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## Über dieses Buch

Selected from papers presented at the 8th Scientific Computation in Electrical Engineering conference in Toulouse in 2010, the contributions to this volume cover every angle of numerically modelling electronic and electrical systems, including computational electromagnetics, circuit theory and simulation and device modelling.

On computational electromagnetics, the chapters examine cutting-edge material ranging from low-frequency electrical machine modelling problems to issues in high-frequency scattering. Regarding circuit theory and simulation, the book details the most advanced techniques for modelling networks with many thousands of components. Modelling devices at microscopic levels is covered by a number of fundamental mathematical physics papers, while numerous papers on model order reduction help engineers and systems designers to bring their modelling of industrial-scale systems within the reach of present-day computational power. Complementing these more specific papers, the volume also contains a selection of mathematical methods which can be used in any application domain.

## Inhaltsverzeichnis

### Sparse Matrix Methods for Circuit Simulation Problems

Differential algebraic equations used for circuit simulation give rise to sequences of sparse linear systems. The matrices have very peculiar characteristics as compared to sparse matrices arising in other scientific applications. The matrices are extremely sparse and remain so when factorized. They are permutable to block triangular form, which breaks the sparse LU factorization problem into many smaller subproblems. Sparse methods based on operations on dense submatrices (supernodal and multifrontal methods) are not effective because of the extreme sparsity. KLU is a software package specifically written to exploit the properties of sparse circuit matrices. It relies on a permutation to block triangular form (BTF), several methods for finding a fill-reducing ordering (variants of approximate minimum degree and nested dissection), and Gilbert/Peierls’ sparse left-looking LU factorization algorithm to factorize each block. The package is written in C and includes a MATLAB interface. Performance results comparing KLU with SuperLU, Sparse 1.3, and UMFPACK on circuit simulation matrices are presented. KLU is the default sparse direct solver in the Xyce

TM

circuit simulation package developed by Sandia National Laboratories.

Timothy A. Davis, E. Palamadai Natarajan

### Some Remarks on A Priori Error Estimation for ESVDMOR

In previous work it is shown how to numerically improve the ESVDMOR method of Feldmann and Liu to be really applicable to linear, sparse, very large scale, and continuous-time descriptor systems. Stability and passivity preservation of this algorithm is also already proven. This work presents some steps towards a global a priori error estimation for this algorithm, which is necessary for a fully automatic application of this approach.

Peter Benner, André Schneider

### Block Preconditioning Strategies for High Order Finite Element Discretization of the Time-Harmonic Maxwell Equations

We study block preconditioning strategies for the solution of large sparse complex coefficients linear systems resulting from the discretization of the time-harmonic Maxwell equations by a high order discontinuous finite element method formulated on unstructured simplicial meshes. The proposed strategies are based on principles from incomplete factorization methods. Moreover, a complex shift is applied to the diagonal entries of the underlying matrices, a technique that has recently been exploited successfully in similar contexts and in particular for the multigrid solution of the scalar Helmholtz equation. Numerical results are presented for 2D and 3D electromagnetic wave propagation problems in homogeneous and heterogeneous media.

Matthias Bollhöfer, Stéphane Lanteri

### From Sizing over Design Centering and Pareto Optimization to Tolerance Pareto Optimization of Electronic Circuits

This paper presents an overview of sizing tasks in electronic circuit design and their corresponding formulations as optimization problems. We will start with the general multi-objective sizing problem. Then, the inclusion of statistically distributed parameters and of range-valued parameters into the scalar problems of yield optimization and design centering will be described. Finally, a problem formulation for considering these parameter tolerances by multi-objective Pareto optimization will be presented.

Helmut Gräb

### Importance Sampling for Determining SRAM Yield and Optimization with Statistical Constraint

Importance Sampling allows for efficient Monte Carlo sampling that also properly covers tails of distributions. From Large Deviation Theory we derive an optimal upper bound for the number of samples to efficiently sample for an accurate fail probability

P

fail

≤ 10

− 10

. We apply this to accurately and efficiently minimize the access time of Static Random Access Memory (SRAM), while guaranteeing a statistical constraint on the yield target.

