Wave finite element-based superelements for forced ...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING

Wave finite element-based superelements for forced response analysis of coupled systems via dynamic substructuring P.B. Silva1,J.-M. Mencik2∗ and J.R.F. Arruda1 1 Departamento

de Mecˆanica Computacional, Faculdade de Engenharia Mecˆanica, Universidade Estadual de Campinas, R. Mendeleyev, 200, 13083-860, Campinas-SP, Brazil 2 INSA Centre Val de Loire, Universit´ e Franc¸ois Rabelais de Tours, LMR EA 2640, Campus de Blois, 3 rue de la chocolaterie, CS 23410, 41034 Blois Cedex, France

SUMMARY The wave finite element (WFE) method is used for assessing the harmonic response of coupled mechanical systems that involve one-dimensional periodic structures and coupling elastic junctions. The periodic structures under concern are composed of complex heterogeneous substructures like those encountered in real engineering applications. A strategy is proposed which uses the concept of numerical wave modes to express the dynamic stiffness matrix (DSM), or the receptance matrix (RM), of each periodic structure. Also, the Craig-Bampton (CB) method is used to model each coupling junction by means of static modes and fixed-interface modes. An efficient WFE-based criterion is considered to select the junction modes that are of primary importance. The consideration of several periodic structures and coupling junctions is achieved through classic finite element assembly procedures, or domain decomposition techniques. Numerical experiments are carried out to highlight the relevance of the WFE-based DSM and RM approaches in terms of accuracy and CPU time savings, in comparison with the conventional FE and CB methods. The following test cases are considered: a 2D frame structure under plane stresses; a 3D aircraft fuselage involving stiffened cylindrical shells. Received . . .

KEY WORDS: WFE method; periodic structures; superelements; dynamic substructuring; midfrequencies.

1. INTRODUCTION In this paper, the harmonic response of complex mechanical systems involving one-dimensional periodic structures and coupling elastic junctions is analyzed within the framework of the wave finite element (WFE) method. Those mechanical systems are commonly encountered in various fields of engineering, such as automotive and aeronautic industries (e.g., chassis frames, aircraft fuselages). The reduction in CPU time involved in the numerical simulation of those systems constitutes a tough and open challenge. The issue may be viewed as circumventing the computational cost of the conventional finite element (FE) method when large-sized numerical models are to be solved at many discrete frequencies, while keeping a high level of accuracy. The challenge especially concerns the development of efficient numerical approaches that outperform, in terms of accuracy and CPU time savings, the usual substructuring techniques like the Craig-Bampton (CB) method [1]. The present paper investigates the development of WFE-based superelements for the modeling ∗ Correspondence to: INSA Centre Val de Loire, Universit´ e Franc¸ois Rabelais de Tours, LMR EA 2640, Campus de Blois, 3 rue de la chocolaterie, CS 23410, 41034 Blois Cedex, France. E-mail: [email protected]

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2 of periodic structures. This yields the consideration of dynamic stiffness matrices or receptance matrices that can be obtained in a computationally efficient way. Originally, the WFE method has been developed to predict the wave modes which travel in right and left directions along infinite periodic structures [2, 3]. Those structures are periodic in the sense that they are built from identical substructures, which are modeled by means of the same FE mesh and assembled along a certain straight direction. High-order wave modes can be obtained by discretizing the left and right boundaries of the substructures with a sufficient number of degrees of freedom. Later on, the WFE method has been applied to address the wave propagation along multi-layered systems [4], fluid-filled pipes [5, 6] and frame structures [7]. Also, the WFE method has been investigated to describe the harmonic response of bounded straight periodic structures [8, 9, 10]. Wave-based matrix formulations which use the concept of wave mode amplitudes have been developed in [11] to compute the frequency response functions (FRFs) of assemblies involving flat shells and coupling junctions at a very low computational cost compared to the conventional FE method. Within that framework, the coupling junctions are modeled by means of the CB method [1], i.e, in terms of static modes and a reduced number of fixed-interface modes, which are selected via a wave-based procedure [12]. By combining low and high-order wave modes for modeling periodic structures, and fixed-interface modes for modeling coupling junctions, the WFE method constitutes an efficient numerical tool for assessing the midfrequency (MF) dynamics of coupled systems, i.e., in frequency ranges where modal densities exhibit large variations [13]. A qualitative comparison between the WFE method and other wavebased MF techniques such as analytical Trefftz techniques [14, 15] and enrichment techniques [16] has been proposed in [11]. One of the main features of the WFE method when compared to those MF techniques concerns the use of matrix equations which are likely to be well-conditioned. Also, a quantitative comparison between the WFE method and the conventional CB method — i.e., when all the periodic structures are modeled in terms of fixed-interface modes instead of wave modes — has been performed in [11] . It should be emphasized that, unlike the CB method or CB-like techniques such as the automated multilevel substructuring (AMLS) method [17], the WFE method does not require either the truncation of (wave) mode bases or the reduction of interface degrees of freedom (DOFs) for assessing the FRFs of periodic structures at a low computational cost. This can be explained by the fact that the WFE method usually deals with full sets of wave modes whose number appears to be much smaller than the total number of DOFs involved for modeling periodic structures. Thus, the WFE method appears to be more accurate than the CB and AMLS methods, while considering matrix equations of similar sizes. It should be noticed, however, that the WFE method remains applicable to the study of periodic systems, even though those periodic systems can be connected by means of junctions of arbitrary shapes, which makes it a less general method than the conventional CB method and other wave-based MF techniques. Although efficient, the WFE approach that uses the concept of wave mode amplitudes is, however, difficult to handle as based on complex matrix formulations that need to be expressed on a case-bycase basis. On the other hand, the WFE method has been investigated to express the condensed dynamic stiffness matrices of straight periodic structures [18, 19]. This approach appears to be an efficient alternative to the spectral element methods [20, 21] that make use of analytical waves for expressing the condensed dynamic stiffness matrices of waveguides. This is understood because the WFE method works fine in the MF range, as opposed to the analytical methods which are constraint by low-frequency assumptions. Such a WFE-based approach appears to be simple to use in the framework of substructuring techniques, i.e., when dealing with several superelements connected to each other. So far, it has never been applied, however, to the study of truly periodic structures — i.e., structures involving heterogeneous substructures which contain many internal DOFs — that may be coupled to complex elastic junctions. Thus, the motivation of the present paper may be viewed as to propose efficient and ease-of-use superelement based approaches, which take benefit of the aforementioned WFE and CB procedures, that can compete with other conventional model reduction techniques for predicting the harmonic response of complex engineering mechanical systems. The rest of the paper is organized as follows. In Sec. 2, the basics of the WFE method for computing the wave modes traveling along periodic structures are recalled. In Sec. 3, the

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Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

3 superelement modeling of a periodic structure is proposed, which consists in expressing its dynamic stiffness matrix (DSM) or receptance matrix (RM) in terms of wave modes. In Sec. 4, the CB-based modeling of a coupling junction is investigated. In Sec. 5, the modeling of coupled systems is investigated by means of the so-called DSM and RM approaches. The DSM approach uses the concept of classic FE assembly procedures for connecting subsystems to each other, while the RM approach involves domain decomposition techniques by means of Lagrange multipliers. In Sec. 6, numerical experiments are carried out regarding the following test cases: (1) a 2D frame structures under plane stresses that involves beam-like structures and curved junctions; (ii) a 3D aircraft fuselage that involves stiffened cylindrical shells, doors and a conical head. The relevance of the WFE-based DSM and RM approaches, in terms of accuracy and CPU time savings, is highlighted in comparison with the conventional FE and CB methods.

