Handbook of Numerical Analysis. Special Volume: Numerical methods in electromagnetics

913 Pages·2005·6.496 MB·English

by  W.H.A. SCHILDERS, E.J.W. TER MATEN, Philippe G. Ciarlet

Special Volume:
Numerical Methods in Electromagnetics
Guest Editors: W.H.A. Schilders and E.J.W. ter Maten
Handbook of
Numerical Analysis
GeneralEditor:
P.G. Ciarlet
LaboratoireJacques-LouisLions
UniversitéPierreetMarieCurie
4PlaceJussieu
75005PARIS,France
and
DepartmentofMathematics
CityUniversityofHongKong
TatCheeAvenue
KOWLOON,HongKong
NORTH
HOLLAND
Amsterdam•Boston•Heidelberg•London•NewYork•Oxford•Paris
SanDiego•SanFrancisco•Singapore•Sydney•Tokyo
Volume XIII
Special Volume:
Numerical Methods
in Electromagnetics
GuestEditors:
W.H.A. Schilders
PhilipsResearchLaboratories,ICDesign
Prof.Holstlaan4,5656AA,Eindhoven
TheNetherlands
E.J.W. ter Maten
PhilipsResearchLaboratories
ElectronicDesign&Tools/AnalogueSimulation
Prof.Holstlaan4,5656AA,Eindhoven
TheNetherlands
2005
NORTH
HOLLAND
Amsterdam•Boston•Heidelberg•London•NewYork•Oxford•Paris
SanDiego•SanFrancisco•Singapore•Sydney•Tokyo
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General Preface
In the early eighties, when Jacques-Louis Lions and I considered the idea of a Hand-
book of Numerical Analysis, we carefully laid out specific objectives, outlined in the
followingexcerptsfromthe“GeneralPreface”whichhasappearedatthebeginningof
eachofthevolumespublishedsofar:
Duringthepastdecades,giantneedsforevermoresophisticatedmathe-
matical models and increasingly complex and extensive computer simula-
tionshavearisen.Inthisfashion,twoindissociableactivities,mathematical
modelingandcomputersimulation,havegainedamajorstatusinallaspects
ofscience,technologyandindustry.
In order that these two sciences be established on the safest possible
grounds,mathematicalrigorisindispensable.Forthisreason,twocompan-
ionsciences,NumericalAnalysisandScientificSoftware,haveemergedas
essential steps for validating the mathematical models and the computer
simulationsthatarebasedonthem.
NumericalAnalysisishereunderstoodasthepartofMathematicsthatde-
scribesandanalyzesallthenumericalschemesthatareusedoncomputers;
itsobjectiveconsistsinobtainingaclear,precise,andfaithful,representa-
tionofallthe“information”containedinamathematicalmodel;assuch,it
is the natural extension of more classical tools, such as analytic solutions,
special transforms, functionalanalysis, as wellas stabilityand asymptotic
analysis.
The various volumes comprising the Handbook of Numerical Analysis
willthoroughlycoverallthemajoraspectsofNumericalAnalysis,bypre-
senting accessible and in-depth surveys, which include the most recent
trends.
Moreprecisely,theHandbookwillcoverthebasicmethodsofNumerical
Analysis,gatheredunderthefollowinggeneralheadings:
– SolutionofEquationsinRn,
– FiniteDifferenceMethods,
– FiniteElementMethods,
– TechniquesofScientificComputing.
v
vi GeneralPreface
Itwillalsocoverthenumericalsolutionofactualproblemsofcontempo-
raryinterestinAppliedMathematics,gatheredunderthefollowinggeneral
headings:
– NumericalMethodsforFluids,
– NumericalMethodsforSolids.
In retrospect, it can be safely asserted that Volumes I to IX, which were edited by
bothofus,fulfilledmostoftheseobjectives,thankstotheeminenceoftheauthorsand
thequalityoftheircontributions.
AfterJacques-LouisLions’tragiclossin2001,itbecameclearthatVolumeIXwould
bethelastoneofthetypepublishedsofar,i.e.,editedbybothofusanddevotedtosome
ofthegeneralheadingsdefinedabove.Itwasthendecided,inconsultationwiththepub-
lisher,thateachfuturevolumewillinsteadbedevotedtoasingle“specificapplication”
andcalledforthisreasona“SpecialVolume”.“Specificapplications”willincludeMath-
ematicalFinance,Meteorology,CelestialMechanics,ComputationalChemistry,Living
Systems,Electromagnetism,ComputationalMathematicsetc.Itisworthnotingthatthe
inclusion of such “specific applications” in the Handbook of Numerical Analysis was
partofourinitialproject.
Toensurethecontinuityofthisenterprise,IwillcontinuetoactasEditorofeachSpe-
cial Volume, whose conception will be jointly coordinated and supervised by a Guest
Editor.
