Mathematics – Key Technology for the Future

Willi Jäger · Hans-Joachim KrebsEditors

Mathematics –Key Technologyfor the Future

Joint Projects Between Universitiesand Industry 2004–2007

123

Willi JägerIWR, Universität HeidelbergIm Neuenheimer Feld 36869120 HeidelbergGermanyjaeger@iwr.uni-heidelberg.de

Hans-Joachim KrebsProjektträger JülichForschungszentrum Jülich GmbH52425 JülichGermanyh.-j.krebs@fz-juelich.de

ISBN 978-3-540-77202-6 e-ISBN 978-3-540-77203-3

DOI 10.1007/978-3-540-77203-3

Library of Congress Control Number: 2008923103

Mathematics Subject Classification (2000): 34-XX, 35-XX, 45-XX, 46-XX, 49-XX, 60-XX, 62-XX,65-XX, 74-XX, 76-XX, 78-XX, 80-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX

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Solving real problems requires teams – mathematicians,engineers, scientists and users from various fields, and – last but

not least – it requires money!

Preface

In 1993, the Federal Ministry of Research and Education of the Federal Re-public of Germany (BMBF) started the first of now five periods of fundingmathematics for industry and services. To date, its efforts have supportedapproximately 280 projects investigating complex problems in industry andservices. Whereas standard problems can be solved using standard mathemat-ical methods and software off the shelf, complex problems arising from e.g.industrial research and developments require advanced and innovative math-ematical approaches and methods. Therefore, the BMBF funding programmefocuses on the transfer of the latest developments in mathematical researchto industrial applications. This initiative has proved to be highly successfulin promoting mathematical modelling, simulation and optimization in scienceand technology.

Substantial contributions to the solution of complex problems have beenmade in several areas of industry and services.

Results from the first funding period were published in “Mathematik –Schlüsseltechnologie für die Zukunft, Verbundprojekte zwischen Universitätund Industrie” (K.-H. Hoffmann, W. Jäger, T. Lohmann, H. Schunck (Edi-tors), Springer 1996). The second publication “Mathematics – Key Technologyfor the Future, Joint Projects between Universities and Industry” (W. Jäger,H.-J. Krebs (Editors), Springer 2003) covered the period 1997 to 2000. Bothbooks were out of print shortly after publication.

This volume presents the results from the BMBF’s fourth funding period(2004 to 2007) and contains a similar spectrum of industrial and mathematicalproblems as described in the previous publications, but with one additionalnew topic in the funding programme: risk management in finance and insur-ance.

Other topics covered are mathematical modelling and numerical simulationin microelectronics, thin films, biochemical reactions and transport, computer-aided medicine, transport, traffic and energy.

As in the preceding funding periods, novel mathematical theories andmethods as well as a close cooperation with industrial partners are essen-

VIII Preface

tial components of the funded projects. In order to strengthen and documentthis cooperation with industry, the projects starting in the 5th period arebased on consortium agreements between universities, industry and services,though there is no direct funding for the industrial partners.

The BMBF has declared the year 2008 as “The Year of Mathematics” inGermany. This book is an excellent contribution to this special year, demon-strating the usefulness of and prospects for mathematics in technology andeconomics. It is widely acknowledged that the support provided by the Min-istry has been substantial in advancing the transfer of mathematics to indus-try and services and in opening academic mathematical research for challengesarising in industrial and economic applications.

In all areas of society, global aspects are becoming more and more im-portant. Therefore it is extremely important to promote mathematics and itstransfer as a key technology to sciences and industry both on the nationaland on the international level. Here, we take the opportunity to acknowledgethe assistance of the BMBF, providing not only necessary national funding,but also emphasizing the importance of mathematics and the necessity ofpromoting its applications in national and international cooperation. By ini-tiating an OECD Global Science Forum on “Mathematics for Industry” andby dedicating the scientific year 2008 to mathematics, the BMBF has madea clear statement. Special thanks for their support go to Dr. H.-F. Wagner,Dr. R. Koepke and Professor Dr. J. Richter from the BMBF. Last but notleast, we would also like to acknowledge the valued support of Dr. StefanMichalowski, representing the OECD in the Global Science Forum.

We also greatly appreciate the excellent cooperation with Springer Ver-lag, Heidelberg in publishing this volume – the third one to collect andpresent results from the BMBF mathematics programme – in its interna-tionally renowned series of scientific publications.

Heidelberg, Jülich Willi JägerMarch 2008 Hans-Joachim Krebs

Contents

Part I Microelectronics

Numerical Simulation of Multiscale Modelsfor Radio Frequency Circuits in the Time DomainU. Feldmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Numerical Simulation of High-Frequency Circuitsin TelecommunicationM. Bodestedt, C. Tischendorf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Wavelet-Collocation of Multirate PDAEs for the Simulationof Radio Frequency CircuitsS. Knorr, R. Pulch, M. Günther . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Numerical Simulation of Thermal Effectsin Coupled Optoelectronic Device-Circuit SystemsM. Brunk, A. Jüngel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Efficient Transient Noise Analysis in Circuit SimulationG. Denk, W. Römisch, T. Sickenberger, R. Winkler . . . . . . . . . . . . . . . . . . 39

Part II Thin Films

Numerical Methods for the Simulation of Epitaxial Growthand Their Application in the Study of a Meander InstabilityF. Haußer, F. Otto, P. Penzler, A. Voigt . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Micro Structures in Thin Coating Layers:Micro Structure Evolution and Macroscopic Contact AngleJ. Dohmen, N. Grunewald, F. Otto, M. Rumpf . . . . . . . . . . . . . . . . . . . . . . 75