E. J. W. ter Maten, O. Wittich, A. Di Bucchianico, T. S. Doorn, T. G. J. Beelen

### Effective Numerical Computation of Parameter Dependent Problems

We analyse parameter dependent differential-algebraic-equations (DAEs)

$$Ad\prime(x,t,p) + b(x,t,p) = 0.$$

For these systems one is interested in the relation between the numerical solutions

x

and some associated parameters

p

. The standard approach is to discretise the equations with respect to the parameters and solve the parameter independent equations afterwards. This approach forces a calculation of the differential equations multiple times (for a huge number of parameter values

p

). This may lead to high computational costs. By using the already computed solutions to calculate the remaining ones and thus exploiting the smoothness of the solution with respect to the parameters, it is possible to save the majority of the computational cost.

Lennart Jansen, Caren Tischendorf

### Analytical Properties of Circuits with Memristors

The memristor is a new lumped circuit element defined by a nonlinear charge-flux characteristic. The recent design of such a device has motivated a lot of research on this topic. In this communication we address certain analytical properties of semistate models of memristive circuits formulated in terms of differential-algebraic equations (DAEs). Specifically, we focus on the characterization of the

geometric index

of the DAEs arising in so-called branch-oriented analysis methods, which cover in particular tree-based techniques. Some related results involving nodal models and non-passive problems are discussed in less detail.

Ricardo Riaza

### Scattering Problems in Periodic Media with Local Perturbations

Within this paper we consider scattering problems with periodic exterior domains, modeled by the Helmholtz equation. Most current works on this subject make specific assumptions on the geometry of the periodic cell, e.g. special symmetries or shapes, and cannot be generalized to higher space dimensions in an easy way. In contrast our goal is the realization of an easy dimension independent concept which is valid for all kinds of periodic structures with local defects. We will first give a general analytical formulation and then present an algorithmic realization. At the end of the paper we will also depict a 1D and 2D example.

Therese Pollok, Lin Zschiedrich, Frank Schmidt

### From Quasi-static to High Frequencies : An Overview of Numerical Simulation at EADS

EADS IW produces mathematical methods, numerical schemes and softwares in the field of electromagnetic simulation for the various needs of all EADS Business Units. Hence, we have produced over the years a wide range of tools for time domain and frequency domain EM problems. The aim of this talk is to give an overview of this work, underlining its most remarkable aspects, the recent developments and future perspectives.

Guillaume Sylvand

### Transient Full Maxwell Computation of Slow Processes

This article deals with finite element solution of the full linear Maxwell’s equations. The focus lies on the transient simulation of slow processes, i.e. of processes, where wave propagation does not play a role. We employ an implicit Euler method for time discretization of the

A

, φ-based Galerkin-formulation with Coulomb-gauge. We propose a novel stabilization technique that makes possible the use of very large timesteps. This is of supreme importance for efficient simulation of slow processes in order to keep the number of timesteps reasonably small. The greatly improved robustness in comparison with a standard formulation is demonstrated through numerical experiments. As an example we simulate the

lightning impulse test

of an industrial dry-type transformer.

J. Ostrowski, R. Hiptmair, F. Krämer, J. Smajic, T. Steinmetz

### A Frequency-Robust Solver for the Time-Harmonic Eddy Current Problem

This work is devoted to fast and parameter-robust iterative solvers for frequency domain finite element equations, approximating the eddy current problem with harmonic excitation. We construct a preconditioned MinRes solver for the frequency domain equations, that is robust (= parameter-independent) in both the discretization parameter

h

and the frequency ω.

Michael Kolmbauer, Ulrich Langer

### Depolarization of Electromagnetic Waves from Bare Soil Surfaces

An improved Two Scale Model (TSM) has been investigated for the depolarization of electromagnetic waves from bare soil surfaces. Classical TSM produces depolarized results due to the tilt of reflecting plane. To include the contribution of actual phenomenon, we add the second order scattering effects at small scale and develop an improved TSM. The performance of the new TSM is assessed by comparing the simulation results in backscattering configuration with the measured data, Advanced Integral Equation Model and Second order Small Slope Approximation at L-, S-, C- and X-band frequencies for a variety of roughness conditions. Finally, we use the new TSM to predict the bistatic scattering and compare the results with classical TSM.

