Interpolation-Based Parameterized Model Order Reduction of Delayed Systems

Brake squeal describes noise with different frequencies that can be emitted during the braking process. Typically, the frequencies are in the range of 1 to 16 kHz. Although the noise has virtually no effect on braking performance, strong attempts are made to identify and eliminate the noise as it can be very unpleasant and annoying. In the field of numerical simulation, the brake is typically modeled using the Finite Element method, and this results in a high‐dimensional equation of motion. For the analysis of brake squeal, gyroscopic and circulatory effects, as well as damping and friction, must be considered correctly. For the subsequent analysis, the high‐dimensional damped nonlinear equation system is linearized. This results in terms that are non‐symmetric and dependent on the rotational frequency of the brake rotor. Many parameter points to be evaluated implies many evaluations to determine the relevant parameters of the unstable system. In order to increase the efficiency of the process, the system is typically reduced with a truncated modal transformation. However, with this method the damping and the velocity‐dependent terms, which have a significant influence on the system, are neglected for the calculation of the eigenmodes, and this can lead to inaccurate reduced models. In this paper, we present results of other methods of model order reduction applied on an industrial high‐dimensional brake model. Using moment matching methods combined with parametric model order reduction, both the damping and the various parameter‐dependent terms of the brake model can be taken into account in the reduction step. Thus, better results in the frequency domain can be obtained. On the one hand, as usual in brake analysis, the complex eigenvalues are evaluated, but on the other hand also the transfer behavior in terms of the frequency response. In each case, the classical and the new reduction method are compared with each other.

Many new promising model order reduction (MOR) methods and algorithms were developed during the last decade. Industry and academic research institutions intend to test, validate, compare, and use these new promising MOR techniques with their own models. Therefore, an MOR toolbox bridging the gap between theoretical, algorithmic, and numerical developments to an end-user-oriented program, usable by non-experts, was developed called ‘Model Order Reduction of Elastic Multibody Systems’ (Morembs). A C++ implementation as well as a Matlab implementation including an intuitive graphical user interface is available. Import from various FE programs is possible, and the reduced elastic bodies can be exported to a variety of programs to simulate the compact models. In the course of the various projects, many improvements on the algorithmic side were added. As we learned over the years, there is not one ‘optimal’ MOR method. ‘Optimal’ MOR depends on circumstances, like boundary conditions, excitation spectra, further model usage. The toolbox is now used, e.g., in solid mechanics, biomechanics, vehicle dynamics, control of flexible structures, or crash simulations. In all these use cases, the toolbox allows the user to facilitate their well-known modeling and simulation environment. Only the critical MOR process during preprocessing is performed with Morembs, which helps to compare the various MOR techniques to find the most suited one.

Parametric model order reduction (PMOR) can effectively reduce analysis time and memory cost for finite element models. However, the calculation precision of the previous methods is not usually high enough. In this paper, an improved method is presented to approximate parametric models with higher accuracy. First of all, Krylov subspace projection-based method generally adopted in model order reduction (MOR) is introduced. Based on that, the method to adjust the local reduced-order models (ROMs) is described briefly. Then an approach to interpolate the compatible ROMs on the tangent space is presented in detail and the calculation flow chart for parametric model order reduction (PMOR) of finite element models is provided. In the end, the analysis results for the moving coil of electrical-dynamic shaker demonstrate that the generated parameterized ROM can approximate the direct ROM with higher fidelity than the original method.

We discuss methods for model order reduction (MOR) of linear systems with many input and output variables, arising in the modeling of linear (sub) circuits with a huge number of nodes and a large number of terminals, like power grids. Our work is based on the approaches SVDMOR and ESVDMOR proposed in recent publications (1; 2; 3; 4; 5). In particular, we discuss efficient numerical algorithms for their implementation. Only by using efficient tools from numerical linear algebra, these methods become applicable for truly large-scale problems.

A new mixed method for relative error model order reduction is proposed. In the proposed method the frequency domain balanced stochastic truncation method is improved by applying the generalized singular perturbation method to the frequency domain balanced system in the reduction procedure. The frequency domain balanced stochastic truncation method, which was proposed in [15] and [17] by the author, is based on two recently developed methods, namely frequency domain balanced truncation within a desired frequency bound and inner-outer factorization techniques. The proposed method in this paper is a carry over of the frequency-domain balanced stochastic truncation and is of interest for practical model order reduction because in this context it shows to keep the accuracy of the approximation as high as possible without sacrificing the computational efficiency and important system properties. It is shown that some important properties of the frequency domain stochastic balanced reduction technique are extended to the proposed reduction method by using the concept and properties of the reciprocal systems. Numerical results show the accuracy, simplicity and flexibility enhancement of the method.

A new method for model order reduction for an important class of non-linear dynamical systems is proposed. The reduction technique is based on the frequency balanced reduction method within a desired frequency band and a developed Trajectory Piecewise-Linear (TPWL) model order reduction approach. Both the methods are recently developed techniques and they can increase the accuracy and efficiency of the reduction method. The computational load and memory are significantly reduced when the method is used in the hybrid reduction method. The accuracy enhanced hybrid method is applicable to very large scale non-linear models efficiently and accurately. The numerical results show the accuracy and efficiency enhancement of the proposed method.

