Structural shape parametric optimization for an internal structural-acoustic problem

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Structural shape parametric optimization for an internal structural-acoustic problem

Christian Soize, J. C. Michelucci

To cite this version:

Christian Soize, J. C. Michelucci. Structural shape parametric optimization for an internal structural-acoustic problem. Aerospace Science and Technology, Elsevier, 2000, 4 (-), pp.263-275.

�10.1016/S1270-9638(00)00135-8�. �hal-00765562�

Structural shape parametric optimization for an internal structural-acoustic problem

Christian Soize ^a, * , Jean-Christophe Michelucci ^a

a

ONERA, Structural Dynamics and Coupled Systems Department, BP 72, 92322 Châtillon cedex, France Received 11 May 1999; revised 28 February 2000; accepted 9 March 2000

Abstract

A structural shape optimization problem, with respect to the structural aspect ratio is developed in the context of an axisymmetric structural-acoustic system, consisting of an elastic dome coupled with an internal acoustic cavity, is analyzed in the low- and medium-frequency ranges. The dome is a thin shell considered as a three-dimensional continuum with a dissipative constitutive equation. The internal f uid is a dissipative acoustic fl id. The dome is submitted to an external wall pressure fi ld modeled by a stochastic fi ld. The cost function is related to the pressure fi ld over an internal axisymmetric observation surface inside the acoustic cavity. We are interested in minimizing the internal noise over the observation surface with respect to the structural aspect ratio def ning the geometric shape of the dome. This paper develops an analysis of the structural-acoustic shape optimization problem to determine wheter or not there exist values of the dome aspect ratio for which the internal noise is a minimum. The frequency response functions of the structural- acoustic system are calculated to construct the cost function. In this context, the Fourier series expansions of the structural displacement f eld and the internal f uid velocity potential are carried out with respect to the polar angle variable. For each f xed circumferential wave number, a reduced matrix model is constructed using the structural modes of the structure in vacuo and the acoustic modes of the internal acoustic cavity with rigid wall. The structural modes and the acoustic modes are computed by the f nite element method. The optimization parameter is the aspect ratio of the structure. The analysis presented shows that the structural shape optimization problem of the dome with respect to its aspect ratio parameter has a clear solution which minimizes internal noise in the low- and medium-frequency ranges.

shape optimization / structural acoustics / internal noise / acoustics / dynamics / vibration

1. Introduction

This paper deals with a structural shape parametric op- timization in the structural-acoustic area. The structure is an elastic dome constituted by an axisymmetric thin shell structure considered as a three-dimensional contin- uum with a dissipative constitutive equation. The shape of the dome is define by its aspect ratio which is a scalar parameter. We are interested in the optimization problem with respect to this scalar parameter and not in the gen- eral shape optimization [5]. The dome is coupled with an axisymmetric internal dissipative acoustic fl id. The dome is excited by an external random wall pressure f eld which is stationary in time, such as a wall pressure in- duced by a turbulent boundary layer due to an external fl w. The objective of this paper is only devoted to ana- lyzing the inf uence of the dome curvature on the cou- pling mechanism between the structure and the inter- nal acoustic cavity (see below). Consequently, the cou- pling effects of the structure with the unbounded external acoustic f uid is neglected in order to simplify the para- metric analysis. It should be noted that the effects of the external f uid on the structure have two effects [7,12]. The f rst one is an additional damping for the structure, in- duced by the acoustic radiation at inf nity in the external f uid. This additional damping, which depends on the fre- quency is smaller than the structural damping and, con- sequently, does not modify the coupling mechanism be- tween the structure and the internal acoustic cavity. The second effect, which is induced by the external acoustic f uid, is an added mass for the structure. This added mass, which depends on the frequency, generally produces a de- creasing of the eigenfrequencies of the structure in a vac- uum when the external acoustic f uid is a liquid. This shift effect is significa t for the f rst structural eigenmodes in the LF range. When the modal density of the structure is high enough [10,17], which is the case for the superior part of the LF range and for the MF range of the struc- ture under consideration, the modal density of the struc- ture is not signif cantly modif ed, and def nitely not suf- ficie tly modifie to distort the performed analysis of the coupling mechanism between the structure and the inter- nal acoustic cavity. The observation made on this axisym- metric structural-acoustic system is the internal pressure f eld over an internal axisymmetric observation surface inside the acoustic cavity. We are interested in minimiz- ing the internal noise over the observation surface with

respect to the geometric shape of the dome defi ed by its aspect ratio and consequently, the optimization parameter is the aspect ratio of the dome (axisymmetric structure).

The fundamental mechanism induced by the curvature of the dome on the coupling between the structure and the internal acoustic cavity, is the following. The structural membrane waves and the structural f exural waves in the dome are coupled by the curvature of the dome. Structural f exural waves excited by the external wall pressure are converted into structural membrane waves. These structural membrane waves induce a piston movement of the dome nose. This type of structural displacement is associated with a variation in volume of the internal acoustic cavity which induces a high level of internal noise (presently, the observation surface inside the acoustic cavity is not located in the near fiel of the wall but is located in the far fiel ). This paper develops an analysis of the structural-acoustic shape optimization problem to determine whether or not there exist values of the dome aspect ratio for which the internal noise is a minimum (without, however, developing an automatic optimization algorithm).

Because we are interested in the low- and medium-

frequency ranges and taking into account the fact that

the dome geometry is not a ‘simple shape geometry’,

the above parametric optimization problem is diff cult to

solve by analytical methods [3,7,9,13]. This is why the

numerical approach proposed in reference [12] is used to

solve this internal structural-acoustic optimization prob-

lem. According to this reference, in the general case,

the modal approach, which is perfectly adapted to the

low-frequency range, cannot be extended to the medium-

frequency range [12]. However, because the structural-

acoustic system considered is axisymmetric, a Fourier se-

ries expansion of the response can be made and therefore,

the modal approach can still be used for the medium-

frequency range which is considered in the application

presented in this paper. It should be noted that this

method could not be used for a general three-dimensional

structural-acoustic system. In addition, in reference [12],

the formulation proposed for internal structural-acoustic

systems in the low-frequency range differs from the for-

mulation adapted to the medium-frequency range. Since

we are not interested in calculation of the structural-

acoustic modes and since we wish to use a single formu-

lation for the optimization problem related to the low- and

medium-frequency ranges, we chose to use the medium-

frequency model presented in [12]. A reduced matrix model for each fi ed circumferential wave number is con- structed by a Ritz–Galerkin projection based on the use of the structural modes of the structure in vacuo and the acoustic modes of the internal acoustic cavity with rigid wall. The structural modes and the acoustic modes are computed by the f nite element method.

