WAVE FINITE ELEMENT METHOD FOR VIBRATION OF PERIODIC STRUCTURES SUBJECTED TO EXTERNAL LOADS

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1 6th European Conference on Computational Mechanics (ECCM 6) 7th European Conference on Computational luid Dynamics (ECD 7) June 2018, Glasgow, UK WAVE NTE EEMENT METHOD O VATON O PEODC STUCTUES SUJECTED TO EXTENA OADS T. HOANG 1, D. DUHAME 1 AND G. OET 1 1 aboratoire Navier, UM 8205, Ecole des Ponts ParisTech, STTA, CNS, UPE Champs-sur-Marne, rance tien.hoang@enpc.fr, denis.duhamel@enpc.fr and gilles.foret@enpc.fr Key words: Wave inite Element, Vibration, Periodic, Waveguide, Computing Methods Abstract. The wave finite element method (WE) for the vibration of waveguides and periodic structures bases on the decomposition of vectors of degree of freedom (DO) into left and right waves. This techniue permits to reduce all DO inside the periodic structure. However, this method cannot be applied easily if the periodic structure is subjected to complex or density loads. This article presents an extended WE for any type of loads. irstly, the dynamic euation is rewritten to separate the vectors of loads and DO. Then, by using the same wave base as for the free-loaded structure, we can obtain a decomposition of DO with a new component which corresponds to the loads. inally, this decomposition is applied to the classical approaches of WE. or the dynamic stiffness matrix (DSM approach), it is shown that the external loads have no contribution to the global matrix but they lead to an euivalent load in the dynamic euation. Otherwise, the wave analysis (WA approach) is represented by a new component which is the wave amplitudes of the loads. Some computations on simple structures show the efficiency of the method. 1 ASC AMEWOK The wave finite element method has been developed to calculate the dynamic behavior of periodic structures and waveguides 1. ecently, this techniue has been used as a reduced model for a complex structure 2, 3. However, this method can not be applied easily for periodic structures subjected to external loads and this is objective of this article. We consider a periodic structure which contains N substructures as shown in igure 1. y using the finite element method, the dynamic euation of a substructure can be written in matrix form D = with D = K + jωc ω 2 M is the dynamic stiffness matrix, and, are the vectors of degrees of freedom and nodal loads. Then, we can

2 igure 1: Structure with a periodic substructure rewrite the dynamic euation to separate the boundaries and inner DO as follows D D D D D D = (1) D D D where, and denote for the left, right boundaries and inner nodes of the substructure. Then, we can reduce the inner nodes by rewriting the first row of euation (1) as follows = D D (2) D 1 Then, by substituting the aforementioned euation into the second and the third rows of euation (1), we obtain D D 1 D D + D + D = D D + D + D (3) = D D 1 n the other way, we can rewrite D D D + D D D = (4) where D = D D D 1 D D = D D D 1 D D = D D 1 D = D D D 1 D D = D D D 1 D D = D D 1 We see that euation (4) presents a relation between the DO and nodal loads and at the left and right boundaries of a substructure, it contains a term of which is zero when the substructure is free-loaded. When the structure is periodic, this euation holds for all substructure. or the two consecutive connected substructures (n) and (n + 1), we have (5) (n) (n) = (n+1) + (n+1) = (n) (6) 2

3 where (n) is the external nodal load at the right boundary of the cell (n). y combining euations (5) and (6), we obtain (n+1) (n+1) = D (n) D f (n) (n) + S (n) (n) (7) where S = D 1 D D D D 1 D D 1 D D 1 D, D f = D 1 D D D D 1 D (8) We can rewrite euation (7) as follows where u (n) = (n) (n) u (n+1) = Su (n) + b (n) (9), b (n) = D (n) D f (n) (n) Euation (9) presents a relation between the substructure (n) and its next substructure (n + 1). Here b (n) presents the external loads on the substructures (n) (when the substructure is free, b (n) = 0). Euation (9) presents also a relation of a geometric series with regard to (n). Therefore, for a structure including a series of N periodic substructures, we have (10) n 1 u (n) = S n 1 u (1) + S n k 1 b (k) (11) u (N+1) = S N n+1 u (n) + S N k b (k) (12) The aforementioned euations are the relation between the substructure (n) and the first and the last substructures. Next, we will develop these expressions with a wave base decomposition. emark: (n) in euation (6) is the external nodal load at the right boundary of the substructure. Therefore, this load is considered applying in the right end of the periodic structure but it is not included on the left end. This will explain the different expressions for the left and right boundary in the DSM and WA approches presented in section 3. 2 WAVE DECOMPOSTON We are looking for wavemodes {(µ j, φ j )} j which are the eigenvalues and eigenvectors of the matrix S such that Sφ j = µ j φ j (13) k=n 3

