Abstract Summary
The dynamical models of many complex engineering systems, such as Finite Element (FE) models of structural dynamics systems, consist of a (very) large number of degrees of freedom (DOF). For such systems, model order reduction (MOR) is required to make simulation and analysis computationally feasible. The well-known component mode synthesis (CMS) MOR methods rely on the partitioning of the overall structure into subsystems, also called substructures or components. With CMS methods, each of the subsystem models is reduced individually before the reduced subsystem models are coupled to create the model of the interconnected system. This approach is completely modular. Dividing the large MOR problem into several smaller problems leads to several advantages: 1. The computationally challenging reduction of one high-dimensional model is avoided. 2. The interconnection structure of the original high-order system is preserved. 3. Subsystems can be developed, modelled, and analyzed in parallel by distinct teams. These teams can independently reduce the subsystem models before the reduced interconnected system model is derived and analyzed. 4. If design changes are made to a single subsystem, only the reduced-order model (ROM) of this subsystem needs to be updated. In general, we are interested in obtaining a reduced-order model that provides an accurate representation of the external input-to-output behavior of the interconnected system. However, by reducing the number of DOFs of a (sub)system model, generally, an error is introduced in the reduced-order (sub)system model, which in turn introduces an error in the overall system model. Generally, it is not trivial to determine a priori how these errors will propagate to the overall reduced-order interconnected model. Therefore, in this work, we introduce a mathematical approach that allows for the computation of (frequency-dependent) requirements on the maximum error introduced by subsystem reduction, given (frequency-dependent) accuracy requirements for the reduced-order interconnected model. With this approach, we allow for the independent reduction of subsystem models using CMS methods while guaranteeing the desired accuracy of the overall system model. The main idea relies on defining the error dynamics introduced by the MOR of the subsystems as block-diagonal structured uncertainties. Then, the interconnected system model can be reformulated into the framework of a robust performance problem as studied in control theory. This allows for a direct computation of a relation between a given upper bound on the error of the reduced-order interconnected model and upper bounds on the error of the reduced-order subsystem models using tools from the field of robust control. More specifically, the structured singular value, μ, is used, which is a mathematical tool defined to efficiently compute the worst-case behavior of a (controlled) system under uncertainty. To illustrate the proposed framework, we apply it to a model of a structural dynamics system consisting of three interconnected beams and show that the subsystem models can be significantly reduced while guaranteeing the accuracy of the interconnected model.
