Showing papers on "Model order reduction published in 2002"

Guaranteed passive balancing transformations for model order reduction

Abstract: The major concerns in state-of-the-art model reduction algorithms are: achieving accurate models of sufficiently small size, numerically stable and efficient generation of the models, and preservation of system properties such as passivity. Algorithms such as PRIMA generate guaranteed-passive models, for systems with special internal structure, using numerically stable and efficient Krylov-subspace iterations. Truncated balanced realization (TBR) algorithms, as used to date in the design automation community, can achieve smaller models with better error control, but do not necessarily preserve passivity. In this paper we show how to construct TBR-like methods that guarantee passive reduced models and in addition are applicable to state-space systems with arbitrary internal structure.

Review: Automatic Model Reduction for Transient Simulation of MEMS‐based Devices

Abstract: The rapid development of MEMS-based devices requires 3D time-dependent simulations for coupled physical domains (thermal, mechanical, electrical, etc.). This in turn requires the solution of high-dimensional ordinary differential equations (ODEs) that result from space discretization of the device. However, instead of a “brute force” approach to integrate a large system of ODEs, one can use modern mathematical methods to reduce the system's dimension. The goal of the present paper is to review them from an engineering perspective. It is shown that in many cases important for practice the order of ODEs can be reduced by several orders of magnitude almost without sacrificing precision.

On projection-based algorithms for model-order reduction of interconnects

Abstract: Model-order reduction is a key technique to do fast simulation of interconnect networks. Among many model-order reduction algorithms, those based on projection methods work quite well. In this paper, we review the projection-based algorithms in two categories. The first one is the coefficient matching algorithms. We generalize the Krylov subspace method on moment matching at a single point, to multipoint moment-matching methods with matching points located anywhere in the closed right-hand side (RHS) of the complex plane, and we provide algorithms matching the coefficients of series expansion-based on orthonormal polynomials and generalized orthonormal basis functions in Hilbert and Hardy space. The second category belongs to the grammian-based algorithms, where we provide efficient algorithm for the computation of grammians and new approximate grammian-based approaches. We summarize some important properties of projection-based algorithms so that they may be used more flexibly.

HiPRIME: hierarchical and passivity reserved interconnect macromodeling engine for RLKC power delivery

Abstract: This paper proposes a general hierarchical analysis methodology, HiPRIME, to efficiently analyze RLKC power delivery systems. After partitioning the circuits into blocks, we develop and apply the IEKS (improved extended Krylov subspace) method to build the multi-port Norton equivalent circuits which transform all the internal sources to Norton current sources at ports. Since there is no active elements inside the Norton circuits, passive or realizable model order reduction techniques such as PRIMA can be applied. To further reduce the top-level hierarchy runtime, we develop a second-level model reduction algorithm and prove its passivity. Experimental results show 400-700/spl times/ runtime improvement with less than 0.2% error.

Efficient model order reduction via multi-node moment matching

Abstract: The new concept of multi-node moment matching (MMM) is introduced in this paper. The MMM technique simultaneously matches the moments at several nodes of a circuit using explicit moment matching around s=0. As compared to the well-known single-point moment matching (SMM) techniques (such as AWE), MMM has several advantages. First, the number of moments required by MMM is significantly lower than SMM for a reduced order model of the same accuracy, which directly translates into computational efficiency. This higher computational efficiency of MMM as compared to SMM increases with the number of inputs to the circuit. Second, MMM has much better numerical stability as compared to SMM. This characteristic allows MMM to calculate an arbitrarily high order approximation of a linear system, achieving the required accuracy for systems with complex responses. Finally, MMM is highly suitable for parallel processing techniques especially for higher order approximations while SMM has to calculate the moments sequentially and cannot be adapted to parallel processing techniques.

