Publikationen

Beddig, Rebekka S.; Benner, Peter; Dorschky, Ines; Reis, Timo; Schwerdtner, Paul; Voigt, Matthias; Werner, Steffen W. R.
Structure-preserving model reduction for dissipative mechanical systems. - In: Calm, smooth and smart, (2024), S. 209-230

Suppressing vibrations in mechanical systems, usually described by second-order dynamical models, is a challenging task in mechanical engineering in terms of computational resources even nowadays. One remedy is structure-preserving model order reduction to construct easy-to-evaluate surrogates for the original dynamical system having the same structure. In our work, we present an overview of recently developed structure-preserving model reduction methods for second-order systems. These methods are based on modal and balanced truncation in different variants, as well as on rational interpolation. Numerical examples are used to illustrate the effectiveness of all described methods.

https://doi.org/10.1007/978-3-031-36143-2_11

Reis, Timo; Stykel, Tatjana
Passivity, port-Hamiltonian formulation and solution estimates for a coupled magneto-quasistatic system. - In: Evolution equations and control theory, ISSN 2163-2480, Bd. 12 (2023), 4, S. 1208-1232

In this paper, we study a quasilinear coupled magneto-quasistatic model from a systems theoretic perspective. First, by taking the injected voltages as input and the associated currents as output, we prove that the magneto-quasistatic system is passive. Moreover, by defining suitable Dirac and resistive structures, we show that it admits a representation as a port-Hamiltonian system. Thereafter, we consider dependence of the solution on initial and input data. We show that the current and the magnetic vector potential can be estimated by means of the initial magnetic vector potential and the voltage. We also analyse the free dynamics of the system and study the asymptotic behavior of the solutions for $ t\to\infty $.

https://doi.org/10.3934/eect.2023008

Berger, Thomas; Snoo, Hendrik S. V. de; Trunk, Carsten; Winkler, Henrik
A Jordan-like decomposition for linear relations in finite-dimensional spaces. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2022. - 1 Online-Ressource (34 Seiten). - (Preprint ; M22,05)

A square matrix A has the usual Jordan canonical form that describes the structure of A via eigenvalues and the corresponding Jordan blocks. If A is a linear relation in a finite-dimensional linear space H (i.e., A is a linear subspace of H × H and can be considered as a multivalued linear operator), then there is a richer structure. In addition to the classical Jordan chains (interpreted in the Cartesian product H × H), there occur three more classes of chains: chains starting at zero (the chains for the eigenvalue infinity), chains starting at zero and also ending at zero (the singular chains), and chains with linearly independent entries (the shift chains). These four types of chains give rise to a direct sum decomposition (a Jordan-like decomposition) of the linear relation A. In this decomposition there is a completely singular part that has the extended complex plane as eigenvalues; a usual Jordan part that corresponds to the finite proper eigenvalues; a Jordan part that corresponds to the eigenvalue infinity; and a multishift, i.e., a part that has no eigenvalues at all. Furthermore, the Jordan-like decomposition exhibits a certain uniqueness, closing a gap in earlier results. The presentation is purely algebraic, only the structure of linear spaces is used. Moreover, the presentation has a uniform character: each of the above types is constructed via an appropriately chosen sequence of quotient spaces. The dimensions of the spaces are the Weyr characteristics, which uniquely determine the Jordan-like decomposition of the linear relation.

https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2022200249

Dorschky, Ines; Reis, Timo; Voigt, Matthias
Balanced truncation model reduction for symmetric second order systems - a passivity-based approach. - In: SIAM journal on matrix analysis and applications, ISSN 1095-7162, Bd. 42 (2021), 4, S. 1602-1635

We introduce a model reduction approach for linear time-invariant second order systems based on positive real balanced truncation. Our method guarantees to preserve asymptotic stability and passivity of the reduced order model as well as the positive definiteness of the mass and stiffness matrices. Moreover, we receive an a priori gap metric error bound. Finally we show that our method based on positive real balanced truncation preserves the structure of overdamped second order systems.

https://doi.org/10.1137/20M1346109