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Stochastic wave finite element for random periodic media through first-order perturbation
Comput. Methods Appl. Mech. Engrg. 200 (2011) 2805–2813
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Stochastic wave finite element for random periodic media through first-order perturbation M.N. Ichchou a,⇑, F. Bouchoucha a,b, M.A. Ben Souf a,b, O. Dessombz a, M. Haddar b a b
LTDS, Ecole Centrale de Lyon, 36 Avenue Guy de Collongue, F-69134 Ecully Cedex, France Unité de Mécanique, Modélisation et Productique, Ecole nationale d’Ingénieurs de Sfax, National School of Engineers of Sfax, BP. W3038 Sfax, Tunisia
a r t i c l e
i n f o
Article history: Received 20 August 2010 Received in revised form 12 May 2011 Accepted 16 May 2011 Available online 1 June 2011 Keywords: Guided waves Random media Stochastic finite element Parametric probabilistic model Dispersion
a b s t r a c t Wave propagation features in random guided elastic media are considered in this paper. A formulation named stochastic wave finite element (SWFE) is offered. This formulation allows wave characteristics definition by means of a stochastic finite element model. The case of spatially homogeneous random properties is dealt with, through a parametric probabilistic technique. Numerical experiments show the effectiveness of the proposed approach to predict the statistics of guided wave propagation characteristics (means and standard deviations) when compared to Monte Carlo calculations. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction To deal with guided wave propagation in elastic media, semianalytical finite element (SAFE) and wave finite element (WFE) methods have been investigated by a number of researchers. Such numerical techniques are extensively employed in many wave dynamic issues. Structural health monitoring (SHM) and nondestructive testing (NDT) are among the fields of application of these numerical tools. Both SAFE and WFE, for instance, can be used for wave propagation predictions and wave scattering estimations. An assessment of damage signature effects can thus be reached. The SAFE technique considers wave propagation by means of specific shape functions. It necessitates the development of a relatively new FE code for specific elements. The WFE technique considers the wave propagation problems of periodic structures. The WFE method regards the homogeneous waveguide structure as a periodic system assembled by identical substructures. The WFE method is easy to apply due to its direct connection with the standard FE method. This method has been used for the structural vibration analysis [1–5], the wave propagation in elastic waveguides [6–8], and the calculations of Green’s functions for homogeneous acoustic and elastic media [9]. Houillon et al. [5] studied the dynamic problems of homogeneous thin-walled structures using this method, which has been shown to be successful when applied to structures with uniform cross-section. Duhamel ⇑ Corresponding author. Tel.: +33 04 72 18 62 30; fax: +33 04 72 18 91 44. E-mail address: [email protected] (M.N. Ichchou). 0045-7825/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2011.05.004
et al. [4] used this method to investigate the vibrations of uniform waveguide structures, where it has been proved to be accurate with relatively low computational cost in comparison with the standard FE method. Mencik and Ichchou [6] proposed a hybrid approach to study the diffusion of multiple wave modes based on this method. Ichchou et al. [10] investigated the numerical sensitivity of this method. The energy propagation features in rib-stiffened panels over a wide frequency range were studied by using this method in [3], where the comparisons of numerical and experimental results are provided. Chen and Wilcox [8] used this method to investigate the effect of load on guided wave propagating properties in rails. The method was also implemented for wave propagation and dynamic problems in the homogeneous structures with internal fluid [11–13]. In the open literature, however, most of what can be found in the numerical issues of wave propagation simulations is mainly limited to deterministic media. Viscoelastic media characteristics are in this case fixed to nominal values. To the authors’ knowledge, the extension of such numerical techniques to non-deterministic problems is not yet considered. From the applications and the academic point of view such considerations are of real and pertinent added value. The paper novelty is then to consider numerical guided wave techniques characterization in spatially homogeneous random media. This work’s main objectives are twofold: to formulate and validate a stochastic wave finite element technique for structural members. To deal with uncertainties in structural dynamics extensive research has been performed over the last decade. This issue is extensively studied in the literature [14,15] and a significant num-
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ber of studies have been performed in this context by many authors among them Soize et al. [16–20]. The uncertainties can be linked to the geometric properties, material characteristics and boundary conditions among other possibilities. This behaviour is taken into account mainly according to two approaches. The non-parametric approach [16,17] introduces the uncertainties directly into the global matrices of the model or into the reduced matrices of the nominal condensed model. The probabilistic description of resulting matrices is governed by the maximum entropy principle. Parametric approaches considers the uncertain parameters (geometrical, material properties etc.) as random quantities. Specific discretization approaches such as the stochastic finite element method (SFEM) which combines finite elements and probabilistic ways of thinking can be employed [14]. Due to the efficient representation of random fields, the SFEM has been successfully applied to various kinds of stochastic problems. Improvements and further consideration of parametric strategies for treating uncertainties are still under investigation. The current study extends the WFE techniques to stochastic media. In this context the concept of a random field cannot be ignored. The variability of wave characteristics that arises from material and geometric variability is desired in many engineering problems. For instance, such variability is expected to play a fundamental role in the medium and high frequency range [21] in the structural–acoustic area. The uncertainties considered in what follows are mainly linked to the geometric properties and to the material characteristics of the propagation media. A parametric approach for uncertainties treatments is considered and combined to the WFE technique. A similar process, such as the stochastic finite element method, is used to develop the stochastic wave finite element (SWFE) approach based on the probabilistic tools. The idea is to consider the random fields as a supplementary dimension of the problem through the spatial discretization using the finite elements process. The dimension associated to the random fields is thus considered using two ways: perturbation and spectral methods. It should be noted that the strategy can be easily adapted to the well-known SAFE techniques. The paper scope is limited to the extension of the WFE to the stochastic case. The paper contains four sections. In Section 2, the formulation of the stochastic wave finite element approach is presented. The deterministic WFE is first very briefly reminded, for the sake of clarity. Then, the stochastic treatment is detailed. Section 3 provides some expressions deduced from the SWFE formulation for wave properties analysis. Section 4 gives mainly numerical experiments. The formulation is general, but the validations were limited to basic longitudinal and flexural wave propagation problems. A first comparison was established analytically in these cases. Ultimately, a comparison of the SWFE prediction with Monte Carlo simulations confirms
the effectiveness of the proposed techniques in the studied cases. A conclusion together with a description of the work in progress is ultimately given. 2. Formulation of the stochastic wave finite element approach In this section, the formulation of the SWFE approach is given. For the sake of clarity, the deterministic case is first concisely presented [2,10]. The introduction of uncertainties and specific developments leads to two different spectral problems. The first stochastic formulation involves a spectral quadratic nature. The second demonstration uses a state space-like process and leads to a first order formulated spectral problem. Properties of each one are also described. 2.1. WFE in the deterministic case The dynamics of straight elastic and dissipative structures are studied. A sample straight structure is illustrated in Fig. 1: in the present framework, the system is assimilated to be a set of identical subsystems, connected along the principal direction, say axis x, and whose left and right cross-sections (x-axis description) are denoted as L and R, respectively. The length of each subsystem, along axis x, is denoted as d. The formulation is based on the finite element model of a typical subsystem, as illustrated in Fig. 1, and whose kinematic variables, displacements and forces, are written as q and forces F, respectively. Mesh compatibility at coupling interfaces between subsystems is assumed, implying that the left and right cross-sections of the given subsystem contains the same number of degrees of freedom, say n. The dynamic equilibrium equation of this subsystem, at frequency x/2p, can be stated as follows [22]
DLL DRL
DLR DRR
qL qR
¼
FL ; FR
ð1Þ
where (n n) matrix Dij = Kij x2Mij ({i, j} 2 {L, R}) stands for the ij component of the dynamic stiffness operator condensed on the left and right cross-sections, namely D [22]. Here, K and M stand for the stiffness and mass matrices, respectively. Dissipation can be considered through standard FEM models. According to Bloch’s theorem, the dynamics of the global waveguide can be expanded on specific wave solutions of the form
qR ¼ lqL
ð2Þ
and
FR ¼ lFL ;
Fig. 1. Typical thin walled structure.
ð3Þ
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where l denotes the propagation coefficient. Expressions (2) and (3) lead to an eigenvalue problem. Indeed, inserting Eqs. (2) and (3) into Eq. (1) leads to the following spectral problem [5]:
DRL þ li ðDLL þ DRR Þ þ l2i DLR ðUq Þi ¼ 0;
ð4Þ
where {(li, (Uq)i)}i=1,. . .,2n stands for the wave modes of the global system. It’remarkable that DTRL ¼ DLR and (DLL + DRR)T = (DLL + DRR). Using these properties we can prove that ðUq ÞTi is the left eigenvectors associated to the left eigenvalue l1 [6]. i A modified and well conditioned format of the spectral problem can also be obtained. For this purpose, the use of a state vector representation is an interesting alternative to the spectral analysis which must be performed in the context of the numerical dispersion curve extraction (see Refs. [22,11,10] for detailed discussions). Indeed, given the following matrices:
" S¼
#
D1 LR DLL
D1 LR
DRL DRR D1 LR DLL
DRR D1 LR
ð5Þ
and matrix Jn, defined by:
Jn ¼
0
In
In
0
ð6Þ
;
to be spatially homogeneous. This guarantees the assumed periodicity will be respected in the non-deterministic case as in the deterministic situation. The considered random sample is assumed to be modelled through a SFEM-like approach in order to describe e will designate random the element equilibrium. In what follows D e is the random dynamic operator of the quantities (for instance D considered sample structure). The stochastic equation of motion for the kth sample is then the following:
eq ~¼e D F:
This expression extends Eq. (1) to the non-deterministic case. In what follows the first order development of stochastic variables is adopted, such that:
þ rq Þ ¼ ðF þ rF Þ; ðD þ rD Þðq
e LL D e RL D
ð7Þ
meaning that S is symplectic [10]. Ultimately, a spectral problem can be established as:
J n Ui ¼ li ST J n Ui ;
ð8Þ
ð9Þ
and thus, considering that matrix S is symplectic (ST ¼ Jn SJ 1 n [22]),
SUi ¼ li Ui :
ðUq ÞTi ðUF ÞTi
ð10Þ
T
Here Ui ¼ stands for the ith eigenvector of operator S, which is decomposed into (n 1) displacements q and forces F wave components. The frequency response of the global system can be expressed by expanding the kinematic variables of the considered subsystem on the basis of eigenvectors:
qL
¼
FL
Uq
UF
QL
and
qR FR
¼
Uq UF
Q R;
ð11Þ
where Uq and UF stand for the matrices of eigenvector {(Uq)i}i and {(UF)i}i, respectively, and where QL and QR stand for the (2n 1) generalized coordinates evaluated for the left and right boundaries of the subsystem, respectively. It has been shown in Refs. [23,6] that generalized coordinates QL and QR can be related in this way
QR ¼
l
0
0
l1
Q L;
ð12Þ
where l stands for the matrix of eigenvalues {li}i=1,. . .,n. Note that the description provided by Eq. (12) is based on the classification of the eigenvectors into incident and reflected waves (see Ref. [6]). Eqs. (12) and (11) will be used for the estimation of energy velocities. 2.2. Stochastic wave finite element method: quadratic spectral formulation In this subsection, a stochastic medium is considered. As previously done, the straight studied structure is partitioned in a number of sample of length d. The uncertainties are assumed to be mostly on the material properties. Such uncertainties are assumed
e LR D e RR D
!
