Michael Striebel

Model order reduction for nonlinear IC models

Model order reduction is a mathematical technique to transform nonlinear dynamical models into smaller ones, that are easier to analyze. In this paper we demonstrate how model order reduction can be applied to nonlinear electronic circuits. First we give an introduction to this important topic. For linear time-invariant systems there exist already some well-known techniques, like Truncated Balanced Realization. Afterwards we deal with some typical problems for model order reduction of electronic circuits. Because electronic circuits are highly nonlinear, it is impossible to use the methods for linear systems directly. Three reduction methods, which are suitable for nonlinear differential algebraic equation systems are summarized, the Trajectory piecewise Linear approach, Empirical Balanced Truncation, and the Proper Orthogonal Decomposition. The last two methods have the Galerkin projection in common. Because Galerkin projection does not decrease the evaluation costs of a reduced model, some interpolation techniques are discussed (Missing Point Estimation, and Adapted POD). Finally we show an application of model order reduction to a nonlinear academic model of a diode chain.

Progress in industrial mathematics at ECMI 2010

ECMI is the brand associated with European mathematics for industry and organizes successful biannual conferences. In this series, the 16th conference was held in the Historical City Hall of Wuppertal (Germany). It covered mathematics in a wide range of applications and methods, from Circuit and Electromagnetic Device Simulation, Model Order Reduction for Chip Design, Uncertainties and Stochastics, Production, Fluids, Life and Environmental Sciences to Dedicated and Versatile Methods. These proceedings of ECMI 2010 emphasize mathematics as an innovation enabler for industry and business, and as an absolutely essential pre-requiste for Europe on its way to becoming the leading knowledge-based economy in the world.We present a new adaptive circuit simulation algorithm base d on spline wavelets. The unknown voltages and currents are expanded in to a wavelet representation, which is determined as solution of nonlinear equ ations derived from the circuit equations by a Galerkin discretization. The spline wavelet representation is adaptively refined during the Newton iteration. The resulti ng approximation requires an almost minimal number of degrees of freedom, and in additi on the grid refinement approach enables very efficient numerical computation s. Initial numerical tests on various types of electronic circuits show promising resu lts when compared to the standard transient analysis.

Proper orthogonal decomposition model order reduction of nonlinear IC models

We demonstrate Model Order Reduction for a nonlinear system of differential-algebraic equations of a diode chain by Proper Orthogonal Decomposition with Adapted Missing Point Estimation. The collected time snapshots also allow for an efficient impression of the sensitivity of objective functions.

Model order reduction for nonlinear IC models with POD

Due to refined modelling of semiconductor devices and increasing packing densities, reduced order modelling of large nonlinear systems is of great importance in the design of integrated circuits (ICs). Despite the linear case, methodologies for nonlinear problems are only beginning to develop. The most practical approaches rely either on linearisation, making techniques from linear model order reduction applicable, or on proper orthogonal decomposition (POD), preserving the nonlinear characteristic. In this paper we focus on POD. We demonstrate the missing point estimation and propose a new adaption of POD to reduce both dimension of the problem under consideration and cost for evaluating the full nonlinear system.

Parameterized model order reduction

This Chapter introduces parameterized, or parametric, Model Order Reduction (pMOR). The Sections are offered in a prefered order for reading, but can be read independently. Section 5.1, written by Jorge Fernandez Villena, L. Miguel Silveira, Wil H.A. Schilders, Gabriela Ciuprina, Daniel Ioan and Sebastian Kula, overviews the basic principles for pMOR. Due to higher integration and increasing frequency-based effects, large, full Electromagnetic Models (EM) are needed for accurate prediction of the real behavior of integrated passives and interconnects. Furthermore, these structures are subject to parametric effects due to small variations of the geometric and physical properties of the inherent materials and manufacturing process. Accuracy requirements lead to huge models, which are expensive to simulate and this cost is increased when parameters and their effects are taken into account. This Section introduces the framework of pMOR, which aims at generating reduced models for systems depending on a set of parameters.

Domain Decomposition Based Multirating and its Perspective in Circuit Simulation

Based on domain decomposition, multirate time integration takes into account largely different timescales. In this class, a mixed multirate scheme and its application to an arbitrary number of subsystems is outlined. Moreover, the matter of activity change and the connection to model order reduction is discussed.