Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for wh...

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A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

With the rapid developments in very large-scale integration (VLSI) technology, design and computer-aided design (CAD) techniques, at both the chip and package level, the operating frequencies are fast reaching the vicinity of gigahertz and switching times are getting to the subnanosecond levels. The ever increasing quest for high-speed applications is placing higher demands on interconnect performance and highlighted the previously negligible effects of interconnects such as ringing, signal delay, distortion, reflections, and crosstalk. In this review paper various high-speed interconnect effects are briefly discussed. In addition, recent advances in transmission line macromodeling techniques are presented. Also, simulation of high-speed interconnects using model-reduction-based algorithms is discussed in detail.

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Simulation of high-speed interconnects

We provide a unifying projection-based framework for structure-preserving interpolatory model reduction of parameterized linear dynamical systems, ie, systems having a structured dependence on parameters that we wish to retain in the reduced-order model The parameter dependence may be linear or nonlinear and is retained in the reduced-order model Moreover, we are able to give conditions under which the gradient and Hessian of the system response with respect to the system parameters is matched in the reduced-order model We provide a systematic approach built on established interpolatory $\mathcal{H}_2$ optimal model reduction methods that will produce parameterized reduced-order models having high fidelity throughout a parameter range of interest For single input/single output systems with parameters in the input/output maps, we provide reduced-order models that are optimal with respect to an $\mathcal{H}_2\otimes\mathcal{L}_2$ joint error measure The capabilities of these approaches are illustrated by several numerical examples from technical applications

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Interpolatory Projection Methods for Parameterized Model Reduction

Large scale dynamical systems are a common framework for the modeling and control of many complex phenomena of scientific interest and industrial value, with examples of diverse origin that include signal propagation and interference in electric circuits, storm surge prediction before an advancing hurricane, vibration suppression in large structures, temperature control in various media, neurotransmission in the nervous system, and behavior of micro-electro-mechanical systems. Direct numerical simulation of underlying mathematical models is one of few available means for accurate prediction and control of these complex phenomena. The need for ever greater accuracy compels inclusion of greater detail in the model and potential coupling to other complex systems leading inevitably to very large-scale and complex dynamical models. Simulations in such large-scale settings can make untenable demands on computational resources and efficient model utilization becomes necessary. Model reduction is one response to this challenge, wherein one seeks a simpler (typically lower order) model that nearly replicates the behavior of the original model. When high fidelity is achieved with a reduced-order model, it can then be used reliably as an efficient surrogate to the original, perhaps replacing it as a component in larger simulations or in allied contexts such as development of simpler, faster controllers suitable for real time applications.

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Interpolatory Model Reduction of Large-Scale Dynamical Systems

A dual-polarized aperture-coupled magneto-electric (ME) dipole antenna is proposed. Two separate substrate-integrated waveguides (SIWs) implemented in two printed circuit board (PCB) laminates are used to feed the antenna. The simulated $- {10}$ -dB impedance bandwidth of the antenna is 21% together with an isolation of over 45 dB between the two input ports. Good radiation characteristics, including almost identical unidirectional radiation patterns in two orthogonal planes, front-to-back ratio larger than 20 dB, cross-polarization levels less than $- {23}\;{\text{dB}}$ , and a stable gain around 8 dBi over the operating band, are achieved. By employing the proposed radiating element, a ${2} \times {2}$ wideband antenna array working at the 60-GHz band is designed, fabricated, and tested, which can generate two-dimensional (2-D) multiple beams with dual polarization. A measured $- {10}\;\text{dB}$ impedance bandwidth wider than 22% and a gain up to 12.5 dBi are obtained. Owing to the superiority of the ME dipole, the radiation pattern of the array is also stable over the operating frequencies and nearly identical in two orthogonal planes for both of the polarizations. With advantages of desirable performance, convenience of fabrication and integration, and low cost, the proposed antenna and array are attractive for millimeter-wave wireless communication systems.

60-GHz Dual-Polarized Two-Dimensional Switch-Beam Wideband Antenna Array of Aperture-Coupled Magneto-Electric Dipoles

There is a significant need for efficient and accurate macromodels of components during the design of microwave circuits. Increased integration levels in microwave devices and higher signal speeds have produced the need to include effects previously neglected during circuit simulations. Accurate prediction of these effects involve solution of large systems of equations, the direct simulation of which is prohibitively CPU expensive. In this paper, an algorithm is proposed to form passive parametrized macromodels of large linear networks that match the characteristics of the original network in time, as well as other design parameters of the circuit. A novel feature of the algorithm is the ability to incorporate a set of design parameters within the reduced model. The size of the reduced models obtained using the proposed algorithm were less than 5% when compared to the original circuit. A speedup of an order of magnitude was observed for typical high-speed transmission-line networks. The algorithm is general and can be applied to other disciplines such as thermal analysis.

Passive parameterized time-domain macromodels for high-speed transmission-line networks

This paper presents a new technique to reduce the order of transmission line circuits simultaneously with respect to multiple parameters. The reduction is based on multi-dimensional congruence transformation. The proposed algorithm provides efficient means to estimate the response of large distributed circuits simultaneously as a function of frequency and other design parameters.

Analysis of transmission line circuits using multi-dimensional model reduction techniques

In this work, a simple and compact transition from substrate integrated waveguide (SIW) to traditional rectangular waveguide is proposed and demonstrated. The substrate of SIW can be easily surface-mounted to the standard flange of the waveguide by creating a flange on the substrate. A longitudinal slot window etched on the broad wall of SIW couples energy between SIW and rectangular waveguide. An example of the transition structure is realized at 35 GHz with substrate of RT/Duroid 5880. HFSS simulated result of the transition shows a return loss less than −15 dB over a frequency range of 800 MHz. A back to back connected transition has been fabricated, and the measured results confirm well with the anticipated ones.

A transition from substrate integrated waveguide (SIW) to rectangular waveguide

Computing the steady-state response of large nonlinear circuits is becoming a key simulation requirement due to the rapid market growth of RF silicon integrated circuits. In this paper, we describe a nonlinear circuit reduction algorithm for finding the steady-state response. The proposed algorithm uses a congruent transformation-based technique to reduce the harmonic-balance equations into a much smaller set of equations. The main feature of the reduced circuit is that it shares with the original one a certain number of the derivatives with respect to the RF input power, steady-state analysis is then carried out on the reduced circuit instead of the original circuit.

A circuit reduction technique for finding the steady-state solution of nonlinear circuits

This paper describes an efficient algorithm based on moment-matching techniques for simulation of high-speed circuits in the presence of electromagnetic interference (EMI). The proposed method is based on the recently developed complex frequency hopping (CFH) technique for interconnect analysis. The new technique is useful for susceptibility analysis and is two to three orders of magnitude faster than conventional simulation techniques. In addition, it can be extended to the analysis of interconnects with frequency-dependent parameters and nonlinear terminations.

Roni Khazaka

Papers

Passive parameterized time-domain macromodels for high-speed transmission-line networks

Analysis of transmission line circuits using multi-dimensional model reduction techniques

A transition from substrate integrated waveguide (SIW) to rectangular waveguide

A circuit reduction technique for finding the steady-state solution of nonlinear circuits

Analysis of high-speed interconnects in the presence of electromagnetic interference