Section 2 is devoted to (1) the geometric description of the three-dimensional axisymmetric structural-acoustic system, (2) its mechanical modeling and the associated three-dimensional boundary value problem, and (3) the Fourier series expansion and the two-dimensional bound- ary value problems associated with each fi ed circumfer- ential wave number. In section 3, we give the variational formulation of the two-dimensional boundary value prob- lem associated with a f xed circumferential wave num- ber. Section 4 is devoted to the construction of the re- duced matrix model using the Ritz–Galerkin method. In section 5, we defin a normalized power spectral den- sity function of the observation, which is directly used to construct the cost function of the optimization problem.

In section 6, we defi e the optimization problem and we present the method for constructing its solution. Finally, in section 7 we describe a complete numerical applica- tion.

2. Three-dimensional axisymmetric

structural-acoustic system boundary value problem

2.1. Structural-acoustic system geometry

The geometry of the structural-acoustic system is de- fi ed in figu e 1. The internal acoustic fl id occupies a bounded domain D

F

of three-dimensional physical space R

³

with boundary Σ

F

= Σ

⁻

∪ Σ

_F⁻

∪ Σ

_F⁰

in which Σ

_F⁻

and Σ

_F⁰

are rigid walls. The structure occupies a bounded domain D

S

of R

³

with boundary Σ

S

= Σ

⁻

∪ Σ

⁺

∪ Σ

_S⁰

in which Σ

_S⁰

is a rigid wall whereas boundary Σ

⁻

is the coupling interface between the elastic structure and the internal acoustic f uid. The unit normal to Σ

_S

ex- ternal to D

S

and the unit normal to Σ

F

external to D

F

are denoted as n and n

⁰

respectively. Therefore, n = −n

⁰

on Σ

⁻

. Space R

³

is referenced to a cartesian reference sys- tem ( 0 , x

₁

, x

₂

, x

₃

) also written ( 0 , x, y, z) . The generic point of R

³

is denoted as x = (x

₁

, x

₂

, x

₃

) . We introduce the cylindrical coordinates (θ , r, z) ∈ [0 , 2 π [×R

⁺

× R such that x

₁

= − r sin θ , x

₂

= r cos θ et x

₃

= z . The as- sociated local cylindrical orthonormal basis is (e

θ

, e

r

, e

z

) as shown in figu e 2. The coupled system is axisymmetric around axis ( 0 z) as is the internal observation surface Σ . The generating plane is defi ed by {x | x = 0 , y > 0}.

Parts Σ

⁺

and Σ

⁻

are denoted as Σ

ⁱ

for i in {+ , −}.

The generatrix of Σ

ⁱ

is denoted as Γ

ⁱ

and is define by the function z 7→ R

ⁱ

(z) . The generatrix Γ of internal ob- servation surface Σ is def ned by the function z 7→ R(z) .

Figure 1.

Geometry of the structural-acoustic system.

Figure 2.

Associated local cylindrical orthonormal basis.

The generating planes of the structure and the internal acoustic flui are denoted as P

_S

and P

_F

respectively.

Boundaries Σ

F

and Σ

_S^o

are generated by curves Γ

F

and Γ

_S⁰

respectively. Let s

ⁱ

be the curvilinear abscissa of gen- eratrix Γ

ⁱ

pointing positively in the direction of increas- ing z . We introduce the curvilinear measure ds

ⁱ

related to Γ

ⁱ

and measures dΓ

ⁱ

and dΣ

ⁱ

such that

ds

ⁱ

(z) = 1 + dR

ⁱ

(z)/dz

₂

₁/2

dz, (1) dΓ

ⁱ

= R

ⁱ

(z) ds

ⁱ

(z), dΣ

ⁱ

= R

ⁱ

(z) dθ ds

ⁱ

(z). (2) For all i in {+ , −} and all x in Σ

ⁱ

, we defin a local physical reference system ( b

ⁱ₁

= e

θ|_Σi

, b

ⁱ₂

= n

ⁱ

, b

ⁱ₃

= e

θ|_Σi

∧ n

ⁱ

) attached to point x where n

⁺

= n on Σ

⁺

and n

⁻

= −n on Σ

⁻

. The orthogonal 3 × 3 matrix transforming the local physical reference system into the local cylindrical reference system depends only on z and is written as

Θ

ⁱ

(z)

=



 1 0 0 0 α

ⁱ

(z) β

ⁱ

(z) 0 − β

ⁱ

(z) α

ⁱ

(z)



 , (3)

α

ⁱ

(z) = 1 + dR

ⁱ

(z)/dz

₂

_−1/2

, β

ⁱ

(z) = α

ⁱ

(z) dR

ⁱ

(z)

dz . (4)

2.2. Modeling and boundary value problem of the three-dimensional axisymmetric

structural-acoustic system

We consider linear vibrations of the structural-acoustic system around a static equilibrium position without pre- stresses and taken as reference conf guration. The for- mulation is written in the frequency domain ω (angu- lar frequency in rad/s) for which the Fourier transform convention used is such that, if t 7→ f (t) is a function from R into C, then its Fourier transform is written as f (ω) = R

R

e

^−iωt

f (t) dt . 2.2.1. External excitation

The external mechanical excitation applied to the structure is the vector f eld x 7→ f ( x , ω) = − p

⁺

( x , ω) n ( x ) def ned on Σ

⁺

with values in C

³

.