2. WAVE FINITE ELEMENT (WFE) METHOD The WFE method aims at describing waves traveling along one-dimensional periodic structures like the one displayed in Fig. 1. Those structures are periodic in the sense that they are made up of identical substructures having a length of ∆, which are assembled to each other along a certain straight direction (say, axis x). Each substructure is supposed to be linear elastic and damped with a constant loss factor η . Also, it undergoes coupling actions (i.e., external loads) which are supposed to be confined on its left and right boundaries only.

Figure 1. FE model of a one-dimensional periodic structure composed of N substructures; FE model of a related substructure.

The WFE method makes use of the FE model of a substructure, see Fig. 1. Here, the left and right boundaries of the substructure are meshed in a same way, i.e., by means of a same number n of DOFs. The stiffness and mass matrices (K, M) of the substructure follow from this FE model, which yields the so-called dynamic stiffness matrix D = −ω 2 M + (1 + iη)K, where ω is the angular frequency. As a result, the dynamic equilibrium equation of the substructure is expressed in the frequency domain as Dq = F,

(1)

where q and F are vectors of nodal displacements/rotations and forces/moments, respectively. Those vectors may be partitioned as q = [(qL )T (qI )T (qR )T ]T and F = [(FL )T (FI )T (FR )T ]T where the subscripts L and R refer to the DOFs on the left and right boundaries of the substructure (respectively), while the subscript I refers to the internal DOFs. Notice that FI = 0 since the internal DOFs are not subject to external loads, by assumption. By condensing the matrix D on the left and Prepared using nmeauth.cls

Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

4 right substructure boundaries, the dynamic equilibrium equation can be readily expressed as      ∗ DLL D∗LR qL FL = , D∗RL D∗RR qR FR

(2)

where D∗ denotes the condensed dynamic stiffness matrix, which is expressed as D∗ = DBB − DBI D−1 II DIB , where the subscript B refers to the DOFs on the left and right boundaries of the substructure. According to Eq. (2), the vectors of displacements/rotations and forces/moments on the left (or right) boundaries of two adjacent substructures k − 1 and k are linked as [22] u(k) = Su(k−1) ,

(3)

where u is a 2n × 1 state vector, given by   q . u= ±F

(4)

Here, the sign ahead of F depends on the kind of substructure boundary considered, i.e., either left (minus sign) or right (positive sign). In Eq. (3), S is a 2n × 2n symplectic matrix expressed as   ∗ −D∗−1 −D∗−1 LR DLL LR (5) S= ∗−1 . ∗ ∗ D∗RL − D∗RR D∗−1 LR DLL −DRR DLR As a result of Bloch’s theorem, the state vector u(k) can be expanded as u(k) =

X

(k)

φj Qj

=

X

(k−1)

φj e−iβj ∆ Qj

,

(6)

j

j

where {φj }j and {Qj }j have the meaning of wave shapes and wave amplitudes, respectively. Eq. (6) means that the wave amplitudes between two consecutive substructures k and k − 1 are linked as (k)

Qj

(k−1)

= µj Q j

with µj = e−iβj ∆ ,

(7)

where {βj }j have the meaning of wavenumbers. Also, the set of parameters {µj }j are referred to as the eigenvalues of the matrix S. As a result, an eigenproblem can be considered Sφj = µj φj .

(8)

In this framework, the wave shapes {φj }j appear to be the right eigenvectors of S, which can be partitioned into n × 1 vectors of displacement/rotation and force/moment components {φqj }j and {φFj }j , respectively. In fact, those eigenvectors provide a spatial description for the wave motion over the substructure boundaries. The eigensolutions {(µj , φj )}j are referred to as the wave modes of the structure. In fact, there are twice as many wave modes as the number of DOFs used to discretize each substructure boundary. Regarding the representation of the wavenumbers {βj }j in the complex plane, the wave modes can be classified as: purely propagating (i.e., the imaginary parts of the wavenumbers are close to zero), purely evanescent (i.e., the real parts of the wavenumbers are close to zero), or complex (i.e., the real and imaginary parts of the wavenumbers are of the same order of magnitude). It is important to notice that, due to the fact that the matrix S is symplectic, its eigenvalues come in pairs as (µj , 1/µj ), which correspond to n right-going wave shapes {φj }j and n left-going wave shapes {φ⋆j }j . As a convention, those right-going and left-going wave shapes are associated to eigenvalues whose magnitudes are, respectively, less and greater than one. If one considers the family {φj }j ∪ {φ⋆j }j as a wave basis, it becomes possible by considering Eq. (7) to express the vectors of nodal displacements/rotations and forces/moments of a structure Prepared using nmeauth.cls

Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

5 composed of N substructures, as follows (k)

qL

= Φq µk−1 Q + Φ⋆q µN +1−k Q⋆

(k)

,

qR

= Φq µk Q + Φ⋆q µN −k Q⋆ k = 1, . . . , N,

(k)

−FL

= ΦF µk−1 Q + Φ⋆F µN +1−k Q⋆

,

(k)

FR

= ΦF µk Q + Φ⋆F µN −k Q⋆ k = 1, . . . , N,

(9)

(10)

where Φq , Φ⋆q , ΦF and Φ⋆F are n × n matrices expressed as Φq = [φq1 · · · φqn ], Φ⋆q = [φ⋆q1 · · · φ⋆qn ], ΦF = [φF1 · · · φFn ] and Φ⋆F = [φ⋆F1 · · · φ⋆Fn ]. Also Q = [Q1 · · · Qn ]T and Q⋆ = [Q⋆1 · · · Q⋆n ]T denote the vectors of wave amplitudes at the left and right ends of the global structure, respectively. Finally, µ is the n × n diagonal matrix whose components are the eigenvalues {µj }j whose magnitudes {|µj |}j are less than one. Remark 1. The eigenvalue problem involving the matrix S, Eq. (8), is usually subject to numerical ill-conditioning, as early discussed by Zhong and Williams [3]. This occurs because the matrix of eigenvectors has largely disparate components involving displacement/rotation and force/moment components. In order to solve this issue, Zhong and Williams considered a better conditioned generalized eigenproblem, as follows: Nwj = µj Lwj ,

(11)

where 

0 N= −DRL

In −DRR



,



In L= DLL

 0 . DLR

(12)

Notice that the relation between the matrix S and the matrices L and N is given by S = NL−1 . In Eq. (11), {wj }j refer to eigenvectors which involve displacement/rotation components, only. Once these eigenvectors are computed by means of the eigenproblem in Eq. (11), the wave shapes {φj }j can be easily retrieved as φj = Lwj . Remark 2. For periodic systems that exhibit symmetric substructures — i.e., the left and right parts of each substructure are symmetric with respect to a plane perpendicular to the main axis x —, the right- and left-going wave shapes are linked as [8] Φ⋆q = RΦq

,

Φ⋆F = −RΦF ,

(13)

where R is a diagonal symmetry transformation matrix. Eq. (13) provides an analytical means to strictly enforce the coherence between the right- and left-going wave shapes. It is used to circumvent the issue of numerical dispersion which results from the computation of the eigensolutions of the matrix S, and which can be detrimental for the computation of the forced response of periodic structures.

