Efficient matrix exponential method based on extended Krylov subspace for transient simulation of large-scale linear circuits

Paper 3C-3Matrix exponential (MEXP) method has been demonstrated to be a competitive candidate for transient simulation of very large-scale integrated circuits. Nevertheless, the performance of MEXP based on ordinary Krylov subspace is unsatisfactory for stiff circuits, wherein the underlying Arnoldi process tends to oversample the high magnitude part of the system spectrum while undersampling the low magnitude part that is important to the final accuracy. In this work we explore the use of extended Krylov subspace to generate more accurate and efficient approximation for MEXP. We also develop a formulation that allows unequal positive and negative dimensions in the generated Krylov subspace for better performance. Numerical results demonstrate the efficacy of the proposed method. © 2014 IEEE.published_or_final_versio

Globally stable, highly parallelizable fast transient circuit simulation via faber series

Time-domain circuit simulation based on matrix exponential has attracted renewed interested, owing to its explicit nature and global stability that enable millionth-order circuit simulation. The matrix exponential is commonly computed by Krylov subspace methods, which become inefficient when the circuit is stiff, namely, when the time constants of the circuit differ by several orders. In this paper, we utilize the truncated Faber Series for accurate evaluation of the matrix exponential even under a highly stiff system matrix arising from practical circuits. Experiments have shown that the proposed approach is globally stable, highly accurate and parallelizable, and avoids excessive memory storage demanded by Krylov subspace methods. © 2012 IEEE.published_or_final_versio

Addressing Computational Complexity of High Speed Distributed Circuits Using Model Order Reduction

Advanced in the fabrication technology of integrated circuits (ICs) over the last couple of years has resulted in an unparalleled expansion of the functionality of microelectronic systems. Today’s ICs feature complex deep-submicron mixed-signal designs and have found numerous applications in industry due to their lower manufacturing costs and higher performance levels. The tendency towards smaller feature sizes and increasing clock rates is placing higher demands on signal integrity design by highlighting previously negligible interconnect effects such as distortion, reflection, ringing, delay, and crosstalk. These effects if not predicted in the early stages of the design cycle can severely degrade circuit performance and reliability. The objective of this thesis is to develop new model order reduction (MOR) techniques to minimize the computational complexity of non-linear circuits and electronic systems that have delay elements. MOR techniques provide a mechanism to generate reduced order models from the detailed description of the original modified nodal analysis (MNA) formulation. The following contributions are made in this thesis: 1. The first project presents a methodology for reduction of Partial Element Equivalent Circuit (PEEC) models. PEEC method is widely used in electromagnetic compatibility and signal integrity problems in both the time and frequency domains. The PEEC model with retardation has been applied to 3-D analysis but often result in large and dense matrices, which are computationally expensive to solve. In this thesis, a new moment matching technique based on Multi-order Arnoldi is described to model PEEC networks with retardation. 2. The second project deals with developing an efficient model order reduction algorithm for simulating large interconnect networks with nonlinear elements. The proposed methodology is based on a multidimensional subspace method and uses constraint equations to link the nonlinear elements and biasing sources to the reduced order model. This approach significantly improves the simulation time of distributed nonlinear systems, since additional ports are not required to link the nonlinear elements to the reduced order model, yielding appreciable savings in the size of the reduced order model and computational time. 3. A parameterized reduction technique for nonlinear systems is presented. The proposed method uses multidimensional subspace and variational analysis to capture the variances of design parameters and approximates the weakly nonlinear functions as a Taylor series. An SVD approach is presented to address the efficiency of reduced order model. The proposed methodology significantly improves the simulation time of weakly nonlinear systems since the size of the reduced system is smaller than the original system and a new reduced model is not required each time a design parameter is changed

A new tightly-coupled transient electro-thermal simulation method for power electronics

Paper no. 224This paper presents a new transient electro-thermal (ET) simulation method for fast 3D chip-level analysis of power electronics with field solver accuracy. The metallization stacks are meshed and solved with 3D field solver using nonlinear temperature-dependent parameters, and the active devices are modeled with nonlinear tabular compact models to avoid time-consuming TCAD simulation. The main contributions include: 1) A tightly-coupled formulation that solves the electrical and thermal responses simultaneously for better convergence property; 2) Explicit account of capacitive effects, including interconnect parasitic capacitance and gate capacitance of power devices, to improve modeling accuracy in highfrequency applications; 3) A specialized transient solver based on the matrix exponential method (MEXP) to address the multi-scale problem caused by the considerably different time scales in electrical and thermal dynamics. Numerical experiments have demonstrated the advantages of the proposed co-simulation framework.postprin

Model reduction and simulation of nonlinear circuits via tensor decomposition

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A trajectory piecewise-linear approach to model order reduction of nonlinear dynamical systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2003.Includes bibliographical references (p. 117-126).(cont.) Finally, we present projection schemes which result in improved accuracy of the reduced order TPWL models, as well as discuss approaches leading to guaranteed stable and passive TPWL reduced-order models.In this study we discuss the problem of Model Order Reduction (MOR) for a class of nonlinear dynamical systems. In particular, we consider reduction schemes based on projection of the original state-space to a lower-dimensional space e.g. by using Krylov methods. In the nonlinear case, however, applying a projection-based MOR scheme does not immediately yield computationally efficient macromodels. In order to overcome this fundamental problem, we propose to first approximate the original nonlinear system with a weighted combination of a small set of linearized models of this system, and then reduce each of the models with an appropriate projection method. The linearized models are generated about a state trajectory of the nonlinear system corresponding to a certain 'training' input. As demonstrated by results of numerical tests, the obtained trajectory quasi-piecewise-linear reduced order models are very cost-efficient, while providing superior accuracy as compared to existing MOR schemes, based on single-state Taylor's expansions. In this dissertation, the proposed MOR approach is tested for a number of examples of nonlinear dynamical systems, including micromachined devices, analog circuits (discrete transmission line models, operational amplifiers), and fluid flow problems. The tests validate the extracted models and indicate that the proposed approach can be effectively used to obtain system-level models for strongly nonlinear devices. This dissertation also shows an inexpensive method of generating trajectory piecewise-linear (TPWL) models based on constructing the reduced models 'on-the-fly', which accelerates simulation of the system response. Moreover, we propose a procedure for estimating simulation errors, which can be used to determine accuracy of the extracted trajectory piecewise-linear reduced order models.by MichaÅ Jerzy RewieÅski.Ph.D