Naheed Sajjad, Ali Khenchaf, Arnaud Coatanhay

### Two Finite-Element Thin-Sheet Approaches in the Electro-Quasistatic Formulation

Two finite-element approaches to cope with thin sheets in the electro-quasistatic formulation are presented. Both rely on the well-known strategy to reduce the sheet volume to a surface. In the first approach, polynomials in the lateral direction are used to allow for a field variation across the sheet. Using the second method, the presence of the thin sheet is modeled by a modification of the local discretization. In contrast to the first approach, here, no additional degrees of freedom are introduced. The different methods are compared to a conventional solution based on simple test examples.

Jens Trommler, Stephan Koch, Thomas Weiland

### Mode Selecting Eigensolvers for 3D Computational Models

For the computation of interior eigenpairs an educated initial guess on the eigenvalue is mandatory in general. The convergence behavior of eigensolvers can be improved by using a starting vector, which should be a reasonable approximation of the searched eigenvector. However, these two provisions do not lead necessarily to the searched eigenpair. We propose an extended selection strategy for the Ritz pairs occurring within the Jacobi-Davidson eigensolver algorithm and compare its performance with the Rayleigh quotient iteration. A complex unsymmetric standard eigenvalue problem resulting from a finite integration discretization of a dielectric disk in free-space serves for numerical experiments.

Bastian Bandlow, Rolf Schuhmann

### Magnetic Model Refinement via a Coupling of Finite Element Subproblems

Model refinements of magnetic circuits are performed via a subdomain finite element method. A complete problem is split into subproblems with overlapping meshes, to allow a progression from source to reaction fields, ideal to real flux tubes, 1-D to 3-D models, perfect to real materials, statics to dynamics, with any coupling of these changes. Its solution is then the sum of the subproblem solutions. The procedure simplifies both meshing and solving processes, and quantifies the gain given by each refinement on both local fields and global quantities.

Patrick Dular, Ruth V. Sabariego, Laurent Krähenbühl, Christophe Geuzaine

### Substrate Modeling Based on Hierarchical Sparse Circuits

In this paper, a new modeling approach appropriate for the substrate modeling is proposed. More generally, this technique can be applied for any homogeneous layer for which an exponential decay of the field variation can be assumed. The main idea is to perform a hierarchical modeling based on an exponential partitioning scheme conducing to a circuit model of linear complexity which is extracted with a low computational effort. The model obtained is further coupled with the models of the other parts in which the integrated circuit is decomposed or its sparse matrix is used as a boundary condition for field in SiO2 domain.

Daniel Ioan, Gabriela Ciuprina, Ioan-Alexandru Lazăr

### A Boundary Conformal DG Approach for Electro-Quasistatics Problems

A boundary conformal technique for solving three dimensional electro-quasistatic problems with a high order Discontinuous Galerkin method on Cartesian grids is proposed. The method is based on a cut-cell approach which is applied only on elements intersected by curved material boundaries. A particular numerical quadrature technique is applied which allows for an accurate integration of the finite element operators taking into account the exact geometry of the cut-cells. Two numerical examples are presented which demonstrate the optimal convergence rate of the method for arbitrary geometry.

A. Fröhlcke, E. Gjonaj, T. Weiland

### Optimization of the Current Density Distribution in Electrochemical Reactors

This paper proposes to investigate, analyze and compare two practical optimization approaches for smoothing the side effects of electrodeposited layers in electrochemical reactors. The study case consists in a hydraulic component protected by a thin chromium (Cr) layer. Both optimization approaches are investigated by using a 3D finite element software for solving the Laplace equation. The obtained results using these approaches are compared with the numerical results for an electrodepositing process without any additional thief current systems. The uniformity of the chromium deposition on the test component is greatly improved.

Florin Muntean, Alexandru Avram, Johan Deconinck, Marius Purcar, Vasile Topa, Calin Munteanu, Laura Grindei, Ovidiu Garvasuc

### Streamer Line Modeling

After reviewing some basics of dielectric withstand of air insulation, we introduce two procedures for an improved prediction of streamer paths in complex geometries. Although based on the electric background field, we generalize conventional models that usually consider paths starting at a field maximum and traveling along field lines. The new approaches are able to explain both streamer inception points different from field maxima as well as deviations of the streamer path from field lines, and may help to further optimize dielectric withstand of high voltage devices.