X Contents

Part III Biochemical Reactions and Transport

Modeling and Simulation of Hairy Root GrowthP. Bastian, J. Bauer, A. Chavarría-Krauser, C. Engwer, W. Jäger,S. Marnach, M. Ptashnyk, B. Wetterauer . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Simulation and Optimization of Bio-Chemical MicroreactorsR. Rannacher, M. Schmich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Part IV Computeraided Medicine

Modeling and Optimization of Correction Measuresfor Human ExtremitiesR. Brandenberg, T. Gerken, P. Gritzmann, L. Roth . . . . . . . . . . . . . . . . . . 131

Image Segmentation for the Investigationof Scattered-Light Images when Laser-Optically DiagnosingRheumatoid ArthritisH. Gajewski, J. A. Griepentrog, A. Mielke, J. Beuthan, U. Zabarylo,O. Minet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Part V Transport, Traffic, Energy

Dynamic Routing of Automated Guided Vehicles inReal-TimeE. Gawrilow, E. Köhler, R. H. Möhring, B. Stenzel . . . . . . . . . . . . . . . . . . 165

Optimization of Signalized Traffic NetworksE. Köhler, R. H. Möhring, K. Nökel, G. Wünsch . . . . . . . . . . . . . . . . . . . . 179

Optimal Sorting of Rolling Stock at Hump YardsR. S. Hansmann, U. T. Zimmermann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Stochastic Models and Algorithms for the Optimal Operationof a Dispersed Generation System Under UncertaintyE. Handschin, F. Neise, H. Neumann, R. Schultz . . . . . . . . . . . . . . . . . . . . 205

Parallel Adaptive Simulation of PEM Fuel CellsR. Klöfkorn, D. Kröner, M. Ohlberger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Contents XI

Part VI Risk Management in Finance and Insurance

Advanced Credit Portfolio Modeling and CDO PricingE. Eberlein, R. Frey, E. A. von Hammerstein . . . . . . . . . . . . . . . . . . . . . . . . 253

Contributions to Multivariate Structural Approachesin Credit Risk ModelingS. Becker, S. Kammer, L. Overbeck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Economic Capital Modelling and Basel II Compliancein the Banking IndustryK. Böcker, C. Klüppelberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Numerical Simulation for Asset-Liability Managementin Life InsuranceT. Gerstner, M. Griebel, M. Holtz, R. Goschnick, M. Haep . . . . . . . . . . . . 319

On the Dynamics of the Forward Interest Rate Curveand the Evaluation of Interest Rate Derivativesand Their SensitivitiesC. Croitoru, C. Fries, W. Jäger, J. Kampen, D.-J. Nonnenmacher . . . . . 343

List of Contributors

Peter BastianUniversität Stuttgart, IPVSUniversitätsstraße 38D-70569 Stuttgartpeter.bastian@ipvs.uni-stuttgart.de

Jenny BauerUniversität Heidelberg, IPMBINF 364D-69120 Heidelbergjenny_bauerbiol@yahoo.de

Swantje BeckerUniversität GießenArndtstraße 2D-35392 Gießenswantje.becker@math.uni-giessen.de

Jürgen BeuthanCharité Universitätsmedizin BerlinCampus Benjamin FranklinInstitut für Medizinische Physik undLasermedizinFabeckstraße 60-62D-14195 Berlinjuergen.beuthan@charite.de

Klaus BöckerRisk Integration, Reporting &PoliciesRisk Analytics and MethodsUniCredit Group, Munich Branchklaus.boecker@hvb.de

Martin BodestedtUniversität zu KölnWeyertal 86–90D-50931 Kölnmbodeste@math.uni-koeln.de

René BrandenbergTechnische Universität MünchenZentrum für MathematikBoltzmannstr. 3D-85747 Garching bei Münchenbrandenb@ma.tum.de

Markus BrunkUniversität MainzStaudingerweg 9D-55099 Mainzbrunk@mathematik.uni-mainz.de

Andrés Chavarría-KrauserUniversität Heidelberg, IAMINF 294D-69120 Heidelbergandres.chavarria.Krauser@iwr.uni-heidelberg.de

Cristian CroitoruUniversität Heidelberg, IWRINF 368D-69120 Heidelbergcristian.croitoru@iwr.uni-heidelberg.de

XIV List of Contributors

Georg DenkQimonda AGAm Campeon 1–12D-85579 Neubiberggeorg.denk@qimoda.com

Julia DohmenUniversität BonnInstitut für Numerische SimulationNussallee 15D-53115 Bonnjulia.dohmen@ins.uni-bonn.de

Ernst EberleinUniversität FreiburgAbteilung für MathematischeStochastikEckerstraße 1D-79104 Freiburgeberlein@stochastik.uni-freiburg.de

Christian EngwerUniversität Stuttgart, IPVSUniversitätsstraße 38D-70569 Stuttgartchristian.engwer@ipvs.uni-stuttgart.de

Rüdiger FreyUniversität LeipzigAbteilung für MathematikD-04081 Leipzigruediger.frey@math.uni-leipzig.de

Uwe FeldmannQimoda AGAm Campeon 1–12D-85579 Neubiberguwe.feldmann@online.de

Christian FriesDZBank AGPlatz der RepublikD-60265 Frankfurt am Mainmail@christian-fries.de

Herbert GajewskiForschungsverbund Berlin e.V.WIASMohrenstraße 39D-10117 Berlingajewski@wias-berlin.de

Ewgenij GawrilowTechnische Universität BerlinInstitut für Mathematik, MA 6-1Straße des 17. Juni 136D-10623 Berlingawrilow@math.tu-berlin.de

Jens A. GriepentrogForschungsverbund Berlin e.V.WIASMohrenstraße 39D-10117 Berlingriepent@wias-berlin.de

Ralf GoschnikZürich Gruppe DeutschlandPoppelsdorfer Allee 25–33D-53115 Bonnr.goschnik@zurich.com

