3 Optimal Selection of Artificial Boundary Conditions for Model Update and Damage Detection – Part 1: Theory 21 and Eq. (3.3) is satisfied for any choice of the vector fu2g, which contains the non-basic parameters. Throughout, we will focus on the choice, fu2gDf0g (3.8) and which results in the following solution: ˚uQR DŒS1 1 ˚ !2 (3.9) We denote this basic solution with the subscript “QR” as this solution will be calculated from the QR decomposition of [S] with column pivoting [27]. As we will see, the set of pivot columns selected by the QR algorithm determines the ability of an ABC set to provide a localization solution. 3.5.1 Least-Norm Versus Basic Solutions We first illustrate some distinguishing characteristics of the two important classes of solution to Eq. (3.3) described above. We will illustrate the difference betweenfulng andfuQRg with a simple numerical example. For [S], we will assume a random 3 by 4 matrix, corresponding to an updating problem using three modes and four updating parameters: ŒS D 2 4 0:6557 0:9340 0:7431 0:1712 0:0357 0:6787 0:3922 0:7060 0:8491 0:7577 0:6555 0:0318 3 5 (3.10) We assume that a single error exists in the first updating parameter, and hence the “true” updating solution vector is: fugD 1 0 0 0 T (3.11) We first calculate the associated “true” mode frequency changes (i.e. the right-hand side), which is fıƒg D ŒS fug D Œ0:6557; 0:0357; 0:8491 T. Assuming now that fug is unknown, the problem to be solved is 8 < : 0.6557 0.0357 0.8491 9 = ; D 2 4 0:6557 0:9340 0:7431 0:1712 0:0357 0:6787 0:3922 0:7060 0:8491 0:7577 0:6555 0:0318 3 5 8 ˆ < ˆ : u1 u2 u3 u4 9 > = > ; (3.12) We will compare the results of the two solutions, uln anduQR. The two solutions are: fulngD 0.9928 0.0503 0.0669 0.0116 T ˚uQR D 1 0 0 0 T (3.13) and jjulnjj D0:9964 and jjuQRjj D1. Comparing these two solutions to the true solution of Eq. (3.11), it is seen that in this case of a well-conditioned, full rank S, the least-norm solution uln fails to provide a localization (or magnitude) solution, while the basic solution uQR (calculated using the QR decomposition with column pivoting) provides an exact localization (and magnitude) solution. We will now repeat this calculation, with the only change being the location of the error. If we place the error in the third update parameter, i.e. fugD 0 0 1 0 T (3.14) We again first calculate the associated mode frequency errors, which arefıƒgDŒS fugDŒ0:7431 0:3922 0:655 T.Assuming that fug is unknown, the system to be solved is, 8 < : 0.7431 0.3922 0.6555 9 = ; D 2 4 0:6557 0:9340 0:7431 0:1712 0:0357 0:6787 0:3922 0:7060 0:8491 0:7577 0:6555 0:0318 3 5 8 ˆ < ˆ : u1 u2 u3 u4 9 > = > ; (3.15)
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