2.2.2. Internal acoustic flu d

We assume that there is no acoustic source inside acoustic f uid domain D

_F

. The internal fl id is a dissi- pative acoustic fl id. Let ρ

₀

be the mass density and c

₀

the speed of sound in the equilibrium state. The pressure fiel p( x , ω) and the velocity potential ψ( x , ω) are such that [11,12]

p( x , ω) = − iωρ

₀

ψ( x , ω) − κπ

₂

( u ), (5) κ = ρ

₀

c

₀²

| D

_F

| , π

₂

( u ) = Z

Σ⁻

u ( x , ω), n

⁰

( x )

dΣ

⁻

, (6) in which i denotes the pure imaginary complex number, hu, vi = u

₁

v

₁

+ u

₂

v

₂

+ u

₃

v

₃

, | D

F

| = R

DF

d x is the vol- ume of the internal acoustic cavity and where ψ must sat- isfy the following constraint equation R

D_F

ψ( x , ω) d x = 0. It should be noted that in this model, ψ is not exactly a velocity potential because the velocity vector is written as v ( x , ω) = ( 1 + iωτ ) ∇ ψ( x , ω) . The three-dimensional equations for the internal acoustic fl id are written as in [12]

− ω

²

ρ

₀

c

²₀

ψ − iωλ

₀

ρ

₀

∇

²

ψ − ρ

₀

∇

²

ψ

= − iωκ

c

²₀

π

₂

( u ) in D

_F

, (7) ρ

₀

( 1 + iωλ

₀

) ∂ψ

∂ n = iωρ

₀

hu , ni on Σ

⁻

, (8)

∂ψ

∂ n = 0 on Σ

_F^o

∪ Σ

_F⁻

, (9) Z

D_F

ψ dx = 0 , (10)

in which λ

₀

is a damping coeff cient which may depend on ω . Constraint equation (10) shows that ψ cannot be a constant f eld.

2.2.3. Structure

The structure is an axisymmetric three-dimensional solid continuum with a linear viscoelastic constitutive equation without memory. Its mass density and its dis- placement f eld with values in C

³

are denoted as ρ

_S

( x ) >

0 and u ( x , ω) respectively. The constitutive equation is written as in [12], σ

_{j k}

( x , ω) = σ

_{j k}^e

( x , ω) + iωσ

_{j k}^d

( x , ω) in which σ

j k

is the stress tensor, σ

_{j k}^e

( x , ω) = a

j kh`

( x ) × ε

h`

( u ) is the elastic part of the stress tensor, σ

_{j k}^d

( x , ω) = b

j kh`

( x )ε

hl

( u ) is the damping part of the stress tensor and ε

j k

=

¹₂

(∂

k

u

j

+ ∂

j

u

k

) is the linearized strain ten- sor in which ∂

_k

denotes the partial derivative with re- spect to x

_k

. Elastic coefficie ts a

_{j kh`}

( x ) and damping co- efficient b

j kh`

( x ) are real and are assumed to be inde- pendent of ω (linear viscoelasticity without memory) in the context of the present shape optimization problem in order to simplify the formulation. These coeff cients de- pend on x and satisfy the usual symmetry and positivity properties. It is assumed that no external body force f eld is applied to the structure. In the cartesian reference sys- tem and for j and k in {1 , 2 , 3}, the elastodynamic equa- tion is written as

− ω

²

ρ

S

u

j

− ∂

k

σ

j k

(u) = 0 in D

S

, (11)

u

_j

= 0 on Σ

_S⁰

, (12)

σ

j k

( u )n

k

= − p

_Σ−

n

k

on Σ

⁻

, (13) σ

_{j k}

( u )n

_k

= f

_j

on Σ

⁺

. (14) 2.3. Fourier series expansion and two-dimensional

boundary value problem associated with each fi ed circumferential wave number

Since the three-dimensional boundary value problem defi ed in section 2.2 is axisymmetric, a Fourier series expansion of the solution can be made with respect to polar angle θ . This yields a sequence of two-dimensional problems indexed by the circumferential wave number denoted as n . For all n and n

⁰

in N, the orthogonality properties are written as

2π

Z

0

sin nθ cos n

⁰

θ dθ = 0 , (15) Z

2π

0

sin nθ sin n

⁰

θ dθ = δ

_nn0

( 1 − δ

₀n

)π, (16) Z

2π

0

cos nθ cos n

⁰

θ dθ = δ

_nn0

( 1 + δ

_0n

)π, (17)

in which δ

_nn0

= 0 if n 6= n

⁰

and δ

_nn0

= 1 if n = n

⁰

.

2.3.1. Structural displacement f eld

Since the coupled system is axisymmetric, for any fixe r, z and ω , function θ 7→ u (θ , r, z, ω) is periodic with period 2 π and has the following Fourier series expansion

u

θ

( x , ω) =

+∞

X

n=0

U

_θ^(n,s)

(r, z, ω) sin nθ + U

_θ^(n,as)

(r, z, ω) cos nθ

, (18) u

r

( x , ω) =

+∞

X

n=0

U

_r^(n,s)

(r, z, ω) cos nθ

− U

_r^(n,as)

(r, z, ω) sin nθ

, (19)

u

_z

( x , ω) =

^+∞

X

n=0

U

_z^(n,s)

(r, z, ω) cos nθ

− U

_z^(n,as)

(r, z, ω) sin nθ

, (20)

in which u

θ

, u

r

and u

z

are the components of f eld u in the cylindrical basis and where for j in { θ , r, z }, n in N and I in { s, as }, U

_j^{(n,I )}

(r, z, ω) is the complex-valued fiel defi ed on P

_S

× R. Functions U

_θ^(0,s)

, U

_r^(0,as)

and U

_z^(0,as)

are equal to zero. Introducing the vector f eld U

^{(n,I )}

= (U

_θ^{(n,I )}

, U

_r^{(n,I )}

, U

_z^{(n,I )}

) , equations (18)–(20) can be rewritten as

u ( x , ω) =

+∞

X

n=0

X

I∈{s,as}

F

^{(n,I )}

(θ )

U

^{(n,I )}

(r, z, ω), (21)

in which, for all θ in [0 , 2 π ], matrices [ F

^(n,s)

(θ ) ] and [ F

^(n,as)

(θ ) ] are def ned by equations (18)–(20). Since we have [ Θ

ⁱ

(z) ]

^T

[ F

^{(n,I )}

(θ ) ][ Θ

ⁱ

(z) ] = [ F

^{(n,I )}

(θ ) ], for any f xed point on surface Σ

⁺

or Σ

⁻

, structural displacement fiel u is also given by equation (21) if it is expressed in the local physical reference system.