Michael GüntherBergische Universität WuppertalFachbereich C, Gaußstr. 20D-42119 Wuppertalguenther@math.uni-wuppertal.de

Michael GriebelUniversität BonnInstitut für Numerische SimulationNussallee 15D-53115 Bonngriebel@ins.uni-bonn.de

Natalie GrunewaldUniversität Bonn, IAMWegelerstr. 10D-53115 Bonngrunewald@ins.uni-bonn.de

List of Contributors XV

Tobias GerkenTechnische Universität MünchenZentrum für MathematikBoltzmannstr. 3D-85747 Garching bei Münchenbrandenb@ma.tum.de

Peter GritzmannTechnische Universität MünchenZentrum für MathematikBoltzmannstr. 3D-85747 Garching bei Münchenbrandenb@ma.tum.de

Thomas GerstnerUniversität BonnInstitut für Numerische SimulationNussallee 15D-53115 Bonngerstner@ins.uni-bonn.de

Ronny HansmannTechnische Universität BraunschweigInstitut für MathematischeOptimierungPockelstraße 14D-38106 Braunschweigr.hansmann@tu-bs.de

Edmund HandschinUniversität DortmundInstitut für Energiesysteme undEnergiewirtschaftEmil-Figge-Straße 70D-44227 Dortmundedmund.handschin@udo.edu

Markus HoltzUniversität BonnInstitut für Numerische SimulationNussallee 15D-53115 Bonnholtz@ins.uni-bonn.de

Frank HaußerTechnische Fachhochschule Berlin,FB IILuxemburger Str. 10D-13353 Berlinhausser@tfh-berlin.de

Marcus HaepZürich Gruppe DeutschlandPoppelsdorfer Allee 25–33D-53115 Bonnmarcus.haep@zurich.com

Ernst August von HammersteinUniversität FreiburgAbteilung für MathematischeStochastikEckerstraße 1D-79104 Freiburghammer@stochastik.uni-freiburg.de

Willi JägerIWR, Universität HeidelbergIm Neuenheimer Feld 368D-69120 Heidelbergjaeger@iwr.uni-heidelberg.de

Ansgar JüngelTechnische Universität WienInstitut für Analysis und ScientificComputingWiedner Hauptstr. 8–101040 Wien, Austriajuengel@anum.tuwien.ac.at

Jörg KampenForschungsverbund Berlin e.V.WIASMohrenstraße 39D-10117 Berlinkampen@wias-berlin.de

Stefanie KammerUniversität GießenArndtstraße 2D-35392 Gießenstefanie.kammer@math.uni-giessen.de

XVI List of Contributors

Robert KlöfkornMathematisches InstitutAbteilung für AngewandteMathematikUniversität FreiburgHermann-Herder-Str. 10D-79104 Freiburg i. Br.robertk@mathematik.uni-freiburg.de

Claudia KlüppelbergTechnische Universität MünchenZentrum für MathematischeWissenschaftenD-85747 Garching bei Münchencklu@ma-tum.de

Stephanie KnorrBergische Universität WuppertalFachbereich C, Gaußstr. 20D-42119 Wuppertalknorr@math.uni-wuppertal.de

Ekkehard KöhlerBrandenburgische TechnischeUniversität CottbusInstitut für MathematikPostfach 10 13 44D-03013 Cottbusekoehler@math.tu-cottbus.de

Dietmar KrönerMathematisches InstitutAbteilung für AngewandteMathematikUniversität FreiburgHermann-Herder-Str. 10D-79104 Freiburg i. Br.dietmar@mathematik.uni-freiburg.de

Sven MarnachUniversität Stuttgart, IPVSUniversitätsstraße 38D-70569 Stuttgartsven.marnach@ipvs.uni-stuttgart.de

Alexander MielkeForschungsverbund Berlin e.V.WIASMohrenstraße 39D-10117 Berlinmielke@wias-berlin.de

Olaf MinetCharité Universitätsmedizin BerlinCampus Benjamin FranklinInstitut für Medizinische Physik undLasermedizinFabeckstraße 60-62D-14195 Berlinolaf.minet@charite.de

Rolf MöhringTechnische Universität BerlinInstitut für Mathematik, MA 6-1Straße des 17. Juni 136D-10623 Berlinmoehring@math.tu-berlin.de

Klaus NökelPTV AGStumpfstraße 1D-76131 Karlsruheklaus.noekel@ptv.de

Frederike NeiseUniversität Duisburg-EssenAbteilung für MathematikForsthausweg 2D-47048 Duisburgneise@math.uni-duisburg.de

Hendrik NeumannUniversität DortmundInstitut für Energiesysteme undEnergiewirtschaftEmil-Figge-Straße 70D-44227 Dortmundhendrik.neumann@udo.edu

List of Contributors XVII

Dirk-Jens NonnenmacherHSH Nordbank AGGerhart-Hauptmann-Platz 50D-20095 Hamburgdirk.jens.nonnenmacher@hsh-nordbank.com

Mario OhlbergerInstitut für Numerischeund Angewandte MathematikUniversität MünsterEinsteinstraße 62D-48149 Münstermario.ohlberger@math.uni-muenster.de

Felix OttoUniversität Bonn, IAMWegelerstr. 10D-53115 Bonnotto@iam.uni-bonn.de

Ludger OverbeckUniversität GießenArndtstraße 2D-35392 Gießenludger.overbeck@math.uni-giessen.de

Mariya PtashnykUniversität Heidelberg, IAMINF 294D-69120 Heidelbergbernhard.wetterauer@urz.uni-heidelberg.de

Patrick PenzlerUniversität Bonn, IAMWegelerstr. 10D-53115 Bonnpenzler@iam.uni-bonn.de