Two-step Lanczos algorithm for model order reduction

Abstract: The Pade-Via-Lanczos (PVL) algorithm proved to be a reliable technique for obtaining reduced-order models of electromagnetic devices. Its computational complexity is, however, quite large, since it involves inversion or factorization of a matrix which can be, for complex devices, on the order of hundreds of thousands. The present paper proposes a two-step approach based entirely on the Lanczos algorithm, meant to drastically reduce the computational complexity. In the first step, a Lanczos-based projection technique is used to reduce the un-inverted matrix to a manageable size, which can be dealt with by the PVL method in the second step. The computing time was thus reduced by a factor of ten, as compared to the classical PVL.

Reduced Order Nonlinear Navier-Stokes Models for Synthetic Jets

Abstract: While the potential for the use of synthetic jet actuators to achieve flow control has been noted for some time, most of such flow control studies have been empirical or experimental in nature. Several technical issues must be resolved to achieve rigorous, model-based, closed-loop control methodologies for this class of actuators. First, we seek to derive and evaluate model order reduction methods based on proper orthogonal decomposition that are suitable for synthetic jet actuators. Second, we seek to derive rigorously stable feedback control laws for the derived reduced order models

Robust and passive model order reduction for circuits containing susceptance elements

Abstract: Numerous approaches have been proposed to address the overwhelming modeling problems that result from the emergence of magnetic coupling as a dominant performance factor for ICsand packaging. Firstly, model order reduction (MOR) methods have been extended to robustly capture very high frequency behaviors for large RLC systems via methods such as PRIMA[8] with guaranteed passivity. In addition, new models of the magnetic couplings in terms of susceptance (inverse of inductance) have shown great promise for robust sparsification of otherwise intractable inductance coupling-matrix problems [3--5]. However, model order reduction via PRIMA for circuits that include susceptance elements does not guarantee passivity. Moreover, susceptance elements are incompatible with the path tracing algorithms that provide the fundamental runtime efficiency of RICE [10]. In this paper a novel MOR algorithm, SMOR, is proposed as an extension of ENOR [11] which exploits the matrix properties of susceptance-based circuits for runtime efficiency, and provides for a numerically stable, provably passive MOR using a new or-thonormalization strategy.

Model order reduction for strictly passive and causal distributed systems

Abstract: This paper presents a class of algorithms suitable for model reduction of distributed systems Distributed systems are not suitable for treatment by standard model-reduction algorithms such as PRIMA, PVL, and the Arnoldi schemes because they generate matrices that are dependent on frequency (or other parameters) and cannot be put in a lumped or state-space form Our algorithms build on well-known projection-based reduction techniques, and so require only matrix-vector product operations and are thus suitable for operation in conjunction with electromagnetic analysis codes that use iterative solution methods and fast-multipole acceleration techniques Under the condition that the starting systems satisfy system-theoretic properties required of physical systems, the reduced systems can be guaranteed to be passive For distributed systems, we argue that causality of the underlying representation is as important a consideration as passivity has become

A comprehensive 2-D inductance modeling approach for VLSI interconnects: frequency-dependent extraction and compact circuit model synthesis

Abstract: Although three-dimensional (3-D) partial inductance modeling costs have decreased with stable, sparse approximations of the inductance matrix and its inverse, 3-D models are still intractable when applied to full chip timing or crosstalk analysis. The 3-D partial inductance matrix (or its inverse) is too large to be extracted or simulated when power-grid cross-sections are made wide to capture proximity effect and wires are discretized finely to capture skin effect. Fortunately, 3-D inductance models are unnecessary in VLSI interconnect analysis. Because return currents follow interconnect wires, long interconnect wires can be accurately modeled as two-dimensional (2-D) transmission lines and frequency-dependent loop impedances extracted using 2-D methods . Furthermore, this frequency dependence can be approximated with compact circuit models for both uncoupled and coupled lines. Three-dimensional inductance models are only necessary to handle worst case effects such as simultaneous switching in the end regions. This paper begins by explaining and defending the 2-D modeling approach. It then extends the extraction algorithm to efficiently include distant return paths. Finally, a novel synthesis technique is described that approximates the frequency-dependent series impedance of VLSI interconnects with compact circuit models suitable for timing and noise analysis.

