## meet the team

THE PEOPLE BEHIND VIMS

Zoran Tomljanović

principal investigator

## Abstract

keywords: mechanical system, vibration reduction, quadratic eigenvalue problem, frequency isolation, reduced-order model, parameter optimization

Vibration analysis and vibration reduction for mechanical systems are prominent problems in numerous research fields. Although the vibrations analysis is an intensively studied topic in recent decades, many problems still remain open. While the case without external excitation leads to the study of homogeneous systems, presence of an external forcing leads to the study of nonhomogeneous systems. Depending on the presence of an external excitation and applications, we will consider four different research themes.

Within the first research theme, we will study theoretical results that are relevant for vibration reduction. We plan to develop theoretical results that characterize important properties of the quadratic eigenvalue problem (QEP) arising from vibration analysis of mechanical systems. Within the second research theme we will develop new methods for frequency isolation and utilize methods which are based on algorithms for non-smooth optimization. For this case we will derive new algorithms that preserve the structure of the matrices and structural properties of the considered QEP. In the third research theme, we will consider vibration reduction based on criteria that use system norms (e.g. H2 and Hinf) for Multiple-Input Multiple-Output case. We will also study approaches for approximating the full-order model with a reduced-order model that retains the structure of parametric dependence. The new approaches will be well suited for computationally efficient parameter optimization and the study of important system properties. In the fourth research theme we will consider integrating research themes I-III and applications in real world examples.

Moreover, we will apply obtained new approaches and algorithms in various academic examples, but also in real life examples that arise, e.g., in car industry (such as disc brake problem) and civil engineering (such as beams, civil buildings), etc. Therefore, the results from this project could have wide applications.

## Research themes

I - IV

RT I: homogeneous case

We develop theoretical results that characterize important properties of the Quadratic Eigenvalue Problem (QEP) that arise from vibration analysis of mechanical systems. In relative perturbation bounds typically relative gaps appear, which makes such bounds demanding to compute. We will approximate these perturbation bounds and develop new bounds that can be used for efficient vibration reduction.

RT II: case with given external force

We develop criterion that considers vibration isolation in order to avoid resonance in vibrational system. For this case we will derive new algorithms that preserve the structure of the matrices and structural properties of the considered QEP.

RT III: MIMO case

We consider a new measures that can ensure better robustness, stability and other important system properties. Within this case, to make the optimization algorithm computationally feasible, we will employ a novel approach to parametric model order reduction. We will also study approaches for approximating the full-order model with a reduced-order model that retains the structure of parametric dependence. The new approach will be well suited for computationally efficient parameter optimization and the study of important system properties.

RT IV: integrating research themes and applications

This is integrating research themes I-III and applications in real world examples. Firstly, we will employ the theoretical results obtained for the homogeneous case in the setting of the nonhomogeneous case as well. We propose new approaches that are based on criteria that are nonsmooth, such as Hinf criterion or criterion based on frequency isolation. We will test our new approaches and algorithms in various academic examples, but also in real life examples that arise, for example, in car industry (such as disc break problem) and civil engineering (such as beams, civil buildings), etc.

## Publications

Sampling-free parametric model reduction for structured systems

Christopher Beattie, Serkan Gugercin, Zoran Tomljanović

- Advances in Computational Mathematics 46/6, 1-34, (2020)

Semi-active H∞ damping optimization by adaptive interpolation

Zoran Tomljanović, Matthias Voigt

- Numerical Linear Algebra with Applications 27/4, 1-17, (2020)

Relative Perturbation Theory for Quadratic Hermitian Eigenvalue Problem

Peter Benner, Xin Liang, Suzana Miodragović, Ninoslav Truhar

- Linear Algebra and its Applications, 618, 97-128 (2021)

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

Nevena Jakovčević Stor, Tim Mitchell, Zoran Tomljanović, Matea Ugrica

- submitted

The tan Θ theorem for definite matrix pairs

Ivana Kuzmanović Ivičić, Suzana Miodragović, Matea Ugrica

- submitted

## Dissemination

COST Action 18232 Zagreb meeting, 24-27 February 2020

Z. Tomljanović, Sampling-free parametric model reduction for structured systems

ApplMath20, Brijuni, 14-18 September 2020

S. Miodragović, Frequency isolation for the hyperbolic guadratic eigenvalue problem

M. Ugrica, Frequency-weighted damping via nonsmooth optimization and fast computation of QEPs with low-rank updates

Magdeburg Lectures on Optimization and Control, Magdeburg, 25 September 2020

Z. Tomljanović, Damping optimization in mechanical systems using sampling-free model reduction

Optimization and application seminar, Department of Mathematics, Osijek, 25 November 2020

Z. Tomljanović, Sampling-free model reduction of systems with low-rank parameterization

Seminar za primijenjenu matematiku i teoriju upravljanja, University of Dubrovnik, 5 February 2021

Z. Tomljanović, Damping optimization in mechanical systems using parametric model reduction

Optimization and application seminar, Department of Mathematics, Osijek, 3 March 2021

M. Ugrica, Frequency-weighted damping via non smooth optimization and fast computation of QEPs

91st GAMM Annual Meeting in Kassel, 15-19 March 2021

Z. Tomljanović, Frequency-weighted damping via non smooth optimization and fast computation of QEPs

Croatian German meeting on analysis and mathematical physics, 22nd to 25th March 2021

Z. Tomljanović, Sampling-free model reduction of systems with low-rank parameterization

8th European Congress of Mathematics, 20 - 26 June 2021, Portorož, Slovenia

Suzana Miodragović, Hyperbolic quadratic eigenvalue problem and frequency isolation

Ivana Kuzmanović Ivičić, Optimization of parameter dependent structured Sylvester and T-Sylvester equations

Workshop on Control of Dynamical Systems 14-16 June 2021, Dubrovnik, Croatia

M. Ugrica, Frequency-weighted damping via nonsmooth optimization and fast computation of QEPs with low-rank updates

## News

Invited lectures & visiting fellows

Christopher BeattieVirginia Tech, USAProfessor Christopher Beattie is our guest at the Mathematical Colloquium in Osijek on 20th May 2021. On-line lecture will start at 7pm. He will give a talk on Balancing and the reduction of dynamical systems based on data.

Ren-Cang LiUniversity of Texas at Arlington, USAProfessor Ren-Cang Li is our guest at the Mathematical Colloquium in Osijek on 10th December 2020. He will give a talk on SCF Iteration for Orthogonal Canonical Correlation Analysis.

Serkan GugercinVirginia Tech, USAProfessor Serkan Gugercin is our guest at the Optimization and application seminar on 10th June 2020. He will give a talk on An Introduction to Interpolatory Methods for Model Reduction.