Roland PulchBergische Universität WuppertalFachbereich C, Gaußstr. 20D-42119 Wuppertalpulch@math.uni-wuppertal.de

Werner RömischHumboldt-Universität zu BerlinInstitute of MathematicsD-10099 Berlinromisch@math.hu-berlin.de

Martin RumpfUniversität BonnInstitut für Numerische SimulationNussallee 15D-53115 Bonnmartin.rumpf@ins.uni-bonn.de

Lucia RothTechnische Universität MünchenZentrum für MathematikBoltzmannstr. 3D-85747 Garching bei Münchenbrandenb@ma.tum.de

Rolf RannacherUniversität Heidelberg, IAMIm Neuenheimer Feld 293/294D-69120 Heidelbergrolf.rannacher@iwr.uni-heidelberg.de

Michael SchmichUniversität Heidelberg, IAMIm Neuenheimer Feld 293/294D-69120 Heidelbergmichael.schmich@iwr.uni-heidelberg.de

Rüdiger SchulzUniversität Duisburg-EssenAbteilung für MathematikForsthausweg 2D-47048 Duisburgschultz@math.uni-duisburg.de

Thorsten SickenbergerHumboldt-Universität zu BerlinInstitute of MathematicsD-10099 Berlinsickenberger@math.hu-berlin.de

XVIII List of Contributors

Björn StenzelTechnische Universität BerlinInstitut für Mathematik, MA 6-1Straße des 17. Juni 136D-10623 Berlinstenzel@math.tu-berlin.de

Caren TischendorfUniversität zu KölnWeyertal 86-90D-50931 Kölnctischen@math.uni-koeln.de

Axel VoigtTechnische Universität DresdenIWRZellescher Weg 12–14D-01062 Dresdenaxel.voigt@tu-dresden.de

Renate WinklerHumboldt-Universität zu BerlinInstitute of MathematicsD-10099 Berlinwinkler@math.hu-berlin.de

Bernhard WetterauerUniversität Heidelberg, IPMBINF 364D-69120 Heidelbergbernhard.wetterauer@urz.uni-heidelberg.de

Gregor WünschTechnische Universität BerlinInstitut für Mathematik, MA 6-1Straße des 17. Juni 136D-10623 Berlinwuensch@math.tu-berlin.de

Urszula ZabaryloCharité Universitätsmedizin BerlinCampus Benjamin FranklinInstitut für Medizinische Physik undLasermedizinFabeckstraße 60–62D-14195 Berlinurszula.zabarylo@charite.de

Uwe T. ZimmermannTechnische Universität BraunschweigInstitut für MathematischeOptimierungPockelstraße 14D-38106 Braunschweigu.zimmermann@tu-bs.de

Part I

Microelectronics

Numerical Simulation of Multiscale Models

for Radio Frequency Circuitsin the Time Domain

Uwe Feldmann

Qimonda AG, Am Campeon 1–12, 85579 Neubiberg, Germanyuwe.feldmann@online.de

1 Introduction

Broadband data communication via high frequent (RF) carrier signals hasbecome a prerequisite for successful introduction of new applications and ser-vices in the hightech domain, like cellular phones, broadband internet services,GPS, and radar sensors for automotive collision control. It is driven by theprogress in microelectronics, i. e. by scaling down from micrometer dimen-sions into the nanometer range. Due to decreasing feature size and increasingoperating frequency very powerful electronic systems can be realized on inte-grated circuits, which can be produced for mass applications at very moderatecost. However, technological progress also opens clearly a design gap, and inparticular a simulation gap:

• Systems get much more complex, with stronger interaction between digitaland analog parts.

• Parasitic effects become predominant, and neither mutual interactions northe spatial extension of circuit elements can be further neglected.

• The signal-to-noise ratio decreases and statistical fluctuations in the fab-rication lines increase, thus enhancing the risk of circuit failures and yieldreduction.

Currently available industrial simulation tools can not cope with all of thesechallenges, since they are mostly decoupled, and adequate models are not yetcompletey established, or too expensive to evaluate. The purpose of this paperis to demonstrate that joint mathematical research can significantly contributeto improve simulation capabilities, by proper mathematical modelling anddevelopment of numerical methods which exploit the particular structure ofthe problems. The depth of mathematical research for achieving such progressis beyond industrial capabilities; so academic research groups are stronglyinvolved. However, industry takes care that the problems being solved are ofindustrial relevance, and that the project results are driven into industrial use.

4 U. Feldmann

The challenging simulation problems mentioned above accumulate in RFtransmitter-receiver pairs (transceivers), which constitute the core function-ality of most RF communication systems. Although not being very large interms of transistor count, transceivers are often extremely hard to simulatein practice. Mathematically, they exhibit widely separated timescales, theyrequire coupling of semiconductor device equations with thermal equationsand standard circuit equations, and they are sensitive with respect to de-vice noise, which is stochastic by nature. For the purpose of this project, asimplified, but typical representative of a CMOS transceiver was chosen asa common benchmark. So this transceiver constitutes a common frameworkfor the research activities within this project. The global objective was toextend standard simulation methods towards more accurate, comprehensiveand efficient simulation of this transceiver in a large digital circuit environ-ment.

Since in this setting frequency domain methods are not very helpful, re-search is focused on time domain models and methods.

For a further discussion of the mathematical problems involved, thetransceiver is shortly described in the next Subsect.

1.1 RF Transceiver Blocks

Systems for RF data transmission usually have comprehensive parts for digitalsystem processing which work on a sufficiently large number of parallel bits atconventional clock rates. For data transmission the signals are condensed bymultiplexing and modulation onto a very high frequent analog carrier signal.This is illustrated in the upper part of Fig. 11.