~ LðkÞ q ~ RðkÞ q
! ¼
! ðkÞ e FL : e ðkÞ F
ð15Þ
R
Following the same process as in the previous subsection, one can readily obtain a quadratic spectral problem formulated in the stochastic area, which is (16):
which leads to T J 1 n S J n Ui ¼ l i Ui ;
ð14Þ
where is Gaussian variable centred and reduced. This equation is a simplified application for the stochastic finite element method used by many contributors, see Ghanem and Spanos [14] for instance. In and F represent the mean quantities of the dythis expression, D; q namic operator, the displacement vector and the load; rD, rq and rF their standard deviations. As done in the deterministic case, the stochastic problem stated in Eq. (13) can be partitioned as follows:
it can be readily shown that:
ST J n S ¼ J n
ð13Þ
e RL þ l e LR ð U e LL þ D e RR Þ þ l e q Þ ¼ 0: ~ iðD ~ 2i D D i
ð16Þ
~ i and the corresponding stochastic The stochastic eigenvalue l e i are solutions of this established quadratic equation. eigenvector U The zero order and the first order development of this relationship can be readily performed and leads to:
i ðDLL þ DRR Þ þ l 2i DLR ðUq Þi ¼ 0; DRL þ l
ð17Þ
which is the ‘zero order’ expression. This is exactly the deterministic spectral problem already obtained in the previous section for deterministic cases. The first order relation is also derived as follows:
i ðDLL þ DRR Þ þ l 2i DLR rðUq Þi þ ½rDRL þ l i ðrDRR þ rDLL Þ DRL þ l 2 þrli ðDRR þ DLL Þ þ li rDLR þ 2li rli DLR ðUq Þi ¼ 0: ð18Þ In order to express the standard deviation of li ðrli Þ, let us multiply Eq. (18) by ðUq ÞHi , an analytical development leads then to:
rli ¼
ðUq ÞHi
rDRL þ l i ðrDRR þ rDLL Þ þ l 2i rDLR ðUq Þi ; DLR ðUq Þi ðUq ÞHi ½DRR þ DLL þ 2l
ð19Þ
where H is the hermitian transpose. Similarly, the standard deviation of the eigenvectors rUq can be written as:
þ
rðUq Þi ¼ DRL þ l i ðDRR þ DLL Þ þ l 2i DLR ½rDRL þ l i ðrDRR þ rDLL Þ i rli DLR ðUq Þi ; 2i rDLR þ 2l þrli DRR þ DLL þ l ð20Þ where + is the pseudo inverse. Eqs. (19) and (20) constitute the close expression of the means and standard deviations of wave properties in the general elastodynamics case. Then from the knowledge of means and deviations of the dynamic operator, which are associated, to the considered uncertainties in the studied media, one can easily express the statistics of the guided waves characteristics. The ’relatively’ strong coupling between the statistics of the propagation constants and the corresponding eigenvectors is to be noted.
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2.3. Stochastic wave finite element method: state space formulation As explained in Section 2.2, from a numerical point of view, the state space formulation is often preferred. This is especially true in the Hamiltonien [11] case where the properties of the expressed operators allow important properties of the wave problems to be ~ L ¼ ððq ~ L ÞT ð e exhibited. So, let us define u F L ÞT ÞT and ~ R ¼ ððq ~ R ÞT ð e u F R ÞT ÞT as the representation of kinematic variables through stochastic state vectors. The dynamic of the representative kth cell is then expressed as follows:
~ L ¼ ½u ~ R : ðS þ rS Þ½u
SRL þ rSRL SRR þ rSRR
!
L þ rqL q
! ¼
FL rFL
R þ rqR q FR þ rFR
! :
ð22Þ
The zero order development of the expression leads to the definition of S as follows:
!
D1 LR DLL
D1 LR
DRL DRR D1 LR DLL
DRR D1 LR
ð23Þ
;
which is simply the S matrix employed in the deterministic case. Equivalently, the first order expression leads to:
rS ¼
0
DRR
1
!1
rSLL rSLR rSRL rSRR
1 D1 LR DLL
: D1 LR 0
ð24Þ
This equation will be used to obtain the statistics of estimated wave numbers. The stochastic eigenvalue problem can be written:
e ei ¼ l e i; ~iU SU ~ i I2n j ¼ 0; je Sl
ð25Þ ð26Þ
e i Þ are the solutions of the eigenvalue problem. If the ~ i; U where ðl stochastic eigensolutions are expressed as:
l~ ¼ ðl þ rl Þ;
ð32Þ
for the standard deviation of wave modes. Eqs. (31) and (32) provide the statistics of guided wave in the considered random viscoelastic media. The equivalence between Eqs. (19) and (31) associated to the quadratic spectral problem and Eqs. (20) and (32) associated to the state space problem is not obvious in the general case. It will be considered in the numerical validations offered in Section 4 of this paper. 3. Statistics of energies and energy and group velocities
SLL þ rSLL SLR þ rSLR
DLR
rUi ¼ ½S l i I2n þ ½rS rli I2n Ui
ð21Þ
Or equivalently:
S¼
for the standard deviation of propagation constants and
e ¼ ðU þ rU Þ; U
then the zero order (Deterministic case) terms are given by:
i I2n ÞUi ¼ 0; ðS l
ð27Þ
From the definition of standard deviation of propagation constants, it can be of interest, in many applicative engineering cases, to consider wave numbers, energies and energy velocities statistics. Indeed, from the knowledge of the zero order and first order terms and considering the wave number to be written as ~¼k þ r , one can easily express the statistics of k ~ as follows k k (j2 = 1):
¼ j Logðl Þ; k d
ð33Þ
which is the mean of the wave number and developing the first order leads to:
rk ¼
j rl : d l
ð34Þ
Given the statistics of computed wave numbers, it can be easily shown that the general expression for standard deviation of group velocity is:
@r rcg ¼ cg k ; @k
ð35Þ
where cg is the deterministic group velocity. Similarly, definition of the energy or group velocity in the multimodal case provided in Ref. [10] in the deterministic case, can be extended thanks to the SWFE formulation. Indeed, the analytical expression of the energy velocity [10] associated with each propagating branch i can be written in the stochastic case as:
ei P
whilst the first order contribution, couples the standard deviation of propagation constants and associated eigenvectors, as follows:
~cgi ðxÞ ¼
i I2n ÞrUi þ ðrS rli I2n ÞUi ¼ 0: ðS l
e i is the stochastic energy flow contribution of the ith wave where P e i and U e i stand for the stochastic kinetic and potential mode, and T energy densities, respectively. Energies of the stochastic media can be expressed thanks to the stochastic wave modes basis given in Section 2.1. Indeed, given an excitation described by the force vector e F L , the wavemode amplitudes associated to this external input in the case of an infinite structure can be written as:
ð28Þ
e T J n is a left eigenvector of e ~ i. As U S associated to the eigenvalue 1=l i
e T Jn e e T Jn : ~i U S ¼ 1=l U i i
ð29Þ
The expansion and the polynomial chaos projection of Eq. (29) gives the first order term as follows:
T 2 1 ðrUi Þ J n S l i I 2n þ ðUi Þ J n rS þ rli li I2n ¼ 0: T
ð30Þ
Eqs. (28) and (30) can then be readily solved in order to calculate the standard deviation of eigenvalues rli and the eigenvectors rUi . After some analytical treatments, statistics of wave characteristics can be expressed as:
h
T i rli ¼ ðUi ÞT rTS ðST l i I2n Þ1 Jn S ðl i Þ1 I2n Ui Jn rS 1 1 i I2n i Þ1 I2n þ ðl i Þ2 ðUi ÞT J n J n S ðl ðUi ÞT ST l ð31Þ
ei þ U ei T
ð36Þ
;
1 e ex ¼ U e e inc Q FL; F
1
ð37Þ
e inc is the stochastic component associated to the SWFE where U F spectral problem. Expanding (37) allows the mean to be ex Eq. 1 pressed simply as Q ex ¼ Uinc FL and the standard deviation as F
rQ ex ¼ Uinc F
ex rFL rinc : UF Q
ð38Þ
For the sake of clarity this standard deviation will be denoted,
rQ ex ¼ aðQ ex Þ; where a is a matrix.
ð39Þ
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Using Eq. (11) and developing it for stochastic media allows the resultant displacement standard deviation to be expressed as follows:
rqL rqR
rUinc q
¼
!
rUrefq
kinc þ Uinc r inc r ref kref þ Uref r ref rUinc q q k k Uq q Uinc q inc Uinc q k
þ
Uref q ref Uref q k
!
