Governing equations

Shape optimisation

Shape optimisation

Shape optimisation

Shape optimisation

Shape optimisation

Shape optimisation

Shape optimisation

Impedance matching

Impedance matching

Impedance matching

Impedance matching

Numerical implementation

Results

Results

Results

Results

Results

Results

Results

Results

Conclusion

Thank you for your attention.