CMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, PolandCMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, PolandCMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, Poland

Error-controlled Model Reduction in Elastic Multibody Dynamics

Peter Eberhard, Jörg Fehr, Christine NowakowskiInstitute of Engineering and Computational Mechanics, University of Stuttgart,

Pfaffenwaldring 9, 70569 Stuttgart, Germanye-mail: [peter.eberhard,joerg.fehr,christine.nowakowski]@itm.uni-stuttgart.de

One important issue for the simulation of flexible multibody systems is the reduction of the flexible body’s degrees of freedom. In recentyears the authors developed a new pre-processor Morembs based on advanced model reduction techniques like Krylov-subspaces. SVD-based reduction techniques and combination of those. Not only the fact that the distribution of the loads is taken into account a prioribut also that very accurate models can be obtained within a predefined frequency range have to be considered as advantages of modernreduction techniques. Furthermore, a priori error bounds or efficient error estimators are available. Only the load distribution, thefrequency range of interest and a measure for the desired accuracy have to be provided by the user which makes the method especiallyattractive for optimization or error-controlled computations. We evaluate and compare these methods in the frequency as well as in thetime domain by reducing different models from different application areas to show the generality of the reduction methods.

Keywords: elastic multibody systems, model reduction, Krylov-subspace, SVD, floating frame of reference, error control

1. Introduction

The simulation of mechanical systems nowadays has to beconsidered as an essential part of the development process of newproducts. By the use of simulation, a multitude of trials and pro-totypes can be cut down. That is why simulation can be con-sidered as a third discipline along with theory and experiments[1]. The trend towards the simulation of coupled systems fromdifferent physical domains is explained in [2] and its importanceis increasing. Especially as far as numerical solutions of linkedsystems are concerned, consisting of mechanical, electrical, hy-draulic or pneumatic components, in combination with control al-gorithms, the efficient simulation of the subcomponents and theiradequate coupling are significant issues.

For the modeling of the mechanical subsystems the methodof elastic multibody systems (EMBS) is frequently used. Multiphysics systems with an EMBS part are used, e.g. to examinethe dynamic behavior of fuel injectors, see Fig. 1, robot-arms orelasto hydro-dynamic engine crank trains. Considering the ele-mentary form of this method, rigid bodies and elastic bodies areconnected with each other and the environment by the use of idealjoints and coupling elements.

Figure 1: Magnetic fuel injector (right). FE-model of armatureand valve which are simulated as EMBS (left).

The concept of flexible multibody systems is described e.g.in [3]. In the floating frame of reference formulation the motionr of a flexible body is separated into an usually nonlinear motionof the reference frame Ki and into a linear elastic deformation uwith respect to the reference frame. The linear elastic deforma-tions are approximated with u = Φ·q, where q is referring to thenodal displacements of a finite element (FE) model and Φ are theelastic shape functions. This modeling approach leads to 6 + Ndegrees of freedom per elastic body. Three translational and threerotational degrees of freedom of the rigid body dynamics plus theN elastic degrees of freedom which is equal to the number ofdegrees of freedom (dof) of the FE-model. In the following, allflexible bodies are considered simultaneously which yields thesame structure of the equation with merely different dimensions.As a consequence, transient simulations, endurance tests, designby optimization of such huge systems are often hardly feasible,especially if they are a subsystem of a bigger simulation system.Because of this, it is necessary to reduce N by approximatingq with a projection matrix V ∈ RN×n by q = V · q̄, withn = dim(q̄) � dim(q) = N , and requiring the residual tobe orthogonal to a second projection matrix W . This procedureleads to the reduced equations of motion[

Mr MTer · V

W T ·Mer W T ·Me · V

]·[aq̈

]=

[hr

W T · he

]+[

0−W T ·Ke · V · q −W T ·De · V · q̇

] (1)

where the submatrix Mr corresponds to the mass matrix knownfrom rigid multibody dynamics, Me, De and Ke are the flexi-ble mass, damping and stiffness matrices, whereas Mer providesthe coupling between the rigid body movement and the elastic de-formation. The vector a contains the global accelerations of thefloating frame of reference, vectors hr and he collect generalizedinertia forces, gravitational forces and forces acting on the body’ssurface. In this work, we concentrate on the reduction process ofthe elastic degrees of freedom.

CMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, PolandCMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, PolandCMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, Poland

2. Comparison of the Different Model Reduction Tech-niques

For the model reduction process only the linear elastic partneeds to be considered to ensure that the large nonlinear be-havior is kept in the reduced system. All the reaction and ap-plied forces acting on the elastic body are considered as inputsf(t) = Be · u(t) and the most important displacements of theelastic body as outputs y(t) = Ce · q(t). Then the elastic partof the body can be considered as a linear time-invariant secondorder multi input multi output (MIMO)-system

Me · q̈(t) + De · q̇(t) + Ke · q(t) = Be · u(t),

y(t) = Ce · q(t).(2)