2.3.2. Velocity potential of the internal acoustic flu d The Fourier series expansion of periodic function θ 7→

ψ (θ , r, z, ω) with period 2 π is written as ψ (θ , r, z, ω) = X

^+∞

n=0

Ψ

^(n,s)

(r, z, ω) cos nθ

− Ψ

^(n,as)

(r, z, ω) sin nθ

, (22) where for n ∈ N and I ∈ {s, as}, the C-valued two- dimensional f eld (r, z) 7→ Ψ

^{(n,I )}

(r, z, ω) is def ned on P

F

× R. Functions Ψ

^(0,as)

are equal to zero. Equa- tion (22) can be rewritten as

ψ( x , ω) =

+∞

X

n=0

X

I∈{s,as}

F

^{(n,I )}

(θ )Ψ

^{(n,I )}

(r, z, ω), (23)

with F

^(n,s)

(θ ) = cos nθ and F

^(n,as)

(θ ) = − sin nθ . Let Ψ

^(n,s)

and Ψ

^(n,as)

be the vector-valued f elds def ned by Ψ

^(n,s)

= ( −

ⁿ_r

Ψ

^(n,s)

, ∂

_r

Ψ

^(n,s)

, ∂

_z

Ψ

^(n,s)

) and Ψ

^(n,as)

= (

ⁿ_r

Ψ

^(n,as)

, − ∂

_r

Ψ

^(n,as)

, − ∂

_z

Ψ

^(n,as)

) , depending on vari- ables (r, z, ω) . Substituting equations (21) and (23) into equation (5) and using equations (15)–(17) yields

p( x , ω) = p

⁽⁰⁾

(r, z, ω) +

+∞

X

n=1

p

^(n,s)

(r, z, ω) cos nθ

− p

^(n,as)

(r, z, ω) sin nθ

, (24)

p

⁽⁰⁾

(r, z, ω) = − iωρ

₀

Ψ

^(0,s)

(r, z, ω)

− κπ

₂

U

^(0,s)

+ U

^(0,as)

if n = 0 , (25) p

^{(n,I )}

(r, z, ω) = − iωρ

₀

Ψ

^{(n,I )}

(r, z, ω) if n > 0 . (25

⁰

) Substituting equations (21) and (23) into equations (7)–

(14) and using equations (15)–(17) yields a sequence of two-dimensional problems indexed by n whose varia- tional formulation is derived below.

3. Variational formulation of the two-dimensional boundary value problem associated with a f xed circumferential wave number

The admissible function space of displacement f eld U

^{(n,I )}

is the complex vector space W

U

of ‘suff ciently differentiable’ function U def ned on P

_S

with values in C

³

such that U = 0 on Γ

_S⁰

. The admissible function space of velocity potential Ψ

^{(n,I )}

is the complex vector space W

Ψ

of ‘suff ciently differentiable’ function Ψ def ned on P

F

with values in C such that ∂Ψ/∂ n = 0 on Γ

F

and R

PF

Ψ (r, z)r dr dz = 0. For a f xed circumferential wave number n and ω fixe in R, given f

^(n,a)

and f

^(n,as)

, the variational formulation of the two-dimensional problem indexed by n consists in f nding U

^{(n,I )}

denoted as U in W

U

and Ψ

^{(n,I )}

denoted as Ψ in W

Ψ

such that, for all δ U in W

U

and for all δΨ in W

Ψ

, we have

− ω

²

m

⁽ⁿ⁾_S

( U , δ U ) + iωc

⁽ⁿ⁾_S

( U , δ U ) + iωa

_F⁽ⁿ⁾

(Ψ, δ U ) + k

⁽ⁿ⁾_S

( U , δ U )

+ κj

⁽ⁿ⁾

( U , δ U ) = f

^{(n,I )}

(ω ; δ U ), (26)

− ω

²

m

⁽ⁿ⁾_F

(Ψ, δΨ ) + iωc

⁽ⁿ⁾_F

(ω ; Ψ, δΨ )

− iωa

_F⁽ⁿ⁾

(δΨ, U ) + k

⁽ⁿ⁾_F

(Ψ, δΨ ) = 0 , (27) in which the mass, dissipation and stiffness structural bilinear forms are defi ed by

m

⁽ⁿ⁾_S

( U , δ U ) = ( 1 + δ

₀n

)π

× Z

P_S

ρ

_S

hU , δ Ui r dr dz, (28)

c

⁽ⁿ⁾_S

( U , δ U ) = ( 1 + δ

₀n

)π Z

P_S

σ

_{j k}^{(n,I ),d}

( U )

× ε

^{(n,I )}_{j k}

(δ U )r dr dz, (29) k

⁽ⁿ⁾_S

( U , δ U ) = ( 1 + δ

_0n

)π

Z

DS

σ

_{j k}^{(n,I ),e}

( U )

× ε

^{(n,I )}_{j k}

(δ U )r dr dz, (30) and where ε

^{(n,I )}

, σ

^{(n,I ),d}

and σ

^{(n,I ),e}

denote the restric- tion of the strain tensor, the dissipative part and the elastic part of the stress tensor to W

U

. Concerning the internal acoustic fl id, we have

m

⁽ⁿ⁾_F

(Ψ, δΨ ) = ( 1 + δ

_0n

)π ρ

₀

c

²₀

Z

PF

Ψ δΨ r dr dz, (31)

c

_F⁽ⁿ⁾

(ω ; Ψ, δΨ ) = λ

₀

(ω)k

⁽ⁿ⁾_F

(Ψ, δΨ ), (32) k

⁽ⁿ⁾_F

(Ψ, δΨ ) = ( 1 + δ

_0n

)πρ

₀

Z

PF

hΨ , δΨ i r dr dz, (33) in which vectors Ψ and δΨ are the vectors associated with Ψ and δΨ as define section 2.3.2. The fluid structure coupling bilinear form a