3. WFE-BASED SUPERELEMENT MODELING FOR PERIODIC STRUCTURES 3.1. Dynamic stiffness matrix One of the purpose of the present study is to formulate the condensed dynamic stiffness matrix D(s) of a given periodic structure — namely (Ps ) — in terms of WFE wave modes . This dynamic stiffness matrix links the vector of nodal displacements/rotations to the vector of nodal forces/moments on the structure ends — i.e., the left end of the substructure 1 and the right end of Prepared using nmeauth.cls

Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

6 the substructure N (see Sec. 2) —, as follows D(s) q(s) = F(s) ,

(14)

where (s)

q

"

(1)

qL = (N ) qR

#

,

(s)

F

"

# (1) FL = (N ) . FR

(15)

The motivation behind the use of WFE wave modes is to compute the matrix D(s) , as well as its inverse, at a low computational cost compared to the conventional approaches. The WFE procedure is achieved in this way. According to Eqs. (9) and (10), the vectors of nodal displacements/rotations and forces/moments on the left and right ends of the periodic structure are expressed as (1)

qL = Φq Q + Φ⋆q µN Q⋆

(N )

,

(1)

−FL = ΦF Q + Φ⋆F µN Q⋆

qR ,

= Φq µN Q + Φ⋆q Q⋆ ,

(N )

FR

= ΦF µN Q + Φ⋆F Q⋆ .

In matrix form, this yields " #    (1) Φq Φ⋆q µN Q qL , = (N ) Q⋆ Φq µN Φ⋆q qR

(16) (17)

(18)

and "

#  (1) −ΦF FL (N ) = Φ µN F FR

−Φ⋆F µN Φ⋆F



 Q . Q⋆

(19)

To derive the condensed dynamic stiffness matrix D(s) of the periodic structure, Eq. (18) is left multiplied by the following matrix   −1 Φq 0 , (20) 0 Φ⋆−1 q which yields  −1  " (1) #  Φq 0 I qL ⋆−1 (N ) = 0 Φ⋆−1 Φ Φq µ N qR q q

⋆ N Φ−1 q Φq µ I

 Q . Q⋆



(21)

The motivation behind the use of the matrix (20) is to provide a matrix, on the right hand side of Eq. (21), that is well-conditioned [8]. The size of this matrix is 2n × 2n — i.e., it is related to the number of DOFs on the left and right ends of the structure, only — which appears to be very small compared to the number of DOFs used to model the whole periodic structure. Also, note that this matrix is symmetric due to the symmetry property of the wave shapes, see Eq. (13), while it is sparse, as it has identity matrices as diagonal block terms. In other words, it can be inverted without numerical difficulties, at a low computational cost. Thus the vectors of wave amplitudes Q and Q⋆ can be expressed, from Eq. (21), as  " (1) #    ⋆ N −1  −1 Φq 0 I Φ−1 qL Q q Φq µ (22) = ⋆−1 ⋆−1 (N) . N Q⋆ 0 Φq Φq Φq µ I qR The condensed dynamic stiffness matrix of the periodic structure (Eq. (14)) is readily derived from Eqs. (19) and (22). This yields    −1   ⋆ N −1 Φq 0 I Φ−1 −ΦF −Φ⋆F µN q Φq µ . (23) D(s) = ΦF µ N Φ⋆F 0 Φ⋆−1 Φ⋆−1 Φq µN I q q

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Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

7 3.2. Receptance matrix One interesting feature behind the use of wave modes is to formulate the inverse of the dynamic stiffness matrix D(s) without the need of explicitly inverting the matrix appearing on the right hand side of Eq. (23). Such an inverse is usually called receptance matrix and can be derived as follows. Eq. (19) is left multiplied by the following matrix   −Φ−1 0 F , (24) 0 Φ⋆−1 F which yields  −Φ−1 F 0

0 Φ⋆−1 F

"

#  (1) FL I ⋆−1 (N ) = Φ ΦF µ N FR F

⋆ N Φ−1 F ΦF µ I

 Q . Q⋆



(25)

In a similar way as in Eq. (21), the matrix appearing on the right hand side of Eq. (25) is symmetric and sparse with identity matrices as diagonal block terms, i.e., it is likely to be invertible. Thus it turns out that the vectors of wave amplitudes Q and Q⋆ can be expressed as  " (1) #      ⋆ N −1 −Φ−1 0 FL I Φ−1 Q F F ΦF µ = (26) (N) . Q⋆ 0 Φ⋆−1 Φ⋆−1 ΦF µ N I FR F F The receptance matrix of the periodic structure — namely R(s) — is defined as q(s) = R(s) F(s) .

(27)

Its expression follows from Eqs. (18) and (26), i.e. R

(s)



Φq = Φq µN

Φ⋆q µN Φ⋆q



I Φ⋆−1 ΦF µ N F

⋆ N Φ−1 F ΦF µ I

−1  −Φ−1 F 0

0 Φ⋆−1 F



.

(28)

4. CB-BASED SUPERELEMENT MODELING FOR NON-PERIODIC COUPLING JUNCTIONS The problem of predicting the dynamic behavior of several periodic structures coupled to elastic junctions is addressed. For the sake of clarity, two periodic structures (e.g., curved stiffened panels) — namely (P1 ) and (P2 ) — which are connected to an elastic junction (P3 ) (e.g., stiffened panel with a hole) are shown in Fig. 2. Here, the vectors of wave amplitudes for the modes traveling along the periodic structures (P1 ) and (P2 ) towards the junction (P3 ) are denoted as Q1 and Q2 , respectively, while those for the modes traveling outward from the junction are denoted as Q⋆1 and Q⋆2 . A WFE-based superelement modeling of periodic structures has been proposed in Sec. 3. In the present section, the FE method combined with the Craig-Bampton (CB) method [1] is used to model the coupling junctions. In this framework, a coupling junction is modeled by means of static modes and fixed-interface modes. The key idea behind the proposed approach is to select among all the fixed-interface modes those which actually contribute to the system forced response. As just a few modes are involved, the proposed procedure yields significant computational savings, which improves the performance of the CB method for MF analysis. Within the WFE framework, a normwise selection criterion of contributing fixed-interface modes has been developed by Mencik [12]. The procedure is briefly recalled hereafter. Consider a coupling junction (Pc ) that is meshed by means of internal and boundary DOFs. Here, the internal DOFs are free from external excitation sources, while the boundary DOFs relate Prepared using nmeauth.cls

Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

8

Figure 2. Illustration of three periodic structures (P1 ) and (P2 ), which are connected to a coupling junction (P3 ).

the coupling interfaces with periodic structures. The dynamic equilibrium equation of the junction, which links the vectors of displacements/rotations q(c) and forces/moments F(c) on the boundary DOFs, is expressed as D(c) q(c) = F(c) ,

(29)

where D(c) is the dynamic stiffness matrix of the junction condensed at the boundary DOFs. Within the CB framework, the matrix D(c) is expressed in terms of static modes and fixed-interface modes, i.e. ˜ (c) , D(c) ≈ D

(30)

˜ (c) = Dst−st − D ˜T ˜ −1 ˜ D el−st Del−el Del−st ,

(31)

where

(c)

(c)

(c)

(c)

(c)

(c)

Dst−st = −ω 2 (XTst MII Xst + MBI Xst + XTst MIB + MBB ) + (1 + iη (c) )(KBI Xst + KBB ), (32) (c) (c) ˜ el−st = −ω 2 X ˜ Tel (MII D Xst + MIB ),

(33)

n o ˜ el−el = diag γ˜j (−ω 2 + ω D ˜ j2 (1 + iη (c) )) .

