Zoran Tomljanović


Associate Professor
Department of Mathematics
Josip Juraj Strossmayer University of Osijek
Trg Ljudevita Gaja 6
Osijek, HR-31000, Croatia¸
phone: +385-31-224-847
fax: +385-31-224-801
email:  ztomljan @ mathos.hr
office:  E (ground floor), near student office/pored referade


Research Interests

Numerical linear algebra
Damping optimization in mechanical systems
Control Theory

Matrix Equations


  • PhD in Mathematics, Department of Mathematics, University of Zagreb, May 2011.

PhD Thesis: Optimal damping for vibrating systems using dimension reduction  (PDF)

  • BSc in Mathematics, Department of Mathematics, University of Zagreb, Croatia, December 2005.
  • 1997-2001 Mathematical Gymnasium at high school in Našice


Journal Publications

  1. N. Truhar, Z. Tomljanović, M. Puvača, Approximation of damped quadratic eigenvalue problem by dimension reduction, Applied mathematics and computation 347 (2019), 40-53
    This paper presents an approach to the efficient calculation of all or just one important part of the eigenvalues of the parameter dependent quadratic eigenvalue problem $(lambda^2(mathbf{;v};) M + lambda(mathbf{;v};) D(mathbf{;v};) + K) x(mathbf{;v};) = 0$, where $M, K$ are positive definite Hermitian $ntimes n$ matrices and $D(mathbf{;v};)$ is an $ntimes n$ Hermitian semidefinite matrix which depends on a damping parameter vector $mathbf{;v};= begin{;bmatrix}; v_1 & ldots & v_k end{;bmatrix};in mathbb{;R};_+^k$. With the new approach one can efficiently (and accurately enough) calculate all (or just part of the) eigenvalues even for the case when the parameters $v_i$, which in this paper represent damping viscosities, are of the modest magnitude. Moreover, we derive two types of approximations with corresponding error bounds. The quality of error bounds as well as the performance of the achieved eigenvalue tracking are illustrated in several numerical experiments.
  2. I. Nakić, Z. Tomljanović, N. Truhar, Mixed control of vibrational systems, Journal of Applied Mathematics and Mechanics 99/9 (2019), 1-15
    We consider new performance measures for vibrational systems based on the $H_2$ norm of linear time invariant systems. New measures will be used as an optimization criterion for the optimal damping of vibrational systems. We consider both theoretical and concrete cases in order to show how new measures stack up against the standard measures. The quality and advantages of new measures as well as the behaviour of optimal damping positions and corresponding damping viscosities are illustrated in numerical experiments.
  3. Y. Kanno, M. Puvača, Z. Tomljanović, N. Truhar, Optimization Of Damping Positions In A Mechanical System, Rad HAZU, Matematičke znanosti. 23 (2019), 141-157
    This paper deals with damping optimization of the mechanical system based on the minimization of the so-called "average displacement amplitude". Further, we propose three different approaches to solving this minimization problems, and present their performance on two examples.
  4. Z. Tomljanović, C. Beattie, S. Gugercin, Damping optimization of parameter dependent mechanical systems by rational interpolation, Advances in Computational Mathematics 44/6 (2018), 1797-1820
    We consider an optimization problem related to semi-active damping of vibrating systems. The main problem is to determine the best damping matrix able to minimize influence of the input on the output of the system. We use a minimization criteria based on the $mathcal{H}_2$ system norm. The objective function is non-convex and the associated optimization problem typically requires a large number of objective function evaluations. We propose an optimization approach that calculates `interpolatory' reduced order models, allowing for significant acceleration of the optimization process. In our approach, we use parametric model reduction (PMOR) based on the Iterative Rational Krylov Algorithm, which ensures good approximations relative to the $mathcal{H}_2$ system norm, aligning well with the underlying damping design objectives. For the parameter sampling that occurs within each PMOR cycle, we consider approaches with predetermined sampling and approaches using adaptive sampling, and each of these approaches may be combined with three possible strategies for internal reduction. In order to preserve important system properties, we maintain second-order structure, which through the use of modal coordinates, allows for very efficient implementation. The methodology proposed here provides a significant acceleration of the optimization process; the gain in efficiency is illustrated in numerical experiments.
  5. N. Truhar, Z. Tomljanović, M. Puvača, An Efficient Approximation For Optimal Damping In Mechanical Systems, International journal of numerical analysis and modeling 14/2 (2017), 201-217
    This paper is concerned with an efficient algorithm for damping optimization in mechanical systems with a prescribed structure. Our approach is based on the minimization of the total energy of the system which is equivalent to the minimization of the trace of the corresponding Lyapunov equation. Thus, the prescribed structure in our case means that a mechanical system is close to a modally damped system. Although our approach is very efficient (as expected) for mechanical systems close to modally damped system, our experiments show that for some cases when systems are not modally damped, the proposed approach provides efficient approximation of optimal damping.