Modulation is done by the multiplexer. The latter gets the high frequentcarrier signal from an RF clock generator, which is usually built as a phaselocked loop (PLL). The RF signal being modulated with the data is fed intothe RF transmitter, which – for optical data transmission – may be a laserdiode or – for wireless data transmission – an amplifier with a resonator andantenna.

A rough scheme for the clock generating PLL is given in Fig. 2. Its core isa voltage controlled oscillator (VCO), which generates the high frequent har-monic clock pulses. For stabilization, this RF clock is frequency divided downonto system frequency and fed into the phase detector. The latter comparesit with the system clock and generates a controlling signal for the VCO: Ifthe VCO is ahead then its frequency is reduced, and if the VCO lags behindthen its frequency is increased. The number of cycles for the PLL to ‘lock in’is usually rather large (up to 104 . . . 105), which gives rise to very challengingsimulation tasks.

1 In the Figs. bold lines denote RF signals (with frequencies in the range of10 . . . 50 GHz), while the thinner lines denote signals at standard frequencies(currently about 1 . . . 2 GHz).

Numerical Simulation of Multiscale Models for RF Circuits 5

Fig. 1. RF data transmission with a pair of transmitter (top) and receiver (bottom)

Fig. 2. PLL for generating RF clock

The receiver part of a transceiver system first has to amplify the signaland to synchronize itself with the incoming signal, before it can re-extractthe digital information for further processing at standard clock rates, see thebottom part of Fig. 1.

The RF receiver will be a photo diode in case of optical transmission, andan antenna with resonator in case of electromagnetic transmission. The highfrequent and noisy low power input signal is amplified in a transimpedanceamplifier and then fed into the demultiplexer, which extracts the digital datafrom it and puts them on a low frequent parallel bus for the digital part. Asecond PLL takes care that the incoming signal and the receiver’s system clockare well synchronized. Typically, in this PLL both VCO and phase detectoroperate at high frequency, see Fig. 3.

Finally, the VCO’s output signal is down converted onto the base frequencyfor delivering a synchronized system clock into the digital part.

Usually, transmitter and receiver are realized on one single chip, in orderto enable a handshaking mode between the server and the host system. Fur-

6 U. Feldmann

Fig. 3. PLL for clock synchronization

Fig. 4. A GSM transceiver circuit

thermore, both make use of the same building blocks, such that e. g. only oneVCO is needed. Figure 4 shows a photo of a GSM transceiver circuit, inclusivedigital signal processing. The inductor windings of the VCO can be clearlyidentified in the bottom right corner.

1.2 New Mathematics for Simulating Transceivers

Traditionally, transceivers have been simulated with a standard circuit sim-ulator, which employs Kirchhoffs equations to set up a system of ordinarynonlinear differential algebraic equations of DAE index 2, and solves themeither in the time or in the nonlinear frequency domain. However, technolog-ical progress with decreasing feature sizes and increasing frequency bandwithdrives this approach beyond its limits.

Efficient solvers for multiscale problems in the time domain.Due to widely separated frequencies of the VCO and the digital systemclock we have a multiscale system, which has to be analyzed over many

Numerical Simulation of Multiscale Models for RF Circuits 7

clock cycles for complete verification e. g. of the lock in of the PLL. Withsingle rate integration, this may require weeks of simulation time. Recentlydeveloped schemes for separating time scales are more promising, but theycannot directly be used due to very nonharmonic shape of the digital clocksignals.

Coupling of device and circuit simulation.Spatial extension of some of the devices (transistors, diodes, . . . ) oper-ating in the critical path at their technological limits can no longer beneglected. Hence it is advisable to substitute their formerly used compactmodels by sets of semiconductor equations, thus requiring coupled cir-cuit device simulation. The latter is already available in some commercialpackages, but here the focus is different: The new objective is to simulatesome few transistors and diodes on the device level efficiently togetherwith thousands of circuit level transistors. This requires extremely robustcoupling schemes.

Interaction with thermal effects in device modelling.In particular for the emitting laser diode and the receiving photo diode inopto-electronic data transmission there is interaction with thermal effects,whose impact on the transceiver chain has not yet been taken into account.So thermal feedback has to be incorporated into device simulation modelsand schemes.

Efficient solvers for stochastic differential algebraic equations.Widely opened eye-diagrams of the noisy signals on the receiver side areessential for reliable signal detection and low bit error rates. Hence noiseeffects are to be considered very carefully in particular on the receiverpart. Up to now, only frequency domain methods have been developedfor this purpose. However, time domain based methods should be moregenerally applicable and hence more reliable in the setting to be consideredhere. Therefore, efficient numerical solvers for large systems of stochasticdifferential algebraic equations of index 2 are necessary. The focus is hereon efficient calculation of some hundred or even thousand solution pathes,as are necessary for getting sufficient numerical confidence about openingof eye-diagrams, etc.

Although the items of mathematical activity look rather widespread here, allof them serve just to improve simulation capabilities for the RF transceivercircuitry in advanced CMOS technologies. Therefore all of their results willbe combined in one single simulation environment which is part of or linkedto industrial circuit simulation.

Numerical Simulation of High-Frequency

Circuits in Telecommunication

Martin Bodestedt and Caren Tischendorf

Universitat zu Koln, Mathematisches Institut, Weyertal 86–90, 50931 Koln,Germany, {mbodeste,ctischen}@math.uni-koeln.de

Summary. Driving circuits with very high frequencies may influence the standardswitching behavior of the incorporated semiconductor devices. This motivates us touse refined semiconductor models within circuit simulation. Lumped circuit modelsare combined with distributed semiconductor models which leads to coupled sys-tems of differential-algebraic equations and partial differential equations (PDAEs).Firstly, we present results of a detailed perturbation analysis for such PDAE sys-tems. Secondly, we explain a robust simulation strategy for such coupled systems.Finally, a multiphysical electric circuit simulation package (MECS) is introducedincluding results for typical circuit structures used in chip design.

