# Preprint No. MPIMD/12-07

#### Author(s): Christine Nowakowski, Patrick Kürschner, Peter Eberhard, Peter Benner

#### Email: kuerschner@mpi-magdeburg.mpg.de

#### Date: 2012-03-19

#### Abstract:

System analysis and optimization of combustion engines and engine components
are increasingly supported by digital simulations. In the simulation process of
combustion engines multi physics simulations are used. As an example, in the
simulation of a crank drive the mechanical subsystem is coupled to a
hydrodynamic subsystem. As far as the modeling of the mechanical subsystems is
concerned, elastic multibody systems are frequently used. During the simulation
many equations must be solved simultaneously, the hydrodynamic equations as well
as the equations of motion of each body in the elastic multibody system. Since
the discretization of the elastic bodies, e.g with the help of the finite
element method, introduces a large number of elastic degrees of freedom, an
efficient simulation of the system becomes difficult. The linear model reduction
of the elastic degrees of freedom is a key step for using flexible bodies in
multibody systems and turning simulations more efficient from a computational
point of view. In recent years a variety of new reduction methods alongside the
traditional techniques were developed in applied mathematics. Some of these
methods are introduced and compared for reducing the equations of motion of an
elastic multibody system. The special focus of this work is on balanced
truncation model order reduction which is a singular value
based reduction technique using the Gramian matrices of the system. We
investigate a version of this method that is adapted to the structure of a
special class of second order dynamical systems which is important for
the particular application discussed here. The main
computational task in balanced truncation is the solution or large-scale
Lyapunov equations for which we apply a modified variant of the low-rank ADI
method. The simulation of a crank drive with a flexible crankshaft is taken as
technically relevant example. The results are compared to other methods like
Krylov approaches or modal reduction.

#### BibTeX:

@TECHREPORT{MPIMD12-07,

author = {Christine Nowakowski and Patrick Kürschner and Peter Eberhard and Peter Benner},

title = {Model Reduction of an Elastic Crankshaft for Elastic Multibody Simulations},

number = {MPIMD/12-07},

month = mar,

year = 2012,

institution = {Max Planck Institute Magdeburg},

type = {Preprint},

note = {Available from \url{http://www.mpi-magdeburg.mpg.de/preprints/}},

}