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By Douady R., Douady A.

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Claeys et al. a b c d d H L b a e c Fig. 2 Unit cell of a honeycomb core with added resonating structure. 1 Dimension of the unit cell in Fig. 2 L H a 1 30 mm 15 mm 15 L b c d e 1 8L 1 5L 2 15 L 2 15 L Fig. 4 The unit cell of the honeycomb core with added resonating structure is shown in Fig. 2. 1 lists the dimensions of the unit cell. The resonating structure comprises two parts with different thickness: the centre and the wings (Fig. 5 mm of thickness, while the wings have a 9 mm thickness. The resonator structure can be seen as a mass-spring structure where the centre is a three-legged spring and the wings act as mass.

The main idea is to use available factorization information from the eigenvalue solution context and try to use this information as a preconditioner to improve the performance of the iterative solution process. 29), along with the projection matrix, Pm, is convenient at this point, namely, Pm ¼ I À fm fTm M; ðK À lMÞ ¼ nunc n X X o2s À l o2s À l Mfs fTs M þ Mfs fTs M: m m s s s¼1 s¼n þ1 unc And use of this specific projection matrix, Pm, transforms the operator matrix to ~ ¼ PT ðK À lMÞPm ; A m ~¼ A n X s¼nunc þ1 o2s À l Mfs fTs M; ms 30 U.

Repeating the equations from Sect. 1, namely,   Ks À o2f Ms ds ¼ Ào2f Ksf ff ; À Á Kf À o2s Mf df ¼ ÀKfs fs : These two equations are basically standard linear equations of the form, Ax ¼ b. However, the challenge is that, for each mode vector, the associated correction has to be solved for a shifted operator matrix which differs for different values of the shift frequency. Use of direct solution methods puts a heavy burden on the computations. The reason is that every time a different correction vector is solved for, a new factorization and a forward-backward substitution is necessary.

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