Two algorithms for the solution of generalised eigenvalue problems arising in structural dynamics

Mante Zemaityte (The University of Manchester)

ATB Frank Adams 1,

 To accurately describe the behaviour of a structure under dynamic 
loading, structural engineers require the smallest number \(\ell\) of 
eigenvectors \(x_j\), \(j=1, \ldots, \ell\), of the symmetric definite 
generalized eigenvalue problem \(Kx = \lambda Mx\) with stiffness matrix 
\(K\) and mass matrix \(M\), that satisfy

 \(\sum\limits_{j=1}^\ell \phi(x_j)>0.9,\quad\phi(x_j) = \frac{(x_j^TMr)^2}{r^TMr},\)

where \(r\) is the rigid body vector.
I will present two state-of-the-art algorithms, the shift-and-invert 
Lanczos (SIL) and the automated multi-level substructuring (AMLS) 
algorithm, for the solution of generalized eigenvalue problems arising 
from structural dynamics, in an attempt to balance the efficiency of the 
algorithms and the accuracy of the dominant eigenpairs.

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