Q inc Q ref
rQ inc rQ ref ;
where:
l inc
0 l ref
0
!
rk ¼
and
rlinc
0
0
rlref
!
rqL rqR
ð41Þ
:
In the case of an infinite structure Q ref ¼ 0; and rQ inc ¼ rQ ex and then:
rQ ref ¼ 0; Q inc ¼ Q ex
¼ AðQ ex Þ þ B rQ ex ;
ð42Þ
where:
!
rUinc q
A¼
! ð44Þ
:
q
T¼
x
4d
rT ¼
ðQ ex ÞH BH MBðQ ex Þ;
x2 4d
ex H
ð45Þ
H
H
ðQ Þ ½B MðA þ BaÞ þ B
H
ex
rM B þ ðA þ BaÞ MBðQ Þ: ð46Þ
The potential energy density readily expressed as:
1 ðQ ex ÞH BH KBðQ ex Þ; 4d 1 ¼ ðQ ex ÞH ½BH KðA þ BaÞ þ BH rK B þ ðA þ BaÞH KBðQ ex Þ: 4d
U¼
rU
e ex ÞH P e ex Þ: e ¼ ðQ e UðQ P
ð48Þ
ð49Þ
After expanding this relationship, the mean and standard deviation of P are expressed as follows: ex H
ex
U
P ¼ ðQ Þ P ðQ Þ; ex H
H
ð50Þ
rP ¼ ðQ Þ ½a P þ P a þ r U
U
U ex P ðQ Þ;
ð51Þ
Uq UF
!H Jn
Uq
!
UF
ð52Þ
for the zero order and
2
rPU
let us consider a two node rod element. The structure properties are the following: E is the Young modulus, S is the cross section area, q is the mass density. d is the length of the considered element. The mass and stiffness matrices are:
ES d
1
1
1
1
!H !3 rUq rUq H Uq 5 jx 4 Uq : þ ¼ Jn Jn 4 rUF rUF UF UF
ð53Þ
Finally, the standard deviation of the energy velocity can be expressed as:
;
M¼
qSd 2 1 6
1 2
ð55Þ
:
The dynamic stiffness matrix, D = K x2M becomes then:
0 ðk dÞ2 ES @ 1 L3 D¼ d sym
1 ðkL6dÞ 1 ðkL3dÞ
2
2
1 A;
ð56Þ
qffiffiffi where, kL ¼ qE x is the ‘‘exact’’ longitudinal wave number and the target of the spectral WFE problem. The transfer matrix can be written as:
S¼
0
1 1 þ ðkL6dÞ
2
@
1 ðkL3dÞ
2
d ES
1
A: 4 2 L dÞ 1 ðkL3dÞ ES ðkL dÞ2 ðk12 d
ð57Þ
Terms involved in the state space spectral problem Eq. (10) are thus simply made expressed. The analytical solution of the problem in terms of eigenvalues (propagation constants) and eigenvectors (wave modes) can thus be obtained as follows:
l 1;2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 ðk dÞ ðkL dÞ2 5 41 L : ¼ jðkL dÞ 1 2 3 12 1 þ ðkL dÞ 2
1
ð58Þ
6
For weak kLd, Eq. (58) can be rewritten as:
l 1;2 ¼ 1 jkL d
where:
jx 4
In this section the validation of the SWFE finding is the main issue. The formulation is general and can be applied to complex geometries. For the sake of clarity, validations proposed in this paper allow analytical treatments. So, the choice of one dimensional finite elements is considered. Precisely, the two node rod linear element is first investigated. This element allows treatments of compressional and longitudinal wave. A second example is also detailed. It is the four node beam element representative of simple flexural wave. In both cases, analytical expressions of mass and stiffness matrices facilitate the validation issues. As a generalization, a multi-modal example is presented.
ð47Þ
The energy flow matrix can also be expressed through the notations provided. Indeed, the stochastic energy flow matrix is expressed as:
PU ¼
Eq. (45) up to (54) allow the statistics of energy densities, energy flow, energy and group velocities to be estimated. These information can be of important use in many problems involving wave dynamics.
K¼
Finally, the kinetic energy density for unbounded media under a multimodal propagation behaviour can be closely given by: 2
ð54Þ
:
4.1. Longitudinal waves case study
and
Uinc q Uinc kinc
Tii þ Uii
ð43Þ
kinc þ Uinc r inc rUinc q k q
B¼
rPii cgi ðrTii þ rUii Þ
4. Validations of the SWFE
ð40Þ
k ¼
rcgi ¼
ðkL dÞ2 5ðkL dÞ3 j þ 2 24
ð59Þ
It should be noted, that the wave number associated to this propagation constant is approximately equal to the target value kL. Indeed, it can be shown that the estimated wave number 1;2 is: corresponding to the propagation constant l
" # 7ðkL dÞ2 jðkL dÞ3 5ðkL dÞ4 k1;2 ¼ kL 1 : 24 3 48
ð60Þ
Moreover, the right eigenvectors associated with the eigenvalues provided in expression (58) can be written as:
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U1;2 ¼
q ¼ F
"
#
jESkL
q1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 L dÞ 1 ðk12
ð61Þ
Hence, the deterministic problem is completely solved through expressions (58) and (61). Such values are as shown before the mean quantities. Let us consider now a random medium. To simplify the analytical treatment let us assume that the Young modulus is a random parameter with rE as a dispersion. From the expression of the dynamic operator, it can be readily established that rD is:
rD ¼
rE S
d
1
1
1
1
4.2. Flexural waves case study
:
ð62Þ
From the expressions (31) and (32) the standard deviation of the eigenvalues can be analytically and simply obtained as follows:
rl1 ¼
dispersion equation computed using the non-intrusive Monte Carlo scheme. It should be noted that only positive values of the Young’s modulus were considered. Figs. 2–4, offer such comparisons. Good concordance is shown between Monte Carlo and SWFE in all considered cases and throughout a wide frequency range. The non-dispersive nature of the propagation can be observed for the standard deviation. In Figs. 2–4 the analytical estimations were also reported for the sake of consistency.
Let us now consider a simple dispersive media. A two node beam element is considered. The mass and stiffness matrices of the beam are given simply as:
0
rE
12
6d
12
4d EI B B 3B d @ sym
6d
2
E
h i 5 25 ðkL dÞ 1 þ jðkL dÞ j 24 ðkL dÞ3 23 ðkL dÞ2 þ 576 ðkL dÞ4 h i 1 5 ðkL dÞ2 þ 16 ðkL dÞ3 þ j 72 ðkL dÞ4 2j þ j 12
K¼
12
6d
1
2 2d C C C; 6d A 2
ð63Þ
0
and the standard deviation of the eigenvector expressed as: 3 ðkL d rE SkL Þ 2j þ 2ðkL dÞ þ j 17 ðkL dÞ2 13 ðkL dÞ3 þ j 32 ðkL dÞ4 12 kU1 k: rU1 ¼ 1 53 ESkL ðkL dÞ3 13 þ j 13 ðkL dÞ 12 ðkL dÞ2 þ j 6912 ðkL dÞ3 96
M¼
156
qSd B B
B 420 @ sym
4d 22d 54 2
4d
13d
1
3d C C C; 156 22d A 2
13d
2
4d
ð64Þ
rl1 ¼ kL rkL d2 þ jdrkL ; where rkL ¼
kL rE : 2 E
ð65Þ
So, the standard deviation of the group velocity is expressed as follows:
1 rE rcg ¼ pffiffiffiffiffiffi 2 Eq
ð66Þ
It is then feasible in this simple case to compare the statistics of wave numbers and propagation constants when Young’s modulus is assumed to be random. In order to check the generality of the proposed SWFE formulation, further comparisons were achieved assuming, respectively: Young’s modulus, mass density and the element length d to be random. Validations of SWFE results in terms of
0.35
Standard deviation of wave number (m−1)
Eq. (63) can be compared analytically to the standard deviation of the exact propagation constant qffiffiffi estimated from the wavenumber basic expression kL ¼ qE x . After some calculations, it can be shown that:
0.2
0.15
0.1
0.05
0
1000
2000
3000
4000
5000
6000
7000
8000
Frequency (Hz) Fig. 3. Standard deviation of wave number (q stochastic), (–): SWFE, (): analytical solution and (⁄): Monte Carlo simulation.
0.7
Standard deviation of wave number (m−1)
Standard deviation of wave number (m−1)
0.25
0
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0.3
0.6
0.5
0.4
0.3
0.2
0.1
0 0
1000
2000
3000
4000
5000
6000
7000
8000
Frequency (Hz) Fig. 2. Standard deviation of wave number (E stochastic), (–): SWFE, (): analytical solution and (⁄): Monte Carlo simulation.
0
1000
2000
3000
4000
5000
6000
7000
8000
Frequency (Hz) Fig. 4. Standard deviation of wave number (d stochastic), (–): SWFE, (): analytical solution and (⁄): Monte Carlo simulation.