  • Control of Dynamical Systems -- scientific project (IP-2016-06-2468, ConDyS). This project has been fully supported by Croatian Science Foundation for the period 01.03.2017.--28.02.2021. (investigator)

  • Robustness optimization of damped mechanical systems, -- scientific project; supported by the DAAD for period 2017--2018 (principal investigator with Matthias Voigt); cooperation with TU Berlin, Germany

  • Optimization of parameter dependent mechanical systems  -- scientific project (IP-2014-09-9540; OptPDMechSys). This project has been fully supported by Croatian Science Foundation for the period 01.07.2015.--30.06.2019. (investigator)

  • Damping optimization in mechanical systems excited with external force -- scientific project; supported by the J. J. Strossmayer University of Osijek for period 2015 (principal investigator)

  • Mixed Integer Nonlinear Programming (MINLP) for damper optimization -- scientific project; supported by the DAAD for period 2015--2016 (investigator); cooperation with Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg 

  • European Model Reduction Network (EU-MORNET). Funded by: COST (European Cooperation in Science and Technology) (investigator).

  • Optimization of semi-active damping in vibrational systems -- scientific project; supported by the J. J. Strossmayer University of Osijek for period 2014 (principal investigator); cooperation with Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg

  • Optimal Damping of Vibrating Systems -- scientific project; supported by the DAAD for period 2013--2014 (investigator) 

  • Passive control of mechanical models -- scientific project No.235-2352818-1042 of the Croatian Ministry of Science, Education and Sports for period 2007.-- (investigator) 

  • Optimization algorithms for determination of optimal damping in mechanical systems -- scientific project; supported by the Croatian Science Foundation for period 2008--2009 (principal investigator)  

Professional Activities

Professional Societiey Membership
  • International Association of Applied Mathematics and Mechanics, GAMM
  • GAMM Activity Group Applied and Numerical Linear Algebra, GAMM ANLA
  • Croatian Mathematical Society, HMD
  • Croatian Operational Research Society, CRORS
  • Society for Industrial and Applied Mathematics, SIAM



Committee Memberships and organization





  • Mathematical Communications
  • Osječki matematički list


Workshop and Conference Talks


Seminar talks


  • Hinf damping optimization by adaptive interpolation, Applied Numerical Analysis Seminar, 26 April 2019, Departments of Mathematics at Virginia Tech, USA
  • Approximation bounds for parameter dependent quadratic eigenvalue problem, Applied Numerical Analysis Seminar, 4 Oct 2017, Departments of Mathematics at Virginia Tech, USA
  • Optimization of semi-active damping and viscous damping in excited mechanical systems, Matrix Computation Seminar,   10 November 2015, Departments of Mathematics at Virginia Tech, USA
  • Optimization of semi-active damping and external damping in mechanical systems with external force, Absolventen-Seminar  - Numerische Mathematik, 30 June 2015, Berlin, Germany
  • Damping optimization in mechanical systems with external force, CSC 