Thomas Christen, Helmut Böhme, Atle Pedersen, Andreas Blaszczyk

### A Discontinuous Galerkin Formalism to Solve the Maxwell-Vlasov Equations. Application to High Power Microwave Sources

In this paper, we present a

Particle-In-Cell

(PIC) method based on a Discontinuous Galerkin (DG) scheme to solve the Maxwell-Vlasov equations in time-domain. Comparisons with an other industrial software are given to validate the method.

Laura Pebernet, Xavier Ferrieres, Vincent Mouysset, François Rogier, Pierre Degond

### Tonti Diagrams and Algebraic Methods for the Solution of Coupled Problems

Tonti diagrams highlight a common structure of several physical laws describing different phenomena. From a computational viewpoint, this underlying common structure allows to build topological operators (discrete counterpart of differential operators) only once, and they can be used to easily assemble the stiffness matrices and the coupling terms of the various problems. An application of this concept to the coupled electromagnetic-thermal problem of induction heating is presented in this work, taking into account the nonlinear effects of temperature on the magnetic characteristic beyond the Curie point.

Fabio Freschi, Maurizio Repetto

### Soliton Collision in Biomembranes and Nerves- A Stability Study

Collision of moving solitons is an interesting phenomena which is closely related to the stability of solitons. We study the head-on collision of solitons in a recently introduced model for biomembranes and nerves. We conduct simulations for pairs of solitons moving in opposite directions with the same velocity. It is found that these stable solitons collide elastically and it results a small amplitude noise traveling with higher velocity. We have also examined the energy loss of the solitons after collision.

Revathi Appali, Benny Lautrup, Thomas Heimburg, Ursula van Rienen

### Nonlinear Characterization and Simulation of Zinc-Oxide Surge Arresters

A combined experimental and numerical procedure to model zinc-oxide varistor based surge arresters is presented. In a series of experiments, measurements on single varistor disks exposed to two millisecond current pulses are taken. Thereafter, the measurement data are used to establish the nonlinear electro-thermal characteristics of the ZnO ceramics under electrical stress. Using this information, an accurate finite element model with coupled thermal and electric fields can be constructed. This approach is applied to calculate the transient voltage and temperature distribution within a complete surge arrester unit.

Frank Denz, Erion Gjonaj, Thomas Weiland

### Behavioural Electro-Thermal Modelling of SiC Merged PiN Schottky Diodes

This paper presents a new accurate behavioural static model of SiC Merged PiN Schottky (MPS) diode. This model is dedicated to static and quasi-static electro-thermal simulations of MPS diodes for industrial applications. The model parameters were extracted using the Weighted Least Square (WLS) method for a few selected commercially available SiC MPS diodes. Additionally, the PSPICE Analogue Behavioural Model (ABM) model implementation is also given. The relevance of the model has been statistically proven. The thermal behaviour of the devices was taken into account using the lumped Cauer canonical networks extracted from electro-thermal measurements.

M. Zubert, M. Janicki, M. Napieralska, G. Jablonski, L. Starzak, A. Napieralski

### A Convergent Iteration Scheme for Semiconductor/Circuit Coupled Problems

A dynamic iteration scheme is proposed for a coupled system of electric circuit and distributed semiconductor (

pn

-diode) model equations. The device is modelled by the drift-diffusion (DD) equations and the circuit by MNA-equations. The dynamic iteration scheme is investigated on the basis of discrete models and coupling via sources and compact models. The analytic divergence and analytic convergence results are verified numerically.

Giuseppe Alì, Andreas Bartel, Markus Brunk, Sebastian Schöps

### Multirate Time Integration of Field/Circuit Coupled Problems by Schur Complements

When using distributed magnetoquasistatic field models as additional elements in electric circuit simulation, the field equations contribute with large symmetric linear systems that have to be solved. The naive coupling and solving (using direct solvers) is not always efficient, since the electric circuit is coupled only via coils, which are often represented only by a small subset of the unknowns. We revisit the Schur complement approach, give a physical interpretation and show that a heuristics for bypassing Newton iterations allow for efficient multirate time-integration for the field/circuit coupled model.