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where EI is the bending stiffness, q is the mass density and S is the cross section. The dynamic stiffness matrix D = K x2M becomes:
modulus E is random. Fig. 6 corresponds to the mass density random configuration whilst Fig. 7 describes the case where d is
1 0 22 54 13 12 156 ðkB dÞ4 ðkB dÞ4 ðkB dÞ4 ðkB dÞ4 d 6 420 12 420 d 6 þ 420 420 B C C B 2 2 4 4 4 4 13 3 C B 4 ðk dÞ ðk dÞ 2 þ ðk dÞ d 6 d d B B B C 420 420 420 EI B B C D¼ 3B C; 4 4 156 22 d B sym 12 420 ðkB dÞ d 6 þ 420 ðkB dÞ C C B @ A 2 4 4 d 4 420 ðkB dÞ qffiffiffiffiffiffiffiffi 2 4 where, kB ¼ qSEIx is the bending wavenumber. The transfer matrix can thus be written as [24]:
1 4
302400 þ 720ðkB dÞ þ ðkB dÞ
8
SLL SRL
SLR ; SRR
ð69Þ
SLL, SLR, SRL and SRR are 2 2 matrices and can be analytically expressed. In [24] an analysis of the associated spectral problem in the deterministic case and comparisons with analytical expected formulas are discussed. Indeed, the approximate solution for the characteristic equation are [24]:
l1;2
1 j 1 ðkB dÞ4 ¼ 1 jðkB dÞ ðkB dÞ2 ðkB dÞ3 þ 2 6 24 23 1 ðkB dÞ5 ðkB dÞ6 . . . ; 2880 960
0.4
Standard deviation of wave number (m−1)
S¼
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
1 6
1 ðkB dÞ4 24
where l1,2 are related to the propagating waves and l3,4 are the near field waves. The eigenvectors associated with l1,2 are analytical written as:
1
1
0
1 w1;2 4 Bh C B jk ð1 ðk dÞ =2880 ðkB dÞ6 =10800...Þ C C B B B B 1;2 C B C; ¼B C¼B 3 4 6 C A @ f1;2 F @ jEIkB ð1 ðkB dÞ =960 ðkB dÞ =302400...Þ A 2 4 6 m1;2 EIkB ð1 ðkB dÞ =1440 ðkB dÞ =18900...Þ
q
ð70Þ where w1,2, h1,2, f1,2, m1,2 are the translational displacement, the rotational displacement, the shear force and the flexural moment. In the case of random media, if the Young modulus is considered to be a random parameter as an example, the standard deviation of the dynamic stiffness matrix is expressed as the following:
0
rD ¼
rE I B B
12
B d @ sym 3
6d
12
2
6d
4d
12
6d
2000
3000
4000
5000
6000
7000
8000
0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
1
2 2d C C C: 6d A
1000
Fig. 5. Standard deviation of wave number (E stochastic), (–): SWFE, (): analytical solution and (⁄): Monte Carlo simulation.
23 1 ðkB dÞ5 þ ðkB dÞ6 . . . ; 2880 960
0
0
Frequency (Hz)
Standard deviation of wave number (m−1)
1 2
l3;4 ¼ 1 ðkB dÞ þ ðkB dÞ2 ðkB dÞ3 þ
ð68Þ
0
1000
2000
3000
4000
5000
6000
7000
8000
Frequency (Hz)
ð71Þ
Fig. 6. Standard deviation of wave number (q stochastic), (–): SWFE, (): analytical solution and (⁄): Monte Carlo simulation.
2
4d
The standard deviation of the eigenvalues can be analytically obtained through Eq. (31) as:
rl1 ¼
rE E
h i 5 7 ðkB dÞ 1 þ jðkB dÞ 12 ðkB dÞ2 j 30 ðkB dÞ3 þ 180 ðkB dÞ4 h i : 5 j B dÞ ðkB dÞ4 ðk720 4j þ 60j ðkB dÞ2 700
assumed to be a random quantity. Figs. 8 and 9 give the statistics of energy velocities estimation when compared to Monte Carlo computations. The analytical standard deviation of the wave number is written,
ð72Þ
Figs. 5–7 compare the wave numbers analytical standard deviation with Monte Carlo simulations and WFE predictions in different random situations. Precisely, Fig. 5 gives the comparison when Young ’s
rkB ¼
k B rE : 4 E
ð73Þ
So, the standard deviation of the group velocity is expressed as follows:
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Standard deviation of wave number (m−1)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
1000
2000
3000
4000
5000
6000
7000
8000
Frequency (Hz) Fig. 7. Standard deviation of wave number (d stochastic), (–): SWFE, (): analytical solution and (⁄): Monte Carlo simulation.
Standard deviation of wavenumber (1/m)
Standard deviation of group velocity (m/s)
3
2.5
2
1.5
1
0.5
0
1000
2000
3000
4000
5000
6000
7000
0.2
0.15
0.1
0.05
0
2000
3
2.5
2
1.5
1
0.5
6000
8000
10000
Fig. 11. Standard deviation of wavenumber (E stochastic), (–): SWFE, (): Monte Carlo simulation.
Standard deviation of wavenumber (1/m)
Fig. 8. Standard deviation of group velocity (E stochastic), (–): SWFE, (): analytical solution and (⁄): Monte Carlo simulation.
4000
Frequency (Hz)
8000
Frequence(Hz)
Standard deviation of group velocity (m/s)
0.25
0 0
0.25
0.2
0.15
0.1
0.05
0
0
2000
4000
6000
8000
10000
Frequency (Hz) 0 0
1000
2000
3000
4000
5000
6000
7000
8000
Frequence(Hz) Fig. 9. Standard deviation of group velocity (q stochastic), (–): SWFE, (): analytical solution and (⁄): Monte Carlo simulation.
rc g ¼
Fig. 10. Finite element discretization of one subsystem.
1 xI1=4 rE : 2 ðqSx2 Þ1=4 E3=4
ð74Þ
Good concordance is shown between Monte Carlo and SWFE in all considered cases and over a wide frequency range. The dispersive
Fig. 12. Standard deviation of wavenumber (q stochastic), (–): SWFE, (): Monte Carlo simulation.
nature of the propagation can be observed for the standard deviation. 4.3. A complex structure case study In this section, a validation of the proposed model for a complex structure is offered. Let’s consider a multi-modal wave guide with these properties: Young’s modulus E = 2 1011 Pa mass density
M.N. Ichchou et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2805–2813
2813
q = 7800 kg m3, length of the considered element d = 0.005 m, Poisson’s ratio m = 0.3, loss factor g = 0.01, and cross section
References
area = 0.05 m 0.03 m. Fig. 10 represents the discretized studied subsystem. The extraction of the mass and stiffness matrices is performed using a commercial finite element software. 3D Solid elements were employed in the numerical scheme. Figs. 11 and 12 represent the standard deviation of extracted wave numbers in the multi-modal structure with 2% of standard deviation (E stochastic; q stochastic). This representation of statistics of dispersion curves prove the efficiency of the proposed model. It is remarkable that the SWFE and the MC simulations (2000samples) results are very close in all frequency band.
[1] B.R. Mace, D. Duhamel, M.J. Brennan, L. Hinke, Finite element prediction of wave motion in structural waveguides, J. Acoust. Soc. Am. 117 (2005) 2835– 2843. [2] M.N. Ichchou, J. Berthaut, M. Collet, Multi-mode wave propagation in ribbed plates: Part i, wavenumber-space characteristics; Part ii, predictions and comparisons, Int. J. Solids Struct. 45 (5) (2008) 1179–1216. [3] M.N. Ichchou, J. Berthaut, M. Collet, Multi-mode wave propagation in ribbed plates. Part ii: Predictions and comparisons, Int. J. Solids Struct. 45 (5) (2008) 1196–1216. [4] D. Duhamel, B.R. Mace, M.J. Brennan, Finite element analysis of the vibrations of waveguides and periodic structures, J. Sound Vib. 294 (2006) 205–220. [5] L. Houillon, M.N. Ichchou, L. Jezequel, Dispersion curves of fluid filled elastic pipes by standard fe models and eigenpath analysis, J. Sound Vib. 281 (2005) 483–507. [6] J.M. Mencik, M.N. Ichchou, Multi-mode propagation and diffusion in structures through finite elements, Eur. J. Mech. A – Solids 24 (5) (2005) 877–898. [7] M. Maess, Methods for efficient acoustic-structure simulation of piping systems, PhD Thesis, University of Stuttgart, 2006. [8] F. Chen, P.D. Wilcox, The effect of load on guided wave propagation, Ultrasonics 47 (2007) 111–122. [9] D. Duhamel, Finite element computation of green’s functions, Engrg. Anal. Bound. Elem. 31 (2007) 919–930. [10] M.N. Ichchou, S. Akrout, J.M. Mencik, Guided waves group and energy velocities via finite elements, J. Sound Vib. 305 (2007) 931–944. [11] A. Bocquillet, M.N. Ichchou, L. Jezequel, Energetics of axisymmetric fluid-filled pipes up to high frequencies, J. Fluids Struct. 17 (2003) 491–510. [12] J.M. Mencik, M.N. Ichchou, Wave finite elements in guided elastodynamics with internal fluid, Int. J. Solids Struct. 44 (2007) 2148–2167. [13] Matthias Maess, Nils Wagner, Lothar Gaul, Dispersion curves of fluid filled elastic pipes by standard fe models and eigenpath analysis, J. Sound Vib. 296 (2006) 264–276. [14] R. Ghanem, P. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer, New York, NY, 1991. [15] G.I. Schudller, H.J. Pradlwafer, Uncertain linear systems in dynamics:retrospective and recent developments by stochastic approaches, Engrg. Struct. 31 (2009) 2507–2517. [16] C. Soize, A nonparametric model of random uncertainties for reduced matrix models in structural dynamics, Probab. Engrg. Mech. 15 (3) (2000) 277–294. [17] C. Soize, A comprehensive overview of a non-parametric probabilistic approach of model uncertainties for predictive models in structural dynamics, J. Sound Vib. 288 (3) (2005) 623–652. [18] C. Soize, E. Capiez-Lernout, J.-F. Durand, C. Fernandez, L. Gagliardini, Probabilistic model identification of uncertainties in computational models for dynamical systems and experimental validation, Comput. Methods Appl. Mech. Engrg. 198 (1) (2008) 150–163. [19] C. Soize, Probabilistic model identification of uncertainties in computational models for dynamical systems and experimental validation, Comput. Methods Appl. Mech. Engrg. 194 (12–16) (2005) 1333–1366. [20] G.I. Schudller, H.J. Pradlwafer, Computational methods in optimization considering uncertainties an overview, Comput. Methods Appl. Mech. Engrg. 198 (1) (2008) 2–13. [21] N.S. Ferguson, E. Hills, B.R. Mace, Acoustic response variability in automotive vehicles, J. Sound Vib. 321 (1–2) (2009) 286–304. [22] W.X. Zhong, F.W. Williams, On the direct solution of wave propagation for repetitive structures, J. Sound Vib. 181 (1995) 485–501. [23] M.N. Ichchou, J.-M. Mencik, W.J. Zhou, Wave finite elements for low and midfrequency description of coupled structures with damage, Comput. Methods Appl. Mech. Engrg. 198 (15–16) (2009) 1311–1326. [24] Y. Waki, On the application of finite element analysis to wave motion in onedimensional waveguides, PhD Thesis, Thesis of Doctor of Philosophy, University of Southampton, 2007. [25] M.N. Ichchou, B. Jemai, L. Bellon, Active rubber mounts by means of piezoelectric actuators, experimental work, Smart Mater. Struct. 10 (5) (2001) 1095–1099. [26] B. Jemai, M. Ichchou, L. Jezequel, An assembled plate active control damping set-up: optimization and control, J. Sound Vib. 225 (2) (1999) 327–343.
