A stochastic perturbation finite element-least square point interpolation method for the analysis of uncertain structural-acoustics problems with random variables

Applied Acoustics 137 (2018) 18–26

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A stochastic perturbation finite element-least square point interpolation method for the analysis of uncertain structural-acoustics problems with random variables

T

⁎

G.Q. Jiang, L.Y. Yao , F. Wu College of Engineering and Technology, Southwest University, Chongqing 400715, PR China

A R T I C LE I N FO

A B S T R A C T

Keywords: Stochastic analysis SP-FEM Monte Carlo SP-FE-LSPIM Structural-acoustics problems

In this paper, a novel stochastic perturbation finite element-least square point interpolation method (SP-FELSPIM) is introduced to improve the calculation accuracy for analyzing structural-acoustics problems. Inherited the element-compatibility of finite element method and the quadratic polynomial completeness of LSPIM, the present method obtains the global shape function for partition of unity (PU) and the least square point interpolation for local approximation. Besides, a first-order perturbation technique is also introduced into this theory for probabilistic analyzing. Thus, the response of the coupled systems can be expressed simply as a linear function of all pre-defined input variables by using the change-of-variable techniques. Due to the linear relationships between variables and the response, the probability density function and the cumulative probability density function of response can be obtained based on a simple mathematical transformation of probability theory. So the proposed approach not only improves the numerical accuracy of deterministic output quantities with respect to a given random variable, but also handles the randomness well in the systems. One numerical example for frequency response analysis of random structural-acoustics is presented and verified by Monte Carlo (MC) simulation and stochastic perturbation finite element method (SP-FEM) to demonstrate the effectiveness of the present method.

1. Introduction When a dynamic load is applied, engineering systems such as vehicles, sea vessels, and air crafts encounter structural-acoustics problems, which range mostly from the “low-frequency” to “high-frequency” regime. In order to cover different frequency ranges, different types of numerical simulation techniques have been developed for the structural-acoustic predictions. Most can be categorized into two major groups: deterministic and statistical. The former is suitable for the lower frequency range, while the latter is suitable for the higher. Finite element method (FEM) [1–4], a typical deterministic method, is successfully used to model the low frequency vibrational and acoustical behaviors of structural-acoustic systems. However, it will suffer from numerical dispersion when solving the acoustic wave equation, which means that the phase error of numerical wave will disperse significantly for high wave number problems. There are two ways to obtain reliable results in solving the high wave number problems: discretizing the problem domain into smaller and more accurate elements and using higher order polynomial approximation function, but these will lead to the increase of computing time and memory ⁎

space. The meshfree method [5–8], with the characteristic of high precision and with not needing to divide the grid, can effectively avoid some disadvantages of FEM. Nevertheless, it cannot impose boundary conditions directly because of lack of the Kronecker properties, which results in a decrease in computational efficiency. Melenk et al. [9,10] proposed a mixed finite element-meshfree method to analyze various mechanical problems by combining FEM and mesh-free technique. The basic principle is to construct high order global finite element formula by increasing the order of local support function without increasing the support point. Cui and co-workers proposed some novel numerical methods to significantly improve the computation accuracy for solving acoustic problems [11–14]. Zhang et al. [15,16] proposed a finite element-least square point interpolation method (FE-LSPIM) to effectively analyze the statics and dynamics. Later, Yao et al. [17] obtained good results by using FE-LSPIM for analyzing the two-dimension acoustic. Recently, the focus of exploiting a suitable prediction technique for structural-acoustic problems has shifted to the mid-frequency regime, in which the critical factor is uncertainty. Owing to the difficulties involved in developing a reliable deterministic method for the prediction of higher frequency domain, researchers now focus on developing

Corresponding author. E-mail address: [email protected] (L.Y. Yao).

https://doi.org/10.1016/j.apacoust.2018.03.003 Received 10 March 2017; Received in revised form 6 December 2017; Accepted 4 March 2018 Available online 09 March 2018 0003-682X/ © 2018 Elsevier Ltd. All rights reserved.

