No abstract

A practical procedure based on implicit time integration methods applied to the differential Lyapunov equations arising in the square root balanced truncation method is presented. The application of high order time integrators results in indefinite right-hand sides of the algebraic Lyapunov equations that have to be solved within every time step. Therefore, classical methods exploiting the inherent low-rank structure often observed for practical applications end up in complex data and arithmetic. Avoiding the additional effort treating complex quantities, a symmetric indefinite factorization of both the right-hand side and the solution of the differential Lyapunov equations is applied.

In this contribution we present two approaches allowing to find a reduced order approximant of a full order model featuring a moving load term. First, we apply the Balanced Truncation (BT) method to a switched linear system (SLS) using the special structure given in the spatially discretized model. The second approach treats the variability as a continuous parameter dependence and uses the iterative rational Krylov algorithm (IRKA) to compute a parameter preserving reduced order model. Zusammenfassung:In diesem Beitrag werden zwei Ansätze vorgestellt, welche es erlauben, ein reduziertes Modell eines Originalsystems mit beweglichem Lastterm zu bestimmen. Der erste Ansatz verwendet die Methode des balancierten Abschneidens (BT) zur Reduktion eines geschalteten, linearen Systems (SLS), welches sich aus der speziellen diskreten Struktur des Modells ergibt. Der zweite Ansatz behandelt die Variabilität als eine stetige Parameterabhängigkeit und verwendet den iterativen, rationalen Krylov Algorithmus (IRKA) zur Berechnung eines parametererhaltenden, reduzierten Modells.Schlüsselwörter: Modellordnungsreduktion, parametrisch, linear zeitinvariante Systeme, geschaltete lineare Systeme.

In this article an optimal sensor placement problem for a thermo-elastic solid body model is considered. Temperature sensors are placed in a nearoptimal way so that their measurements allow an accurate prediction of the thermally induced displacement of a point of interest (POI). Lowdimensional approximations of the transient thermal field are used which allows for efficient calculations. Four model order reduction (MOR) methods are applied and subsequently compared with respect to the accuracy of the estimated POI displacement and the location of the sensors obtained.

Modern machine tools are highly optimized with respect to their design and the production processes they are capable to. Now for further advances, especially a detailed knowledge about the thermo-elastic behavior is needed, because the nowadays still existing deficits are mainly related to this. That is why, endeavors in improvement, like the optimization of the design, the evaluation of new materials and the regulation of the production process, particularly rely on accurate computed thermal deformations. One possible approach to increase their quality is to also include the relevant structural variabilities of the machine tools as well as the resulting interactions between the coupled parts within the calculations. In this article, three different numerical methods are presented, which include structural motions in thermoelastic analyses. Thereby, several conflicting criteria, like real-time capability, memory saving issues and accuracy are fulfilled each time in a different manner. Those methods are afterwards compared with respect to their runtime and accuracy. Finally, the paper concludes with a classification of the usability of the methods in real-time control and optimization tasks.