engleski

Approximation of quadratic eigenvalue problem and application to damping optimization

In this thesis we study the parameter dependent Hermitian quadratic eigenvalue problem (PEQP) given by $(\lambda^2(\mathbf p) M(\mathbf p)+\lambda(\mathbf p) C(\mathbf p) +K(\mathbf p)) x(\mathbf p)=0$, where $\mathbf p \in \mathbb R^m$ is a vector of parameters. Through this thesis matrices $M(\mathbf p), \, C(\mathbf p), \, K(\mathbf p)\in \mathbb R^n $ arise from corresponding (vibrational) mechanical system represented by $M (\mathbf p)\ddot{; ; ; q}; ; ; (\mathbf p ; t) + C(\mathbf p)\dot q(\mathbf p ; t) + K(\mathbf p) q(\mathbf p ; t) = 0$ and represent mass, damping, and stiffness, respectively. Usually matrices $M(\mathbf p)$, $K(\mathbf p)$ are Hermitian positive definite and $C(\mathbf p)$ is Hermitian positive semidefinite matrix. After a brief introduction and problem formulation we give three approximation approaches for efficient computation of eigenpairs. These approaches preserve structure and allow computation of eigenpairs, for different sets of parameters, that is computationally efficient and at the same time they provide satisfactory relative accuracy. The first approach is based on dimension reduction, the second on first order approximation, while the third one uses modified Rayleigh quotient iterations for structured damping matrices. For the first and the third approach we need to linearize PQEP. Within the first approach we distinguish two very important cases. In the first case we consider efficient approximation for eigenvalues for the selected part of the undamped spectrum. In the second case, we consider efficient approximation of all eigenvalues. The first order approximation considered in second approach is based on Taylor’s theorem and it is efficient for eigenvalue computation when the change in parameter is small enough. In the third approach we provide an efficient method of eigenvalue computation of the diagonal-plus- rank-one matrices (DPR1), and show how one can apply this method on corresponding linearized eigenvalue problem, by exploiting the structure of the damping matrix. Numerical experiments confirm efficiency and accuracy of these approahes. Further on, we use obtained first order approximation bounds for efficient estimations of the gap functions that appear in different perturbation bounds for the quadratic eigenvalue problem. These estimations of the gap functions are based on removing perturbed quantities from them. Accuracy and efficiency of these estimations are given in numerical examples. Last we focus on damping optimization. Two different optimization criteria are considered: minimization of total average energy, and frequency isolation, which will determine the damping matrix which ensures vibration decay is as fast as possible. While dealing with the minimization of total average energy we provide an approximation of the solution of the structured Lyapunov equation, which can be efficiently computed. Frequency isolation is eigenvalue based criterion so we use obtained eigenvalue approximations to determine the optimal damping, while the areas from which we isolate the frequencies are ellipses with centers on the imaginary axis. Both approaches are illustrated on numerical examples.

quadratic eigenvalue problem ; vibrational mechanical systems ; eigenvalue approximation ; dimension reduction ; first order approximation ; modified RQI ; estimation of gap function ; damping optimization ; frequency isolation ; total average energy minimization

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