Applied Acoustics 137 (2018) 18–26

G.Q. Jiang et al.

efficient methods that take into account model uncertainties and variability [18]. Probabilistic methods are mostly chosen with acceptable efficiency to handle the uncertainties, which are expressed by using different types of random variables [19]. The Monte Carlo simulation method (MCM) [20–23], the most robust and simplest probabilistic method, can always offer the most direct and reliable results at the expense of the large computational load. Therefore, the MCM results are chosen as the reference results in this work. The spectral stochastic finite element method (SSFEM) [24–26] and stochastic perturbation finite element method approach (SPFEM) [27–30] are another two classical probabilistic methods. In SSFEM, several random Hermite polynomials are used to approximate the response quantity of the systems [26]. In SPFEM, the response vectors or matrixes related to the pre-defined random input variables must be expanded at the first order or higher order approximations according to the Taylor series. Cui’s group successfully applied this technique into acoustic problems [27–30]. In this work, the stochastic perturbation technique is introduced into FE-LSPIM. Like SPFEM, the response vectors or matrixes related to the predefined random input variables are expanded based on the Taylor series expansion in hybrid stochastic perturbation FE-LSIPM (SP-FE-LSPIM). For a low variability level of the design parameters, the result will still be accurate if only the first order expansion is applied [31,32]. As the first order approximations are applied, the acoustic pressure response can be approximated as a linear function of the random variable. Thus the computational procedure can be largely simplified, and the expectations and the standard deviations of the response can be easily obtained through the SP-FE-LSIPM. Moreover, a change-of-variable technique [33–37] is introduced to observe the distribution of random responses. There are two significant steps for the calculation of probability density functions (PDF) and cumulative density functions (CDF): firstly, the first order Taylor expansion technique produce the linear functions between the inputs and responses; secondly, the change-ofvariable technique can help to produce the PDF and CDF of responses. Thus the promising approach for the comprehensive analysis of uncertainties can be acquired by combining SP-FE-LSPIM with the changeof-variable technique. Considering the superior performance of FE-LSPIM for the structural-acoustic problem, one is quite sure that the change-of-variable technique can be introduced into the FE-LSPIM frame and a hybrid stochastic perturbation FE-LSPIM frame (SP-FE-LSPIM) can be presented as a promising method to solve the structural-acoustic problem. The newly constructed SP-FE-LSPIM will be expected to efficiently and effectively evaluate the response probability density function and cumulative density functions.

Fig. 1. The structural-acoustic coupling system.

Fig. 2. Schematic of shell element.

⎧u (x ,y,z ) = u 0 (x ,y ) + zθx (x ,y ) v (x ,y,z ) = v0 (x ,y ) + zθy (x ,y ) , ⎨ w (x ,y,z ) = w0 (x ,y ) ⎩

(1)

where u 0 , v0 and w0 denote the displacements of three direction x, y and z of the middle surface of shell, respectively. θx and θy are the rotation angles of XY and YZ surface, respectively. The strain vector ε can be written as:

2. Structural-acoustic coupling system As shown in Fig. 1, the structural-acoustics system is composed of the structural domain Ωs , the acoustic domain Ωf and the coupling interface ∂Ωsf . The boundary of structural domain Ωs includes the essential boundary Γu and the natural boundary Γt . The boundary of acoustic domain Ωf includes the structural domain Ωs and a rigid boundary.

2.1. FE-LSPIM formulations for shell structural In this section, the FE-LSPIM formulations for the shell structural are briefly introduced.

zεb εm 0 ε = [εxx ,εyy,γxy,γxz ,γyz ]T = ⎡ ⎤ + ⎡ ⎤ + ⎡ ε ⎤ ⎣ 0 ⎦ ⎣ 0 ⎦ ⎣ s⎦

(2)

T ⎧ εm = [∂u 0 / ∂x ∂v0/ ∂y ∂u 0 / ∂y + ∂v0/ ∂x ] ⎪ T εb = [∂θ0 / ∂x ∂θ0 / ∂y ∂θ0 / ∂y−∂θ0 / ∂x ] , ⎨ ⎪ εs = [∂w0/ ∂x + θx ∂w0/ ∂y + θy]T ⎩

(3)

where εm is the membrane stress vector, εb is the bending stress vector, and εs is the shear stress vector. The bending stiffness constitutive matrix Db , the transverse shear stiffness constitutive matrix Ds , and the membrane stiffness constitutive matrix Dm can be expressed as:

2.1.1. Finite element-least square point interpolation of shell structural According to the First shear deformation theory shown in Fig. 2, the displacement components u, v and w of the shell structural can be written as below: 19

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G.Q. Jiang et al.