5. Conclusions and current developments In this paper, the issue of wave propagation parameters estimations in random guided viscoelastic media was dealt with. Guided wave properties are an important physical notion for use in many engineering problems, the paper results are expected to be helpful in non-deterministic problems. To that end, a formulation named stochastic wave finite element (SWFE) was offered. This formulation allows wave characteristics to be defined by means of a stochastic finite element model. The case of spatially homogeneous random properties is dealt with through a parametric probabilistic approach. The formulation is general and can be applied to various structural shapes. The formulation is also easy to use as it considers a finite element model of a sample structural element. The statistics of wave propagation constants are thus described by a spectral problem. The statistics of wave modes are also obtained from the spectral problem. Two different eigenvalue problems were formulated and discussed. Ultimately, numerical and analytical comparisons were given in the case of a simple non-dispersive media and a dispersive one. Such analytical developments and numerical experiments showed the effectiveness of the proposed approach to predict the statistics of guided wave propagation characteristics (means and standard deviations) when compared to non-intrusive Monte Carlo calculations. The SWFE offers some interesting research perspectives. Among future work, the evaluation of the statistics of waves diffusions and scattering at notch and defects is an important task. Theses statistics are expected to be of use in the sensitivity analysis of guided waves techniques for damage detection. Among the investigations in progress, the use of SWFE for energy issues in structural complex wave guide is considered. The mid-high frequency behaviour is the main target in this case. Further investigations are under progress in order to use such numerical methods in the context of smart materials and structures [25,26]. Acknowledgments Part of this work is funded under the ‘‘Mid-Frequency’’ Marie-Curie contract. Support from the EC is greatly acknowledged.
Contents lists available at ScienceDirect
Comput. Methods Appl. Mech. Engrg. journal homepage: www.elsevier.com/locate/cma
Stochastic wave finite element for random periodic media through first-order perturbation M.N. Ichchou a,⇑, F. Bouchoucha a,b, M.A. Ben Souf a,b, O. Dessombz a, M. Haddar b a b
LTDS, Ecole Centrale de Lyon, 36 Avenue Guy de Collongue, F-69134 Ecully Cedex, France Unité de Mécanique, Modélisation et Productique, Ecole nationale d’Ingénieurs de Sfax, National School of Engineers of Sfax, BP. W3038 Sfax, Tunisia
a r t i c l e
i n f o
Article history: Received 20 August 2010 Received in revised form 12 May 2011 Accepted 16 May 2011 Available online 1 June 2011 Keywords: Guided waves Random media Stochastic finite element Parametric probabilistic model Dispersion
a b s t r a c t Wave propagation features in random guided elastic media are considered in this paper. A formulation named stochastic wave finite element (SWFE) is offered. This formulation allows wave characteristics definition by means of a stochastic finite element model. The case of spatially homogeneous random properties is dealt with, through a parametric probabilistic technique. Numerical experiments show the effectiveness of the proposed approach to predict the statistics of guided wave propagation characteristics (means and standard deviations) when compared to Monte Carlo calculations. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction To deal with guided wave propagation in elastic media, semianalytical finite element (SAFE) and wave finite element (WFE) methods have been investigated by a number of researchers. Such numerical techniques are extensively employed in many wave dynamic issues. Structural health monitoring (SHM) and nondestructive testing (NDT) are among the fields of application of these numerical tools. Both SAFE and WFE, for instance, can be used for wave propagation predictions and wave scattering estimations. An assessment of damage signature effects can thus be reached. The SAFE technique considers wave propagation by means of specific shape functions. It necessitates the development of a relatively new FE code for specific elements. The WFE technique considers the wave propagation problems of periodic structures. The WFE method regards the homogeneous waveguide structure as a periodic system assembled by identical substructures. The WFE method is easy to apply due to its direct connection with the standard FE method. This method has been used for the structural vibration analysis [1–5], the wave propagation in elastic waveguides [6–8], and the calculations of Green’s functions for homogeneous acoustic and elastic media [9]. Houillon et al. [5] studied the dynamic problems of homogeneous thin-walled structures using this method, which has been shown to be successful when applied to structures with uniform cross-section. Duhamel ⇑ Corresponding author. Tel.: +33 04 72 18 62 30; fax: +33 04 72 18 91 44. E-mail address: [email protected] (M.N. Ichchou). 0045-7825/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2011.05.004
et al. [4] used this method to investigate the vibrations of uniform waveguide structures, where it has been proved to be accurate with relatively low computational cost in comparison with the standard FE method. Mencik and Ichchou [6] proposed a hybrid approach to study the diffusion of multiple wave modes based on this method. Ichchou et al. [10] investigated the numerical sensitivity of this method. The energy propagation features in rib-stiffened panels over a wide frequency range were studied by using this method in [3], where the comparisons of numerical and experimental results are provided. Chen and Wilcox [8] used this method to investigate the effect of load on guided wave propagating properties in rails. The method was also implemented for wave propagation and dynamic problems in the homogeneous structures with internal fluid [11–13]. In the open literature, however, most of what can be found in the numerical issues of wave propagation simulations is mainly limited to deterministic media. Viscoelastic media characteristics are in this case fixed to nominal values. To the authors’ knowledge, the extension of such numerical techniques to non-deterministic problems is not yet considered. From the applications and the academic point of view such considerations are of real and pertinent added value. The paper novelty is then to consider numerical guided wave techniques characterization in spatially homogeneous random media. This work’s main objectives are twofold: to formulate and validate a stochastic wave finite element technique for structural members. To deal with uncertainties in structural dynamics extensive research has been performed over the last decade. This issue is extensively studied in the literature [14,15] and a significant num-
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ber of studies have been performed in this context by many authors among them Soize et al. [16–20]. The uncertainties can be linked to the geometric properties, material characteristics and boundary conditions among other possibilities. This behaviour is taken into account mainly according to two approaches. The non-parametric approach [16,17] introduces the uncertainties directly into the global matrices of the model or into the reduced matrices of the nominal condensed model. The probabilistic description of resulting matrices is governed by the maximum entropy principle. Parametric approaches considers the uncertain parameters (geometrical, material properties etc.) as random quantities. Specific discretization approaches such as the stochastic finite element method (SFEM) which combines finite elements and probabilistic ways of thinking can be employed [14]. Due to the efficient representation of random fields, the SFEM has been successfully applied to various kinds of stochastic problems. Improvements and further consideration of parametric strategies for treating uncertainties are still under investigation. The current study extends the WFE techniques to stochastic media. In this context the concept of a random field cannot be ignored. The variability of wave characteristics that arises from material and geometric variability is desired in many engineering problems. For instance, such variability is expected to play a fundamental role in the medium and high frequency range [21] in the structural–acoustic area. The uncertainties considered in what follows are mainly linked to the geometric properties and to the material characteristics of the propagation media. A parametric approach for uncertainties treatments is considered and combined to the WFE technique. A similar process, such as the stochastic finite element method, is used to develop the stochastic wave finite element (SWFE) approach based on the probabilistic tools. The idea is to consider the random fields as a supplementary dimension of the problem through the spatial discretization using the finite elements process. The dimension associated to the random fields is thus considered using two ways: perturbation and spectral methods. It should be noted that the strategy can be easily adapted to the well-known SAFE techniques. The paper scope is limited to the extension of the WFE to the stochastic case. The paper contains four sections. In Section 2, the formulation of the stochastic wave finite element approach is presented. The deterministic WFE is first very briefly reminded, for the sake of clarity. Then, the stochastic treatment is detailed. Section 3 provides some expressions deduced from the SWFE formulation for wave properties analysis. Section 4 gives mainly numerical experiments. The formulation is general, but the validations were limited to basic longitudinal and flexural wave propagation problems. A first comparison was established analytically in these cases. Ultimately, a comparison of the SWFE prediction with Monte Carlo simulations confirms
the effectiveness of the proposed techniques in the studied cases. A conclusion together with a description of the work in progress is ultimately given. 2. Formulation of the stochastic wave finite element approach In this section, the formulation of the SWFE approach is given. For the sake of clarity, the deterministic case is first concisely presented [2,10]. The introduction of uncertainties and specific developments leads to two different spectral problems. The first stochastic formulation involves a spectral quadratic nature. The second demonstration uses a state space-like process and leads to a first order formulated spectral problem. Properties of each one are also described. 2.1. WFE in the deterministic case The dynamics of straight elastic and dissipative structures are studied. A sample straight structure is illustrated in Fig. 1: in the present framework, the system is assimilated to be a set of identical subsystems, connected along the principal direction, say axis x, and whose left and right cross-sections (x-axis description) are denoted as L and R, respectively. The length of each subsystem, along axis x, is denoted as d. The formulation is based on the finite element model of a typical subsystem, as illustrated in Fig. 1, and whose kinematic variables, displacements and forces, are written as q and forces F, respectively. Mesh compatibility at coupling interfaces between subsystems is assumed, implying that the left and right cross-sections of the given subsystem contains the same number of degrees of freedom, say n. The dynamic equilibrium equation of this subsystem, at frequency x/2p, can be stated as follows [22]
DLL DRL
DLR DRR
qL qR
¼
FL ; FR
ð1Þ
where (n n) matrix Dij = Kij x2Mij ({i, j} 2 {L, R}) stands for the ij component of the dynamic stiffness operator condensed on the left and right cross-sections, namely D [22]. Here, K and M stand for the stiffness and mass matrices, respectively. Dissipation can be considered through standard FEM models. According to Bloch’s theorem, the dynamics of the global waveguide can be expanded on specific wave solutions of the form
qR ¼ lqL
ð2Þ
and
FR ¼ lFL ;
Fig. 1. Typical thin walled structure.
