$$\varepsilon $$-Embedding Model Reduction Method for Time-Delay Differential Algebra Systems

Three-dimensional electromagnetic methods are fundamental tools for the analysis and design of high-speed systems. These methods often generate large systems of equations, and model order reduction (MOR) methods are used to reduce such a high complexity. When the geometric dimensions become electrically large or signal waveform rise times decrease, time delays must be included in the modeling. Design space optimization and exploration are usually performed during a typical design process that consequently requires repeated simulations for different design parameter values. Efficient performing of these design activities calls for parameterized model order reduction (PMOR) methods, which are able to reduce large systems of equations with respect to frequency and other design parameters of the circuit, such as layout or substrate features. We propose a novel PMOR method for neutral delayed differential systems, which is based on an efficient and reliable combination of univariate model order reduction methods, a procedure to find scaling and frequency shifting coefficients and positive interpolation schemes. The proposed scaling and frequency shifting coefficients enhance and improve the modeling capability of standard positive interpolation schemes and allow accurate modeling of highly dynamic systems with a limited amount of initial univariate models in the design space. The proposed method is able to provide parameterized reduced order models passive by construction over the design space of interest. Pertinent numerical examples validate the proposed PMOR approach.

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.

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.

Sample selection based on sensitivity analysis in parameterized model order reduction

Arnoldi-based model order reduction for linear systems with inhomogeneous initial conditions

Many applications concerning physical and technical processes employ dynamical systems for simulation purposes. The increasing demand for a more accurate and detailed description of realistic phenomena leads to high dimensional dynamical systems and hence, simulation often yields an increased computational effort. An approximation, e.g. with model order reduction techniques, of these large-scale systems becomes therefore crucial for a cost efficient simulation. This paper focuses on a model order reduction method for linear time in-variant (LTI) systems based on modal approximation via dominant poles. There, the original large-scale LTI system is projected onto the left and right eigenspaces corresponding to a specific subset of the eigenvalues of the system matrices, namely the dominant poles of the system's transfer function. Since these dominant poles can lie anywhere in the spectrum, specialized eigenvalue algorithms that can compute eigentriplets of large and sparse matrices are required. The Jacobi-Davidson method has proven to be a suitable and competitive candidate for the solution of various eigenvalue problems and hence, we discuss how it can be incorporated into this modal truncation approach. Generalizations of the reduction technique and the application of the algorithms to second-order systems are also investigated. The computed reduced order models obtained with this modal approximation can be combined with the ones constructed with Krylov subspace or balanced truncation based model order reduction methods to get even higher accuracies.

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.

A method of nested feedback loop decomposition of a given system is developed based on a special coordinate basis which reveals the finite and infinite zero structure of the system. This decomposition leads to a model order reduction procedure by which any arbitrary subset of the system invariant zeros can be retained in the model while the infinite zero structure of the model and the system are one and the same. Thus this method of model order reduction is a method of reducing the dynamic order via the zero structure of the given system. Several examples illustrate the method

Registration-based nonlinear model reduction of parametrized aerodynamics problems with applications to transonic Euler and RANS flows

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.

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.

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

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.

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.