Finite element‐based model order reduction of electromagnetic devices

Abstract: In this paper an efficient algorithm is presented for the development of compact and passive macro-models of electromagnetic devices through the systematic reduction of the order of discrete models for these devices obtained through the use of finite elements. The proposed methodology is founded on a new finite element formulation that casts Maxwell's curl equations in a state-space form. Such state-space representations are very compatible with a class of robust model order reduction techniques based on Krylov subspaces. However, the advantage of this compatibility appears to be hindered by the fact that the state matrix of the discretized Maxwellian system is of dimension almost twice that obtained from the finite element approximation of the electromagnetic vector wave equation. It is shown in this paper that the apparent penalty in both memory and computation efficiency can be avoided by a proper selection of the finite element expansion functions used for the discretization of the electromagnetic fields. More specifically, it is shown that the proper selection of expansion functions renders the state-space form of the Maxwellian system equivalent to the discrete problem obtained from the approximation of the vector wave equation using tangentially continuous vector finite elements. This equivalence is then used to effect Krylov-based model order reduction directly from the finite element approximation of the vector wave equation. In particular, a passive model order reduction algorithm is used for this purpose. The proposed reduced order macro-modelling algorithm is demonstrated through its application to a variety of microwave passive components. Copyright © 2002 John Wiley & Sons, Ltd.

Fast frequency sweep model order reduction of polynomial matrix equations resulting from finite element discretizations

Abstract: The frequency domain finite element method (FEM) results in matrix equations that have polynomial dependence (or transcendental dependence which can be written as a polynomial via a Taylor series) on the frequency of excitation. For a wide-band fast frequency sweep technique based on a moment-matching model order reduc­ tion (MORe) process, researchers generally take one of two approaches. The first is to linearize the polynomial dependence (which will either limit the bandwidth of accuracy or require the introduction of extra degrees of freedom) and then use a wellconditioned Krylov subspace technique such as the projection via Arnoldi (PVA) or the Pade via Lanczos (PVL) processes. The second approach is to work directly with the polynomial matrix equation and use one of the available, but ill-conditioned, asymptotic waveform evaluation (AWE) methods. For large-scale FEM simulations, introducing extra degrees of freedom, and therefore increasing the length of the MORe vectors and the amount of memory required, is not desirable: therefore, the first approach is not alluring. On the other hand, an ill-conditioned AWE process is unattractive. This dissertation presents two MORe techniques for polynomial matrix equations. The first, an automated multipoint Galerkin AWE (MGAWE) process, is capable of producing a reduced order model (ROM) with a relatively small subspace. The second novel process presented, well-conditioned AWE (WCAWE), is capable of producing an accurate, robust, wide-band simulation with just one expansion point. ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. These novel processes are able to circumvent the problematic issues that arise from the traditional PVA, PVL or AWE techniques. First, these novel processes do not require any additional unknowns and can operate directly on the polynomial matrix equation. Second, these processes are wide-band, and in the case of WCAWE, very well-conditioned even for a large approximation order. Along with the presentation of these algorithms, numerical examples modeled using the FEM axe given throughout the work to illustrate their accuracy, efficiency and robustness. Finally, this disser­ tation closes with a detailed description of many possible areas of further research including an extension of the methods to a block and/or multivariable versions, and applications of the methods to problems in which the system matrix has exponential variations in the ROM varying parameter.

A Comparison of Some Model Order Reduction Techniques

Abstract: In this paper, an analysis of some model order reduction (MORe) techniques is presented. More precisely, this paper considers asymptotic waveform evaluation (AWE), Galerkin asymptotic waveform evaluation (GAWE) with a short-term vector recurrence relation, multipoint Galerkin asymptotic waveform evaluation (MGAWE) also using a short-term recurrence, and matrix-Pade via Lanczos (MPVL). These techniques are applied to matrix equations resulting when the enite element method (FEM) is used to model electromagnetic wave propagation problems. The reduced order model equations can then be solved repeatedly to obtain a wideband frequency simulation with a reduction in total computation time. The analysis contained herein compares and contrasts the MORe techniques by not only considering the nature of the individual algorithms, but also solving several illustrative numerical examples. These examples show how, for a MORe technique, a radiation and scattering problem might have to be treated very differently. In addition, it is noted that the unknown(s) desired as output(s) from the FEM mesh can ineuence which MORe technique is more efecient. The solutions obtained through the MORe techniques are compared to an LU decomposition at each frequency point of interest to benchmark their accuracy and efeciency.