The projection matrices V and W which reduce the elastic bodyto the reduced equation of motion (1) are the same projectionmatrices which reduce the second order MIMO-system in Eqn(2). In state of the art reduction methods like modal reduction,the projection matrices V = W consist of the dominant eigenvectors plus some additional modes. However, the convergenceof modal reduction can be slow, e.g., because the spatial distri-bution of loads is not considered. In addition, no informationabout the error introduced by model reduction can be gained ifmodal reduction techniques are used and a tuning of the reducedmodel for certain frequency ranges is not possible. The mainnon-modal model reduction techniques used for the reduction ofsecond order MIMO-system are, moment-matching by projectionon Krylov-subspaces, SVD-based reduction techniques and com-binations of those. It must be emphasized, that here V and W nolonger contain eigen modes belonging to certain eigen frequen-cies, but mathematically determined ansatz functions with con-vincing properties. However, each of these methods implies itsspecific advantages and disadvantages. The following criteria areof special interest for the user: computability for large scale sys-tems, stability preservation, quality of the reduced order model,knowledge about the error induced through reduction methods,preservation of the second order mechanical structure during thereduction, emphasizing a certain frequency range and the possi-bility to automate the reduction process.

2.1. Krylov-subspace based reduction techniques

By using a Krylov-subspace as projection space span(V ),certain moments of the original transfer function matrix and thereduced transfer function matrix of the flexible body match atspecific expansion points. The fact that Krylov-subspace basedreduction methods are iterative methods and can be applied tolarge scale models represents their decisive advantage. The per-formance of Krylov-subspace based reduction methods clearlydepends on the choice of expansion points. Due to that fact meth-ods for an automated selection of expansion points were devel-oped in recent years. In addition, error estimation is nowadayspossible, see [4]. However, the method lacks flexibility with re-spect to systems with many inputs where a very small reducedmodel is required. As a result, one current research topic is thein- and output reduction by ESVDMOR prior to the reductionprocess as explained in [5].

2.2. SVD-based reduction techniques

The controllability and observability Gramian matrices of asystem, P and Q, are strongly related to SVD-based reduc-tion methods. By the use of second order frequency weightedGramian matrix based reduction techniques the distribution of theloads is taken into account a priori and very accurate models canbe obtained within a predefined frequency range. Furthermore, anenergy interpretation of the reduction procedure and a priori errorbounds are available [6]. Hence, using this method only the loaddistribution, the frequency range of interest and a measure for thedesired accuracy have to be provided by the user. That is why themethod is especially attractive as far optimization applications are

concerned. For reduction with frequency weighted Gramian ma-trices the number and location of the snapshots, which are neededto calculate the Gramian matrices, have a big influence on the re-duction results. In recent years sophisticated snapshot-selectionmethods from the Reduced Basis methods are used for basis con-struction. The method can now be viewed as an automatic de-termination of optimal frequency weighting and allows a totallyautomated reduction process.

3. Used Programs and Summary

During the last years, a large number of new model order re-duction techniques have been developed, implemented and testedat our institute. Each is characterized by certain advantages anddisadvantages. By giving the user an easy to use tool for testingseveral model order reduction techniques and giving assistancehow to choose an appropriate reduction space, the simulation pro-cess of EMBS can be improved. The advanced model reductiontechniques are now used in the structural optimization processof several technical systems and helped to speedup the simula-tion process of different models from different application areas.The methods are implemented in the software package Morembswhich has a C++ and a Matlab implementation. Morembs isable to handle data from several FE programs. Up to now, datafrom the FE programs Ansys, Abaqus and Permas can be usedfor model reduction. Morembs provides either the system matri-ces of the reduced system or the calculated projection matrices Vand W as output. The reduced system matrices can be directlyused in Neweul-M2 or a conversion into an SID-File for Sim-pack is possible. In addition, the reduced projection matrix canbe written back to the FE programs, e.g. Permas and then the Per-mas converters to various other EMBS programs like VL-Motion,Adams, etc. can be used to provide the non-modal reduced elas-tic body for these EMBS programs. The C++ implementationis based on advanced math libraries leading to short computationtimes and it is examined how with the help of supercomputers realindustrial problems with more than 2 000 000 degrees of freedomcan be reduced.

References

[1] Schilders, W.: Introduction to Model Order Reduction,Model Order Reduction, Schilders, W., van der Vorst, H.and Rommes, R (Eds.), Springer, Berlin, chap. 1, pp. 3–32,2008.

[2] Arnold, M. and Schiehlen, W. (Eds.): Simulation Tech-niques for Applied Dynamics, Vol. 507 of CISM Interna-tional Centre for Mechanical Sciences. Springer, Vienna,2009.

[3] Shabana, A.A.: Flexible Multibody Dynamics: Review ofPast and Recent Developments, Multibody System Dynam-ics, Vol. 1, No. 2, pp. 159–223, 1997.

[4] Fehr, J., Tobias, C. and Eberhard, P.: Automated and Er-ror Controlled Model Reduction for Durability Based Struc-tural Optimization of Mechanical Systems, ACMD 2010:Proceedings 5th Asian Conference on Multibody Dynamics,Kyoto, Japan, 2010.

[5] Benner, P. and Schneider, A.: Model Order and TerminalReduction Approaches via Matrix Decomposition and LowRank Approximation, Scientific Computing in ElectricalEngineering SCEE 2008, Mathematics in Industry, Roos,J., Costa, L. (Eds.), Springer, Berlin, 2009.

[6] Fehr, J.; Eberhard, P.: Error-controlled Model Reduction inFlexible Multibody Dynamics, Journal of Computationaland Nonlinear Dynamics, Vol. 5, No. 3, 2010.