4 Due to the symplectic nature of the matrix S 4, we consider the eigenproblem of the transformation S + S 1 which yields eigenvalues of the form λ j = µ j + 1/µ j given by ( N J T + JN T ) λ j J T z j = 0 (14) where = 0 n D 0, N = D 0 (D + D ) n 0 n, J = n 0 (15) Then, each pair of eigenvalues (µ j, µ j) can be computed analytically by x 2 λ j x + 1 = 0 (16) Also, the eigenvectors corresponding to these eigenvalues are computed by the closed-form expressions n 0 φ j = w D j, φ n 0 j = w j (17) n D n where w j = J( T µ jn T )z j and w j = J( T µ j N T )z j. f we note Φ = φ 1 φ n and Φ = φ 1 φ n, we have a wave base {Φ, Φ } of the transformation S. We can also separate the components of the wave base corresponding to and as follows Φ Φ Φ = Φ = (18) Φ We can decompose each vector of euation (9) in this wave base as follows Φ u (n) = ΦQ (n) + Φ Q (n) b (n) = ΦQ (n) + Φ Q (n) (19) Then, by replacing the aforementioned euation into euations (11) and (12), we obtain n 1 Q (n) = µ n 1 Q + µ n k 1 Q (k) Q (n) = µ N+1 n Q k=n µ k+1 n Q (k) (20) We see that the DO of a substructure are represented by the two wave amplitudes Q (n) and Q (n). These amplitudes depend on the wave amplitudes of the first and the last substructures (Q and Q ) and the wave amplitudes of the external loads on the middle substructures Q (k) and Q (k), which are calculated from euation (19). Q (n) = Φ T Jb (n), Q (n) = Φ T Jb (n) (21) 4

5 y substituting euation (10) into the aforementioned euation, we obtain Φ T Jb (n) = ( ) Φ T D f Φ T (k) D Φ T Jb (n) = ( Φ T D f Φ T D ) (k) Φ T (k) Φ T (k) n addition, we have the relation between the components of the wave base as follows (see 3) Φ = D Φ + D Φ µ = (D Φ + D Φ µ) Φ = D Φ + D Φ µ = ( D Φ + D Φ ) (23) µ y substituting euations (8) and (23) into euation (22), we obtain Q (k) Q (k) = ( µφ T ) D + Φ T (k) D = ( µ Φ T D + Φ T D ) (k) Φ T (k) Φ T (k) emark: n general, the wave base is not normalized automatically after computing the eigenproblem in euation (14). The ill-conditioned matrix S can influence the orthogonality of the wave base. We can resolve this problem by considering the weighting matrix as follows Ψ = Φ T JΦ = Φ T Φ Φ T Φ (25) Ψ = Φ T JΦ = Φ T Φ Φ T Φ t is important to remark that (22) (24) Ψ = Ψ T (26) Using this weighting matrix leads to modify the wave amplitude of the external loads as follows: ΨQ (n) = Φ T Jb (n), Ψ Q (n) = Φ T Jb (n) (27) 3 ANAYSS O A COMPETE STUCTUE 3.1 DSM approach The DSM approach is to establish the link between the DOs and nodal loads of the left and right ends of the periodic structure which are components of u (1) and u (N+1). or the first and the last substructure, we obtain the following results from euations (19) and (20) u (1) = ΦQ + Φ µ N Q Φ u (N+1) = Φµ N Q + Φ Q + Φ N µ k Q (k) µ N k Q (k) The aforementioned euation can lead to the following result (see Appendix A) (1) (1) (N+1) = D T (N+1) + T (29) 5 (28)