This paper proposes a new second-order Model Order Reduction (MOR) method suitable for reducing very large sized RC circuits or RC circuit parts of a non-RC circuit. The MOR is carried out on a partitioned circuit, which enables the use of low-order moments and macromodels of few elements. It is shown that the use of the second-order MOR method in conjunction with lower-order methods improves the overall MOR results and reliability in the case where a circuit contains particularly complex structures.

The Bilinear Interpolatory Rational Krylov Algorithm (BIRKA; P. Benner and T. Breiten, Interpolation-based H2-model reduction of bilinear control systems, SIAM J. Matrix Anal. Appl. 33 (2012), pp. 859–885. doi:10.1137/110836742) is a recently developed method for Model Order Reduction (MOR) of bilinear systems. Here, it is used and further developed for a certain class of parametric systems. As BIRKA does not preserve stability, two different approaches generating stable reduced models are presented. In addition, the convergence for a modified version of BIRKA for large systems is analysed and a method for detecting divergence possibly resulting from this modification is proposed. The behaviour of the algorithm is analysed using a finite element model for the thermal analysis of an electrical motor. The reduction of two different motor models, incorporating seven and thirteen different physical parameters, is performed.

ABSTRACTThe purpose of this paper is the identification of a reduced order model (ROM) from numerical code output by non-intrusive techniques (i.e. not requiring projecting of the governing equations onto the reduced basis modes). In this paper, we perform a comparison between two methods of model order reduction based on dynamic mode decomposition (DMD). The first method is a deterministic (classic) DMD technique endowed with a dynamic filtering criterion of selection of modes used in the ROM model. The second method is an adaptive randomised DMD algorithm (ARDMD) based on a randomised singular value decomposition. This produced an accelerating algorithm, which is endowed with a few additional advantages. In addition, the reduced order model is guaranteed to satisfy the boundary conditions of the full model, which is crucial for surrogate modelling. For numerical illustration, we use the shallow water equations model.

Numerical simulations for flow and transport in subsurface porous media often prove computationally prohibitive due to property data availability at multiple spatial scales that can vary by orders of magnitude. A number of model order reduction approaches are available in the existing literature that alleviate this issue by approximating the solution at a coarse scale preserving fine scale features. We attempt to present a comparison between two such model order reduction techniques, namely (1) adaptive numerical homogenization and (2) generalized multiscale basis functions. We rely upon a non-linear, multi-phase, black-oil model formulation, commonly encountered in the oil and gas industry, as the basis for comparing the aforementioned two approaches. An expanded mixed finite element formulation is used to separate the spatial scales between non-linear, flow, and transport problems. A numerical benchmark is setup using fine scale property information from the 10th SPE comparative project dataset for the purpose of comparing accuracies of these two schemes. An adaptive criterion is employed by both the schemes for local enrichment that allows us to preserve solution accuracy compared to the fine scale benchmark problem. The numerical results indicate that both schemes are able to adequately capture the fine scale features of the model problem at hand.

Sensitivity analysis of random linear differential–algebraic equations using system norms

A new model order reduction (MOR) method for large-scale time-delay differential algebra systems is proposed herein. The presented MOR algorithm is based on parametric moment matching and $$\varepsilon $$ -embedding. By selecting an appropriate projection matrix, this process generates reduced-order models that preserve the structure of the original time-delay differential algebra system. Additionally, moment-matching results and error estimations are provided. Finally, two numerical circuits are tested to illustrate the effectiveness of the proposed MOR method.

This paper deals with a new method for model order reduction of linear continuous time interval system. This new method is based on the Kharitonov’s theorem, the Stability equation method and the error minimization by Differential Evolution. The reduced order interval model is determined by using Kharitonov’s polynomials, which make use of the Kharitonov’s theorem and general form of the stability equation method for denominator, while the numerator is obtained by minimizing the integral square error between the transient responses of original and reduced order models using Differential Evolution algorithm. This method generates stable reduced order interval system if the original higher order system is stable and retains the steady-state value. The proposed method is illustrated with the help of typical numerical example considered from the literature.

Comparative Study of Different Methods of Model Order Reduction

We consider linear dynamical systems including random variables to model uncertainties of physical parameters. The output of the system is expanded into a series with orthogonal basis functions. Our aim is to identify a sparse approximation, where just a low number of basis functions is required for a sufficiently accurate representation. The coefficient functions of the expansion are approximated by a quadrature method or a sampling technique. The performance of a quadrature scheme can be described by a larger linear dynamical system, which is weakly coupled. We apply methods of model order reduction to the coupled system, which results in a sparse approximation of the original expansion. The approximation error is estimated by Hardy norms of transfer functions. Furthermore, we present numerical results for a test example modelling the electric circuit of a band pass filter.