⎧ ⎡1 v 0 ⎤ 3 ⎪ Db = Et ⎢v 1 0 ⎥ 12(1 − v 2) ⎪ ⎢0 0 1 − v ⎥ 2 ⎦ ⎪ ⎣ ⎪ Etk τ 1 0 ⎤ Ds = 2(1 + v) ⎡ , ⎨ ⎣0 1⎦ ⎪ ⎡1 v 0 ⎤ ⎪ Et ⎢v 1 0 ⎥ ⎪ Dm = (1 − v 2) ⎢0 0 1 − v ⎥ ⎪ 2 ⎦ ⎣ ⎩

(4)

where E is elastic modulus, vis Poisson ratio, k τ is the shear correction factor, and t is the thickness of the shell element. According to standard FEM, the structural domain Ω is discretized into Ne quadrilateral element, containing Nd nodes. Considering node displacement vector u equals to (u,v,w,θx ,θy )T , the displacement of middle surface can be approximated as:

uh (x ) = N shellue,

(5)

N shell

where is quadrilateral shell element shape function, displacement vector of nodes, and they can be written as:

N shell = [N1shell

N2shell

N3shell

NIshell = diag { NI NI NI }

ue

N4shell]

I = (1,2,3,4),

is the Fig. 3. Schematic of definition of support domain of nodes and elements.

(6) (7)

Similarly, the rotation angle θx and θy can be expressed as:

in which NI is equal to parameter quadrilateral element shape function in FEM. Setting the vector ue as the displacement approximation function ui (x ,y ), (i = 1,2,3,4) of four nodes in quadrilateral element, it can be written as below: e T ⎧ u = {u1 (x ,y ) u2 (x ,y ) u3 (x ,y ) u4 (x ,y )} , u ( x , y ) = { θ θ w }, ( i = 1,2,3,4) ⎨ i xi yi i ⎩

⎧ θx (x ,y ) = N1 × 4 (Φ4 × M • θxM × 1) = (N Φ)1 × M θxM × 1 ≡ Ψ1 × M • θxM × 1 ⎨ ⎩ θy (x ,y ) = N1 × 4 (Φ4 × M • θyM × 1) = (N Φ)1 × M θyM × 1 ≡ Ψ1 × M • θyM × 1

2.1.2. The kinetic equation of shell According to the first shear deformation theory, the Galerkin weak formulations of the shell structural kinetic equation when the structural damping is ignored can be expressed as:

(8)

The node displacement function ui (x ,y ) can be obtained by interpolating support nodes with LSPIM. Deducing the solving process with deflection of generalized displacement, the deflection w (x ,y ) can be written as:

w (x ,y ) = NI w e,

∫Ω

s

where the deflection wi (x ,y ), (i = 1,2,3,4) can be expressed as: (11)

i i i i ⎧ Φi = [Φ1 Φ2 Φ3 ⋯ Φn] , T ⎨ ⎩ wi = [w1 w2 w3 ⋯ wn]

(12)

s

δεsT Ds εs dΩ +

∫Ω

s

δuT ρs tu¨ dΩ (16)

(17)

where Ns denotes the Lagrange shape function of iso-parametric quadrilateral element, us is the displacement vector of iso-parametric quadrilateral element, u¨ s denotes the acceleration vector of iso-parametric quadrilateral element. The membrane stress vector εm , bending stress vector εb and εs can be approximated as:

εm = Bm um,εb = Bb ub,εs = Bs us,

(18)

where Bm , Bb and Bs are the shape function of membrane stress, bending stress, and shear stress, respectively. Substituting Eqs. (17), (18) into Eq. (16), the kinetic equation of shell structural can be obtained as:

Ks u + Ms u¨ = Fb + Fs,

(13)

(19)

where Ks denotes the stiffness matrix of shell structural, Ms denotes the mass matrix of shell structural, Fb is volume force vector, Fs is surface stress vector. The stiffness matrix Ks of shell structural can be expressed as:

where Ψ is the shape function matrix of FE-LSPIM, its definition is written as:

Ψ1 × M = [Ψ1 Ψ2 Ψ3 ⋯ ΨM] = N1 × 4 Φ4 × M ,

s

u = Ns us,u¨ = Ns u¨ s,

where Φi is the LSPIM shape function of node i, which is combined by the support field point of node i in LSPIM. wi is the deflection vector function of support nodes. N is the number of support domain point of node i. Φ4 × M is assembled with Φi (i = 1,2,3,4) , the columns number of which is equal to the nodes number of element support domain. The node support domain Ω1 = {1,2,3,4,5,6,7,8,9} , Ω2 = {1,2,3,4,8,9,10,11,12} , Ω3 = {1,2,3,4,11,12,13,14,15} , Ω 4 = {1,2,3,4,5,6,14,15,16} . The element support domain Ω = Ω1 ⊕ Ω2 ⊕ Ω3 ⊕ Ω4 = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16} is shown in Fig. 3. Substituting Eqs. (10)–(12) into Eq. (7), the approximate formulation of internal field point deflection variable of the quadrilateral element can be obtained as:

w (x ,y ) = N1 × 4 (Φ4 × M • wM × 1) = (N Φ)1 × M wM × 1 ≡ Ψ1 × M • wM × 1,

s

where δ is variation operator, ρs is the density of shell structural, τ is surface stress vector, Dm , Db and Ds are the membrane stiffness constitutive matrix, bending stiffness matrix and transverse shear stiffness constitutive matrix, respectively. u and u¨ denote the displacement vector and the acceleration, respectively. εm , εb and εs are the membrane stress vector, bending stress vector and shear stress vector, respectively. Discretizing the shell structural into iso-parametric quadrilateral element, then displacement vector u and acceleration vector u¨ can be approximated as:

(10)

wi (x ,y ) = Φi wi, (i = 1,2,3,4)

∫Ω δεbT Db εb dΩ + ∫Ω + ∫ δuT τdΓ− ∫ δuT bs dΩ = 0, ∂Ω Ω

δεmT Dm εm dΩ + sf

(9)

w e is the deflection of each node of the quadrilateral element, and the approximation function wi (x ,y ) , (i = 1,2,3,4) can be expressed as:

w e = {w1 (x ,y ) w2 (x ,y ) w3 (x ,y ) w4 (x ,y )}T ,

(15)

(14) 20

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G.Q. Jiang et al.

Ks =

∫Ω

s

BmT Dm Bm d Ω +

∫Ω

s

BbT Db Bb d Ω +

∫Ω

s

BsT Ds Bs d Ω,

(20)

where Bm , Bb , and Bs can be written as:

0 0 − Ψ2,x 0 0 0 0⎤ − ΨM ,x ⎡− Ψ1,x 0 0 − Ψ1,y 0 − Ψ1,y 0 ⋯ − ΨM ,y 0 ⎥ Bm = ⎢ 0 ⎢ ⎥ − ΨM ,y − ΨM ,x 0 ⎦ ⎣− Ψ1,y − Ψ1,x 0 − Ψ2,y − Ψ2,x 0 0 0 − Ψ2,x 0 0 0 0⎤ − ΨM ,x ⎡− Ψ1,x 0 0 − Ψ1,y 0 − Ψ1,y 0 ⋯ − ΨM ,y 0 ⎥, Bb = ⎢ 0 ⎥ ⎢ − ΨM ,y − ΨM ,x 0 ⎦ ⎣− Ψ1,y − Ψ1,x 0 − Ψ2,y − Ψ2,x 0 Ψ1,x − Ψ2 0 Ψ2,x ⋯ − ΨM 0 ΨM ,x ⎤ − Ψ1 0 Bs = ⎡ ⎢ 0 − Ψ1 Ψ1,y 0 Ψ Ψ 0 Ψ ΨM ,y ⎥ − − 2 2, y M ⎣ ⎦ (21) The mass matrix Ms of shell structural can be expressed as:

Ms =

∫Ω

s

NsT mNs dΩ,

(22)

where m is element mass matrix. The volume force vector Fb and the surface force vector Fs can be expressed as:

Fb =

∫Ω

s

NsT bs dΩ,Fs = −

∫∂Ω

sf

NsT τdΓ,

Fig. 4. Nodes of supporting domain for node i of hexahedron element.

(23)

equation of acoustic domain can be written as:

Kf p + Mf p¨f = Ff + Fq, 2.2. FE-LSPIM formulations for acoustic domain

where Kf denotes the stiffness matrix of acoustic domain, and it can be expressed as below:

For the acoustic domain, the fluid is assumed to be homogeneous, inviscid, compressible, and only undergoing small translational movement. p denotes the acoustic pressure and k represents wave number. Then the well-known governing equation for acoustic domain can be written as:

∇2 p + k 2p = 0,

Kf =

f

1 ⎛δ ∇p • ∇p + δp • p⎞ dΩ−ρ c2 ⎠ ⎝

ΨM ,x ⎤ ⎡ Ψ1,x Ψ2,x Bf = ⎢ Ψ1,y Ψ2,y ⋯ ΨM ,y ⎥, ⎢ ⎥ ΨM ,z ⎦ ⎣ Ψ1,z Ψ2,z

(24)

∫∂Ω

sf

δp • u¨ f dΓ−

∫Ω

f

δp •

∂qf ∂t

Mf =

(30)

1 c2

∫Ω ΨTΨf dΩ,

(31)

p denotes the acoustic pressure vector of nodes, expressed as: dΩ = 0,

p = {p1 ,p2 ,…,pn }T ,

(32)

Ff denotes the surface vector of the acoustic domain, which can be written as:

where u¨ f denotes the normal acceleration of fluid in acoustic domain against with shell structural, qf denotes the additional load of element, p is acoustic pressure. In FE-LSPIM, the acoustic domain is needed to be discretized into iso-parametric hexahedron elements. Thus, the acoustic pressure p can be approximated as:

Ff = ρ

∫∂Ω

sf

BTf u¨ f dΓ,

(33)

Fq denotes the additional force vector of element volume, which can be obtained as:

p = Nf p

Fq = ρ

∫Ω

f

BTf

∂qf ∂t

dΩ,

(34)

(26)

where Nf is the Lagrange shape function of iso-parametric hexahedron element. p denotes the acoustic pressure approximate function of nodes of iso-parametric hexahedron element. pi (x ,y ) (i = 1,2,…,8) is obtained by interpolating support nodes by LSPIM. The interpolation process is the same as the finite element-least square interpolation of the shell element [17], so it will not be reiterated here. The range of support domain of LSPIM can be expressed as shown in Fig. 4. The acoustic pressure approximate function obtained by FE-LSPIM can be written as below:

p (x ,y ) = N1 × 8 (Φ8 × M • pM × 1 ) = (N Φ)1 × M pM × 1 ≡ Ψ1 × M • pM × 1 ,

(29)

Mf denotes the mass matrix of acoustic domain, which are given by:

(25)

p = pi (x ,y )(i = 1,2,⋯,8),

∫Ω BTf Bf dΩ,

in which Bf is acoustic gradient matrix and can be expressed as:

where ∇ denotes the Hamiltonian operator, the wavenumber is calculated by: k = ω/ c , where ω is the angular frequency of the pressure oscillation, and c is the speed of sound traveling in the acoustic fluid. The standard Galerkin weak form for the acoustic problem without acoustical damping can be expressed as:

∫Ω

(28)

2.3. Formulation for FE-LSPIM of structural-acoustic system The structural-acoustic coupling system should satisfy the displacement and pressure continuity conditions at the coupling interface. These conditions can be written as:

us n = uf n, τ|n = −p, on ∂Ωsf ,

(35)

where n denotes the normal direction of coupling surface, us denotes the displacement of shell structural in coupling surface, and uf is the displacement of fluid in acoustic domain in coupling interface. According to the displacement continuity conditions, the loading Ff imposed over the interface of structural-acoustic coupling system by

(27)

Substituting Eqs. (26), (27) into Eq. (25), then the discrete dynamic 21

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G.Q. Jiang et al. n

acoustic can be written as:

Ff = ρ

∫∂Ω

sf

NTf u¨ f

dΓ = ρ (

F (t ) = F (t )|t = E (t ) +

∫∂Ω

sf

NTf nf

i=1

Ns d Γ) u¨ s

(36)

∫∂Ω

sf

NsT τd Γ =

∫∂Ω

s

NsT pnd Γ = (

∫∂Ω

s

NsT nf Nf d Γ) P

∫∂Ω

s

(37)

n

J (t ) = J (t )|t = E (t ) +

(38)

n

F (t ) = F (t )|t = E (t ) +

Substituting Eq. (38) into Eqs. (36) and (37), Ff and Fs can be simplified as:

i=1

j=1

∂J (t ) ∂J (t ) ∂ti ∂t j (44)

∑ i=1

∂J (t ) |t = E (t ) (ti−E (ti )) = J 0 + ΔJ ∂ti

(45)

∂F (t ) |t = E (t ) (ti−E (ti )) = F 0 + ΔF ∂ti

(46)

Substituting Eqs. (45), (46) into Eq. (42), it can be arrived at

Ff = ρH T u¨ s,Fs = Hp

(39)

U = (J 0 + ΔJ )−1F 0 + ΔF

According to the FE-LSPIM forms of shell structural, FE-LSPIM forms of acoustic domain and the structural-acoustic coupling conditions, the form of FE-LSPIM/FE-LSPIM for structural-acoustic coupling system can be expressed as:

{ }

According to Neumann expansions theory, the expression of (J 0 + ΔJ )−1 can be expanded as:

2 − H ⎤ us Fb ⎡ Ks−ω Ms = ⎧ ⎫, 2 ⎢ ρω H T Kf −ω2Mf ⎥ p ⎨ ⎩ Fq ⎬ ⎭ ⎦ ⎣

{ }

(41)

U = U 0 + ΔU1 = (J 0)−1F 0 + (J 0)−1ΔF −(J 0)−1ΔJ (J 0)−1F 0

where ω represents the angular frequency of the harmonic external excitation.

n

∂F (t ) ⎛ ⎞ U = (J (t )|t = E (t ) )−1F (t )|t = E (t ) + (J (t )|t = E (t ) )−1 ⎜∑ |t = E (t ) (ti−E (ti ))⎟ ∂ti ⎝ i=1 ⎠

