Norman Lang

Model Order Reduction for Systems with Moving Loads

Abstract 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.

Balanced truncation model reduction for linear time-varying systems

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 in treating complex quantities, a symmetric indefinite factorization of both the right-hand side and the solution of the differential Lyapunov equations is applied.

Computation of Thermo-Elastic Deformations on Machine Tools - A study of Numerical Methods

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 thermo-elastic 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.

Towards the identification of heat induction in chip removing processes via an optimal control approach

AbstractThis paper presents a linear-quadratic regulator (LQR) approach for solving inverse heat conduction problems (IHCPs) arising in production processes like chip removing or drilling. The inaccessibility of the processed area does not allow the measuring of the induced temperature. Hence the reconstruction of the heat source based on given measurements at accessible regions becomes necessary. Therefore, a short insight into the standard treatment of an IHCP and the related LQR design is provided. The main challenge in applying LQR control to the IHCP is to solve the differential Riccati equation. Here, a model order reduction approach is used in order to reduce the system dimension. The numerical results will show the accuracy of the approach for a problem based on data given by practical measurements.

Comparison of model order reduction methods for optimal sensor placement for thermo-elastic models*

ABSTRACT In this article an optimal sensor placement problem for a thermo-elastic solid body model is considered. Temperature sensors are placed in a near-optimal way so that their measurements allow an accurate prediction of the thermally induced displacement of a point of interest (POI). Low-dimensional 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.

Model Order Reduction for Thermo-Elastic Assembly Group Models

We present two model order reduction approaches based on different modelling strategies for a thermo-elastic assembly group model. Here, we consider the machine stand example given in Chap. 7. The focus is on capturing the structural variability. Therefore, we compare a switched linear systems (SLS) approach based on reduced order models determined by the Balanced Truncation (BT) method and a parametric model order reduction (PMOR) scheme based on an interpolatory projection method via the iterative rational Krylov algorithm (IRKA). In order to avoid the high dimensional coupled thermo-elastic system, additionally a Schur complement representation is applied to exploit the special structure of the one-sided coupling property of the system. The results show that both methods generate relative errors in the range of one per thousand.