ð3Þ
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where l denotes the propagation coefficient. Expressions (2) and (3) lead to an eigenvalue problem. Indeed, inserting Eqs. (2) and (3) into Eq. (1) leads to the following spectral problem [5]:
DRL þ li ðDLL þ DRR Þ þ l2i DLR ðUq Þi ¼ 0;
ð4Þ
where {(li, (Uq)i)}i=1,. . .,2n stands for the wave modes of the global system. It’remarkable that DTRL ¼ DLR and (DLL + DRR)T = (DLL + DRR). Using these properties we can prove that ðUq ÞTi is the left eigenvectors associated to the left eigenvalue l1 [6]. i A modified and well conditioned format of the spectral problem can also be obtained. For this purpose, the use of a state vector representation is an interesting alternative to the spectral analysis which must be performed in the context of the numerical dispersion curve extraction (see Refs. [22,11,10] for detailed discussions). Indeed, given the following matrices:
" S¼
#
D1 LR DLL
D1 LR
DRL DRR D1 LR DLL
DRR D1 LR
ð5Þ
and matrix Jn, defined by:
Jn ¼
0
In
In
0
ð6Þ
;
to be spatially homogeneous. This guarantees the assumed periodicity will be respected in the non-deterministic case as in the deterministic situation. The considered random sample is assumed to be modelled through a SFEM-like approach in order to describe e will designate random the element equilibrium. In what follows D e is the random dynamic operator of the quantities (for instance D considered sample structure). The stochastic equation of motion for the kth sample is then the following:
eq ~¼e D F:
This expression extends Eq. (1) to the non-deterministic case. In what follows the first order development of stochastic variables is adopted, such that:
þ rq Þ ¼ ðF þ rF Þ; ðD þ rD Þðq
e LL D e RL D
ð7Þ
meaning that S is symplectic [10]. Ultimately, a spectral problem can be established as:
J n Ui ¼ li ST J n Ui ;
ð8Þ
ð9Þ
and thus, considering that matrix S is symplectic (ST ¼ Jn SJ 1 n [22]),
SUi ¼ li Ui :
ðUq ÞTi ðUF ÞTi
ð10Þ
T
Here Ui ¼ stands for the ith eigenvector of operator S, which is decomposed into (n 1) displacements q and forces F wave components. The frequency response of the global system can be expressed by expanding the kinematic variables of the considered subsystem on the basis of eigenvectors:
qL
¼
FL
Uq
UF
QL
and
qR FR
¼
Uq UF
Q R;
ð11Þ
where Uq and UF stand for the matrices of eigenvector {(Uq)i}i and {(UF)i}i, respectively, and where QL and QR stand for the (2n 1) generalized coordinates evaluated for the left and right boundaries of the subsystem, respectively. It has been shown in Refs. [23,6] that generalized coordinates QL and QR can be related in this way
QR ¼
l
0
0
l1
Q L;
ð12Þ
where l stands for the matrix of eigenvalues {li}i=1,. . .,n. Note that the description provided by Eq. (12) is based on the classification of the eigenvectors into incident and reflected waves (see Ref. [6]). Eqs. (12) and (11) will be used for the estimation of energy velocities. 2.2. Stochastic wave finite element method: quadratic spectral formulation In this subsection, a stochastic medium is considered. As previously done, the straight studied structure is partitioned in a number of sample of length d. The uncertainties are assumed to be mostly on the material properties. Such uncertainties are assumed
e LR D e RR D
!
~ LðkÞ q ~ RðkÞ q
! ¼
! ðkÞ e FL : e ðkÞ F
ð15Þ
R
Following the same process as in the previous subsection, one can readily obtain a quadratic spectral problem formulated in the stochastic area, which is (16):
which leads to T J 1 n S J n Ui ¼ l i Ui ;
ð14Þ
where is Gaussian variable centred and reduced. This equation is a simplified application for the stochastic finite element method used by many contributors, see Ghanem and Spanos [14] for instance. In and F represent the mean quantities of the dythis expression, D; q namic operator, the displacement vector and the load; rD, rq and rF their standard deviations. As done in the deterministic case, the stochastic problem stated in Eq. (13) can be partitioned as follows:
it can be readily shown that:
ST J n S ¼ J n
ð13Þ
e RL þ l e LR ð U e LL þ D e RR Þ þ l e q Þ ¼ 0: ~ iðD ~ 2i D D i
ð16Þ
~ i and the corresponding stochastic The stochastic eigenvalue l e i are solutions of this established quadratic equation. eigenvector U The zero order and the first order development of this relationship can be readily performed and leads to:
i ðDLL þ DRR Þ þ l 2i DLR ðUq Þi ¼ 0; DRL þ l
ð17Þ
which is the ‘zero order’ expression. This is exactly the deterministic spectral problem already obtained in the previous section for deterministic cases. The first order relation is also derived as follows:
i ðDLL þ DRR Þ þ l 2i DLR rðUq Þi þ ½rDRL þ l i ðrDRR þ rDLL Þ DRL þ l 2 þrli ðDRR þ DLL Þ þ li rDLR þ 2li rli DLR ðUq Þi ¼ 0: ð18Þ In order to express the standard deviation of li ðrli Þ, let us multiply Eq. (18) by ðUq ÞHi , an analytical development leads then to:
rli ¼
ðUq ÞHi
rDRL þ l i ðrDRR þ rDLL Þ þ l 2i rDLR ðUq Þi ; DLR ðUq Þi ðUq ÞHi ½DRR þ DLL þ 2l
ð19Þ
where H is the hermitian transpose. Similarly, the standard deviation of the eigenvectors rUq can be written as:
þ
rðUq Þi ¼ DRL þ l i ðDRR þ DLL Þ þ l 2i DLR ½rDRL þ l i ðrDRR þ rDLL Þ i rli DLR ðUq Þi ; 2i rDLR þ 2l þrli DRR þ DLL þ l ð20Þ where + is the pseudo inverse. Eqs. (19) and (20) constitute the close expression of the means and standard deviations of wave properties in the general elastodynamics case. Then from the knowledge of means and deviations of the dynamic operator, which are associated, to the considered uncertainties in the studied media, one can easily express the statistics of the guided waves characteristics. The ’relatively’ strong coupling between the statistics of the propagation constants and the corresponding eigenvectors is to be noted.
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2.3. Stochastic wave finite element method: state space formulation As explained in Section 2.2, from a numerical point of view, the state space formulation is often preferred. This is especially true in the Hamiltonien [11] case where the properties of the expressed operators allow important properties of the wave problems to be ~ L ¼ ððq ~ L ÞT ð e exhibited. So, let us define u F L ÞT ÞT and ~ R ¼ ððq ~ R ÞT ð e u F R ÞT ÞT as the representation of kinematic variables through stochastic state vectors. The dynamic of the representative kth cell is then expressed as follows:
~ L ¼ ½u ~ R : ðS þ rS Þ½u
SRL þ rSRL SRR þ rSRR
!
L þ rqL q
! ¼
FL rFL
R þ rqR q FR þ rFR
! :
ð22Þ
The zero order development of the expression leads to the definition of S as follows:
!
D1 LR DLL
D1 LR
DRL DRR D1 LR DLL
DRR D1 LR
ð23Þ
;
which is simply the S matrix employed in the deterministic case. Equivalently, the first order expression leads to:
rS ¼
0
DRR
1
!1
rSLL rSLR rSRL rSRR
1 D1 LR DLL
: D1 LR 0
ð24Þ
This equation will be used to obtain the statistics of estimated wave numbers. The stochastic eigenvalue problem can be written:
e ei ¼ l e i; ~iU SU ~ i I2n j ¼ 0; je Sl
ð25Þ ð26Þ
e i Þ are the solutions of the eigenvalue problem. If the ~ i; U where ðl stochastic eigensolutions are expressed as:
l~ ¼ ðl þ rl Þ;
ð32Þ
for the standard deviation of wave modes. Eqs. (31) and (32) provide the statistics of guided wave in the considered random viscoelastic media. The equivalence between Eqs. (19) and (31) associated to the quadratic spectral problem and Eqs. (20) and (32) associated to the state space problem is not obvious in the general case. It will be considered in the numerical validations offered in Section 4 of this paper. 3. Statistics of energies and energy and group velocities
SLL þ rSLL SLR þ rSLR
DLR
rUi ¼ ½S l i I2n þ ½rS rli I2n Ui
ð21Þ
Or equivalently:
S¼
for the standard deviation of propagation constants and
e ¼ ðU þ rU Þ; U
then the zero order (Deterministic case) terms are given by:
i I2n ÞUi ¼ 0; ðS l
ð27Þ
From the definition of standard deviation of propagation constants, it can be of interest, in many applicative engineering cases, to consider wave numbers, energies and energy velocities statistics. Indeed, from the knowledge of the zero order and first order terms and considering the wave number to be written as ~¼k þ r , one can easily express the statistics of k ~ as follows k k (j2 = 1):
¼ j Logðl Þ; k d
ð33Þ
which is the mean of the wave number and developing the first order leads to:
rk ¼
j rl : d l
ð34Þ
Given the statistics of computed wave numbers, it can be easily shown that the general expression for standard deviation of group velocity is:
@r rcg ¼ cg k ; @k
ð35Þ
where cg is the deterministic group velocity. Similarly, definition of the energy or group velocity in the multimodal case provided in Ref. [10] in the deterministic case, can be extended thanks to the SWFE formulation. Indeed, the analytical expression of the energy velocity [10] associated with each propagating branch i can be written in the stochastic case as:
ei P
whilst the first order contribution, couples the standard deviation of propagation constants and associated eigenvectors, as follows:
~cgi ðxÞ ¼
i I2n ÞrUi þ ðrS rli I2n ÞUi ¼ 0: ðS l
e i is the stochastic energy flow contribution of the ith wave where P e i and U e i stand for the stochastic kinetic and potential mode, and T energy densities, respectively. Energies of the stochastic media can be expressed thanks to the stochastic wave modes basis given in Section 2.1. Indeed, given an excitation described by the force vector e F L , the wavemode amplitudes associated to this external input in the case of an infinite structure can be written as:
ð28Þ
e T J n is a left eigenvector of e ~ i. As U S associated to the eigenvalue 1=l i
e T Jn e e T Jn : ~i U S ¼ 1=l U i i
ð29Þ
The expansion and the polynomial chaos projection of Eq. (29) gives the first order term as follows:
T 2 1 ðrUi Þ J n S l i I 2n þ ðUi Þ J n rS þ rli li I2n ¼ 0: T
ð30Þ
Eqs. (28) and (30) can then be readily solved in order to calculate the standard deviation of eigenvalues rli and the eigenvectors rUi . After some analytical treatments, statistics of wave characteristics can be expressed as:
h
T i rli ¼ ðUi ÞT rTS ðST l i I2n Þ1 Jn S ðl i Þ1 I2n Ui Jn rS 1 1 i I2n i Þ1 I2n þ ðl i Þ2 ðUi ÞT J n J n S ðl ðUi ÞT ST l ð31Þ
ei þ U ei T
ð36Þ
;
1 e ex ¼ U e e inc Q FL; F
1
ð37Þ
e inc is the stochastic component associated to the SWFE where U F spectral problem. Expanding (37) allows the mean to be ex Eq. 1 pressed simply as Q ex ¼ Uinc FL and the standard deviation as F
rQ ex ¼ Uinc F
ex rFL rinc : UF Q
ð38Þ
For the sake of clarity this standard deviation will be denoted,
rQ ex ¼ aðQ ex Þ; where a is a matrix.