Application of Model Order Reduction to Compressor Aeroelastic Models

Abstract: A model order reduction technique that yields low-order models of blade row unsteady aerodynamics is introduced. The technique is applied to linearized unsteady Euler CFD solutions in such a way that the resulting blade row models can be linked to their surroundings through their boundary conditions. The technique is applied to a transonic compressor aeroelastic analysis, in which the high-fidelity CFD forced-response results are better captured than with models that use single-frequency influence coefficients. A low-speed compressor stage is also modeled to demonstrate the multistage capability of the method. These examples demonstrate how model order reduction can be used to systematically improve the versatility, fidelity, and range of applicability of the low-order aerodynamic models typically used for incorporation of CFD results into aeroelastic analyses.

Control-oriented modeling and analysis of automotive transcritical AC system dynamics

Abstract: This paper presents a dynamic model of a transcritical air-conditioning system, specifically suited for multivariable controller design. The physically based model retains sufficient detail to accurately predict system dynamic response while also being simple enough to be of value in determining appropriate control strategies. The control focus would be quasi-steady transitions between operating states by modulating flow rates of both air and refrigerant to meet changing constraints on capacity, efficiency, noise, etc. Both nonlinear and linearized versions of the model are compared to experimental data. Possibilities for model order reduction are substantiated.

Robust Control of Linearized Poiseuille Flow

Abstract: An approach to feedback control of linearized planar Poiseuille flow using H\infty control is developed. Surface transpiration is used to control the flow and point measurements of the wall shear stress are assumed to monitor its state. A high-but-finite dimensional model is obtained via a Galerkin procedure, and this model is approximated by a low dimensional one using Hankel-optimal model reduction. For the purposes of control design the flow is modeled as an interconnection of this low dimensional system and a perturbation, reflecting the uncertainty in the model. The goal of control design is to achieve robust stability (i.e. to stabilize any combination of the nominal plant and a feasible perturbation), and to satisfy certain performance requirements. Two different types of surface actuation are considered -- harmonic transpiration and a model of a pair of suction/blowing panels. It is found that the latter is more efficient in suppressing disturbances in terms of the control effort required.

Automatic order reduction of thermo-electric models for MEMS: Arnoldi versus Guyan

Abstract: In this paper we present an automatic order reduction of a linear thermo-electric model describing a novel type of micropropulsion device. Model order reduction is essential to achieve easily to evaluate, yet accurate macromodel of the device. We present numerical simulation results of the full finite element model and the different reduced order models that describe the transient thermo-electric behaviour of the device. The advantages of an Arnoldi-algorithm-based model order reduction over a commercially available reduced order modeling after Guyan are shown.

Karhunen-Loeve based model order reduction of nonlinear systems

Abstract: Despite continual advances in computing power and the emergence of very efficient algorithms, rigorous analysis of many electrodynamic devices is still often computationally prohibitive. To help overcome these obstacles, a considerable amount of research into so-called "model order reduction" (MOR) methods has been conducted in both the circuit and electromagnetic communities. Traditionally, nonlinearities are dealt with by partitioning the system into linear and nonlinear sub-blocks, and applying MOR techniques to the linear sub-blocks. In systems where the nonlinearity is distributed, however, this is not possible. A method for accomplishing MOR in such cases is described. The theoretical basis for the method is discussed, choices which must be made in implementation are highlighted, and the analysis of a pulse-compression nonlinear transmission line is carried out to demonstrate the capability of the scheme.