Sebastian Schöps, Andreas Bartel, Herbert De Gersem

### Advances in Parallel Transistor-Level Circuit Simulation

Parallel transistor-level circuit simulation has the potential to significantly impact the need for reliably determining parasitic effects for modern feature sizes. Incorporating parallelism into a simulator at both coarse and fine-grained levels, through the use of message-passing and threading paradigms, is supported by the advent of inexpensive clusters, as well as multi-core technology. However, its effectiveness is reliant upon the development of efficient parallel algorithms for traditional “true SPICE” circuit simulation. In this paper, we will discuss recent advances in fully parallel transistor-level full-chip circuit simulation, concluding with scaling results from a newer strategy for the parallel preconditioned iterative solution of circuit matrices.

Heidi K. Thornquist, Eric R. Keiter

### Sensitivity-Based Steady-State Mismatch Analysis for RF Circuits

Based on the assumption of small parameter variations a sensitivity–based analysis method is proposed for the computation of the steady–state mismatch deviations of periodic and quasi-periodic circuits. Unlike classical Monte-Carlo or worst-case analyses this approach does not require several (time) periodic steady-state analyses to be performed. In addition it provides detailed information on the relative contributions of the different device model parameters. Some numerical examples illustrate the potential and limitations of the approach.

Fabrice Veersé, Joël Besnard, Hubert Filiol

### Modelling and Simulation of Forced Oscillators with Random Periods

In nanoelectronics, the miniaturisation of circuits causes uncertainties in the components. An uncertainty quantification is achieved by the introduction of random parameters in corresponding mathematical models. We consider forced oscillators described by time-dependent differential algebraic equations, where a random period appears. A corresponding uncertainty quantification results from a modelling based on a transformation to a unit time interval. We apply the technique of the generalised polynomial chaos to resolve the stochastic model. Thereby, a Galerkin approach yields a larger coupled system of differential algebraic equations satisfied by an approximation of the random process. We present numerical simulations of an illustrative example.

Roland Pulch

### Initialization of HB Oscillator Analysis from Transient Data

Oscillation frequency and amplitude of a free-running oscillator are commonly solved with harmonic balance (HB) method using an oscillator probe. This usually requires optimization. Poor initial values of oscillation may lead to unsuccessful optimization or will at least require a great number of optimization cycles. Therefore, two methods to initialize HB oscillator analysis from transient data are presented. These methods improve the initial estimates of oscillator frequency and amplitude. In addition, techniques to improve convergence of the analysis by initializing HB voltages from transient data and using an oscillator probe pulse are discussed. The efficiency of the methods is examined and verified through numerical experiments.

Mikko Hulkkonen, Mikko Honkala, Jarmo Virtanen, Martti Valtonen

### Robust Periodic Steady State Analysis of Autonomous Oscillators Based on Generalized Eigenvalues

In this paper, we present a new gauge technique for the Newton Raphson method to solve the periodic steady state (PSS) analysis of free-running oscillators in the time domain. To find the frequency a new equation is added to the system of equations. Our equation combines a generalized eigenvector with the time derivative of the solution. It is dynamically updated within each Newton–Raphson iteration. The method is applied to an analytic benchmark problem and to an LC oscillator. It provides better convergence properties than when using the popular phase-shift condition. It also does not need additional information about the solution. The method can also easily be implemented within the Harmonic Balance framework.

R. Mirzavand Boroujeni, E. J. W. ter Maten, T. G. J Beelen, W. H. A. Schilders, A. Abdipour

### Mutual Injection Locking of Oscillators under Parasitic Couplings

The method to analyze the mutual injection locking of weakly coupled arbitrary oscillators is proposed. The couplings are defined by frequency-dependent admittance matrices. The algebraic system with respect to phases and common locking frequency is derived. For two oscillators the system is transformed to the single phase equation and explicit expression for the locking frequency. The accuracy comparison with SPICE simulation is presented.

M. M. Gourary, S. G. Rusakov, S. L. Ulyanov, M. M. Zharov

### Time Domain Simulation of Power Systems with Different Time Scales

The time evolution of power systems is modeled by a system of differential and algebraic equations. The variables involved in the system may exhibit different time scales. In standard numerical time integration methods the most active variables impose the time step for the whole system. We present a strategy, which allows the use of different, local time steps over the variables. The partitioning of the components of the system in different classes of activity is performed automatically and is based on the topology of the power system.