P.G.CIARLET
July2002
Preface
The electronics industry has shown extremely rapid advances over the past 50 years,
and it is largely responsible for the economic growth in that period. It all started with
theinventionofthebipolartransistorbasedonsiliconattheendofthe1940s,andsince
thentheindustryhascausedanotherevolutionformankind.Itishardtoimagineaworld
withoutalltheachievementsoftheelectronicsindustry.
In order to be able to continue these rapid developments, it is absolutely necessary
to perform virtual experiments rather than physical experiments. Simulations are in-
dispensable in the electronics industry nowadays. Current electronic circuits are ex-
tremelycomplex,anditsproductionrequireshundredsofstepsthataltogethertakesev-
eralmonthsoffabricationtime.Theadagiois“firsttimeright”,andthishasitsreper-
cussionsforthewaydesignersworkintheelectronicsindustry.Nowadays,theymake
extensiveuseofsoftwaretoolsembeddedinvirtualdesignenvironments.Theso-called
“virtualfab”hasmadeanentry,anditisforeseenthatitsimportancewillonlygrowin
thefuture.
Numericalmethodsareakeyingredientofasimulationenvironment,whenceitisnot
surprisingthattheelectronicsindustryhasbecomeoneofthemostfertileworkingenvi-
ronmentsfornumericalmathematicians.Sincethe1970s,thereisastrongdemandfor
efficientandrobustsoftwaretoolsforelectroniccircuitsimulation.Initially,thisdevel-
opmentstartedwiththeanalysisoflargenetworksofresistors,capacitorsandinductors,
butsoonothercomponentssuchasbipolartransistorsanddiodeswereadded.Special-
istsmademodelsforthesecomponents,buttheproblemsassociatedwiththeextreme
nonlinearities introduced by these models had to be tackled by numerical analysts. It
wasoneofthefirstseriousproblemsthatwereencounteredinthefield,anditinitiated
researchintodampedNewtonmethodsforextremelynonlinearproblems.Inthepast30
years,electroniccircuitsimulationhasbecomeaverymaturesubject,withmanybeauti-
fulresults(bothfromtheengineeringandthemathematicalpointofview),anditstillis
averyactiveareaofmathematicalresearch.Nowadays,hottopicsaretheresearchinto
differential algebraic equations and the efficient calculation of (quasi-)periodic steady
states.
Althoughcircuitsimulationwasoneofthefirsttopicstobeaddressedbynumerical
mathematicians in the electronics industry, the simulation of semiconductor devices
quicklyfollowedattheendofthe1970s.Transistorsrapidlybecamemorecomplex,and
vii
viii Preface
a multitude of different devices was discovered. Transistors of the MOS-type (metal-
oxide-semiconductor)becamemuchmorepopular,andarenowmainlyresponsiblefor
the rapid developments in the industry. In order to be able to simulate the behavior
ofthesedevices,researchintosemiconductordevicesimulationwascarriedout.Soon
it became clear that this was a very demanding problem from the numerical point of
view,andittookmanyyearsandmanyconferencesbeforesomelightwasseenatthe
endofthetunnel.Appliedmathematiciansanalyzedthefamousdrift-diffusionproblem,
andnumericalmathematiciansdevelopedalgorithmsforitsdiscretizationandsolution.
Duringthe1990s,extendedmodelswereintroducedforthemodellingofsemiconductor
devices (hydrodynamic models, quantum effects), and nowadays this development is
stillcontinuing.
Paralleltothesedevelopmentsintheareaofelectroniccircuitsanddevices,themore
classicalelectromagneticsproblemswerealsoaddressed.Designofmagnetsforloud-
speakers and magnet design for MRI (magnetic resonance imaging) were important
tasks, for which we can also observe a tendency towards heavy usage of simulation
toolsandmethods.Thefieldalsogeneratedmanyinterestingmathematicalandnumer-
ical results, whereas the role of the numerical mathematicianwas again indispensable
inthisarea.
Whether it is by coincidence or not, the fields of circuit/device simulation and the
more classical electromagnetics simulation, have come very close to each other in re-
cent years. Traditionally, researchers working in the two areas did not communicate
much,andseparateconferenceswereorganizedwithaminimumofcross-fertilization.