ð39Þ
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Using Eq. (11) and developing it for stochastic media allows the resultant displacement standard deviation to be expressed as follows:
rqL rqR
rUinc q
¼
!
rUrefq
kinc þ Uinc r inc r ref kref þ Uref r ref rUinc q q k k Uq q Uinc q inc Uinc q k
þ
Uref q ref Uref q k
!
Q inc Q ref
rQ inc rQ ref ;
where:
l inc
0 l ref
0
!
rk ¼
and
rlinc
0
0
rlref
!
rqL rqR
ð41Þ
:
In the case of an infinite structure Q ref ¼ 0; and rQ inc ¼ rQ ex and then:
rQ ref ¼ 0; Q inc ¼ Q ex
¼ AðQ ex Þ þ B rQ ex ;
ð42Þ
where:
!
rUinc q
A¼
! ð44Þ
:
q
T¼
x
4d
rT ¼
ðQ ex ÞH BH MBðQ ex Þ;
x2 4d
ex H
ð45Þ
H
H
ðQ Þ ½B MðA þ BaÞ þ B
H
ex
rM B þ ðA þ BaÞ MBðQ Þ: ð46Þ
The potential energy density readily expressed as:
1 ðQ ex ÞH BH KBðQ ex Þ; 4d 1 ¼ ðQ ex ÞH ½BH KðA þ BaÞ þ BH rK B þ ðA þ BaÞH KBðQ ex Þ: 4d
U¼
rU
e ex ÞH P e ex Þ: e ¼ ðQ e UðQ P
ð48Þ
ð49Þ
After expanding this relationship, the mean and standard deviation of P are expressed as follows: ex H
ex
U
P ¼ ðQ Þ P ðQ Þ; ex H
H
ð50Þ
rP ¼ ðQ Þ ½a P þ P a þ r U
U
U ex P ðQ Þ;
ð51Þ
Uq UF
!H Jn
Uq
!
UF
ð52Þ
for the zero order and
2
rPU
let us consider a two node rod element. The structure properties are the following: E is the Young modulus, S is the cross section area, q is the mass density. d is the length of the considered element. The mass and stiffness matrices are:
ES d
1
1
1
1
!H !3 rUq rUq H Uq 5 jx 4 Uq : þ ¼ Jn Jn 4 rUF rUF UF UF
ð53Þ
Finally, the standard deviation of the energy velocity can be expressed as:
;
M¼
qSd 2 1 6
1 2
ð55Þ
:
The dynamic stiffness matrix, D = K x2M becomes then:
0 ðk dÞ2 ES @ 1 L3 D¼ d sym
1 ðkL6dÞ 1 ðkL3dÞ
2
2
1 A;
ð56Þ
qffiffiffi where, kL ¼ qE x is the ‘‘exact’’ longitudinal wave number and the target of the spectral WFE problem. The transfer matrix can be written as:
S¼
0
1 1 þ ðkL6dÞ
2
@
1 ðkL3dÞ
2
d ES
1
A: 4 2 L dÞ 1 ðkL3dÞ ES ðkL dÞ2 ðk12 d
ð57Þ
Terms involved in the state space spectral problem Eq. (10) are thus simply made expressed. The analytical solution of the problem in terms of eigenvalues (propagation constants) and eigenvectors (wave modes) can thus be obtained as follows:
l 1;2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 ðk dÞ ðkL dÞ2 5 41 L : ¼ jðkL dÞ 1 2 3 12 1 þ ðkL dÞ 2
1
ð58Þ
6
For weak kLd, Eq. (58) can be rewritten as:
l 1;2 ¼ 1 jkL d
where:
jx 4
In this section the validation of the SWFE finding is the main issue. The formulation is general and can be applied to complex geometries. For the sake of clarity, validations proposed in this paper allow analytical treatments. So, the choice of one dimensional finite elements is considered. Precisely, the two node rod linear element is first investigated. This element allows treatments of compressional and longitudinal wave. A second example is also detailed. It is the four node beam element representative of simple flexural wave. In both cases, analytical expressions of mass and stiffness matrices facilitate the validation issues. As a generalization, a multi-modal example is presented.
ð47Þ
The energy flow matrix can also be expressed through the notations provided. Indeed, the stochastic energy flow matrix is expressed as:
PU ¼
Eq. (45) up to (54) allow the statistics of energy densities, energy flow, energy and group velocities to be estimated. These information can be of important use in many problems involving wave dynamics.
K¼
Finally, the kinetic energy density for unbounded media under a multimodal propagation behaviour can be closely given by: 2
ð54Þ
:
4.1. Longitudinal waves case study
and
Uinc q Uinc kinc
Tii þ Uii
ð43Þ
kinc þ Uinc r inc rUinc q k q
B¼
rPii cgi ðrTii þ rUii Þ
4. Validations of the SWFE
ð40Þ
k ¼
rcgi ¼
ðkL dÞ2 5ðkL dÞ3 j þ 2 24
ð59Þ
It should be noted, that the wave number associated to this propagation constant is approximately equal to the target value kL. Indeed, it can be shown that the estimated wave number 1;2 is: corresponding to the propagation constant l
" # 7ðkL dÞ2 jðkL dÞ3 5ðkL dÞ4 k1;2 ¼ kL 1 : 24 3 48
ð60Þ
Moreover, the right eigenvectors associated with the eigenvalues provided in expression (58) can be written as:
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U1;2 ¼
q ¼ F
"
#
jESkL
q1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 L dÞ 1 ðk12
ð61Þ
Hence, the deterministic problem is completely solved through expressions (58) and (61). Such values are as shown before the mean quantities. Let us consider now a random medium. To simplify the analytical treatment let us assume that the Young modulus is a random parameter with rE as a dispersion. From the expression of the dynamic operator, it can be readily established that rD is:
rD ¼
rE S
d
1
1
1
1
4.2. Flexural waves case study
:
ð62Þ
From the expressions (31) and (32) the standard deviation of the eigenvalues can be analytically and simply obtained as follows:
rl1 ¼
dispersion equation computed using the non-intrusive Monte Carlo scheme. It should be noted that only positive values of the Young’s modulus were considered. Figs. 2–4, offer such comparisons. Good concordance is shown between Monte Carlo and SWFE in all considered cases and throughout a wide frequency range. The non-dispersive nature of the propagation can be observed for the standard deviation. In Figs. 2–4 the analytical estimations were also reported for the sake of consistency.
Let us now consider a simple dispersive media. A two node beam element is considered. The mass and stiffness matrices of the beam are given simply as:
0
rE
12
6d
12
4d EI B B 3B d @ sym
6d
2
E
h i 5 25 ðkL dÞ 1 þ jðkL dÞ j 24 ðkL dÞ3 23 ðkL dÞ2 þ 576 ðkL dÞ4 h i 1 5 ðkL dÞ2 þ 16 ðkL dÞ3 þ j 72 ðkL dÞ4 2j þ j 12
K¼
12
6d
1
2 2d C C C; 6d A 2
ð63Þ
0
and the standard deviation of the eigenvector expressed as: 3 ðkL d rE SkL Þ 2j þ 2ðkL dÞ þ j 17 ðkL dÞ2 13 ðkL dÞ3 þ j 32 ðkL dÞ4 12 kU1 k: rU1 ¼ 1 53 ESkL ðkL dÞ3 13 þ j 13 ðkL dÞ 12 ðkL dÞ2 þ j 6912 ðkL dÞ3 96
M¼
156
qSd B B
B 420 @ sym
4d 22d 54 2
4d
13d
1
3d C C C; 156 22d A 2
13d
2
4d
ð64Þ
rl1 ¼ kL rkL d2 þ jdrkL ; where rkL ¼
kL rE : 2 E
ð65Þ
So, the standard deviation of the group velocity is expressed as follows:
1 rE rcg ¼ pffiffiffiffiffiffi 2 Eq
ð66Þ
It is then feasible in this simple case to compare the statistics of wave numbers and propagation constants when Young’s modulus is assumed to be random. In order to check the generality of the proposed SWFE formulation, further comparisons were achieved assuming, respectively: Young’s modulus, mass density and the element length d to be random. Validations of SWFE results in terms of
0.35
Standard deviation of wave number (m−1)
Eq. (63) can be compared analytically to the standard deviation of the exact propagation constant qffiffiffi estimated from the wavenumber basic expression kL ¼ qE x . After some calculations, it can be shown that:
0.2
0.15
0.1
0.05
0
1000
2000
3000
4000
5000
6000
7000
8000
Frequency (Hz) Fig. 3. Standard deviation of wave number (q stochastic), (–): SWFE, (): analytical solution and (⁄): Monte Carlo simulation.