_F⁽ⁿ⁾

and the bilinear form j

⁽ⁿ⁾

are define by

a

⁽ⁿ⁾_F

(Ψ, δ U ) = − ( 1 + δ

_0n

)πρ

₀

Z

Γ⁻

Ψ h δ U , ni dΓ

⁻

, (34)

j

⁽ⁿ⁾

( U , δ U ) = δ

₀n

4 π

²

π

₂⁽ⁿ⁾

( U )π

₂⁽ⁿ⁾

(δ U ), (35) with

π

₂⁽ⁿ⁾

(U) = − Z

Γ⁻

hU, ni dΓ

⁻

, (36) where hU , ni denotes the component of vector U along normal n on generatrix Γ

⁺

. The linear form related to the excitation is defi ed by

f

^{(n,I )}

(ω ; δ U ) = − Z

Γ⁺

Z

2π 0

p

⁺

(θ , z, ω)F

^{(n,I )}

(θ ) dθ

× h δ U , ni (z) dΓ

⁺

(z). (37)

4. Symmetric reduced matrix model

As explained in section 1, for each fi ed circum- ferential wave number n , a symmetric reduced matrix model of equations (26) and (27) is constructed using the Ritz–Galerkin projection on a fi ite dimension subspace spanned by a set of structural modes of the structure in vacuo and a set of acoustic modes of the internal acoustic cavity with rigid wall. We then have a sequence of re- duced matrix models indexed by n .

4.1. Structural modes of the structure in vacuo For each fi ed n , the structural modes of the structure in vacuo are constructed by f nding the eigenvalues λ = ω

²

and the associated eigenfunctions U ∈ W

U

such that, for all δ U in W

U

,

k

⁽ⁿ⁾_S

( U , δ U ) = λm

⁽ⁿ⁾_S

( U , δ U ). (38) The spectrum of the eigenvalue problem def ned by equa- tion (38) is the countable set λ

⁽ⁿ⁾_S,α

= (ω

⁽ⁿ⁾_S,α

)

²

with α = 1 , 2 , . . . such that 0 < ω

⁽ⁿ⁾_S,1

6 ω

⁽ⁿ⁾_S,2

6 · · · and the asso- ciated real-valued eigenfunctions U

⁽ⁿ⁾α

constitute a com- plete orthogonal set in W

U

. The normalization of the eigenfunctions are chosen such that m

⁽ⁿ⁾_S

(U

⁽ⁿ⁾α

, U

⁽ⁿ⁾_α0

) = µ

_S

δ

_αα0

in which µ

_S

= R

D_S

ρ

_S

d x is the total structural mass. We keep only eigenvectors U

⁽ⁿ⁾α

whose associ- ated eigenfrequencies ω

⁽ⁿ⁾_S,α

lie in the frequency band of analysis denoted as B

₀

and which are such that R

Γ⁻

hU

⁽ⁿ⁾α

, ni

²

dΓ

⁻

6= 0. Since there is no internal acous- tic excitation but only an external structural excitation, it should be noted that if R

Γ⁻

hU

⁽ⁿ⁾α

, ni

²

dΓ

⁻

= 0, then eigenfunction U

⁽ⁿ⁾α

has a contribution to the reponse of the structural displacement but no contribution to the pressure response of the internal acoustic cavity (we re- call that the cost function of the optimization problem is formulated only in terms of the internal pressure f eld).

For each fi ed n , we denote the index set of such eigen- functions as J

_S⁽ⁿ⁾

= {1 , . . . , N

_S⁽ⁿ⁾

}. Consequently, vector fiel U as a solution of equations (26) and (27) is written as

U

⁽ⁿ⁾

(r, z) '

N_S⁽ⁿ⁾

X

α=1

X

_S,α⁽ⁿ⁾

U

⁽ⁿ⁾_α

(r, z). (39)

4.2. Acoustic modes of the internal acoustic cavity with rigid wall

For each fi ed n , the acoustic modes of the internal acoustic cavity with rigid wall are constructed by f nding the eigenvalues λ = ω

²

and the associated eigenfunctions Ψ ∈ W

Ψ

such that, for all δΨ in W

Ψ

,

k

_F⁽ⁿ⁾

(Ψ, δΨ ) = λm

⁽ⁿ⁾_F

(Ψ, δΨ ). (40)

The spectrum of the eigenvalue problem def ned by equa-

tion (40) is the countable set λ

⁽ⁿ⁾_F,β

= (ω

_F,β⁽ⁿ⁾

)

²

with β =

1 , 2 , . . . , such that 0 < ω

⁽ⁿ⁾_F,1

6 ω

⁽ⁿ⁾_F,2

6 · · · and the asso-

ciated real-valued eigenfunctions Ψ

_β⁽ⁿ⁾

constitute a com-

plete orthogonal set in W

Ψ

. The normalization of the

eigenfunctions is chosen such that m

⁽ⁿ⁾_F

(Ψ

_β⁽ⁿ⁾

, Ψ

_β⁽ⁿ⁾₀

) =

(µ

_F

/c

²₀

)δ

_ββ0

in which µ

_F

= R

DF

ρ

₀

d x = ρ

₀

| D

_F

| is the total mass of the acoustic fl id. We keep only eigen- vectors Ψ

_β⁽ⁿ⁾

whose associated eigenfrequencies ω

_F,β⁽ⁿ⁾

lie in frequency band B

₀

and such that there is an index α in J

_S⁽ⁿ⁾

such that a

_F⁽ⁿ⁾

(Ψ

_β⁽ⁿ⁾

, U

⁽ⁿ⁾α

) 6= 0. If there is no such index α , eigenfunction Ψ

_β⁽ⁿ⁾

has no contribution to the internal acoustic pressure (see section 4.1). For each fixe n , we denote the index set of such eigenfunctions as J

