Second-order control systems are used to describe the dynamics of mechanical and vibrational systems, and in particular, their response to excitations. In this article, we discuss model order reduction (MOR) of such systems. A particular focus is on preserving the second-order structure and physical properties such as stability and passivity.

Suppressing vibrations in mechanical models, usually described by second-order dynamical systems, 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 our 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.

The simulation of fluid dynamic problems often involves solving large-scale saddle-point systems.Their numerical solution with iterative solvers requires efficient preconditioners. Low-rank updates canadapt standard preconditioners to accelerate their convergence. We consider a multiplicative low-rank cor-rection for pressure Schur complement preconditioners that is based on a (randomized) low-rank approxi-mation of the error between the identity and the preconditioned Schur complement. We further introducea relaxation parameter that scales the initial preconditioner. This parameter can improve the initial pre-conditioner as well as the update scheme. We provide an error analysis for the described update method.Numerical results for the linearized Navier-Stokes equations in a model for atmospheric dynamics on twodifferent geometries illustrate the action of the update scheme. We numerically analyze various parametersof the low-rank update with respect to their influence on convergence and computational time. MSC codes. 65F08, 65F10, 65N22, 65F55

The simulation of fluid dynamic problems often involves solving large-scale saddle-point systems. Their numerical solution with iterative solvers requires efficient preconditioners. Low-rank updates can adapt standard preconditioners to accelerate their convergence. We consider a multiplicative low-rank correction for pressure Schur complement preconditioners that is based on a (randomized) low-rank approximation of the error between the identity and the preconditioned Schur complement. We further introduce a relaxation parameter that scales the initial preconditioner. This parameter can improve the initial preconditioner as well as the update scheme. We provide an error analysis for the described update method. Numerical results for the linearized Navier–Stokes equations in a model for atmospheric dynamics on two different geometries illustrate the action of the update scheme. We numerically analyze various parameters of the low-rank update with respect to their influence on convergence and computational time.