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

TL;DR

This paper introduces a fast optimization framework for viscosity tuning in second-order vibrating systems, combining nonsmooth constrained optimization with a novel eigensolver to efficiently damp undesirable frequencies.

Contribution

It presents a new eigensolver exploiting matrix structure and a constrained optimization approach for frequency-weighted damping, improving efficiency and stability in second-order systems.

Findings

The new eigensolver is significantly faster than standard methods.

The optimization framework effectively dampens targeted frequency bands.

Numerical examples demonstrate the method's efficiency and accuracy.

Abstract

We consider frequency-weighted damping optimization for vibrating systems described by a second-order differential equation. The goal is to determine viscosity values such that eigenvalues are kept away from certain undesirable areas on the imaginary axis. To this end, we present two complementary techniques. First, we propose new frameworks using nonsmooth constrained optimization problems, whose solutions both damp undesirable frequency bands and maintain stability of the system. These frameworks also allow us to weight which frequency bands are the most important to damp. Second, we also propose a fast new eigensolver for the structured quadratic eigenvalue problems that appear in such vibrating systems. In order to be efficient, our new eigensolver exploits special properties of diagonal-plus-rank-one complex symmetric matrices, which we leverage by showing how each quadratic eigenvalue problem can be transformed into a short sequence of such linear eigenvalue problems. The result is an eigensolver that is substantially faster than standard techniques. By combining this new solver with our new optimization frameworks, we obtain our overall algorithm for fast computation of optimal viscosities. The efficiency and performance of our new methods are verified and illustrated on several numerical examples.