However,owingtotheincreasedoperatingfrequenciesofdevicesandtheshrinkingdi-
mensionsofelectronicscircuits,electromagneticeffectshavestartedtoplayanimpor-
tant role. These effects influence the behavior of electronic circuits, and it is foreseen
that these effects may be dramatic in the future if they are not understood well and
precautionsaretaken.Hence,recentyearsshowanincreasedinterestincombinedsim-
ulationsofcircuitbehaviorwithelectromagneticsthat,inturn,hasledtonewproblems
fornumericalmathematicians.Oneofthesenewtopicsismodelorderreduction,which
istheartofreducinglargediscretesystemstoamuchsmallermodelthatnevertheless
exhibitsbehaviorsimilartothelargesystem.Modelorderreductionisatopicatmany
workshopsandconferencesnowadays,withamultitudeofapplicationsalsooutsidethe
electronicsindustry.
From the foregoing, it is clear that the electronics industry has always been, and
still is, a very fruitful area for numerical mathematics. On the one hand, numerical
mathematicianshave played an important role in enabling the set-up of virtual design
environments.Ontheotherhand,manynewmethodshavebeendevelopedas aresult
oftheworkinthisspecialistarea.Often,themethodsdevelopedtosolvetheelectronics
problems can also be applied in other application areas. Therefore, the reason for this
specialvolumeistwofold.Thefirstaimistogiveinsightinthewaynumericalmethods
are being used to solve the wide variety of problems in the electronics industry. The
second aim is to give researchers from other fields of application the opportunity to
benefitfromtheresults,whichhavebeenobtainedintheelectronicsindustry.
ThisspecialvolumeoftheHandbookofNumericalAnalysisgivesabroadoverview
of the use of numerical methods in the electronics industry. Since it is not assumed
Preface ix
that all readers are familiar with the concepts being used in the field, Chapter 1 gives
a detailed overview of models being used. The starting point is the set of Maxwell
equations, and from this all models can be derived. The chapter serves as the basis
for the other chapters, so that readers can always go back to Chapter 1 for a physical
explanation,oraderivationofthemodels.
The remaining chapters discuss the use of numerical methods for different applica-
tions within the electronics industry. We have attempted to organize the book in the
same way as numerical analysis is performed in practice:modelling, followed by dis-
cretization, followed by solution of nonlinear and linear systems. Unfortunately, our
attemptstoobtainachapteronnonlinearsolutionstrategieshavefailedintheend.The
correspondingchapterwouldhavebeenaveryinterestingone,withresultsondamped
Newton methods and nonlinear variable transformations. These methods will now be
discussedinaseparatebook,andthereaderisreferredtothisortotheextensiveliter-
atureonthesubject.Fortunately,allotheraspectsofnumericalanalysisarepresentin
thisvolume,andinthefollowingwegiveashortsummaryoftheremainingchapters.
Chapter2isdevotedtothemoreclassicalformofelectromagneticssimulations,but
ascanbeseenfromthechapter,thefieldleadstobeautifulmathematicalresults.Chap-
ter 3 also discusses methods for discretising the Maxwell equations, using the finite
difference time domain method that is extremely popular nowadays. The authors of
thischapterhavewidespreadexperienceinapplyingthemethodtopracticalproblems,
and the chapter discusses a multitude of related topics. Chapters 4 and 5 are devoted
tothesimulationofthebehaviorofsemiconductordevices,withanemphasisagainon
discretizationmethods.Chapter 4discussesthewellknowndrift-diffusionmodeland
someextensions,whereasChapter5concentratesonextendedmodels.
CircuitsimulationisthetopicdiscussedinChapter6,whereboththemodellingand
thediscretizationoftheseproblemsisaddressed.Theconceptofdifferential-algebraic
equationsisdiscussedextensively,togetherwithitsimportancefortheanalysisofcir-
cuits. Furthermore, time discretization and the solution of periodic steady-state prob-
lems can be found in this chapter. In Chapter 7, the first step towards coupled cir-
cuit/device simulations with electromagnetic effects is made by considering the prob-
lem of analyzing the electromagnetic behavior of printed circuit boards. The chapter
discussesindetailtheefficientevaluationoftheinteractionintegrals,andshowstheuse
ofsomenumericaltechniquesthatarenotverywellknown.
Chapters 9 and 10 are of a more theoretical character, which does not mean that
their contents are less important. On the contrary, the solution techniques for linear
systemsdiscussedinChapter9areatthecoreofallsimulationsoftware,andhenceitis
extremelyimportanttoperformthesolutionoflinearsystemsasefficientlyaspossible.
ThemodelorderreductionmethodsdiscussedinChapter10areequallyimportant,since
they provide a sound basis for enabling the coupled simulations required in present-
daydesignenvironments.Strangelyenough,itturnsoutthatthetechniquesusedinthe
areaofmodelorderreduction,areintimatelyrelatedtothesolutionmethodsforlinear
systems.Inthisrespect,thelasttwochaptersarecloselyrelated,thoughverydifferent
incharacter.
Wehopethatthisvolumewillinspirereaders,andthatthepresentationgiveninthe
various chapters is of interest to a large community of researchers and engineers. It