Valeriu Savcenco, Bertrand Haut, E. Jan W. ter Maten, Robert M. M. Mattheij

### Adaptive Wavelet-Based Method for Simulation of Electronic Circuits

In this paper we present an algorithm for analog simulation of electronic circuits involving a spline Galerkin method with wavelet-based adaptive refinement. Numerical tests show that a first algorithm prototype, build within a productively used in-house circuit simulator, is completely able to meet and even surpass the accuracy requirements and has a performance close to classical time-domain simulation methods, with high potential for further improvement.

Kai Bittner, Emira Dautbegovic

### Modeling and Simulation of Organic Solar Cells

A model for polymer Solar Cells is presented consisting of a system of nonlinear diffusion-reaction PDEs with electrostatic convection, coupled to a kinetic ODE. A proof of the existence of both stationary and transient solutions is given and an algorithm for computing them is proposed and numerically validated by comparison with experimentally measured data for a photovoltaic cell.

Carlo de Falco, Antonio Iacchetti, Maddalena Binda, Dario Natali, Riccardo Sacco, Maurizio Verri

### Numerical Simulation of a Hydrodynamic Subband Model for Semiconductors Based on the Maximum Entropy Principle

A hydrodynamic subband model for semiconductors has been formulated in (Mascali and Romano, IL NUOVO CIMENTO 33C:155163, 2010) by closing the moment system derived from the Schrödinger-Poisson-Boltzmann equations on the basis of the maximum entropy principle (MEP). Explicit closure relations for the fluxes and the production terms are obtained taking into account scattering of electrons with acoustic and non-polar optical phonons, as well as surface scattering. Here a suitable numerical scheme is presented for the above model together with simulations of a nanoscale silicon diode.

G. Mascali, V. Romano

### Inverse Doping Profile of MOSFETs via Geometric Programming

In this study, we optimize one-dimensional doping profiles between the interface of semiconductor and oxide to the substrate in metal-oxide-semiconductor field-effect transistors (MOSFETs). For a set of given current-voltage curves, the problem is modelled as a geometric programming (GP) problem. The MOSFET’s DC characteristics including the on- and off-state currents are simultaneously derived as functions of the doping profile in the GP problem.

Yiming Li, Ying-Chieh Chen

### Numerical Simulation of Semiconductor Devices by the MEP Energy-Transport Model with Crystal Heating

A new numerical model of semiconductors including crystal heating effect is presented. The model equations have been obtained with the use of the maximum entropy principle. In the numerical model the iterative scheme is used for obtaining stationary solution of electro-thermal problems. Numerical simulations of a 2D nanoscale MOSFET with the self-heating effect are presented. The difference between MEP and simpler thermal expressions is analyzed.

Vittorio Romano, Alexander Rusakov

### Challenges in Model Order Reduction for Industrial Problems

Mathematical challenges arise in many applications in the electronics industry. Device and circuit simulation are well-known examples, and in industry these are typically crucial for circuit and layout optimization. Model order reduction is one of the available tools, and we show when and how, and when not, to use this. We will give an overview of the challenges we are facing, explain how we try to conquer these, discuss the requirements we have to deal with, and indicate where improvements are needed.

Joost Rommes

### On Approximate Reduction of Multi-Port Resistor Networks

Simulation of the influence of interconnect structures and substrates is essential for a good understanding of modern chip behavior. Sometimes such simulations are not feasible with current circuit simulators. We propose an approach to reduce the large resistor networks obtained from extraction of the parasitic effects that builds upon the work in (Rommes and Schilders, IEEE Trans. CAD Circ. Syst. 29:28–39, 2010). The novelty in our approach is that we obtain improved reductions, by developing error estimations which enable to delete superfluous resistors and to control accuracy. An industrial test case demonstrates the potential of the new method.