1 Introduction

In the development of integrated circuit so-called compact models are usedto describe the transistors in the circuit simulations that are performed bythe chip designer. Compact models are small circuits whose voltage-currentcharacteristics are fitted to the ones of the real device by parameter tuning.Unfortunately, not even the computationally most expensive compact modelsare able to fully capture the switching behavior of field effect transistors whenvery high frequencies are used in RF-transceivers.

A remedy to this problem is to employ a more physical modeling ap-proach and model the critical transistors with distributed device equations.Here we consider the drift-diffusion equations which is a system of mixed ellip-tic/parabolic partial differential equations (PDEs) describing the developmentof the electrostatic potential and the charge carrier densities in the transistorregion.

For the non-critical devices a lumped modeling approach is followed withthe modified nodal analysis (MNA) [13]. The electric network equations arein this case a differential algebraic equation (DAE) with node potentials andsome of the branch currents as unknowns. In this article we discuss the ana-lytical and numerical treatment of the partial differential algebraic equation

10 M. Bodestedt, C. Tischendorf

(PDAE) that is obtained when the MNA network equations are coupled withthe drift-diffusion device equations.

This article is organize as follows. We start by discussing the the refinedcircuit PDAE in particular and the perturbation sensitivity of PDAEs in gen-eral. We state results categorizing the perturbations sensitivity of the PDAEin terms of the circuit topology. Then, we present the software package MECS(Multiphysical Electric Circuit Simulator) which makes it possible to performtransient simulations of circuits with a mixed lumped/distributed modelingapproach for the semiconductor devices. The numerical approximation is ob-tained by the method of lines (MOL) combined with time-step controlledtime-integration scheme especially suited for circuit simulations.

2 The Refined Circuit Model

For brevity the refined circuit model consisting of the MNA network equationscoupled with the drift-diffusion device equations is summarized in Box 1 [2].

Electric Network Equations:

ACqC“ATCe”′+ARg

“ATRe”+ALjL + AV jV + AIc ic

“ATIce”+ ASJS =−AIsis, (1a)

φ(jL)′ − AT

Le = 0, (1b)

ATV e = vs. (1c)

Semiconductor Device Equations:

εmΔψ = q(n − p − N), (1d)

q∂tn − divjn = −qR, where jn = qUTμn gradn − qμnn gradψ, (1e)

q∂tp + divjp = −qR, where jp = −qUTμp grad p − qμpp gradψ. (1f)

Coupling interface:

JS =

ZΩ

(jn + jp − εm∂t gradψ) · grad hi dx (1g)

ψ|ΓD =“hAT

S e + ψbi

”|ΓD , n|ΓD = nD|ΓD , p|ΓD = pD|ΓD , (1h)

gradψ · ν|ΓN = gradn · ν|ΓN = grad p · ν|ΓN = 0. (1i)

Initial conditions:

(e(t0), jL(t0), jV (t0)) = (e0, j0L, j0

V ). (1j)

n(x, t0) = n0(x), p(x, t0) = p0(x), a.e. in Ω, (1k)

Box 1. The refined circuit model

The topology of the circuit is described by a network graph consisting ofnetwork nodes and element branches. It contains the element types: resistorsR, inductors L, capacitors C, independent voltage sources V , independent

Numerical Simulation of High-Frequency Circuits in Telecommunication 11

current sources Is, controlled current sources Ic and semiconductor devices S.The positions of the different elements are described by the incidence matricesA accompanied by the corresponding subscript.

The unknowns in the network equations (1a–1c) are node potentials e(t),inductor currents jL(t) and independent voltage source currents jV (t). Thefunctions qC(AT

Ce), g(ATRe) and ic(AT

Ice) model capacitor charges, resistor cur-

rents and currents through controlled current sources. They are dependent oftheir respective applied voltage. The function φ(jL) describe the electromag-netic fluxes in the inductors. Data are the functions describing the independentvoltage and current sources vs(t) and is(t).

For the drift-diffusion equations (1d–1e), after the insertion of the expres-sions for the charge carrier densities jn(x, t) and jp(x, t) into the two balanceequations we have the unknowns electrostatic potential ψ(x, t), electron den-sity n(x, t) and hole density p(x, t). The function R(n, p) is a source termmodeling the recombination of the charge carriers with the semiconductorsubstrate. The carrier mobilities μn(x) and μp(x) and the doping profile ofthe semiconductor substrate N(x) are considered to be space dependent func-tions. The dielectric permittivity εm, the unit charge q, the intrinsic carrierdensity ni and the thermal potential UT are constants.

The semiconductor region Ω ∈ Rd, d ∈ {1, 2, 3} has a boundary Γ whichis the union of Dirichlet ∪nD

i=1ΓDi and Neumann segments ∪nN

i=1ΓNi . In thesimulations, we additionally to Neumann and Dirichlet conditions considermixed boundary conditions at the so-called gate contacts.

The network and device equations are coupled in two ways at the Dirichletboundaries. The currents flowing over the semiconductor contacts JS(t) areevaluated in (1g) and accounted for in Kirchoff’s current law (1a). There, thefunctions hi fulfill the Laplace equation and homogeneous Neumann respec-tively Dirichlet boundary conditions, except at ΓDi where they equal 1. Wehave a coupling in the other direction as well as the values of electrostaticpotential ψ at the Dirichlet boundaries ΓDi depend on the node potentials ein the network (1h).

