Fast Computation of Optimal Damping Parameters for Linear Vibrational Systems

Abstract

We propose a fast algorithm for computing optimal viscosities of dampers of a linear vibrational system. We are using a standard approach where the vibrational system is first modeled using the second-order structure. This structure yields a quadratic eigenvalue problem which is then linearized. Optimal viscosities are those for which the trace of the solution of the Lyapunov equation with the linearized matrix is minimal. Here, the free term of the Lyapunov equation is a low-rank matrix that depends on the eigenfrequencies that need to be damped. The optimization process in the standard approach requires O(n3) floating-point operations. In our approach, we transform the linearized matrix into an eigenvalue problem of a diagonal-plus-low-rank matrix whose eigenvectors have a Cauchy-like structure. Our algorithm is based on a new fast eigensolver for complex symmetric diagonal-plus-rank-one matrices and fast multiplication of linked Cauchy-like matrices, yielding computation of optimal viscosities for each choice of external dampers in O(kn2) operations, k being the number of dampers. The accuracy of our algorithm is compatible with the accuracy of the standard approach.

Fast optimization of viscosities for frequency‐weighted damping of second‐order systems

“…Moreover, we have also observed that when eigenvalues are close to each other, the method of Jakovčević Stor et al. [19] often gets stuck oscillating between approximations in such clusters of eigenvalues. To address these shortcomings, we propose two key modifications, namely, to completely forgo using standard RQI and to introduce a new dynamic step‐size procedure in order to steer our MRQI‐based procedure towards a single eigenvalue in a cluster.…”

“…Thus, we instead consider an approach for DPR1Csym matrices that is inspired by a different approach for DPR1 real symmetric matrices [19]. The method of Jakovčević Stor et al [19] computed eigenpairs using a combination of standard and modified Rayleigh quotient iterations (RQI and MRQI, respectively), but in our setting, the eigenvalues of Equation (3.1) will be complex (and real axis symmetry is not guaranteed), and we have observed that standard RQI often does not converge. Moreover, we have also observed that when eigenvalues are close to each other, the method of Jakovčević Stor et al [19] often gets stuck oscillating between approximations in such clusters of eigenvalues.…”

“…Of course, the essential properties needed to employ such methods are not present for DPR1Csym matrices, the most important being that the diagonal elements of D and the eigenvalues of A are no longer interlaced for the complex problem, since these values are now in the complex plane as opposed to on the real line. Thus, we instead consider an approach for DPR1Csym matrices that is inspired by a different approach for DPR1 real symmetric matrices [19]. The method of Jakovčević Stor et al.…”

“…Second, as the cost of our optimization process is actually dominated by solving a sequence of related QEPs, where 𝐶(𝑣) is changing as the viscosity parameters are optimized, we also propose a fast algorithm to solve this sequence of QEPs. Our method, which is many times faster than using standard eigensolvers for QEPs and can be considered an extension of Jakovčević et al [19] for computing eigenvalues of diagonal-plus-rank-one (DPR1) complex symmetric (DPR1Csym) matrices, works by exploiting the fact that changing the viscosity parameters is equivalent to making a low-rank update to 𝐶(𝑣). Since such structure is not inherent to our problem, we expect that our technique for solving such sequences of QEPs could be quite beneficial in other applications as well.…”

“…Our method, which is many times faster than using standard eigensolvers for QEPs and can be considered an extension of Jakovčević et al. [19] for computing eigenvalues of diagonal‐plus‐rank‐one (DPR1) complex symmetric (DPR1Csym) matrices, works by exploiting the fact that changing the viscosity parameters is equivalent to making a low‐rank update to $$C(v)$$. Since such structure is not inherent to our problem, we expect that our technique for solving such sequences of QEPs could be quite beneficial in other applications as well.…”