_F⁽ⁿ⁾

= {1 , . . . , N

_F⁽ⁿ⁾

}. Consequently, f eld Ψ of equa- tions (26) and (27) is written as

Ψ

⁽ⁿ⁾

(r, z) '

N

X

_F⁽ⁿ⁾

β=1

X

⁽ⁿ⁾_F,β

Ψ

_β⁽ⁿ⁾

(r, z). (41)

4.3. Reduced matrix model for each circumferential wave number

For each fi ed n , the restriction of equations (26) and (27) to the subspaces of W

U

and W

Ψ

spanned by the fi ite families {U

⁽ⁿ⁾₁

, . . . , U

⁽ⁿ⁾

N_S⁽ⁿ⁾

} and { Ψ

₁⁽ⁿ⁾

, . . . , Ψ

⁽ⁿ⁾

N_F⁽ⁿ⁾

} re- spectively yields the reduced matrix model for circumfer- ential wave number n :

− ω

²

"

[ M

_S⁽ⁿ⁾

] [ O

_SF

] [ O

F S

] −[ M

_F⁽ⁿ⁾

]

#

+ iω

"

[ C

_S⁽ⁿ⁾

] [ A

⁽ⁿ⁾_F

] [ A

⁽ⁿ⁾_F

]

^T

−[ C

_F⁽ⁿ⁾

]

#

+

"

[ K

_S⁽ⁿ⁾

] [ O

_SF

] [ O

_{F S}

] −[ K

_F⁽ⁿ⁾

]

#

+ κ

[ J

⁽ⁿ⁾

] [ O

SF

] [ O

_{F S}

] [ O

_{F F}

]

! " X

^{(n,I )}_S

(ω) X

^{(n,I )}_F

(ω)

#

=

Y

^{(n,I )}_S

(ω) 0

, (42)

in which X

^{(n,I )}_S

= (X

⁽ⁿ⁾_S,1

, . . . , X

⁽ⁿ⁾

S,N_S⁽ⁿ⁾

) and X

^{(n,I )}_F

= (X

⁽ⁿ⁾_F,1

, . . . , X

⁽ⁿ⁾

F,N_F⁽ⁿ⁾

) and with M

_S⁽ⁿ⁾

αα⁰

= m

⁽ⁿ⁾_S

U

⁽ⁿ⁾_α

, U

⁽ⁿ⁾_α0

= µ

_S

δ

_αα0

, (43) C

_S⁽ⁿ⁾

αα⁰

= 2 µ

_S

ξ

_S,α⁽ⁿ⁾

ω

⁽ⁿ⁾_S,α

δ

_αα0

, (44) K

_S⁽ⁿ⁾

αα⁰

= k

_S⁽ⁿ⁾

U

⁽ⁿ⁾α

U

⁽ⁿ⁾_α0

= µ

S

ω

⁽ⁿ⁾_S,α

₂

δ

_αα0

, (45) M

_F⁽ⁿ⁾

ββ⁰

= m

⁽ⁿ⁾_F

Ψ

_β⁽ⁿ⁾

, Ψ

_β⁽ⁿ⁾₀

= µ

_F

/c

²₀

δ

_ββ0

, (46) C

⁽ⁿ⁾_F

(ω)

ββ⁰

= c

⁽ⁿ⁾_F

Ψ

_β⁽ⁿ⁾

, Ψ

_β⁽ⁿ⁾₀

= 2 µ

F

/c

₀²

ξ

_F,β⁽ⁿ⁾

ω

⁽ⁿ⁾_F,β

δ

_ββ0

, (47)

K

_F⁽ⁿ⁾

ββ⁰

= k

_F⁽ⁿ⁾

Ψ

_β⁽ⁿ⁾

, Ψ

_β⁽ⁿ⁾₀

= µ

_F

/c

₀²

ω

⁽ⁿ⁾_F,β

₂

δ

_ββ0

, (48) A

⁽ⁿ⁾_F

α⁰β

= a

⁽ⁿ⁾_F

Ψ

_β⁽ⁿ⁾

, U

⁽ⁿ⁾_α0

, (49)

J

⁽ⁿ⁾

αα⁰

= j

⁽ⁿ⁾

U

⁽ⁿ⁾α

, U

⁽ⁿ⁾_α0

.

From equation (35), we deduce that [ J

⁽ⁿ⁾

] = [0] for all n > 1. In the context of the present shape optimization problem and in order to simplify the formulation, we assume that the dissipation structural bilinear form is diagonalized by the eigenfunctions U

⁽ⁿ⁾α

. According to the eigenfunction properties introduced in sections 4.1 and 4.2, the coupling matrix is such that, for all X

F

∈ C

^NF(n)

, [ A

⁽ⁿ⁾_F

]X

F

= 0 ⇒ X = 0. Finally, the generalized external structural forces Y

^{(n,I )}_S

= (Y

_S,1^{(n,I )}

, . . . , Y

^{(n,I )}

S,N_S⁽ⁿ⁾

) are such that

Y

_S,α^{(n,I )}

= f

^{(n,I )}

ω ; U

⁽ⁿ⁾_α

. (50)

We introduce the generalized dynamic stiffness matrices related to the structure and the internal acoustic f uid:

A

⁽ⁿ⁾_S

(ω)

= − ω

²

M

_S⁽ⁿ⁾

+ iω C

⁽ⁿ⁾_S

+ K

_S⁽ⁿ⁾

, (51) A

⁽ⁿ⁾_F

(ω)

= − ω

²

M

_F⁽ⁿ⁾

+ iω C

⁽ⁿ⁾_F

+ K

_F⁽ⁿ⁾

. (52) 5. Normalized power spectral density function of the

internal noise observation

In this section, we calculate the normalized power spectral density function of the internal noise obser- vation corresponding to the time-stationary random re- sponse of the structural-acoustic system excited by a time-stationary random wall pressure f eld such as a tur- bulent boundary layer induced by an external f ow. This normalized power spectral density function is related to the spatial average of the quadratic mean of the random internal fl id pressure over observation surface Σ . 5.1. Random wall pressure f eld excitation