    Seminar at MPI Magdeburg, Germany, 5 May 2015

  • Optimizacija prigušenja u mehaničkim sustavima sa vanjskom silom, Department of Mathematics, University of Osijek, Optmization and applications seminar, 15 April 2015
  • Optimization of semi-active damping and viscous damping in mechanical systems with external force, October 21, 2014, University in Innsbruck, Department of Mathematics, Austria
  • Optimization of mechanical systems using dimension reduction, October 17, 2013, Forschungsseminar Scientific Computing, Technische Universitat Chemnitz, Germany
  • Optimalno prigušenje kod vibracijskih sistema koristeći redukciju dimenzije, March 17, 2011, Seminar za numeričku matematiku i računarstvo, PMF-Mathematical Departments, University of Zagreb, Croatia
  • Optimizacija prigušenja u vibracijskim sistemima pomoću redukcije dimenzije sistema, April 29, 2009,Optmization and applications seminar, University of Osijek, Department of Mathematics, Croatia
  • An efficient algorithm for solving and optimizing some types of Lyapunov, May 15, 2007, TU Chemnitz, Research Seminar Numerics, Germany


Study Visits Abroad and Professional Improvement:

  • visiting researcher at Department of Mathematics, Carlos III University of Madrid, Spain 06/09/2016 - 15/09/2016
  • visiting researcher at Departments of Mathematics at Virginia Tech, USA,  21/04/2019-27/04/2019, 02/10/2017-14/10/2017, 4/11/2015-20/11/2015
  • visiting researcher at TU Berlin, Germany 11/11/2018-15/11/2018, 16/07/2018-22/07/2018, 07/01/2018-14/01/2018,  21/06/2017 - 29/6/2017, 30/03/2017 - 07/04/2017, 23/02/2016 - 25/02/2016, 29/6/2015 - 9/7/2015
  • visiting researcher at University in Innsbruck, Department of Mathematics 19/10/2014-22/10/2014
  • visiting researcher at Max Planck Institute for Dynamics of Complex Technical Systems in Magdeburg, Germany 12/7/2018-19/7/2017, 22/11 /16 - 26/11/16, 29/06/16 - 09/07/16, 15/02/16 - 23/02/16, 04/27/15 - 05/06/15,02/04/14 - 08/4/14,   5/2/2013 - 20/2/2013, 3/7/2013 - 30/7/2013, 26/8/2012 -29/9/2012, 29/8/2010 - 19/9/2010,
  • visiting researcher at TU Chemnitz, Germany 13/5/2007 - 14/6/2007, 10/4/2008 - 8/5/2008, 10/5/2009 - 10/7/2009, 24/1/2010 - 24/2/2010, 9/10/2013 - 20/10/2013


Service Activities
  • Deputy Head of Department for Education and Students, since October 2017 -
  • Moderator of Optimization and application seminar, since October 2017 -
  • Chairman of the Mathematical Colloquium in Osijek, since January 2017 -
  • Erasmus+ coordinator on Department of Mathematics, since 2013 - 2017


Konzultacije (Office Hours):

Termini sljedećih konzultacija:

  • petak 25.10.2019. u 9 sati


Teme diplomskih i završnih radova:

U nastavku se nalaze nazivi tema i kratki opis, a više informacija studenti mogu dobiti na konzultacijama. Mole se zaniteresirani studenti da se jave ukoliko su zainteresirani za neku od tema.

  • Numeričko rješavanje običnih diferencijalnih jednadžbi
    - obraditi osnovne metode: Eulerova i osnovne Runge Kutta metode
    - implementirati ih u Matlabu ili C-u i ilustrirati efikasnost na primjerima
  • QR dekompozicija s pivotiranjem 
    - obraditi QR dekompoziciju i QR dekompoziciju s pivotiranjem
    - implementirati ju u Matlabu
    - na primjerima pokazati osnovne primjene npr. na određivanje ranga matrice
  • Schurova dekompozicija i primjene
    - definirati definirati Schurovu dekompoziciju 
    - obraditi osnovna svojstva i primjene
    - implementirati i ilustrirati na primjeru
  • Stabilnost sustava
    -obraditi pojam stabilnosti
    -obraditi glavne teorijske rezultate s naglaskom na primjenu u robotici i mehaničkim sustavima
    -implementirati metode za stabilizaciju i na primjerima napraviti usporedbu i ilustriraciju
  • Metoda Gaussovih eliminacija s potpunim pivotiranjem
    - obraditi metodu Gaussovih eliminacija s potpunim pivotiranjem
    - u Matlabu napraviti ilustraciju metode kroz vizualizaciju koraka
    - napraviti vizualizaciju rjesenja dvije jednadzbe s dvije nepoznanice
  • AHP metoda za odlučivanje
    - izgraditi model za rješavanje višekriterijskih problema odlučivanja;
    - primjeniti AHP metodu za hijerarhijsko odlučivanje na primjerima
  • Rezononcija
    - obraditi pojam rezonancije kod mehaničkih sustava
    - na primjerima sa dvije ili tri mase ilustrirati rezonanciju
  • Iterativne metode za rješavanje linearnih sustava
    - napraviti osnovni pregled iterativnih metode za sustave 
    - imlementirati neku od metoda te napraviti ilustraraciju na numeričkim primjerima 
  • NP-potpuni problem i redukcija problema
    - definirati pojam NP-potpnog problema
    - obraditi pojam redukcije i detaljno ilustrirati redukciju na primjeru


