Efficient balancing-based MOR for large-scale second-order systems

Large-scale structure dynamics models arise in all areas where vibrational analysis is performed, ranging from control of machine tools to microsystems simulation. To reduce computational and resource demands and be able to compute solutions and controls in acceptable, that is, applicable, time frames, model order reduction (MOR) is applied. Classically modal truncation is used for this task. The reduced-order models (ROMs) generated are often relatively large and often need manual modification by the addition of certain technically motivated modes. That means they are at least partially heuristic and cannot be generated fully automatic. Here, we will consider the application of fully automatic balancing-based MOR techniques. Our main focus will be on presenting a way to efficiently compute the ROM exploiting the sparsity and second-order structure of the finite element method (FEM) semi-discretization, following a reduction technique originally presented in [V. Chahlaoui, K.A. Gallivan, A. Vandendorpe, and P. Van Dooren, Model reduction of second-order system, in Dimension Reduction of Large-Scale Systems, P. Benner, V. Mehrmann, and D. Sorensen, eds., Lecture Notes in Computer Science and Engineering, Vol. 45, Springer Verlag, Berlin, 2005, pp. 149--172], [Y. Chahlaoui, D. Lemonnier, A. Vandendorpe, and P. Van Dooren, Second-order balanced truncation, Linear Algebra Appl. 415 (2006), pp. 373--384], [T. Reis and T. Stykel, Balanced truncation model reduction of second-order systems, Math. Comput. Model. Dyn. Syst. 14 (2008), pp. 391--406] and [J. Fehr, P. Eberhard, and M. Lehner, Improving the Reduction Process in Flexible Multibody Dynamics by the Use of 2nd Order Position Gramian Matrices, Proceedings ENOC, St. Petersburg, Russia, 2008]. Large-scale sparse solvers for the underlying matrix equations solved in the balancing process are adapted to the second-order structure of the equations and the suitability of our approach is demonstrated for two practical examples.