Beside the dynamic drift-diffusion model (1d–1f) we also consider the sta-tionary version, which is obtained by putting ∂tn = ∂tp = 0 in (1e–1f):

εmΔψ = q(n− p−N), (1.1d’)qdiv(UTμn gradn− μnn gradψ) = qR, (1.1e’)qdiv(UTμp grad p + μpp gradψ) = qR. (1.1f’)

In this case the dynamical term in the current evaluation is neglected. Insteadof (1g) we have

JS =∫

Ω

(jn + jp) · gradhi dx (1.1g’)

The complete circuit model consisting of the MNA equations coupled withthe stationary drift-diffusion model is (1.1a–c, 1.1d’–g’, 1.1h–k).

12 M. Bodestedt, C. Tischendorf

3 Perturbation and Index Analysis

Differential algebraic equations (DAEs), but also PDAEs, which can be seenas abstract DAEs, may have solution operators that contain differential opera-tors. Now, numerical differentiation is an ill-posed problem, wherefore numer-ical differentiation of small errors that arise in the approximation process canlead to large deviations of the approximative solution form the exact one. Inorder to successfully integrate PDAEs in time it is important to have knowl-edge of perturbation sensitivity of the solutions. A measure for this sensitivityis the perturbation index.

Definition 1. Let X, Z and Y be real Hilbert spaces, I = [t0, T ], F : Z ×X × I → Y , δ(t) ∈ Y for all t ∈ I and let wδ and w solve the perturbed,respectively unperturbed problem:

F (∂twδ(t), wδ(t), t) = δ(t), F (∂tw(t), w(t), t) = 0.

F = 0 is said to have the perturbation index ν if it is the lowest integer suchthat an estimate of the form

maxt∈I

||wδ(t)−w(t)||X ≤c

(||wδ(t0)−w(t0)||X+

ν−1∑i=0

maxt∈I

||(∂t)i(wδ(t)−w(t))||Y)

holds for some constant c.

Example 1. Consider the PDAE

∂tu2(t)− ∂2xxu1(x, t) = 0, u2(t) = 0, (x, t) ∈ Ω × I, (2)

where u1(t) ∈ Rn and u2(·, t) ∈ H10 (Ω). If the two equations are perturbed

with δ1 ∈ C(I, H−1(Ω)) and δ2 ∈ C1(I,Rn) we can derive the followingbounds for the deviation from the exact solution

maxt∈I

||u1δ(t)− u1(t)||H10≤ cmax

t∈I(||δ1(t)||H−1 + |∂tδ2(t)|) ,

maxt∈I

|u2δ(t)− u2(t)| = maxt∈I

|δ2(t)|.

According to Definition 1 the perturbation index of (2) is two since a firstorder time-derivative of δ2 appears in the first bound.

Before we turn to the results, we briefly summarize the assumptions neededfor the mathematical analysis of the coupled system.

Assumption 1. The electric network is consistent, its elements are passiveand its data smooth. The controlled sources are shunted parallel with capaci-tors. The recombination is of Shockley-Read-Hall type and the doping fulfillsN ∈ H1

0 (Ω ∩ ΓN ) ∩W 1,4(Ω) ∩H2(Ω).

Numerical Simulation of High-Frequency Circuits in Telecommunication 13

The domain Ω ∈ Rd for d ∈ {1, 2, 3} has a Lipschitz boundary. The Neu-mann boundary ΓN = ∪nN

i=1ΓNi is the union of C2-parameterizable segments.Further, the (d − 1)-dimensional Lebesgue measure of the Dirichlet bound-ary ΓD = ∪nD

i=1ΓDi is positive. The angles between Neumann and Dirichletsegments do not succeed π/2.

For a more exhaustive discussion of Assumption 1 we refer to [2] and thereferences therein. Next we will see that the perturbation sensitivity of therefined network model strongly depends on the topology of the electric networkgraph.

Theorem 1. Let Assumption 1 hold and assume that the contacts of the de-vice is connected by a capacitive paths.Then, the perturbation index of the PDAE (1.1a–c, 1.1d’–g’, 1.1h–k) is 1 ifand only if the network graph contains neither loops of capacitors and at leastone voltage source nor cutsets of inductors and independent current sources.Otherwise, the perturbation index is 2.

The proof can be found in [2]. This result generalizes known index criteria forthe MNA equations [13, 14, 11, 3] of that PDAE with stationary drift-diffusionequations.

Since we are interested in high-frequency applications it is important toaccount for the dynamical behavior of the devices, especially when a stationarydescription is used. This is done by the assumption that the contacts areconnected by a capacitive path, which models the reactive or charge storingbehavior of the device. In our next theorem we give a first perturbation resultfor a refined circuit model with dynamical device equations.

Theorem 2. Let Assumption 1 hold, the network equations be linear and with-out loops of capacitors, semiconductors and at least one voltage source and alsowithout cutsets of inductors and independent current sources. Assume that thedomain Ω is one-dimensional. If the network equations are perturbed by con-tinuous sufficiently small perturbations the perturbed solutions exist and thedeviations fulfill the bounds

maxt∈I

“|y(t)|2 + ||n(t)||2L2 + ||p(t)||2L2

”+

Z T

t0

“||∂xn(τ )||2L2 + ||∂xp(τ )||2L2

”dτ

≤ Cynp

„|δ0y|2 + ||n0

δ ||2L2 + ||p0δ ||2L2 +

Z T

t0

|δP |2 dτ

«maxt∈I

|z(t)|2 ≤ Cz

„|δ0y |2 + ||n0

δ ||2L2 + ||p0δ ||2L2 +

Z T

t0

|δp|2 dτ +maxt∈I

|δQ|2«

maxt∈I

maxx∈Ω

|∂xψ(x, t)| ≤ Cψ

„|δ0y |2 + ||n0

δ ||2L2 + ||p0δ ||2L2 +

Z T

t0

|δp|2 dτ

«.