Simplifying the process of analyzing the acoustic pressure corresponding of structural-acoustic coupling system, Eq. (41) can be rewritten as:

n

∂J (t ) ⎞ ⎛ −(J (t )|t = E (t ) )−1 ⎜∑ |t = E (t ) (ti−E (ti )) ⎟ (J (t )|t = E (t ) )−1F (t )|t = E (t ) ∂ti i = 1 ⎠ ⎝ (50)

(42)

JU = F,

Ks−ω2Ms −H ⎤ where J = ⎡ ⎢ ρω2H T Kf −ω2Mf ⎥ is the dynamic stiffness matrix of ⎦ ⎣ us structural-acoustic coupling system, U = p represents the correFb sponding vector, F = ⎧ ⎫ denotes the total external load vector. ⎨ ⎩ Fq ⎬ ⎭ The uncertainties of parameters are commonly encountered in numerous structural-acoustic systems, and must be carefully handled in computational modeling and simulation. In this subsection, the SP-FELSPIM is constructed based on the traditional stochastic perturbation method, FE-LSPIM, and the change-of variables techniques. A set of random field t(x), with probability density function (PDF) and the expected value of which are pi (tr ) , Ei (tr ) , (r = 1,2,…,R, i = 1,2), respectively, are both introduced here. Firstly, the operations of Taylor expansions are performed on the dynamic stiffness matrix of whole structural-acoustic system and random external load vector at the expectation of the random vector. The whole dynamic stiffness matrix can be rewritten as follow:

Based on Eq. (50), a simple mathematical transformation is carried on, and then the unknown vector U can be written in a matrix and vector forms as follows:

{ }

i=1

(49)

Substituting Eqs. (45), (46) into Eq. (49), U can be rewritten as:

3. Formulation for hybrid SP-FE-LSPIM

∑

(48)

Substituting Eq. (48) into Eq. (47), the different perturbation term of U in Eq. (47) can be further written as: U 0 = (J 0)−1F 0 ; ΔU1 = −(J 0)−1 (ΔJU 0−ΔF ) ; ΔU 2 = −(J 0)−1ΔJU1 ΔU n = −(J 0)−1ΔJU n − 1. To simplify the computational complexity and eliminate the computational cost, only the first order perturbation of Neumann series is kept in the calculation while all the higher order perturbation terms are ignored. Thus, the unknown vector U can be written as:

(40)

Without considering any structural damping, the discrete system of equations for a structural-acoustic system subject to the harmonic external excitations can be written as:

n

(47)

(J 0 + ΔJ )−1 = (J 0)−1−(J 0)−1ΔJ (J 0)−1 + (J 0)−1 (ΔJ (J 0)−1)2−⋯

0 ⎤ u¨ s ⎡ Ms ⎧ ⎫ + ⎡ Ks − H ⎤ us = ⎧ Fb ⎫ 0 Kf ⎥ p ⎢− ρH T Mf ⎥ ⎨ p¨ ⎭ ⎢ ⎨ ⎣ ⎦ ⎩ Fq ⎬ ⎭ ⎦⎩ ⎬ ⎣

J (t ) = J (t )|t = E (t ) +

∑ i=1

NsT nf Nf dΓ

n

∑∑

The higher order of Taylor expansions terms are ignored for simplification. Therefore, the first order of Taylor expansions of matrix and vector can be expressed as:

The spatial coupling matrix H can be expressed as:

H=

n

∂F (t ) 1 |t = E (t ) (ti−E (ti )) + ∂ti 2!

|t = E (t ) (ti−E (ti ))(t j−E (t j )) + …

Similarly, according to the pressure continuity conditions, the loading Fs imposed over the surface of structural-acoustic coupling system by structural can be written as:

Fs = −

∑

∂J (t ) 1 |t = E (t ) (ti−E (ti )) + ∂ti 2!

n

n

∑∑ i=1

j=1

n

U = (J (t )|t = E (t ) )−1F (t )|t = E (t ) − ∑ ( i=1 n

−∑ i=1

n

∂F (t ) ∂J (t ) ⎛ |t = E (t ) )−1 ⎜∑ |t = E (t ) ∂ti ∂ti ⎝ i=1

∂J (t ) ⎞ |t = E (t ) (J (t )|t = E (t ) )−1F (t )|t = E (t ) ⎟ E (ti ) ∂ti ⎠ n

∂F (t ) ⎛ + (J (t )|t = E (t ) )−1 ⎜∑ |t = E (t ) ∂ti ⎝ i=1 n

−∑ i=1

∂J (t ) ⎞ |t = E (t ) (J (t )|t = E (t ) )−1F (t )|t = E (t ) ⎟ ti = T 0 + ∂ti ⎠

Ti ti,

i=1

(51) in which the detailed expressions of T 0 and ti can be written as: n

n

−1 ∂J (t ) ∂F (t ) ⎛ T 0 = (J (t )|t = E (t ) )−1F (t )|t = E (t ) − ∑ ⎛ |t = E (t ) ⎞ ⎜∑ |t = E (t ) ⎠ ⎝ i = 1 ∂ti i = 1 ⎝ ∂ti