Let E be the mathematical expectation. Random wall pressure fi ld p

⁺

applied to external structural sur- face Σ

⁺

is a second-order real-valued stochastic f eld ( x , t) 7→ p

⁺

( x , t) indexed by Σ

⁺

× R which is centered and mean-square stationary with respect to t . In addi- tion, it is assumed that stochastic fiel p

⁺

is statisti- cally axisymmetric with respect to surface Σ

⁺

and we reuse the model introduced in [17]. Let R e

_p+

( x , x

⁰

, τ ) = E { p

⁺

( x , t + τ )p

⁺

( x

⁰

, t) } be its real-valued cross-cor- relation function which is written as R e

_p+

( x , x

⁰

, τ ) = R

R

e

^iωτ

e S

_p+

( x , x

⁰

, ω) dω where e S

_p+

( x , x

⁰

, ω) is the

complex-valued cross-spectral density function [8]. For all x and x

⁰

in Σ

⁺

, we introduce the notation S

_p+

(θ − θ

⁰

, z, z

⁰

, ω) = e S

_p+

( x , x

⁰

, ω) and the power spectral den- sity function of the mean-square stationary stochastic process { p

⁺

( x , t), t ∈ R} is a positive-valued function def ned by Φ(x, ω) e = e S

_p+

(x, x, ω) which is indepen- dent of θ . Consequently, for all x fixe in Σ

⁺

, we have Φ( e x , ω) = Φ(z, ω) = S

_p+

( 0 , z, z, ω) . The cross-spectral density function of p

⁺

is written as [17]

S

_p+

(θ − θ

⁰

, z, z

⁰

, ω)

= p

Φ(z, ω)Φ(z

⁰

, ω)G ξ(z, z

⁰

), η(θ − θ

⁰

, z, z

⁰

), ω , (53) in which ξ(z, z

⁰

) = s

⁺

(z) − s

⁺

(z

⁰

) with s

⁺

(z) the curvi- linear abscissa of generatrix Γ

⁺

introduced in sec- tion 2.1, η(γ , z, z

⁰

) = 0 . 5 (R

⁺

(z) + R

⁺

(z

⁰

))g(γ ) with g(γ ) = γ if − π 6 γ 6 π , g(γ ) = γ − 2 π if π < γ 6 2 π and g(γ ) = γ + 2 π if −2 π 6 γ < − π . The complex- valued coherence function G(ξ, η, ω) is given by the Cor- cos model [2] which is written as

G(ξ, η, ω) = exp

i ξ ω U

_c

− | ξ |

L

₁

(ω) − | η | L

₂

(ω)

, (54) in which U

_c

= 0 . 65 U

_E

is the average convection velocity with U

_E

the average external f ow velocity. The longi- tudinal and lateral correlation scales L

₁

(ω) and L

₂

(ω) are written as L

₁

(ω) = U

c

/( 0 . 115| ω | ) and L

₂

(ω) = U

c

/( 0 . 7| ω | ) . For the application presented in section 7, the model used for the power spectral density function Φ(z, ω) is written as

Φ(z, ω) = 1

4000 ρ

₀²

U

_E⁴

δ(z)

³

× ω

²

U

_E²

+ 25 ω

²

δ(z)

²

₋₃/2

, (55) in which δ(z) is the thickness displacement of the boundary layer.

5.2. Reference power spectral density function Let s

_p+

ref

(ω) be the reference power spectral density function related to surface Σ

⁺

. Let Π

_Σ+

be the spatial average over surface Σ

⁺

of the mean power of stochastic process { p

⁺

( x , t), t ∈ R} for x in Σ

⁺

. We then have Π

_Σ+

= R

R

s

_p+

ref

(ω) dω in which s

_p+

ref

(ω) is the reference power spectral density function def ned by

s

_p+

ref

(ω) = 1

| Σ

⁺

| Z

Σ⁺

Φ( e x , ω) dΣ

⁺

( x ). (56)

5.3. Normalized power spectral density function calculation

From equations (37) and (50), we deduce that

Y

_S,α^{(n,I )}

(t) = − Z

Γ⁺

Z

2π 0

p

⁺

(θ , z, t)

× F

^{(n,I )}

(θ ) dθ U

⁽ⁿ⁾_α

, n

(z) dΓ

⁺

(z), (57) in the time domain. Therefore, Y

^{(n,I )}_S

(t) = (Y

_S,1^{(n,I )}

(t), . . . , Y

^{(n,I )}

S,N_S⁽ⁿ⁾

(t)) is a second-order, centered, mean-square stationary and mean-square continuous stochastic process indexed by R with values in R

^NS(n)

. The cross-correlation function of stochastic processes Y

_S,α^{(n,I )}

(t) and Y

_S,α^(n0,I₀ ⁰⁾

(t) is def ned by

R

^(n,n_Y ^0,I,I0)

S,αY_S,α0

(τ ) = E

Y

_S,α^{(n,I )}

(t + τ )Y

_S,α^(n0,I₀ ⁰⁾

(t) , (58) and can be written as

R

_Y^(n,n0,I,I0)

S,αY_S,α0

(τ ) = Z

R

e

^iωτ

S

_Y^(n,n0,I,I0)

S,αY_S,α0

(ω) dω, (59) where S

_Y^(n,n0,I,I0)

S,αY_S,α0

(ω) is the cross-spectral density function which can be written as

S

_Y^(n,n0,I,I0)

S,αY_S,α0

(ω)

= 2 π δ

_nn0

δ

_{I I}0

( 1 + δ

_0n

)

(60)

× Z

Γ⁺

Z

Γ⁺

Z

π 0

S

_p+

(γ , z, z

⁰

, ω) cos nγ dγ

× U

⁽ⁿ⁾_α

, n

(z) U

⁽ⁿ_α0⁰⁾

, n

(z

⁰

) dΓ

⁺

(z) dΓ

⁺

(z

⁰

).

The matrix-valued spectral density function [ S

_Y⁽ⁿ⁾

S

(ω) ] of vector-valued process Y

^{(n,I )}_S

(t) is then defi ed by [ S

_Y⁽ⁿ⁾

S

(ω) ]

αα⁰

= S

_Y^{(n,n,I,I )}

S,αY_S,α0

(ω) .