(34)

˜ (c) Ψ⋆ + Ψ⋆ ]−1 [D ˜ (c) Ψq + ΨF ], C = −[D q F

(35)

j

In these equations, K(c) and M(c) are the stiffness and mass matrices of the coupling junction; the subscripts I and B relate the internal and boundary DOFs, respectively. Also, Xst is the matrix of (c)−1 (c) ˜ ˜ static modes, defined as Xst = −KII KIB ; X el is the matrix of fixed-interface modes {(Xel )j }j , (c) (c) 2 which are obtained by solving the following eigenproblem KII (Xel )j = ωj MII (Xel )j ({ωj }j being the eigenpulsations). The tilde sign indicates that a reduced number of fixed-interface modes are retained to formulate the matrix D(c) . Finally, η (c) is the loss factor of the junction, while {˜ γj } j (c) ˜ T ˜ denote the modal masses (γ˜j = (Xel )j MII (Xel )j ∀j ). ˜ el )j }j is based on a The procedure to select the contributing fixed-interface modes {(X perturbation analysis of a matrix C which relates the reflection/transmission coefficients of wave modes, traveling along the periodic structures towards the junction. In case where the FE meshes of the periodic structures are compatible with that of the junction over the coupling interfaces, the matrix C is defined as [12]

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Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

9 where 

(c)

L1 (Φq )1  .. Ψq =  . 0



(c)

L1 (ΦF )1  .. ΨF =  . 0

··· .. . ··· ··· .. . ···

0 .. .



(c) LR (Φq )R

 

0 .. .



(c) LR (ΦF )R

 

,

,

(c)

L1 (Φ⋆q )1  .. Ψ⋆q =  . 0



··· .. . ···

(c)

L1 (Φ⋆F )1  .. Ψ⋆F =  .



··· .. . ···

0

 0  .. , . (c) ⋆ LR (Φq )R 0 .. . (c) LR (Φ⋆F )R



 .

(36)

(37)

Here, R is the number of periodic structures that are connected to the junction; the superscript ⋆ (c) relates the wave modes that travel outward from the junction; also, Ls denotes a direction cosine matrix which links the coordinate system of the junction to that of a given periodic structure (Ps ). The selection of the fixed-interface modes is carried out by assessing how much the matrix C is ˜ el )j }j . perturbed when a given mode (Xel )j is not taken into account in the reduced basis {(X (c) Clearly speaking, consider the condensed dynamic stiffness matrix D of the junction that results from the consideration of all the fixed-interface modes. Omitting one mode (Xel )j yields the ˜ (c) = D(c) + ∆j D(c) , ∆j D(c) being expressed as (Eqs (31-34)): perturbed matrix D ∆j D(c) =



T ω4 BT (Xel )j BT (Xel )j , γ˜j (−ω 2 + ωj2 (1 + iη c ))

(c)

(38)

(c)

where B = MII Xst + MIB . Considering D(c) + ∆j D(c) , instead of D(c) , yields the perturbed matrix C + ∆j C. Hence, the proposed selection procedure consists in retaining the fixed-interface modes which satisfy the following criterion, k∆j CkF ≥ǫ k Ck F

∀j,

(39)

where ǫ is a given small tolerance threshold, while k.kF denotes the Frobenius norm. In other words, the selection procedure consists in rejecting the fixed-interface modes for which the relative error k∆j CkF /kCkF is below ǫ, meaning that they weakly contribute to the dynamic behavior of the periodic structures that are connected to the junction. Note that the computation of the relative error k∆j CkF /kCkF , for each mode (Xel )j , may be performed once, i.e., at the maximum discrete frequency involved in the frequency band of interest. This is understood since a maximum number of wave modes are likely to contribute to the dynamic behavior of periodic structures at the maximum frequency, meaning that k∆j CkF /kCkF reaches its maximum value.

5. COUPLED SYSTEM MODELING The WFE-based numerical model of a fully coupled system, which involves several periodic structures and coupling junctions, is presented. Two well-known procedures are recalled hereafter, which use, respectively, the dynamic stiffness matrices and receptance matrices of subsystems.

5.1. Dynamic stiffness matrix (DSM) method Consider a coupled system that involves Ns periodic structures and Nc coupling junctions. Also, denote as Os and Oc (Os ∩ Oc = ∅) the sets of integers used to number the periodic structures Prepared using nmeauth.cls

Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

10 and the coupling junctions, respectively. Within the DSM framework, the global dynamic stiffness matrix of the coupled system — namely Dgl — is constructed from the dynamic stiffness matrices of the periodic structures and coupling junctions (see Secs. 3.1 and 4). The construction of Dgl follows from a classic assembly procedure, i.e., X X ˆ (c) L(c) , ˆ (s) L(s) + L(c)T D (40) L(s)T D Dgl = c∈Oc

s∈Os

ˆ (s) and D ˆ (c) are defined as where L(s) and L(c) are mesh incidence matrices; also, D ˆ (s) = N (s)T D(s) N (s) D

ˆ (c) = N (c)T D(c) N (c) , D

,

(41)

where N (s) and N (c) are direction cosine matrices which link the local coordinate frames of the periodic structures and coupling junctions, respectively, to the global coordinate frame of the coupled system. The dynamic equilibrium equation of a coupled system, free from kinematic constraints, results as Dgl qgl = Fgl , where qgl and Fgl are vectors of nodal displacements/rotations and forces/moments, respectively. More generally, the consideration of kinematic constraints yields the following dynamic equilibrium equation: gl gl gl gl Dgl II qI = FI − DIB qB ,

(42)

where qgl B denotes a vector of prescribed displacements/rotations. Here, the subscript B (resp. I) relates the DOFs subject to (resp. free from) kinematic constraints. The vector of nodal displacements/rotations of the coupled system is finally assessed by solving Eq. (42), i.e. gl gl gl gl gl
(43) qI = (DII )−1 FI − DIB qB .