M. V. Ugryumova, J. Rommes, W. H. A. Schilders

### Improving Model-Order Reduction Methods by Singularity Exclusion

This paper presents a novel stand-alone method for overcoming a singular system matrix in Model-Order Reduction (MOR) algorithms, which would otherwise foil successful algorithm operation and thus reduction. The basic idea of the method is to locate and identify the circuit areas that generate the singularities to the system matrix prior to MOR, and exclude these from the reduction. The method is applicable to any netlist-in–netlist-out type MOR method.

Pekka Miettinen, Mikko Honkala, Janne Roos, Martti Valtonen

### Partitioning-Based Reduction of Circuits with Mutual Inductances

This paper describes a novel model-order reduction (MOR) method to reduce the number of mutual inductances in conjunction with a recently proposed MOR algorithm, PartMOR. As the method produces passive mutual inductances as a reduction realization, it extends the existing RLC-in–RLC-out PartMOR to a RLCM-in–RLCM-out MOR method. The method is verified and compared to a well-known MOR method with test simulations and is shown to produce good reduction results in terms of CPU speed-up and generated error.

Pekka Miettinen, Mikko Honkala, Janne Roos, Martti Valtonen

### Model Order Reduction of Parameterized Nonlinear Systems by Interpolating Input-Output Behavior

In this paper we propose a new approach for model order reduction of parameterized nonlinear systems. Instead of projecting onto the dominant state space, an analog macromodel is constructed for the dominant input-output behavior. This macromodel is suitable for (re)use in analog circuit simulators. The performance of the approach is illustrated for a benchmark nonlinear system.

Michael Striebel, Joost Rommes

### On the Selection of Interpolation Points for Rational Krylov Methods

We suggest a simple and an efficient way of selecting a suitable set of interpolation points for the well-known rational Krylov based model order reduction techniques. To do this, some sampling points from the frequency response of the transfer function are taken. These points correspond to the places where the sign of the numerical derivation of transfer function changes. The suggested method requires a set of linear system’s solutions several times. But, they can be computed concurrently by different processors in a parallel computing environment. Serial performance of the method is compared to the well-known

H

2

optimal method for several benchmark examples. The method achieves acceptable accuracies (the same order of magnitude) compared to that of

H

2

optimal methods and has a better performance than the common selection procedures such as linearly distributed points.

E. Fatih Yetkin, Hasan Dağ

### Discrete Empirical Interpolation in POD Model Order Reduction of Drift-Diffusion Equations in Electrical Networks

We consider model order reduction of integrated circuits with semiconductors modeled by modified nodal analysis and drift-diffusion (DD) equations. The DD-equations are discretized in space using a mixed finite element method. This discretization yields a high dimensional, nonlinear system of differential-algebraic equations. Proper orthogonal decomposition is used to reduce the dimension of this model. Since the computational complexity of the reduced order model through the nonlinearity of the DD equations still depends on the number of variables of the full model we apply the discrete empirical interpolation method to further reduce the computational complexity. We provide numerical comparisons which demonstrate the performance of this approach.

Michael Hinze, Martin Kunkel

### Model Order Reduction for Complex High-Tech Systems

This paper presents a computationally efficient model order reduction (MOR) technique for interconnected systems. This MOR technique preserves block structures and zero blocks and exploits separate MOR approximations for the individual sub-systems in combination with low rank approximations for the interconnection blocks. The reduction is demonstrated to be accurate and efficient for a beam-controller system.

Agnieszka Lutowska, Michiel E. Hochstenbach, Wil H. A. Schilders

### Parametric Model Order Reduction by Neighbouring Subspaces

Electrodynamic field simulations in the frequency domain typically require the solution of large linear systems. Model Order Reduction (MOR) techniques offer a fast approach to approximate the system impedance with respect to the frequency parameter. Most commonly, MOR via projection is applied associated with certain Krylov projection matrices. During the design process it is desirable to vary specified parameters like the frequency, geometry details as well as material parameters, giving rise to multivariate dynamical systems. In this work, a multivariate MOR method is presented for parameterized systems based on the Finite Integration Technique (FIT). It utilizes the observation, that for small parameter variations the matrices associated with the univariate MOR differ only slightly. Thus, the multivariate MOR method is deduced from the usage of specified univariate subspaces.

Kynthia Stavrakakis, Tilmann Wittig, Wolfgang Ackermann, Thomas Weiland

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