6 where D T, T are the euivalent dynamic stiffness matrix and the external loads applied on the structure which are calculated by D 0 Φ D T = + T µ N Φ T 1 0 D Φ T µ N Φ T (30) Φ T µ N 1 Φ T Φ T µφ T D 0 Φ T µφ T Φ T µ N 1 Φ T 0 D with T denotes for the inverse of the transpose matrix, and Φ T = T µ N Φ T 1 Φ T µ N Φ T Φ T µ N k 1 Φ T Φ T µ N k Φ T Φ T µ k+1 Φ T Φ T µ k Φ T D (k) D (k) + (k) (31) Euation (29) presents the relation between the DO and the loads of the periodic structure. When T = 0 (which is deduced by (k) = (k) = 0), we obtain the same relation for the free loaded structure which has been studied in different publications 2. We can combine this relation with the DSM of the rest of the structure to get the reduced DSM of the whole structure. 3.2 WA approach The aim of the WA approach is to calculate the response via the wave amplitudes {Q, Q } by using the boundary conditions at the left and right ends of the structures. y combining euation (19) and (20), we obtain n 1 (n) = Φ µ n 1 Q + Φ µ N+1 n Q + Φ µ n k 1 Q (k) n 1 ± (n) = Φ µ n 1 Q + Φ µ N+1 n Q + Φ µ n k 1 Q (k) Φ k=n Φ k=n µ k+1 n Q (k) µ k+1 n Q (k) rom the dynamic stiffness matrix of the substructures at the left and right ends of the periodic structure, we can write (1) (32) = D(1) + D 0 + D 0 (0) (33) where D, D, D are calculated from the dynamic euation of the first substructure (see 3) and (0) is the external load on the left end of the periodic structure. y using the wave decomposition in euation (32), we obtain (1) = Φ Q + Φ µ N Q Φ µ k Q (k) (1) = Φ Q + Φ µ N Q 6 Φ µ k Q (k) (34)

7 Then, by substituting the aforementioned euations into euation (33), we have (Φ DΦ ) Q = ( ) ( ) DΦ Φ µ N Q Φ µ k Q (k) + D 0 + D 0 (35) are calculated by euation (24). n the similar way, for the ending sub- with Q (k), Q (k) structure, we get (N+1) = D (N+1) + D 0 + D 0 (36) where D, D, D are calculated from the dynamic euation of the last substructure. Then, we obtain the similar result (Φ DΦ ) Q = ( ) ( ) DΦ Φ µ N Q Φ µ k Q (k) + D 0 + D 0 (37) y combining euations (35) and (37), we obtain Q = C µ N Q µ k Q (k) +, (38) where Q = C µ N Q + µ N k Q (k) + (39) C = DΦ Φ 1 DΦ Φ, C = D Φ + Φ 1 D Φ + Φ, = DΦ Φ 1 D 0 + D 0, = D Φ + Φ 1 D 0 + D 0 y combining euations (38) and (39), we obtain AQ = (40) where Q = Q T Q T T and A = n C µ N Cµ N n, = C N µk Q (k) + C N µn k Q (k) (41) Euation (40) permits to calculate the wave amplitudes {Q, Q } and then the response is calculated by euation (32). 4 EXAMPES et s consider a beam of width a = 0.1m, thickness of b = 0.01m and length = 2m. The material parameter is given by the Young modulus of 30GPa and the mass density of 2200kg/m 3. The beam is fixed at the left boundary and it is subjected to a pressure p in 7