Nastavne aktivnosti u zimskom semestru Akademske 2018/2019

Matematička teorija računarstva, predavanja i vježbe

petkom od 8 do 12 sati

Građevinski fakultet Osijek

        Matematika na prvoj godini Arhitekture i urbanizma

utorkom od 10 do 12 sati

Nastavne aktivnosti u ljetnom semestru Akademske 2018/2019

Računarski praktikum, predavanja i vježbe

petkom 8:00 - 11:00

Teorija odlučivanja, predavanja i vježbe

srijedom  8:00 - 10:00

Nastavne aktivnosti u prošlosti (Past Courses)

 PhD study, Parameter Dependent Nonlinear Eigenvalue Problems (with Prof. N. Truhar) (summer semester 2017/2018)

Matematička teorija računarstva (Mathematical theory of computation(exercises and lectures, winter semester  13/14, 14/15, 15/16, 16/17,17/18)

Teorija odlučivanja (Decision theory) (exercises and lectures, winter semester  13/14, 14/15, 15/16, 16/17,17/18)

Metode optimizacije (Optimization methods), (vježbe, winter semestar 08/09, 09/10, 10/11, 11/12, 12/13, 13/14, 16/17)

Numerička linearna algebra (Numerical linear algebra), (exercises, winter semester 06/07, 08/09, 09/10, 10/11, 11/12, 12/13, summer semester 07/08)

Računarski praktikum, (exercises and lectures, ljetni semestar 08/09, 09/10, 11/12, 12/13, 13/14, 14/15,15/16, 16/17, 17/18)

Metodika nastave informatike, (exercises and lectures, winter semester 11/12, 12/13, 13/14, 14/15, 15/16)

Elementarna geometrija (Elementary geometry), (exercises, ljetni semestar 05/06, 06/07)

Obične diferencijalne jednadžbe (Ordinary differential equations), (exercises, zimski semestar 06/07, 07/08)

Numerička matematika (Numerical mathematics), (exercises, zimski semestar 07/08, ljetni semestar 08/09)

Integralni račun (Integral calculus), (vježbe, ljetni semestar 09/10, 10/11)

Odjel za fiziku (Department of Physics)

Diferencijalne jednadžbe (Differential equations), vježbe, ljetni semestar 06/07, 10/11, 11/12

Geometrija ravnine i prostora - uvod u algebru (Geometry of plain and space - introduction to algebra), (vježbe, zimski semestar 06/07, 07/08, 08/09, 09/10, 10/11)

Ekonomski fakultet (Faculty of Economycs)

Matematika (Mathematics), (vježbe, zimski semestar 07/09, 08/09, 09/10, 10/11, 11/12, predavanja, zimski semestar 12/13)

Prehrambeno tehnološki fakultet

Inženjerska matematika , (vježbe 08/09, 10/11, 11/12, 12/13) Primijenjena matematika (vježbe 08/09, 09/10,11/12, 12/13, 13/14)

Građevinski fakultet Osijek

Matematika (na razlikovnoj godini), exercises and lectures 12/13, 13/14,16/17

Matematika (na Arhitekturi i urbanizmu) (winter semester 16/17, 17/18)


Poslovna matematika, predavanja i vježbe 12/13, 13/14