0.7
Standard deviation of wave number (m−1)
Standard deviation of wave number (m−1)
0.25
0
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0.3
0.6
0.5
0.4
0.3
0.2
0.1
0 0
1000
2000
3000
4000
5000
6000
7000
8000
Frequency (Hz) Fig. 2. Standard deviation of wave number (E stochastic), (–): SWFE, (): analytical solution and (⁄): Monte Carlo simulation.
0
1000
2000
3000
4000
5000
6000
7000
8000
Frequency (Hz) Fig. 4. Standard deviation of wave number (d stochastic), (–): SWFE, (): analytical solution and (⁄): Monte Carlo simulation.
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where EI is the bending stiffness, q is the mass density and S is the cross section. The dynamic stiffness matrix D = K x2M becomes:
modulus E is random. Fig. 6 corresponds to the mass density random configuration whilst Fig. 7 describes the case where d is
1 0 22 54 13 12 156 ðkB dÞ4 ðkB dÞ4 ðkB dÞ4 ðkB dÞ4 d 6 420 12 420 d 6 þ 420 420 B C C B 2 2 4 4 4 4 13 3 C B 4 ðk dÞ ðk dÞ 2 þ ðk dÞ d 6 d d B B B C 420 420 420 EI B B C D¼ 3B C; 4 4 156 22 d B sym 12 420 ðkB dÞ d 6 þ 420 ðkB dÞ C C B @ A 2 4 4 d 4 420 ðkB dÞ qffiffiffiffiffiffiffiffi 2 4 where, kB ¼ qSEIx is the bending wavenumber. The transfer matrix can thus be written as [24]:
1 4
302400 þ 720ðkB dÞ þ ðkB dÞ
8
SLL SRL
SLR ; SRR
ð69Þ
SLL, SLR, SRL and SRR are 2 2 matrices and can be analytically expressed. In [24] an analysis of the associated spectral problem in the deterministic case and comparisons with analytical expected formulas are discussed. Indeed, the approximate solution for the characteristic equation are [24]:
l1;2
1 j 1 ðkB dÞ4 ¼ 1 jðkB dÞ ðkB dÞ2 ðkB dÞ3 þ 2 6 24 23 1 ðkB dÞ5 ðkB dÞ6 . . . ; 2880 960
0.4
Standard deviation of wave number (m−1)
S¼
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
1 6
1 ðkB dÞ4 24
where l1,2 are related to the propagating waves and l3,4 are the near field waves. The eigenvectors associated with l1,2 are analytical written as:
1
1
0
1 w1;2 4 Bh C B jk ð1 ðk dÞ =2880 ðkB dÞ6 =10800...Þ C C B B B B 1;2 C B C; ¼B C¼B 3 4 6 C A @ f1;2 F @ jEIkB ð1 ðkB dÞ =960 ðkB dÞ =302400...Þ A 2 4 6 m1;2 EIkB ð1 ðkB dÞ =1440 ðkB dÞ =18900...Þ
q
ð70Þ where w1,2, h1,2, f1,2, m1,2 are the translational displacement, the rotational displacement, the shear force and the flexural moment. In the case of random media, if the Young modulus is considered to be a random parameter as an example, the standard deviation of the dynamic stiffness matrix is expressed as the following:
0
rD ¼
rE I B B
12
B d @ sym 3
6d
12
2
6d
4d
12
6d
2000
3000
4000
5000
6000
7000
8000
0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
1
2 2d C C C: 6d A
1000
Fig. 5. Standard deviation of wave number (E stochastic), (–): SWFE, (): analytical solution and (⁄): Monte Carlo simulation.
23 1 ðkB dÞ5 þ ðkB dÞ6 . . . ; 2880 960
0
0
Frequency (Hz)
Standard deviation of wave number (m−1)
1 2
l3;4 ¼ 1 ðkB dÞ þ ðkB dÞ2 ðkB dÞ3 þ
ð68Þ
0
1000
2000
3000
4000
5000
6000
7000
8000
Frequency (Hz)
ð71Þ
Fig. 6. Standard deviation of wave number (q stochastic), (–): SWFE, (): analytical solution and (⁄): Monte Carlo simulation.
2
4d
The standard deviation of the eigenvalues can be analytically obtained through Eq. (31) as:
rl1 ¼
rE E
h i 5 7 ðkB dÞ 1 þ jðkB dÞ 12 ðkB dÞ2 j 30 ðkB dÞ3 þ 180 ðkB dÞ4 h i : 5 j B dÞ ðkB dÞ4 ðk720 4j þ 60j ðkB dÞ2 700
assumed to be a random quantity. Figs. 8 and 9 give the statistics of energy velocities estimation when compared to Monte Carlo computations. The analytical standard deviation of the wave number is written,
ð72Þ
Figs. 5–7 compare the wave numbers analytical standard deviation with Monte Carlo simulations and WFE predictions in different random situations. Precisely, Fig. 5 gives the comparison when Young ’s
rkB ¼
k B rE : 4 E
ð73Þ
So, the standard deviation of the group velocity is expressed as follows:
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M.N. Ichchou et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2805–2813
Standard deviation of wave number (m−1)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
1000
2000
3000
4000
5000
6000
7000
8000
Frequency (Hz) Fig. 7. Standard deviation of wave number (d stochastic), (–): SWFE, (): analytical solution and (⁄): Monte Carlo simulation.
Standard deviation of wavenumber (1/m)
Standard deviation of group velocity (m/s)
3
2.5
2
1.5
1
0.5
0
1000
2000
3000
4000
5000
6000
7000
0.2
0.15
0.1
0.05
0
2000
3
2.5
2
1.5
1
0.5
6000
8000
10000
Fig. 11. Standard deviation of wavenumber (E stochastic), (–): SWFE, (): Monte Carlo simulation.
Standard deviation of wavenumber (1/m)
Fig. 8. Standard deviation of group velocity (E stochastic), (–): SWFE, (): analytical solution and (⁄): Monte Carlo simulation.
4000
Frequency (Hz)
8000
Frequence(Hz)
Standard deviation of group velocity (m/s)
0.25
0 0
0.25
0.2
0.15
0.1
0.05
0
0
2000
4000
6000
8000
10000
Frequency (Hz) 0 0
1000
2000
3000
4000
5000
6000
7000
8000
Frequence(Hz) Fig. 9. Standard deviation of group velocity (q stochastic), (–): SWFE, (): analytical solution and (⁄): Monte Carlo simulation.
rc g ¼
Fig. 10. Finite element discretization of one subsystem.
1 xI1=4 rE : 2 ðqSx2 Þ1=4 E3=4
ð74Þ
Good concordance is shown between Monte Carlo and SWFE in all considered cases and over a wide frequency range. The dispersive
Fig. 12. Standard deviation of wavenumber (q stochastic), (–): SWFE, (): Monte Carlo simulation.
nature of the propagation can be observed for the standard deviation. 4.3. A complex structure case study In this section, a validation of the proposed model for a complex structure is offered. Let’s consider a multi-modal wave guide with these properties: Young’s modulus E = 2 1011 Pa mass density
M.N. Ichchou et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2805–2813
2813
q = 7800 kg m3, length of the considered element d = 0.005 m, Poisson’s ratio m = 0.3, loss factor g = 0.01, and cross section
References
area = 0.05 m 0.03 m. Fig. 10 represents the discretized studied subsystem. The extraction of the mass and stiffness matrices is performed using a commercial finite element software. 3D Solid elements were employed in the numerical scheme. Figs. 11 and 12 represent the standard deviation of extracted wave numbers in the multi-modal structure with 2% of standard deviation (E stochastic; q stochastic). This representation of statistics of dispersion curves prove the efficiency of the proposed model. It is remarkable that the SWFE and the MC simulations (2000samples) results are very close in all frequency band.
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5. Conclusions and current developments In this paper, the issue of wave propagation parameters estimations in random guided viscoelastic media was dealt with. Guided wave properties are an important physical notion for use in many engineering problems, the paper results are expected to be helpful in non-deterministic problems. To that end, a formulation named stochastic wave finite element (SWFE) was offered. This formulation allows wave characteristics to be defined by means of a stochastic finite element model. The case of spatially homogeneous random properties is dealt with through a parametric probabilistic approach. The formulation is general and can be applied to various structural shapes. The formulation is also easy to use as it considers a finite element model of a sample structural element. The statistics of wave propagation constants are thus described by a spectral problem. The statistics of wave modes are also obtained from the spectral problem. Two different eigenvalue problems were formulated and discussed. Ultimately, numerical and analytical comparisons were given in the case of a simple non-dispersive media and a dispersive one. Such analytical developments and numerical experiments showed the effectiveness of the proposed approach to predict the statistics of guided wave propagation characteristics (means and standard deviations) when compared to non-intrusive Monte Carlo calculations. The SWFE offers some interesting research perspectives. Among future work, the evaluation of the statistics of waves diffusions and scattering at notch and defects is an important task. Theses statistics are expected to be of use in the sensitivity analysis of guided waves techniques for damage detection. Among the investigations in progress, the use of SWFE for energy issues in structural complex wave guide is considered. The mid-high frequency behaviour is the main target in this case. Further investigations are under progress in order to use such numerical methods in the context of smart materials and structures [25,26]. Acknowledgments Part of this work is funded under the ‘‘Mid-Frequency’’ Marie-Curie contract. Support from the EC is greatly acknowledged.