Where 0 < σ<1 is a user-defined limit value. The user must also set the relative weights W1 and W2, and define which norm to use. If taking the ` 2-norm, (9) can be generalized further to the quadratic formJ= εTWε popular in engineering sciences, where εT =[(mG−mL) (MT G−MT L )] andWis a positive definite weighting matrix. 2.2 Reduction of candidate sets The number of index sets I satisfying I ⊂S is vast. If a brute-force method is to be used for the solution of this problem, with n analytical modes under consideration, the number of combinations Nto be compared is N= n ∑k=1 n! (n−k)!k! (11) Even for a relatively moderate number of FE modes, this becomes prohibitive computational-wise. For instance to include up to 30 FE modes as basis would require considering just over one billion combinations. To examine the entire solution space must therefore be considered impractical, if not outright impossible. We propose to circumvent this through considering only combinations of analytical eigenvectors such that the corresponding eigenvalue indices make up an interval I :=[ jmin, jmax] (such that I ⊂S). This reduces the number of combinations needed, and yet it still requires the evaluation of a great many uninteresting sets, such as ones that do not contain the analytical eigenvector which correlates best with the mode to be expanded. Thus, the concept of an iterative method of candidate set construction arose.1 2.2.1 Constructing candidate sets The candidate set construction algorithm proposed here is based on closed frequency intervals; that is, to include all analytical modes in a frequency range. The optimization problem of (3) is then reduced to finding the limits of that frequency band. The algorithm consists of two steps; the first step expands the candidate set, while the second translates the frequency band in the frequency domain. 1 An idea inspired by Kammer’s method of Effective Independence [12] for sensor placement. A candidate set construction algorithm based on a methodology more closely resembling the EFI, in which the analytical eigenvector making the largest impact on the cost function was added without restricting the candidate sets to be intervals, was also tested. In this context, it did not perform as good as the algorithm in the subsequent section, however, and was left out. 6
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