5.2. Receptance matrix (RM) method An alternative strategy is investigated to determine the vector of nodal displacements/rotations of the coupled system. It uses the concept of domain decomposition techniques in which receptance matrices of subsystems are involved. In fact, the receptance matrix R(s) of a periodic structure (Ps ) can be easily computed from Eq. (28), without the need of explicitly inverting a dynamic stiffness matrix. This means CPU time savings compared to the DSM method, i.e., when the inverse of a global matrix Dgl II (Eq. (43)) is estimated. This interesting feature has motivated the development of the present RM method. The latter is detailed as follows. For the sake of simplicity, it will be assumed here that the FE meshes of subsystems are compatible across the coupling interfaces. The dynamic equilibrium equations, for periodic structures and coupling junctions, can be expressed in the following form ˆ (s) q ˆ (s) − B (s)T λ ˆ (s) = F D

,

ˆ (c) q ˆ (c) − B (c)T λ, ˆ (c) = F D

(44)

ˆ relate, respectively, vectors of nodal displacements/rotations and forces/moments in ˆ and F where q the global reference frame of the coupled system; also, B (s) and B (c) are Boolean matrices, while λ is to be understood as a vector of Lagrange multipliers. The consideration of receptance matrices yields     ˆ (s) F ˆ (s) − B (s)T λ ˆ (c) F ˆ (c) − B (c)T λ . ˆ (s) = R ˆ (c) = R q , q (45)

ˆ (s) = N (s)T R(s) N (s) where R(s) is the receptance matrix of a periodic structure (Ps ) in its Here, R local coordinate frame (Eq. (28)), while N (s) is a direction cosine matrix (see after Eq. (41)). Also, Prepared using nmeauth.cls

Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

11 ˆ (c) = (D ˆ (c) )−1 , D ˆ (c) being the receptance matrix of a coupling junction (Pc ) is to be expressed as R the dynamic stiffness matrix of the junction in the global reference frame (Eq. (41)). Besides, the consideration of kinematic constraints, which relate the coupling conditions between subsystems as well as the boundary conditions where prescribed displacements/rotations apply, leads to X X ˆ (s) + ˆ (c) = qgl B (s) q B (c) q (46) 0 , s∈Os

c∈Oc

gl where qgl 0 is a vector of prescribed displacements/rotations, which may be partitioned as q0 = gl T T gl T [0 (qB ) ] , where qB has been defined previously (Eq. (42)). The determination of the vector of Lagrange multipliers λ in Eq. (44) follows from Eq. (46) as [23]:

λ = BRgl B T


−1

where B = [B (1) B (1) · · · B (Ns+Nc ) ], and

Rgl

ˆ (1) R  0  = .  ..



gl


BRglFgl − q0

,

0 ˆ R(2) .. .

··· ··· .. .

0 0 .. .

0

···

ˆ (Ns +Nc ) R

0

(47)

    

,

ˆ (1) F ˆ (2) F .. .



  Fgl =  



ˆ (Ns +Nc ) F

  . 

(48)

The determination of the vector of nodal displacements/rotations of the coupled system finally results from Eq. (45). It is worth pointing out that the computation of the vector of Lagrange multipliers λ involves inverting a matrix (BRgl B T ) whose size relates the number of kinematic constraints only. Thus, the size of this matrix is expected to be smaller than that of the global dynamic stiffness matrix Dgl II appearing in Eq. (43). In other words, the RM method appears to be interesting for improving the computational speed of the proposed WFE approach.

6. NUMERICAL RESULTS The WFE-based DSM and RM methods are applied to compute the frequency forced response of two elastic systems involving periodic structures and elastic junctions. Recall that, within the present framework, the periodic structures are modeled using the WFE method (Sec. 3), while the coupling junctions are modeled using the CB method (Sec. 4). Here, a reduced set of fixed-interface modes, which are selected by means of the WFE-based strategy proposed in Sec. 4, is used to model the junctions. The following test cases are considered: (1) a 2D frame structure under plane stresses (Fig. 4); (2) a 3D aircraft fuselage involving stiffened cylindrical shells (Fig. 9). The first test case is rather simple, while the second one includes complexities of real engineering applications. For each test case, the frequency response functions (FRFs) provided by the WFE-based DSM R and RM approaches are compared with a reference FE solution issued from the ANSYS software, which uses a sparse direct solver. Also, the DSM and RM approaches are compared with the conventional CB method, i.e., when each subsystem (periodic structure, junction) is modeled in terms of static modes and fixed-interface modes. For the sake of clarity, the numerical tasks and platform environments involved when simulating the DSM and RM approaches are shown in Fig. R is used as a means to assess the mass and 3. Regarding the WFE and CB modelings, ANSYS stiffness matrices of substructures and subsystems. Post-treatment of those matrices is achieved R using MATLAB with a view to computing the forced response of the coupled system. All the numerical simulations are performed in double precision using a 64-bit CPU equipped with an Prepared using nmeauth.cls

Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

12 R R Intel Xeon E5-2609 2.40 GHz processor and 32GB of RAM memory. It is worth emphasizing R that the MATLAB environment is used as a means to assess the performances of the DSM and RM approaches in terms of CPU time savings, when compared to the conventional CB method.

Commercial FE package environment (ANSYS® has been used in this paper):

MATLAB® environment:

Frequency domain (to be considered at each discrete frequency

):

at

max

Figure 3. Flowchart illustrating the different numerical steps involved in the DSM and RM approaches.

6.1. 2D frame structure 6.1.1. Problem description This section investigates the harmonic response of a 2D frame structure undergoing plane stresses. The structure can be partitioned into three straight beam-like structures (P1 ), (P2 ) and (P3 ), and two quarters of torus (junctions (P4 ) and (P5 )), as shown in Fig. 4. Here, the beam-like structures are considered periodic, it being understood that their meshes are periodic (as outlined in Sec. 2). The whole structure is clamped over its left bottom edge while being subject to harmonic nodal loads (Fx = 100 N, Fy = 50 N ) over the whole right bottom edge. The periodic structures and junctions exhibit the following characteristics: Young’s modulus E = 2.1 × 1011 Pa, density ρ = 7850 kg.m-3 , Poisson’s ratio ν = 0.3, loss factor η = 0.01, thickness 0.001 m and width 0.4 m. The length of the periodic structures (P1 ) and (P3 ) is 2 m, while that of the periodic structure (P2 ) is 1.5 m. Besides, the coupling junctions (P4 ) and (P5 ) exhibit an internal and external radius of curvature of 0.2 m and 0.4 m, respectively. Within the WFE framework, each periodic structure is modeled by means of identical substructures, as explained in Sec. 2. Here, the substructures used are similar (up to a rotation of Prepared using nmeauth.cls

Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

13

Figure 4. Partitioning of the 2D frame structure into three periodic structures (P1 ), (P2 ) and (P3 ), and two coupling junctions (P4 ) and (P5 ); FE model of a substructure that is used to model the periodic structure (P2 ).

90o ), with a length ∆ = 0.01 m (Fig. 4). Thus, it turns out that the periodic structures (P1 ) and (P3 ) are composed of N1 = N3 = 200 substructures, while the periodic structure (P2 ) is composed of N2 = 150 substructures. Besides, 2D plane stress quadrilateral elements, with four nodes and two translational DOFs per node, are used to model the substructures as well as the coupling junctions (Fig. 4). Each substructure is meshed by means of 20 finite elements over its width, which yields n = 42 DOFs over the left/right boundary and no internal DOFs. Also, each coupling junction is meshed by means of 2, 226 DOFs that incorporate 42 DOFs over each coupling interface. As a result, the number of DOFs involved for modeling the whole coupled system is 25, 284. The harmonic response of the coupled system is analyzed over a frequency band βf = [110 − 10, 000] Hz that involves discrete frequencies that are equally spaced with a step of 10 Hz. This frequency band is supposed to be wide enough so that the periodic structures and the junctions exhibit local resonances. 6.1.2. Junction modeling The modeling of each junction is carried out by considering the CB strategy proposed in Sec. 4. Recall that the standard strategy for selecting the fixed-interface modes of a junction is to retain those for which the eigenfrequencies are below a certain frequency limit, while rejecting the other modes. As a rule of thumb, this frequency limit is classically chosen as twice the maximum frequency of the frequency band of interest, i.e. 20, 000 Hz in the present case. This yields 51 fixed-interface modes to be retained (cf. Fig. 5(a)). In contrast, the selection strategy proposed in Sec. 4 enables one to retain 43 fixed-interface modes only, which means subsequent CPU time savings. This is done by considering a tolerance threshold ǫ = 1% in Eq. (39), when the ratio k∆j CkF /kCkF is calculated for each of the aforementioned 51 modes (see Fig. 5(b)). This result confirms that the contribution of a junction mode is not necessarily linked to its rank. 6.1.3. Forced response computation The FRF of the 2D frame structure is assessed by means of the WFE-based DSM and RM approaches depicted in Sec. 5. In this framework, each periodic structure is modeled using the WFE method (Sec. 3), while the coupling junctions are modeled using the CB method (Sec. 6.1.2). In the present case, the number of right/left-going wave modes involved for modeling each periodic structure is n = 42. Here, the magnitude of the velocity (y -direction) of the periodic structure (P3 ) is analyzed at a corner node on the right excited edge. The WFE solutions are compared with the results issued from the conventional FE method in Fig. 6(a). Also, the relative Prepared using nmeauth.cls

Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

14 (a)

(b)

60

70 Mode rank = 51

60 Error Bound (in %)

Mode rank

50 40 30 20 10 0

50 40 30 20 10

0.25

0.5

0.75 1 1.25 1.5 Frequency (Hz)

1.75

2

1%

0 0

2.25 4 x 10

10

20 30 Mode rank

40

50

Figure 5. Strategies used to select the fixed-interface modes of the coupling junctions (P4 ) and (P5 ) occurring in the 2D frame structure: (a) classic strategy consisting in retaining the modes whose eigenfrequencies are below twice the maximum frequency of the frequency band of interest (this yields 51 modes); (b) WFE-based strategy proposed in Sec. 4 (this yields 43 modes).

error of the WFE solutions, averaged over several sub-frequency bands {βfi }i of same width, is displayed in Fig. 6(b). The relative error is expressed as FE < kqWFE meas k2 − kqmeas k2 >βfi

< kqFE meas k2 >βfi

,

(49)

where qmeas denotes the vector of nodal displacements at the measurement point, while the notation < . >βi denotes the quadratic mean over a sub-frequency band βfi . Here, ten sub-frequency bands {βfi }i are considered which cover the whole frequency band βf (i.e., βf = ∪i βfi ). As it can be seen, the WFE results completely match the reference FE solution over the whole frequency band where the relative error (Eq. (49)) never exceeds 5%. This fully validates the proposed approaches. (a)

(b) 3

20

10

Relative error (in %)

Velocity (dB)

10 0 −10 −20

2

10

1

10

0

10

−30 −40 0

−1

2000

4000 6000 Frequency (Hz)

8000

10000

10

0

2000

4000 6000 Frequency (Hz)

8000

10000

Figure 6. FRF of the 2D frame structure (a) and relative error, Eq. (49) (b): FE solution (—–); WFE-based DSM approach (- - -); WFE-based RM approach (- - -).

Besides, the WFE approaches are compared with the conventional CB method that consists in modeling each subsystem (periodic structure, junction) in terms of static modes and fixed-interface modes. For the purpose of the conventional CB method, each junction is modeled by means of 51 fixed-interface modes, as suggested by the rule of thumb consisting in selecting the modes whose eigenfrequencies are below twice the maximum frequency of the frequency band of interest. Within the CB method, two procedures are investigated to model the periodic structures, i.e.: (1) by considering fixed-interface modes selected by means of the aforementioned rule of thumb — this yields 146 fixed-interface modes for the periodic structures (P1 ) and (P3 ), and 109 fixed-interface Prepared using nmeauth.cls

Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

15 modes for the periodic structure (P2 ); (2) by considering fixed-interface modes whose number is equal to that of the right/left-going wave modes of each periodic structure — this yields 42 fixedinterface modes only. The related FRFs are displayed in Fig. 7(a), along with the following relative error (Fig. 7(b)): FE < kqCB meas k2 − kqmeas k2 >βfi

< kqFE meas k2 >βfi

,

(50)

As it can be seen, the relative error of the CB solutions reaches 20%, even in the case when the fixed-interface modes are selected by means of the classic rule of thumb. In particular, the CB method which uses 42 fixed-interface modes for each periodic structure yields erroneous results, as shown in Fig. 7(a). In this case, the basis of fixed-interface modes and static modes used to model each periodic structure has been chosen so that its dimension is equal to that of the wave mode basis used by the DSM and RM approaches. This confirms the fact that the WFE-based DSM and RM approaches are much more accurate than the CB method. For the sake of clarity, the mean values of the relative errors — i.e., when averaged over all the sub-frequency bands {βfi }i — involved in the WFE and CB solutions are compared as shown in Fig. 8(a). (a)

(b) 3

20

10

Relative error (in %)

Velocity (dB)

10 0 −10 −20

2

10

1

10

0

10

−30 −40 0

−1

2000

4000 6000 Frequency (Hz)

8000

10000

10

0

2000

4000 6000 Frequency (Hz)

8000

10000

Figure 7. FRF of the 2D frame structure (a) and relative error, Eq. (50) (b): FE solution (—–); CB method that uses 146 and 109 fixed-interface modes to model the periodic structures (- - -); CB method that uses 42 fixed-interface modes to model the periodic structures (- - -).

Regarding CPU times, it takes 183 s and 190 s to compute the forced response of the coupled system with the WFE-based DSM and RM approaches, respectively, against 263 s with the conventional CB method that uses 146 and 109 fixed-interface modes for the periodic structures. R It is worth recalling that those CPU times are obtained using MATLAB regarding both the WFE and CB techniques. This yields a computational saving of 30% in benefit of the DSM and RM approaches. This clearly highlights the relevance of the proposed approaches. It should be emphasized that the WFE-based DSM and RM methods give almost the same CPU time, which might be explained since the matrices that are inverted in Eqs. (43) and (47) have the same size. Finally, note that the conventional CB method that uses 42 fixed-interface modes for the periodic structures yields a CPU time of 138 s, which appears small compared to the WFE approaches. However, as outlined previously, such a method appears inefficient to accurately capture the FRF of the system. For the sake of conciseness and clarity, the CPU times involved in the WFE and CB approaches are shown in Fig. 8(b) and Table I. Also, the total number of physical and generalized DOFs managed by each approach is listed in Table I. Regarding the FE method, this number corresponds to the total number of DOFs involved in the full FE model; regarding the conventional CB method, it corresponds to the sum of the numbers of static modes and fixedinterface modes among all the structural components (periodic structures and junctions); regarding the WFE approaches, it corresponds to the sum of the number of boundary DOFs of the periodic structures with the number of static modes and fixed-interface modes of the junctions. This provides Prepared using nmeauth.cls

Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

16 a coarse estimate of the size of the matrix equations which are to be managed by each approach. As it can be seen, the total number of DOFs involved in the WFE-based DSM and RM approaches constitutes a better compromise between accuracy and performance than it is the case with the CB method. (a)

(b)

35

300 250

25 CPU time (s)

Mean Relative Error (%)

30

20 15 10

150 100 50

5 0

200

1

2 3 Approach used

4

0

1

2 3 Approach used

4

Figure 8. Bar plots of the mean relative errors (a), and CPU times (b) involved in the proposed approaches regarding the 2D frame structure: (black) CB method that uses 146 and 109 fixed-interface modes to model the periodic structures; (blue) CB method that uses 42 fixed-interface modes to model the periodic structures; (red) WFE-based DSM approach; (green) WFE-based RM approach.