Model order reduction and equivalent circuit extraction for fit discretized electromagnetic systems

Abstract: In this paper a new approach for model order reduction of electrodynamic devices discretized by the finite integration theory is presented. It has two main applications: The fast frequency sweep of the transfer behaviour of resonant filters and the extraction of general equivalent circuits simplifying coupled circuit–electromagnetic simulations. The proposed two-step Lanczos (TSL) algorithm combines a Lanczos based projection in the first step to pre-reduce the non-inverted system with a PVL-like algorithm in the second step to get an optimal Pade approximation. This algorithm drastically reduces the computational complexity of model order reduction while maintaining the same accuracy as direct PVL. By applying it to a curl–curl formulation of the system, the computational cost is further reduced by a factor of two. By use of a projection method the curl–curl formulation can also be extended to lossy systems. Additionally, the method keeps the system stable, passive, and real throughout the process—an important prerequisite for the interpretation of the model as an equivalent network. Copyright © 2002 John Wiley & Sons, Ltd.

Equivalence of bilinear Routh and Schwarz approximation methods for discrete-time systems

Abstract: Two methods using the bilinear Routh approximations and Schwarz approximations have been proposed in the literature for the order reduction of discrete-time systems. Both methods yield the reduced models which preserve the stability and first few time-moments of the original ones. Recently, improvements have been made so that the impulse response energies of original models are also retained in the reduced models. In this Letter, it is shown that the former two methods and their improved versions are respectively equivalent to each other.

Automatic order reduction of thermo-electric model for micro-ignition unit

Abstract: In this paper we present an automatic, Krylov-subspace-based order reduction of a thermo-electric model, describing a novel type of micropropulsion device. Model order reduction is essential for achieving easily to evaluate, yet accurate macromodel of the device, and Is needed for simulating both the microthruster array and its driving circuitry. We present numerical simulation results of the full finite element model and the reduced order model that describes the transient thermo-electric behavior. A comparison between Krylov-subspace-based order reduction and order reduction using control theoretical approaches, such as Balanced Truncation Approximation (BTA), has been performed. For the first time a Single-Input-Single-Output (SISO) setup for the reduction algorithm was sufficient to approximate the complete time-dependent temperature distribution of the device.

Parameterized Reduced Models for Efficient Optimization of Structural Dynamic Behavior

Abstract: Substructure synthesis is a method of model order reduction which is generally more efficient, computationally speaking, than analyzing the complete structural system. However, these methods are not necessarily well adapted for use within an optimization process since they do not preserve the fidelity of the reduced models when structural modifications within the reduced substructure are introduced. As a result, a costly model reduction must be performed at each iteration step. This paper presents a new method to improve standard reduction methods by taking into account a priori knowledge of the potential structural modifications. Indeed, this information proves to be salutary in creating a single enriched model reduction transformation that preserves the precision of the reduced substructure model throughout the optimization process. The proposed approach consists in extending the standard transformation matrix by a set of static residual vectors which are optimized with respect to the design variables to be modified. The proposed method can be used with a variety component mode synthesis approaches with any type of substructure natural modes: free-free, cantilever or hybrid modes. The proposed methodology is illustrated on the basis of a simulated test case taken from the aero-engine industry. Introduction The ever increasing demand for faster engineering analysis in the design process has resulted in a substantial amount of research and development on faster and more accurate approximate reanalysis method. The difficulty of any model reduction procedure lies in how to complete the representation basis in order to reduce truncation effects. Two main classes of methodologies can be found in the literature. The first class seeks to generate a set of Ritz vectors capable of representing with precision the structural behavior under a wide variety of structural modifications. For example, Balmes 1 10 studied the possibility of using a constant basis of Ritz vectors to create parametric families of reduced models whereas Bouazzouni A., et al 2 9 developed a method for optimally constructing additional vectors by using the dynamic behavior of the structure before modification combined with the a priori knowledge of the design variables. Both of these approaches have already been used effectively in an industrial context. The second class of reduction methods are based on a high order polynomial expansions of model responses about a nominal point in parameter space. This approach proves to be particularly interesting for small number of design variables and can be used for topological optimization. In this paper, we will extend the approach developed in 2 for use with substructure synthesis techniques. The latter represent a economic means for evaluating the structural behavior of complex mechanical assemblies and are known collectively as component mode synthesis methods. This approach represents a structure as an assembly of individually reduced substructures called superelements. For example, the Craig-Bampton (CB) method is one of many techniques of component mode synthesis used intensively in the aerospace industry 3 4 . However, these methods are not always well adapted to the industrial problem. This is especially true when parametric studies are to be performed with respect to design variables contained within individual superelements. The designer has the choice of either re-using the nominal model reduction transformation or performing a new superelement analysis for each modified component. The first option generally leads to inaccurate results while the latter is often impracticable due to cost considerations. We propose in this paper a new method to improve the standard superelement reduction methods by taking into account an a priori knowledge of the potentially modifiable design variables with the objective of constructing a unique reduction transformation matrix which will preserve Copyright  2002 The American Institute of Aeronautics and Astronautics Inc. All rights rserved. 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con 22-25 April 2002, Denver, Colorado AIAA 2002-1392 Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Model reduction of uncertain systems

