The advantage on the use of these matrices is that they are reduced in size and symmetric. Thus they result in cheap computations and standard well-developed symmetric eigensolvers might be employed (QR for instance). Step 5: Convergence check A convergence check is performed with the previous eigenfrequencies, namely, ε= ωj −ωj−1 2 ωj 2 , (21) where ωj is the set of approximate frequencies obtained at iteration j. The iteration is stopped if the frequency error estimate (21) is below the desired tolerance εpre. If not, the iteration continues by going to Step 2. The remaining issue is to decide how to transform the current mode approximates to mono-physical modes needed to restart at Step 2 where corrections are computed. In this work, we take the first Napproximate modes and split them into acoustic and structural parts. The whole algorithm has just been outlined above. However in the context of this research paper, we are looking for some vectors (or pseudovectors) to be able to enrich the standard Craig-Bampton reduction space. To accomplish this task, one has two options in the context of IRCA. The first option is to stop IRCA iterations at a higher tolerance value than the preset tolerance value. The second option is to stop the iterations after a small iteration count (2 or 3, but which is essentially smaller than the iteration count on convergence) to end up with the enrichment vectors. These options are intended to limit the cost of computation of the correction vectors. However, it is still determined on user’s choice. 3 Extension of the Craig-Bampton based substructuring to vibro-acoustic problems Application of the Craig-Bampton method to vibro-acoustic problems has been illustrated in [10]. Extension of this reduction method with an enrichment is going to be outlined in this section. The degrees of freedom (dofs) are separated into interface and internal dofs. The interface dofs are denoted by “b” superscript and the internal dofs are denoted by the “i” superscript. The schematic representation of the problem is provided in Figure 1 on two components, A and B. The source of the enrichment vectors are the outcomes of IRCA that has just been explained in the previous section. The partitioning of the dofs on the component level is represented by Mbb Mbi Mib Mii M ¨qb ¨qi + Kbb Kbi Kib Kii K qb qi = Fb Fi . (22) Globally enriched substructuring techniques for vibro-acoustic simulation 269
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