Table I. Total number of DOFs and CPU times involved in the proposed approaches regarding the 2D frame structure.

Approach FE CB (146 and 109 modes) CB (42 modes) DSM RM

Total number of DOFs 25, 284 923 648 506 506

CPU time (s) — 263 137 184 190

Reduction (%) — — 48 30 28

6.2. Aircraft fuselage 6.2.1. Problem description This section investigates the harmonic response of an aircraft fuselage. The structure under concern consists of a conical head (coupling junction (P3 ), a long cylindrical shell (periodic structure (P1 )), a short cylindrical shell with holes (coupling junction (P4 )) that may correspond to aircraft doors, and a short cylindrical shell (periodic structure (P2 )) (Fig. 9). The cylindrical shells incorporate axial and circumferential flat stiffeners (i.e., stringers and frames, respectively) so as to include true periodicities. The coupled system is subject on its right end to two vertical forces acting in opposite directions, as shown in Fig. 9. The periodic structures and junctions share the same material characteristics, i.e.: Young’s modulus E = 7 × 1010 Pa, density ρ = 2700 kg.m−3 , Poisson’s ratio ν = 0.3 and loss factor η = 0.01. The cylindrical shells exhibit a diameter of 4 m and a thickness of 0.001 m. The circumferential and axial stiffeners have a height of 0.035 m, and respective thicknesses of 0.005 m and 0.002 m. Those stiffeners are periodically distributed over the cylinders (i.e., periodic structures (P1 ) and (P2 ), and coupling junction (P4 )), two consecutive circumferential (resp. axial) stiffeners being spaced with a length of 0.4 m (resp. an angle of 11.25◦). Besides, the conical head (P3 ) has a parabolic curvature and it does not have stiffeners. Prepared using nmeauth.cls

Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

17 Within the WFE method, the periodic structures (P1 ) and (P2 ) are modeled by means of N1 = 30 and N2 = 10 identical substructures, respectively, as shown in Fig. 9. The substructures and coupling junctions are meshed by means of quadrilateral Reissner-Mindlin shell elements with six DOFs per node, i.e., translations in the x, y and z directions and rotations about the same axes. Here, the total number of DOFs used to discretize the whole coupled system is 32, 586. Each substructure is meshed by means of n = 288 DOFs over its left/right boundary, and 384 internal DOFs. Also, the coupling junctions (P3 ) and (P4 ) are meshed by means of 3, 174 and 2, 724 DOFs, respectively.

Figure 9. Partitioning of the 3D aircraft fuselage into two periodic structures (P1 ) and (P2 ), and two coupling junctions (P3 ) and (P4 ); FE model of a substructure (cylindrical shell with longitudinal and circumferential stiffeners) that is used to model the periodic structures.

The harmonic response of the coupled system is analyzed over a frequency band βf = [20 − 200] Hz that involves discrete frequencies that are equally spaced with a step of 0.2 Hz.

6.2.2. Junction modeling The CB strategy proposed in Sec. 4 is used to model the coupling junctions (P3 ) and (P4 ). Again, a tolerance threshold ǫ = 1% is considered to carry out the selection of the fixed-interface modes being involved by means of Eq. (39). Recall that the classic selection strategy consists in retaining the fixed-interface modes for which the eigenfrequencies are smaller than a certain frequency limit, which, in the present case, is chosen as four times the maximum frequency of the frequency band of interest (i.e., 800 Hz). The choice of this frequency limit is justified by the fact that the coupled system under concern is rather complex — i.e, its geometry is locally heterogeneous, due to stiffeners and holes —, meaning that high order modes are likely to contribute to its dynamic behavior, even at low frequency. This classic selection strategy yields, respectively, 350 and 114 fixed-interface modes for the coupling junctions (P3 ) and (P4 ), as shown in Figs. 10(a) and 11(a). On the other hand, the WFE-based CB strategy proposed in Sec. 4 enables 107 and 58 fixed-interface modes to be retained among those aforementioned modes, as shown in Figs. 10(b) and 11(b). Notice that high order junction modes are expected to contribute to the system forced response, hence giving credit to the choice of the extended frequency limit (as discussed previously). To summarize, it appears that the proposed selection strategy yields a large decrease in the number of fixed-interface modes to be retained, which means subsequent CPU time savings.

Prepared using nmeauth.cls

Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

18 (a)

(b) 70

400 Mode rank = 350

350

60 Error Bound (in %)

Mode rank

300 250 200 150 100

40 30 20 10

50 0 0

50

200

400 600 Frequency (Hz)

800

0 0

1000

1% 50

100

150 200 Mode rank

250

300

350

Figure 10. Strategies used to select the fixed-interface modes of the coupling junction (P3 ) occurring in the 3D aircraft fuselage: (a) classic strategy consisting in retaining the modes whose eigenfrequencies are below four times the maximum frequency of the frequency band of interest (this yields 350 modes); (b) WFE-based strategy proposed in Sec. 4 (this yields 107 modes).

(a)

(b)

120

80

Mode rank = 114

70 Error Bound (in %)

Mode rank

100 80 60 40 20 0 0

60 50 40 30 20 10

200

400 600 Frequency (Hz)

800

1000

0 0

1% 20

40

60 Mode rank

80

100

120

Figure 11. Strategies used to select the fixed-interface modes of the coupling junction (P4 ) occurring in the 3D aircraft fuselage: (a) classic strategy consisting in retaining the modes whose eigenfrequencies are below four times the maximum frequency of the frequency band of interest (this yields 114 modes); (b) WFE-based strategy proposed in Sec. 4 (this yields 58 modes).

6.2.3. Forced response computation The FRF of the aircraft fuselage is assessed by means of the WFE-based DSM and RM approaches depicted in Sec. 5. Recall that the periodic structures (P1 ) and (P2 ) (i.e., the stiffened cylindrical shells) are modeled by means of the WFE method (Sec. 3), while the coupling junctions (P3 ) and (P4 ) (conical head and cylindrical part with doors) are modeled by means of static modes and fixed-interface modes (Sec. 6.2.2). Here, the number of right/leftgoing wave modes involved for modeling each periodic structure — i.e., which relates the number of DOFs used to discretize the cross-section — is n = 288, which appears to be large compared to the previous test case. The magnitude of the velocity in the vertical y -direction, at an excited node, is analyzed over the frequency βf = [20 − 200] Hz (Fig 12(a)). Also, the relative error of the WFE solutions is displayed in Fig. 12(b). The relative error is expressed by Eq. (49) when qmeas denotes the vector of displacements/rotations at the excitation point. As it can be seen, the WFE solutions perfectly agree with the FE results. The relative error of the WFE solutions appears smaller than 6% over the whole frequency band, which fully validates the proposed approaches. Again, the WFE-based approaches are compared with the conventional CB method when each subsystem (periodic structure, coupling junction) is modeled by means of static modes and fixedinterface modes. Here, the coupling junctions (P3 ) and (P4 ) are modeled by means of 350 and 114 fixed-interface modes, respectively, as explained in Sec. 6.2.2. Also, two procedures are investigated Prepared using nmeauth.cls

Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

19 (b) 2

10

−30

10

Relative error (in %)

Velocity (dB)

(a) −15

−45

−60

1

0

10

−1

10

−2

−75

50

100 Frequency (Hz)

150

200

10

50

100 Frequency (Hz)

150

200

Figure 12. FRF of the 3D aircraft fuselage (a) and relative error, Eq. (49) (b): FE solution (—–); WFE-based DSM approach (- - -); WFE-based RM approach (- - -).

to model the periodic structures (P1 ) and (P2 ), i.e.: (1) when the fixed-interface modes are selected by means of the rule of thumb that consists in retaining those for which the eigenfrequencies are smaller than four times the maximum frequency of the frequency band of interest — this yields 1, 133 fixed-interface modes for the periodic structure (P1 ) and 350 fixed-interface modes for the periodic structure (P2 ); (2) when the number of fixed-interface modes is equal to that of the right/left-going wave modes of each periodic structure — this yields 288 fixed-interface modes for each periodic structure. The related FRFs are displayed in Fig. 13(a), along with the relative error, Eq. (50) (Fig. 13(b)). In addition, the CPU times and total numbers of physical and generalized DOFs involved in performing the WFE-based approaches and CB methods are listed in Table II. As it can be seen, the WFE-based DSM and RM approaches outperform both CB procedures in terms of CPU times. Also, they appear to be more accurate compared to the CB procedure (2), i.e., when the dimension of the basis of static modes and fixed-interface modes is equivalent to that of the basis of wave modes. The solution provided by the CB procedure (2) exhibits a relative error that reaches 29%, while the related CPU time appears almost 50% greater than those involved in the WFE-based approaches. In other words, the CB method requires a mode basis of large dimension to reach the convergence of its solution. Regarding the CB procedure (1), the relative error is very small, but the related CPU time appears tremendous, i.e., almost 300% greater than those required by the WFE-based approaches. Figure 14 presents a visual comparison of the mean relative errors and CPU times involved in each proposed approach. Again, it is clear that the WFE-based DSM and RM approaches constitute the best compromise between accuracy and performance. To summarize, the WFE-based DSM and RM approaches constitute two efficient numerical tools for assessing the dynamic behavior of coupled systems that involve complex periodic structures. Also note that, in the present case, the CPU time involved in the RM method is smaller than that of the DSM method. This interesting feature has been underlined in Sec. 5.2 and results from the fact that the matrix which is inverted in Eq. (47) has a smaller size than the matrix being inverted in Eq. (43). This highlights the capability of the proposed RM approach for treating complex coupled systems with many DOFs.

As a last advantage of the WFE-based approaches, the spatial distribution of the displacement/rotation fields of periodic structures can be assessed without difficulty at a very small computational cost. This is done by considering the vectors of nodal displacements/rotations on the interfaces, between subsystems, and making use of the wave expansion in Eq. (9). Then, the vector of nodal displacements/rotations, on any substructure boundary k , is expressed as (k)

q



= Φq µ

k−1

Prepared using nmeauth.cls

Φ⋆q µN −k+1





I ⋆−1 Φq Φq µ N

⋆ N Φ−1 q Φq µ I

−1  −1 Φq 0

0 Φ⋆−1 q

"

# (1) qL (N) . (51) qR

Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

20 (b) 2

10

−30

10

Relative error (in %)

Velocity (dB)

(a) −15

−45

−60

−75

1

0

10

−1

10

−2

50

100 Frequency (Hz)

150

10

200

50

100 Frequency (Hz)

150

200

Figure 13. FRF of the 3D aircraft fuselage (a) and relative error, Eq. (50) (b): FE solution (—–); CB method that uses 1, 133 and 350 fixed-interface modes to model the periodic structures (- - -); CB method that uses 288 fixed-interface modes to model the periodic structures (- - -).

(a)

(b) 4

3.5

5

3

4

2.5 CPU time (s)

Mean Relative Error (%)

6

3 2

x 10

2 1.5 1

1 0

0.5

1

2 3 Approach used

4

0

1

2 3 Approach used

4

Figure 14. Bar plots of the mean relative errors (a), and CPU times (b) involved in the proposed approaches regarding the 3D aircraft fuselage: (black) CB method that uses 1, 133 and 350 fixed-interface modes to model the periodic structures; (blue) CB method that uses 288 fixed-interface modes to model the periodic structures; (red) WFE-based DSM approach; (green) WFE-based RM approach. Table II. Total number of DOFs and CPU times involved in the proposed approaches regarding the aircraft fuselage.

Approach FE CB (1, 133 and 350 modes) CB (288 modes) DSM RM

Total number of DOFs 32, 586 3, 867 2, 960 2, 085 2, 085

CPU time (s) — 30, 532 11, 404 7, 577 6, 608

Reduction (%) — – 62.7 75.2 78.4

For instance, the displacement field (y -direction) of the periodic structures (P1 ) and (P2 ), at 123 Hz, is displayed in Fig. 15. The WFE and FE solutions are shown in Fig. 15. As shown, the WFEbased approaches seem to be highly accurate for predicting the spatial dynamics of the coupled system.

Prepared using nmeauth.cls

Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

21 (a)

(b) −6

−6

x 10 6

x 10 6

4

4

2

2

0

0

−2

−2

−4

−4

−6

−6

Figure 15. Displacement field (real part, y -direction) of the periodic structures (P1 ) and (P2 ) occurring in the 3D aircraft fuselage, at 123 Hz: (a) FE solution, (b) WFE-based solution.

7. CONCLUDING REMARKS Two WFE-based approaches have been proposed for modeling complex periodic structures. In this framework, either the dynamic stiffness matrix or the receptance matrix of a periodic structure can be computed in an accurate and very fast way. This has motivated the development of the so-called WFE-based DSM and RM approaches. These procedures use the concept of wave modes that travel in right and left directions along a periodic structure. The consideration of sophisticated systems involving several periodic structures and arbitrary coupling junctions follows from classic FE assembly procedures or domain decomposition techniques, which make the proposed approaches quite simple of use. The CB method has been applied for modeling the coupling junctions in terms of static modes and fixed-interface modes. A WFE-based strategy has been considered for efficiently selecting the fixed-interface modes being retained. This is done by assessing how much they contribute to the system forced response. Such a strategy enables the number of junction modes to be considerably reduced compared to the classic CB strategy that consists in retaining those for which the eigenfrequencies are below a certain frequency limit. The proposed WFE-based DSM and RM approaches have been applied to describe the harmonic response of a simple structure (2D frame) and a complex system (3D aircraft fuselage) that involves truly periodic structures. In both cases, the WFE-based approaches have been proved to be highly accurate in comparison with the conventional FE method. Also, they yield large CPU time savings in comparison with the conventional CB method that makes use of fixed-interface modes for modeling the periodic structures and coupling junctions. The numerical approaches proposed in this paper have the potential to be extended to the analysis of vibroacoustic systems that concern periodic structures with internal fluid, which will be the object of future work.

Acknowledgements The authors are grateful to the government funding agency Fundac¸a˜ o de Amparo a` Pesquisa de S˜ao Paulo, FAPESP, for the financial support provided for the present research work through processes 2010/17317-9 and 2013/23542-3. The authors are also thankful to the Program “C´atedras Francesas na UNICAMP”.

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