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.
Operator splitting based dynamic iteration for linear differential-algebraic port-Hamiltonian systems. - In: Numerische Mathematik, ISSN 0945-3245, Bd. 155 (2023), 1, S. 1-34
A dynamic iteration scheme for linear differential-algebraic port-Hamiltonian systems based on Lions-Mercier-type operator splitting methods is developed. The dynamic iteration is monotone in the sense that the error is decreasing and no stability conditions are required. The developed iteration scheme is even new for linear port-Hamiltonian systems governed by ODEs. The obtained algorithm is applied to a multibody system and an electrical network.
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 $.
Analysis of a quasilinear coupled magneto-quasistatic model: solvability and regularity of solutions. - In: Journal of mathematical analysis and applications, ISSN 1096-0813, Bd. 523 (2023), 2, 127033
We consider a quasilinear model arising from dynamical magnetization. This model is described by a magneto-quasistatic (MQS) approximation of Maxwell's equations. Assuming that the medium consists of a conducting and a non-conducting part, the derivative with respect to time is not fully entering, whence the system can be described by an abstract differential-algebraic equation. Furthermore, via magnetic induction, the system is coupled with an equation which contains the induced electrical currents along the associated voltages, which form the input of the system. The aim of this paper is to study well-posedness of the coupled MQS system and regularity of its solutions. Thereby, we rely on the classical theory of gradient systems on Hilbert spaces combined with the concept of E-subgradients using in particular the magnetic energy. The coupled MQS system precisely fits into this general framework.
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.
Idempotent relations, semi-projections, and generalized inverses. - In: Contributions to mathematics and statistics, (2021), S. 87-110
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.
Some notes on port-Hamiltonian systems on Banach spaces. - In: IFAC-PapersOnLine, ISSN 2405-8963, Bd. 54 (2021), 19, S. 223-229
We consider port-Hamiltonian systems from a functional analytic perspective. Dirac structures and Hamiltonians on Banach spaces are introduced, and an energy balance is proven. Further, we consider port-Hamiltonian systems on Banach manifolds, and we present some physical examples that fit into the presented theory.