Abstract: This research deals with model order reduction of uncertain discrete-time SISO systems. Our main directions of investigation are approximation of the original uncertain high-order system by a fixed-coefficient low-order system and by an uncertain low-order system.

Order Reduction and Closed-Loop Vibration Control in Helicopter Fuselages

Abstract: The problem of vibration reduction in helicopter fuselages using the concept of active control of structural responseis addressed. When the largesize of the coupled gearbox ‐e exible fuselage system dynamicsis considered, e rst a balanced-realization-based order reduction is employed to reduce the size of the problem. Then using the reduced-order model, a closed-loop controller is designed to minimize the vibratory levels in the fuselage with the constraintthatthecontrollerensuresstability oftheoriginalfull-ordersystem.Thecontrollerdesignisbasedonthe concept of disturbance rejection by the internal model principle. When a four-block representation of the problem and doubly coprime factorization theory is employed, a stable controller is designed for this multi-input/multioutput control problem. It is observed that this controller yields a closed-loop transfer function, which rejects the external disturbance not only at the desired frequency but also in its neighborhood. In addition, contrary to open-loop control, the present technique of closed-loop control reduces the vibratory levels both in the fuselage and the gearbox. The ine uence of sensor locations on vibration minimization has also been highlighted.

Macromodeling of Electrical Interconnects and Packages via PEEC Approach

Abstract: This paper deals with model order reduction of multi-input-multi-output structures described by Partial Element Equivalent Circuits (PEEC). A Krylovbased algorithm is adopted, that guarantees the stability and the passivity of the obtained reduced circuit with good accuracy. Our procedure is validated by means of a simple example made of two parallel stacked traces with air filling, representing an ideal interconnect. Comparisons showing the frequency behaviour of the interconnect scattering parameters with analytical expressions based on Telegrapher’s equations are presented. We then apply the proposed modeling process to analyze, both in the time and in the frequency domain, the crosstalk between two microstrip traces either in the case of air dielectric and in the case of relative dielectric constant of 4.7. The influence of the model order on the results is also discussed.

Reduced Order Modeling for RLC Interconnect Tree Using Hurwitz Polynomial

Abstract: We investigate on-chip RLC interconnect reduced order modeling problem A provably realizable and stable model order reduction approach is proposed To guarantee stability of reduced order circuits, we first employ a realizable reduction for load approximation to preserve the first three driving-point admittance coefficients Then, we use Hurwitz polynomials to approximate the denominators of original rational transfer functions We prove that stability can be guaranteed during a hierarchical analysis while circuit response moments can still be matched implicitly We also give some experimental results to show the accuracy and efficiency of the proposed approach

Accurate reduced RL model for frequency dependent transmission lines

Abstract: In this paper, we provide a new algorithm to model the frequency dependent RL parameters of transmission lines. Each set of the parameters is modeled by the input impedance of a single RL port. The modeling process consists of two steps. In the first step, a model with high enough order is generated to fit the given data precisely, and in the second step, model order reduction is applied to reduce the model order while keeping the accuracy of the model high enough. Experiments show that this algorithm generates models with low order and high accuracy.