Let us recall that the symmetrized vibro-acoustic problem (see previous section) is given as, (Msystemτ¨x+Ksystemτx)=0, (11) where τis provided in equation (3). Approximating the solution with Ritz vectors stored in Tresults in x=Tη. (12) The heart of the problem in model reduction studies is the selection of the proper ingredients for the projection matrix, T. The iterative correction algorithm fills T with the uncoupled system modes and their corrections for the first iteration loop and updates the frequencies and modes simultaneously. Important correction vectors are selected on the basis of their relative contribution to the strain energy. The short general outline is presented next. 2.3.1 General outline Let us call Nthe number of modes we want to find. Step 1: Compute the first Nstructural and acoustic modes Ks −ω2 s Ms φs =0 →ω2 s , φs, (13) Kf −ω2 f Mf φf =0 →ω2 f , φf . (14) Step2: Computation of the correction vectors In the vibro-acoustic problem (2), we see that the uncoupled mechanical modes (u= φu, p=0) fully satisfy the mechanical equilibrium (second line of (2)), but not the acoustic balance (first equation of (2)). So let us use the acoustic balance to estimate the acoustical counterpart to φs as Kf −ω2 s Mf s f = ω2 s KT sf φs, (15) where s f is the acoustic correction to the mechanical mode. Obviously with this correction the mode u=φs, p= s f satisfies the acoustic balance, but no longer the mechanical equilibrium. But we hope that the correction so-computed gives valuable information on what behavior for the acoustics should be included in the reduction space. Following the same reasoning for the purely acoustic modes we can compute structural corrections from the mechanical equilibrium: Ks −ω2 f Ms f s =−Ksf φf . (16) Globally enriched substructuring techniques for vibro-acoustic simulation 267
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