In order to split the variables into dynamical and algebraic parts we haveput y := (PCSe, jL) and z := (QCSe, jV ) where QCS is a projector onto

14 M. Bodestedt, C. Tischendorf

ker(ACAS)T and PCS = I−QCS . Here, the bar¯denotes the deviation betweenthe exact and perturbed solution.

The proof can be found in [2]. This result is in good correspondence withthe index criteria in the previous theorem as well as with the ones in [13, 14,11, 3].

If perturbations in the drift-diffusion equations are allowed one cannot(at least in the standard way [5, 1]) prove the non-negativity of the chargecarriers and the charge preservation in the diode anymore. These propertiesare essential for the a priori estimates of the perturbed solutions, which inturn are necessary for the estimation of the nonlinear drift currents.

4 Numerical Simulation

Concerning the existence of well-established circuit simulation and devicesimulation packages alone, the first natural idea would be to couple thesesimulation packages in order to solve the circuit PDAE system described by(1). However, this approach turned out to involve persistent difficulties. Themain problem consists of the adaption of the different time step controls withinboth simulations. This can be handled for low frequency circuits since timeconstants for circuits on the one hand and devices on the other hand differby several magnitudes in such cases. Our main goal is to investigate high fre-quency circuits. Here, the pulsing of the circuit is driven near to the switchingtime of the device elements. Our coupling of circuit and device simulationsoften failed for such cases. Whereas the time step control of the circuit simu-lation works well, the device simulation does not find suitable stepsizes toprovide sufficiently accurate solutions when higher frequencies are applied.

Therefore, we pursue a different strategy to solve the circuit PDAE systemdescribed by (1) numerically. In order to control the stepsize for the wholesystem at once we choose a method of lines approach. First, we discretize thesystem with respect to space. Then we use an integration method for DAEsfor the resulting differential-algebraic equation system. Consequently, we takethe same time discretization for the circuit as for the device part.

Space discretization is needed for the device equations (1d)–(1f) as wellas for the coupling interface equations (1g)–(1i). The former ones representa coupled system of one elliptic and two parabolic equations for each semi-conductor. We use here finite element methods leading to

εmThψh + qSh(nh − ph −Nh) = 0, (3a)

Mn,h∂n

∂t+ gn,h(jn.h, nh, ph) = 0, (3b)

Mp,h∂n

∂t+ gp,h(jp.h, nh, ph) = 0, (3c)

The coupling interface equations are handled as follows. The Neumann bound-ary conditions (1i) are already considered in (3a)–(3c) by choosing proper test

Numerical Simulation of High-Frequency Circuits in Telecommunication 15

functions leading to the discrete defined functions ψh, nh and ph. The Dirich-let boundary conditions (1h) are evaluated at the grid points of the Dirichletboundary. For the approximation of the current equation (1g) we use Gaussquadrature.

Under the assumptions of Theorem 1, we obtain as DAE index for theresulting differential-algebraic equation (after space discretization of the cou-pled problem) exactly the same as the perturbation index for the system(1.1a–c, 1.1d’–g’, 1.1h–k). Furthermore, under the assumptions of Theorem2, we obtain DAE index 1 for the system (1.1a–k) as expected concering theperturbation result given in Theorem 2. For a proof we refer to [11].

4.1 The Software MECS

The multiphysical electric circuit simulator MECS [7] allows the time inte-gration of electrical circuits using different models for semiconductor devices.Beside the standard use of lumped models the user can choose distributedmodels. So far, drift diffusion models are implemented. On the one hand, thestandard model equations are used as described in (1d)–(1f). On the otherhand one can also select the drift diffusion model where the Poisson equation(1d) is replaced by the current conservation equation

div(jn + jp − εm

∂

∂t∇ψ)

= 0. (4)

For the space discretization, the standard finite element method as well asa mixed finite element [8] is implemented. For the time integration of thewhole system, the user can choose between BDF methods [4], Runge Kuttamethods [12] and a general linear method [15].

4.2 Flip Flop Circuitry

Flip flop circuits represent a basic circuit structure of digital circuits servingone bit of memory. Depending on the two input signals (set and reset), a flipflop switches between two stable states. The circuit in Fig. 1 shows a realiza-tion containing four MOSFETs (Metal Oxid Field Effect Transistors).

Fig. 1. Schematic diagram of a flip flop circuit

16 M. Bodestedt, C. Tischendorf

Fig. 2. Flip flop simulation results. On the left, the input signals Set and Reset. Onthe right, the output realising two more or less stable states when applying differentfrequencies

In Fig. 2, we see the output voltage for different applied frequencies de-pending on the two given input signals. It shows that the stable states 0Vand 5V are not so stable when increasing higher frequencies, namely whenapplying 1GHz. The simulation results may provide a more stable behaviorwhen decreasing the gate length or changing the doping profile. The advantageof the simulation here is the possibility to study the influence of the dimen-sions/doping/geometry of the semiconductors onto the switching behavior.

4.3 Voltage Controlled Oscillator (VCO)

As real benchmark we have tested a voltage controlled oscillator from thetransceiver described in the preceding introduction by Uwe Feldmann. It gen-erates a 1.8 GHz signal with the amplitude of 2.5V (see simulation result inFig. 3 on the right). Continuing the simulation over longer time periods showsthe expected behavior to hold the frequency and amplitude as the last periodson the figure indicate. The tested circuitry contained six MOSFETs that havebeen simulated using the drift diffusion model.

On the left of Fig. 3, the electrostatic potential of one of the transistorsis shown at a randomly chosen time point. The simulation package MECSprovides the data at each time discretization point for each semiconductor forthe electrostatic potential and the charge carrier densities.

Numerical Simulation of High-Frequency Circuits in Telecommunication 17

Fig. 3. VCO simulation results. On the left, the electrostatic potential of the thirdMOSFET at a certain time point. On the right, the generated oscillator voltage

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