It should be mentioned at this point that after the first iteration ωf and ωs represent the same values which are the eigenfrequencies of the coupled system representation. So that the mode vectors and eigenfrequencies are updated simultaneously. Step 3: Energy based selection, transformation matrix and Gram-Schmidt orthogonalization The importance of the corrections to properly approximate the coupled modes depends on the level of coupling and geometrical properties. In order to capture only important ingredients, computation of a relative strain energy based factor is proposed per physics. The factors are given as the ratio of the strain energy norm of the correction vectors with respect to their representative mode counterparts. Since in IRCA, at convergence, we expect that these correction vectors complement and correct the starting uncoupled basis vectors, we compare their energy norms to the energy norms of the modes from which these corrections are computed (see (15) and (16)). The importance of corrections can thus be estimated by εp = s f Kf φs Ks , (17) εu = f s Ks φf Kf . (18) We select important correction vectors for the acoustic by setting εp >0.2 and similarly, for the structural side, εu >0.2. The comparison of the relative energy ratios with respect to the value 0.2 is based on current practical observation, however the driving idea is to choose the correction components that trigger important energy levels with respect to their sources. The other correction vectors that do not satisfy the selection criteria are not included in the basis. Note that the basis size is not fixed in this manner, but can vary between iterations. After the energy based selection, the projection matrix Tc is formed with the selected correction vectors (cindicating that it will include the corrections computed in step 2) and is orthogonalized by the modified gram-schmidt orthogonalization algorithm to guarantee the linear independence of the vectors in the subspace where the best estimates of the eigenpairs will be searched for, namely, Tc = φs 0 f s 0 0 φf 0 s f , orth(Tc) −→Tc ⊥, (19) Step 4: Reduce system equations (interaction problem) Using now the orthogonalized reduction basis, Tc ⊥, to reduce the symmetrized eigenvalue problem, one can write the so-called interaction problem [12], Tc ⊥ TMsystemτTc ⊥ Mred ¨η+Tc ⊥ TKsystemTc ⊥ Kred η=0. (20) Umut Tabak and Daniel J. Rixen 268
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