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

Abstract

Large-scale structure dynamics models arise in all areas where vibrational analysis is performed, ranging from control of machine tools to microsystems simulation. These models result from a finite element analysis being applied to their mechanical structure. Therefore they are in general sparse, but very large, since many details have to be resolved. This accounts for unacceptable computational and resource demands in simulation and especially control of these models. To reduce these demands and to be able to compute solutions and controls in acceptable, i.e. applicable, time frames model order reduction is applied. Classically modal truncation is used for this task. The reduced order models (ROMs) generated are normally relatively large and often need manual modification by addition of certain technically motivated modes. That means they are at least partially heuristic and cannot be generated fully automatic. Engineers are therefore searching for alternate reduction methods.Here we will concentrate on the application of balancing based model order reduction techniques. A central topic is to provide a reduced order model for the construction and parameterization of a practicable controller for the application. Our main focus will be on presenting a way to efficiently compute the ROM exploiting the sparsity and second order structure of the FEM semi-discretization, rather than presenting a new reduction technique.

An improved numerical method for balanced truncation for symmetric second-order systems

“…For the generalized case this amounts to AE T + EA T < 0. In our case A, E, are block structured matrices representing a linearization of a second order system and it can be shown that, except for unnatural cases [27], the dissipativity condition is not fulfilled. Consider for instance the system with M = 1, D = α > 0, K = β > 0.…”

Efficient handling of complex shift parameters in the low-rank Cholesky factor ADI method

“…In fact, the final normalized residual was of order 10 6 . One explanation might be that forming AE −1 A in (11) induces for this example enough rounding errors to render the overall process inapplicable.…”

“…end if 17: end for Inserting these formulas into Algorithm 1 leads to the generalized LRCF-ADI method (G-LRCF-ADI) [30], [11], [3] which can be rewritten, using the new realification approach of the previous section, to a generalized version of Algorithm 3. Of course, the other two strategies are applicable as well, but using the complete real formulation of G-LRCF-ADI will result in linear systems involving the matrix…”

“…This is especially the case when M, D, K, B 1 define a second order linear time invariant dynamical system [11] and E, A, B represent a transformation to a first order generalized system. Note that there are other possible choices to carry out this transformation [34], but all of those can be dealt with similarly.…”

“…By assuming again that Λ(A, E) ⊂ C − and by exploiting the structure of M, D, K, B, it is possible to solve (10) by a version of G-LRCF-ADI which works directly using the n × n matrices M, D, K and the right hand side B 1 . This modified version is then called second order LRCF-ADI (SO-LRCF-ADI) [30], [11] and can certainly be equipped with the realification strategy, too.…”