118 F. M. Gruber et al. 1 3 5 7 9 111315171921232527293133 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Number k of eigenvalue λred,k Relative error εrel, ,k of real part Craig and Ni’s method Liu and Zheng’s method de Kraker and van Campen’s method Liu and Zheng’s reduction basis, primal assembly Third order reduction (a) 1 3 5 7 9 111315171921232527293133 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Number k of eigenvalue λred,k Relative error εrel, ,k of imaginary part (b) Fig. 11.4 Relative error of the real and imaginary parts of the approximated eigenvalues λred of the clamped beam of Fig. 11.1. The relative errors of the 34 eigenvalues with the lowest absolute value are shown using Craig and Ni’s method (CN), Liu and Zheng’s method (LZ) and de Kraker and van Campen’s method (KC). Additionally, the combination of Liu and Zheng’s reduction basis and primal assembly (LZ,KC) as well as the third order reduction (TO) are shown. The number of DOFs of the assembled system according to Craig and Ni’s method as well as Liu and Zheng’s method is nred,CN =nred,LZ =38 and according to de Kraker and van Campen’s method, the combination of Liu and Zheng’s reduction basis and primal assembly, as well as the third order reduction is nred,KC =nred,LZ,KC =nred,TO =42. (a) Relative error εrel, ,k of real part to eigenvalue λred,k. (b) Relative error εrel, ,k of imaginary part to eigenvalue λred,k
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