Let Π

_Σ

be the spatial average over surface Σ of the mean power of stochastic process { p( x , t), t ∈ R} for x in Σ . From this def nition, we deduce that Π

_Σ

= E {

_|Σ¹_|

R

Σ

p( x , t)

²

dΣ } which can be rewritten as Π

_Σ

=

Z

R

s

_pΣ

(ω) dω, (61)

where s

_pΣ

(ω) is the power spectral density function of the internal noise observation. It is proved [10,17,18] that

s

p_Σ

(ω) = s

₀

(ω) +

+∞

X

n=0

s

⁽ⁿ⁾_p

Σ

(ω). (62)

The term s

₀

(ω) is an additional function induced by the presence of − κπ

₂

in equation (5), whose contribution exists only for n = 0 and which is written as

s

₀

(ω) = 8 π

²

κ

| Σ | tr π κ

S

⁽ⁿ⁾_X

S

(ω) U

_n⁽ⁿ⁾

− ωρ

₀

= m S

_X⁽ⁿ⁾

SF

(ω)

Υ

_n⁽ⁿ⁾

, (63) in which = m is the imaginary part, tr is the trace operator and [ Υ

_n⁽ⁿ⁾

] is the matrix defi ed by

Υ

_n⁽ⁿ⁾

βα

= Z

Γ

Ψ

_β⁽ⁿ⁾

Z

Γ⁻

U

⁽ⁿ⁾α

, n

dΓ

⁻

dΓ. (64)

The main contributions are the terms s

_p⁽ⁿ⁾_Σ

(ω) which are written as

s

_p⁽ⁿ⁾

Σ

(ω) = 2 π ω

²

ρ

₀²

| Σ | tr S

⁽ⁿ⁾_X

F

(ω)

Ψ

⁽ⁿ⁾

. (65) In equations (63) and (65), we have

S

⁽ⁿ⁾_X

F

(ω)

=

H

_{F S}⁽ⁿ⁾

(ω) S

_Y⁽ⁿ⁾

S

(ω)

H

_{F S}⁽ⁿ⁾

(ω)

_∗

, (66) S

_X⁽ⁿ⁾

S

(ω)

=

H

_S⁽ⁿ⁾

(ω) S

_Y⁽ⁿ⁾

S

(ω)

H

_S⁽ⁿ⁾

(ω)

_∗

, (67) S

_X⁽ⁿ⁾

SF

(ω)

=

H

_S⁽ⁿ⁾

(ω) S

_Y⁽ⁿ⁾

S

(ω)

H

_{F S}⁽ⁿ⁾

(ω)

_∗

, (68) H

_S⁽ⁿ⁾

(ω)

=

A

⁽ⁿ⁾_S

(ω) + κ

J

⁽ⁿ⁾

− ω

²

A

⁽ⁿ⁾_F

×

A

⁽ⁿ⁾_F

(ω)

₋₁

A

⁽ⁿ⁾_F

T

₋₁

, (69)

H

_{F S}⁽ⁿ⁾

(ω)

= iω

A

⁽ⁿ⁾_F

(ω)

₋₁

A

⁽ⁿ⁾_F

T

H

_S⁽ⁿ⁾

(ω) , (70) H

_SF⁽ⁿ⁾

(ω)

_∗

= −

H

_{F S}⁽ⁿ⁾

(ω) , Ψ

⁽ⁿ⁾

ββ⁰

= Z

Γ

Ψ

_β⁽ⁿ⁾

Ψ

_β⁽ⁿ⁾₀

dΓ, (71) U

_n⁽ⁿ⁾

αα⁰

= Z

Γ

Z

Γ⁻

U

⁽ⁿ⁾_α

, n dΓ

⁻

× Z

Γ⁻

U

⁽ⁿ⁾_α0

, n

dΓ

⁻

dΓ. (72)

Finally, the normalized power spectral density function of the internal noise observation is def ned by

s

p_Σ,norm

(ω) = s

p_Σ

(ω) s

_p+

ref

(ω) . (73)

It should be noted that [A

⁽ⁿ⁾_F

(ω) ] is a diagonal matrix and consequently, the inverse matrix which appears in equations (69) and (70) is explicitly known.

6. Shape optimization with respect to the aspect ratio of the dome

6.1. Class of geometry

The objective is to optimize the dome shape in order to minimize the noise related to observation surface Σ located inside the internal acoustic cavity. Since we are looking for the inf uence of the dome curvature, all the other main parameters of the structural acoustic system (volume of the internal acoustic cavity, external structural surface area, dome thickness and constitutive material) have to remain constant when the shape of the dome is modif ed in the dome-shape optimization process. The external power injected in the structural-acoustic system is proportional to the external structural surface area on which the external random wall pressure f eld excitation is applied. Since we are only interested in analyzing the inf uence of the dome curvature on the internal noise, this surface area has to remain a constant in order to not introduce a strong variation of the external power input when the dome shape is modif ed. The volume of the internal acoustic cavity is a constant for that the modal density of the internal acoustic cavity be nearly a constant when the dome shape is modif ed. In addition, since the dome thickness is very small compared to the other structural-acoustic system dimensions, the area of internal structural surface Σ

⁻

is almost equal to the area of external structural surface Σ

⁺

. The internal generatrix Γ

⁻

∪ Γ

_F⁻

is chosen as an arc of an ellipse, possibly extended by a line segment of a line parallel to axis ( 0 z) and belonging to the generative plane. This ellipse is centered in the reference system origin and characterized by its semiminor axis which has a f xed value a

_ref

and its semimajor axis b (see figu es 3 and 4). The characteristic ratio of the ellipse is defi ed by q = b/a

_ref

. Each value of characteristic ratio q define a geometric conf guration of the structural-acoustic system and q is called the structural aspect ratio. Figures 3 and 4 def ne the z -axes, denoted as z

_a

, z

_b

, z

_c

, z

_d

= b and z

_e

, of all

Figure 3.

Geometry of the generative plane: case of an arc of

the ellipse.