∂J (t ) ∂J (t ) ∂ti ∂t j

n

|t = E (t ) (ti−E (ti ))(t j−E (t j )) + ….

n

∑

(43)

−∑ i=1

While the random external force vector can be expressed as: 22

∂J (t ) ⎞ |t = E (t ) (J (t )|t = E (t ) )−1F (t )|t = E (t ) ⎟ E (ti ) ∂ti ⎠

(52)

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G.Q. Jiang et al. n

∂F (t ) ⎛ Ti = (J (t )|t = E (t ) )−1 ⎜∑ |t = E (t ) ∂ti ⎝ i=1 n

∂J (t ) ⎞ |t = E (t ) (J (t )|t = E (t ) )−1F (t )|t = E (t ) ⎟ ∂ti ⎠

−∑ i=1

(53)

The unknown vector U can be rewritten in a simpler and more U = {u1 ⋯ uk ⋯ um}T ; apparent vectors forms as: Ti = {T0,1 ⋯ T0,k ⋯ T0,m}T ; Ti = {Ti,1 ⋯ Ti,k ⋯ Ti,m}T , then U can be expressed as: n

U = T0 +

∑

Ti ti,

(54)

i=1

in which uk can be calculated as follows: n

uk = T0,k +

∑

Ti,k ti

(55)

i=1

As previously deduced, the relationship between the response of U and the input random b(x) has already been transformed to linear functions as shown in Eq. (55). In order to obtain the response probability density function of structural-acoustics, the change-of-variable technique is applied in this work. Due to the linear relationship between the response and inputs, the probability density function of response can be calculated in a step by step manner. If there are only two inputs, according to the change-of-variable technique, the probability density function of the response can be expressed as below:

f y1,r ,y2,r (y1,r ,y2,r ) = f y1,r (y1,r ) f y2,r (y2,r ) = f y1,r (y1,r ) f y2,r (ur −T0,r −y1,r )

Fig. 5. Three randomly chosen response points.

Table 1 The pre-defined random variables of the structural-acoustic coupling model.

(56)

where y1,r = T1,k t1, y2,r = T2,k t1, the f y1,r (y1,r ) f y2,r (y2,r ) are the probability density of the inputs. The detailed derivations of the probability density function are expressed as follows:

∫y

∞

=

∫y

∞

=

1 |T2,r |

fur (ur ) =

∫y

1,r =−∞

∞

2,r =−∞

f y1,r ,y2,r (y1,r ,y2,r ) dy1,r dy2,r

∞

∫t =−∞ ft (t1) ft 1

1

2

⎛⎜ ur −T0,r −y1,r t1 ⎞⎟ dt , r = 1,⋯,L 1 T2,r ⎝ ⎠

(57)

It is assumed that Y1,r = y1,r + y2,r when more than two random inputs are involved. Thus, the probability density function of these two inputs is calculated just as the above case:

fY1,r (Y1,r ) =

1 T2,r

∞

∫t =−∞ ft (t1) ft 1

1

2

⎛ Y1,r −T1,r t1 ⎞⎟ dt 1 ⎝ T2,r ⎠

⎜

(58)

Assuming that Y2,r = Y1,r + y3,r , y3,r = T3,k t3 ,the PDF of Y2,r can be calculated similarly as below based on the change-of variable technique.

fY2,r (Y2,r ) =

∞

∫t =−∞ ft (t3) fY 3

3

1,r

(Y2,r −T3,r t3) dt3

(59)

After several repeat calculations, the final PDF can be expressed as:

fYZ ,r (YZ ,r ) = fur (ur ) =

∫t

∞

Z + 1=−∞

∞

∫t =−∞ ft n

n

ftZ + 1 (tZ + 1) fYZ − 1,r (YZ ,r −TZ + 1,r tZ + 1) dtZ + 1

(tn ) fYn − 2,r (ur −T0,r −Tn,r tn ) dtZ r = 1,…,L

Distribution type

Parameters of random variable

Modulus of elasticity E Poisson ratio υ

Gaussian distribution

μE = 2.1 × e11

σE = 5 × e9

μ υ = −1.2

σE = 0.033

Thickness t Structure density ρs

Log-normal distribution Gaussian distribution Gaussian distribution

μt = 0.001 μ ρs = 7850

σt = 0.0001 σρs = 80

The external force Q

Gumbel distribution

μQ = 10

σQ = 4

in Fig. 5, the structural-acoustics model consists of two parts: a flat shell (plate) structural as roof which is made of steel (ρ = 7800 kg/m3) and an acoustic cavity, which is filled with air (ρ = 1.225 kg/m3,c = 340.5 m/s) .The closed acoustic cavity has dimensions of 414 × 314 × 360 mm, while the plate has dimensions of 414 × 314 mm, with the thickness of 1 mm. The walls (except the coupling wall) of the cavity are assumed to be acoustically rigid with the surface velocity fixed at v = 0. As shown in Fig. 5, three points (Node 1, Node 2, Node 3) are randomly selected in this model. The coordinate positions of these three points are as follow: Node 1 (331.2, 62.8, 90) mm, Node 2 (248.4, 219.8, 225) mm, Node 3(41.4, 62.8, 360) mm. In order to illustrate the effectiveness of SP-FE-LSIPM, the acoustic pressure response for each pointis calculated at a frequency of 300 Hz and compared with SP-FEM and MCM. In random structural-acoustic systems, the modulus of elasticity E, Poisson ratio υ, thickness t of the flat shell and the external force Q are considered as the random variables with some specified probability density distributions, as shown in Table 1. The main numerical procedure provided in Eqs. (44)–(61). Figs. 6 and 7 show the frequency response results of Node 1 at 300 Hz. It is shown in the figures that the PDF or CDF calculated by SP-FELSPIM matched those of MCM better as compared to those of the SPFEM at Node 1. Figs. 8 and 9 show the frequency response results of Node 2 at 300 Hz. Similarly, the result of PDF or CDF obtained by SP-FE-LSPIM matched those of MCM better as compared to those of the SP-FEM at Node 2. Figs. 10 and 11 show the frequency response results of Node 3 at 300 Hz. As shown in Fig. 6 and 7, the proposed SP-FE-LSPIM always provides high accuracy results, better than those of MCM and SP-FEM in any different points. Above all, it can be concluded that the proposed SP-FE-LSPIM can effectively handle the randomness in structuralacoustic systems.

f y1,r (y1,r ) f y2,r (pr −T0,r −y1,r ) dy1,r

1,r =−∞

Random variables

(60) (61)

Once the PDF is obtained, the CDF results can be easily obtained by a simple mathematical transform. 4. Numerical application In this section, the proposed SP-FE-LSPIM approach is applied to a standard example to demonstrate its validity and efficiency. As shown 23

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Fig. 6. The probability density functions of the frequency response (300 Hz) of Node 1 calculated by MCM, SP-FEM, and SP-FE-LSPIM.

Fig. 7. The cumulative density functions of the frequency response (300 Hz) of Node 1 calculated by MCM, SP-FEM, and SP-FE-LSPIM.

Fig. 8. The probability density functions of the frequency response (300 Hz) of Node 2 calculated by MCM, SP-FEM, and SP-FE-LSPIM.

Fig. 9. The cumulative density functions of the frequency response (300 Hz) of Node 2 calculated by MCM, SP-FEM, and SP-FE-LSPIM.

5. Conclusion

acoustic problems. Furthermore, the imbedded stochastic perturbation technique has properly dealt with the uncertainties. Numerical example has demonstrated the following features of the present method.

In this work, a hybrid SP-FE-LSPIM framework is proposed to predict the average response of a structural-acoustic system in the midfrequency regime. The proposed approach utilizes the advantages of FELSPIM to reduce the numerical dispersion error in the process of solving

(1) Like FE-LSPIM, SP-FE-LSPIM uses the quadrilateral element for 2D problems (hexahedron element for 3D problems), thus no 24

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Fig. 10. The probability density functions of the frequency response (300 Hz) of Node 3 calculated by MCM, SP-FEM, and SP-FE-LSPIM.

Fig. 11. The cumulative density functions of the frequency response (300 Hz) of Node 3 calculated by MCM, SP-FEM, and SP-FE-LSPIM.

additional parameters or degrees of freedom need be introduced, and it can be straightforwardly implemented with some modifications to the FEM code. (2) In probabilistic analysis, the probability density functions and cumulative distribution functions of structural-acoustic systems calculated by SP-FE-LSPIM more closely match MCM as compared to SP-FEM. Compared to MCM, SP-FE-LSPIM offers significant advantages in computation efficiency without any significant lose in accuracy. (3) SP-FE-LSPIM is constructed based on the first order of Taylor expansions. If a further increase in accuracy is required, then the higher order perturbation technique is recommended, however, the computational cost will increase.

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