8 igure 2: ixed-free beam subjected to dynamic loads an interval between 0.3 and 0.4m as shown in igure 2. n this structure, the boundary condition is the following (1) = 0, (N+1) = 0 (42) We can use the DMS or WA approaches presented in section 3 by excluding the first and the last substructures. However, we can also include them to the wave analysis by substituting the boundary condition into euation (32) and obtain Φ Q + Φ µ N Q Φ Φ µ N Q + Φ Q + Φ N µ k Q (k) = 0 µ N k Q (k) = 0 (43) Then, we can rewrite to obtain euation (40) with A = n Φ 1 Φ µ N Φ 1 Φ µ N, = n Φ 1 Φ Φ 1 Φ N µk Q (k) N µn k Q (k) (44) The beam is considered of 20 substructures which are suares of dimension a = 0.1m and the same thickness. With the external load at the third substructure, we have Q (k) Q (k) = = 0, k 4. y using the finite element method, we obtain the DMS of the substructure with the mesh of 10x10 elements. igure 3 show that the WE has the same uality as the EM. Moreover, the calculation time for WE is 4.08s while 58.22s for EM, euivalent to 93% of time reduction. We take another example of a pipeline under pressure as shown in igure 4. The pipeline is a cylindre of radius 1m and thickness 5mm, made of steel with Young modulus 210GPa and Poisson coefficient of 0.3. The cylindre has two fixed ends and it is subjected to a dynamic pressure. igure 5 presents a comparison of the results between the finite element method and the wave finite element method. The calculation time is reduced from 1277,7s with EM to 332,8s with WE, that means a reduction of 76% of calculation time. 8

9 Modulus of displacement (m) EM WE reuency (Hz) igure 3: esponse of the fixed-free beam igure 4: ixed-fixed pipeline subjected to dynamic pressures Modulus of radical displacement (m) EM WE reuency (Hz) igure 5: esponse of the pipeline 9

10 5 CONCUSON This article presents a new development of WE for the periodic structure subjected to external loads. t is significant because it open more application domain for WE. The numerical application once again conclure the efficience of WE in term of calculation time in comparing with classical EM. ACKNOWEDGMENTS This work is supported by a cooperation with Eurotunnel Group. EEENCES A 1 Zhong W.X and Willians. W. On the direct solution of wave propagation for repetitive structures, J Sound Vib. (1995), 181(3): Duhamel D., Mace.., rennan M.J. inite element analysis of the vibrations of waveguides and periodic structures. J Sound Vib. (2006), 294(1 2): Mencik J-M, Duhamel D. A wave finite element-based approach for the modeling of periodic structures with local perturbations. inite Elem Anal Des (2016) 121: Mencik JM, Duhamel D. A wave-based model reduction techniue for the description of the dynamic behavior of periodic structures involving arbitrary-shaped substructures and large-sized finite element models. inite Elem Anal Des (2015) 101:1 14. CACUATON O DMS APPOACHE y using the othogonality of the wave base, we obtain the following result from euation (28) Φ T Ju (N+1) = µ N Φ T Ju (1) + µ N k Q (k) Φ T Ju (1) = µ N Φ T Ju (N+1) µ k Q (k) y decomposing the components corresponding to and of u (1) and u (N+1), we get µ N Φ T (1) + Φ T (N+1) = µ N Φ T (1) + Φ T (N+1) + µ N k Q (k) Φ T (1) Then, we can write µ N Φ T Φ T (1) Φ T µ N Φ T (N+1) + µn Φ T (N+1) = Φ T (1) + µn Φ T (N+1) + = µ N Φ T Φ T 10 Φ T µ N Φ T (1) (N+1) + µ k Q (k) (45) (46) µ N k Q (k) µ k Q (k) (47)

11 y substituting euation (24) into euation (47), we obtain µ N k Q (k) ( ) µ µ k Q (k) = N k µ 1 Φ T D + Φ T D µ N k Φ T µ ( ) k µφ T D + Φ T D µ k Φ T µ = N k 1 Φ T µ N k Φ T D 0 µ k+1 Φ T µ k Φ T D (k) (k) (k) (k) (48) n a similar way, we can write (see 2) µ N Φ T Φ T Φ T µ N Φ T = + µ N Φ T Φ T µ N 1 Φ T Φ T µ N Φ T µφ T µφ T µ N 1 Φ T D 0 0 D D 0 0 D (49) y substituting euations (48), (49) into euation (47), we obtain the result in euation (29). 11