The Theory of Laser Materials Processing

Theuseoflasersinmaterialsprocessinghasbecomewidespreadinrecent years,sothatanunderstandingofthenatureofheatandmasstransferin thisbranchofmoderntechnologyisofincreasingimportance. Theaimofthe authorsofthisbookistoconcentrateonthephysicalprocesses;thesecanbe developedfromamathematicalpointofview,orfromdirectexperimental- derivedobservation. Thetwoapproachesarecomplementary;eachcanprovide insightsandthesynthesisofthetwocanleadtoaverypowerfulunderstanding oftheprocessesinvolved. Mathematicalmodellingofphysicalprocesseshas hadanimportantroletoplayinthedevelopmentoftechnologyoverthe centuriesandparticularlysointhelastonehundredand?ftyyearsorso. Itcanbearguedthatitismoreimportanttodaythaneverbeforesincethe availabilityofhigh-speedcomputersallowsaccuratenumericalsimulationof industrialprocessesatafractionofthecostofthecorrespondingexperiments. Thisisoneaspectofmathematicalmodelling,highpro?leandmuchvalued, butitisnottheonlyone. Inthepastmathematicalmodellinghadtorelyonqualitativeinves- gation,veryspecialanalyticalsolutions,orinaccurateandtime-consuming calculationsperformedwithlittleinthewayoftabulatedormechanical assistance. Logtablesandsliderulesarestillrememberedbypeopleworking today,thoughtherearesurelyfewwhoregrettheirdisappearance. Thevalueanddistinctivefunctionofmethodsbasedontheanalytical approachisnowbecomingmuchclearer,nowthattheyarenolongerexpected toproducedetailedimitationsofwhathappensinrealexperimentsofind- trialprocesses,afunctionnowful?lledmostlybynumericalmethods,c- sideredbelow. Theemphasistodayisontheirabilitytocon?rmandextend ourunderstandingofthebasicphysicalmechanismsinvolvedintheprocesses of interest. These are essential for any intelligent use of numerical simulation. Theargumentaboutthevalueofteachingpeoplehowtodoarithmetic themselveswithouttheaidofacalculatorseemstobepassingintohistory, vi Preface butitisanimportantoneandprovidesasimpleanalogy. Ifsomeonedoes nothaveafeelingfornumbersandthewayarithmeticworks,theywillalltoo easilyfailtospotanerrorproducedbyamachine. Computersarenotinfallible –andneitherarethosewhobuildorprogramthem. Computersarenow takingonlessmundanemathematicaltasksandthesamecontroversiesare appearinginconnectionwithalgebraicmanipulation. Equally,andwitheven greaterpenaltiesintermsofcostintheeventoferrors,thesameconsiderations applytonumericalsimulationofmajorindustrialprocesses. Awarenessofthe analyticalsolutionscanbeinvaluableindistinguishingtherightfromthe wrong,i. e. forthepractitionertounderstandthebasisofthework,andto haveanideaofthekindsofoutcomesthatareplausible–andtorecognise thosewhicharenot. Thephrase“mathematicalmodelling”is,however,ambiguous,perhaps morenowthanithaseverbeen. Thereisanenormousamountofworkdone todayonsimulationbasedontheuseofverypowerfulcomputerprograms, anditisquitecorrectlyreferredtoasmathematicalmodelling. Theprograms aresometimesconstructedin-housebutareusuallycommercialpackages. This isanentirelyvalidapproachwithspeci?c(generallycommercial)objectives. Ingeneraltherearetwouses. Thedominantobjectiveisnumericalagreement withaparticularexperimentinthe?rstinstance,leadingtopredictivec- mercialuseinthesecondinstance. Thesecondobjectiveistheclari?cation ofphysicalmechanisms,aimedatthegenerationofunderstandingofcomplex interconnectedprocesses,ratherthantheexactreproductionofaparticular experiment. Itissometimesoverlookedthat,withsu?cientcare,anum- icalapproachisequallyvalidintheinvestigationofphysicalfundamentals. Numericalsimulationisnotacentraltopicofthisbook,butbecauseofits crucialimportancetoeachofthetwousestowhichnumericalmodellingcan beput,itisvitalthatthecomputationalbasisoftheworkshouldbec- pletelysound. Inaddition,thelevelofprocessdetailwhichcanbeconsidered bythenumericalapproachusuallyexceedswhatispossiblewiththeanaly- calapproachsigni?cantly,leavinglittlechoicebuttoreverttothenumerical treatmentwheninvestigatingtheinterconnectionsbetweenprocesses. Itis forthesereasonsthatthebookconcludeswithachapteroncomprehensive numericalsimulation. Inmanyways,theapproachadoptedhereiscomplementarytothemore phenomenologicalapproach. Itisalwaysimportantina?eldwhichhasvery directindustrialapplicationstobearinmindhowtechniquessuchasthose describedherewillbeused,butitisessentialnottolosesightofthef- damentals. Thereareserioussafetyimplications;therearecostimplications; therearemoralimplications;thereareconsiderationsoftheappropriateness ofthetechnologytotheapplicationunderconsideration. Aproperrespectfor alltheserequiresanunderstandingofthefundamentals. Wearealltoowellawarethatthisbookdoeslittlemorethanscratch thesurfaceoftheproblemsinvolvedinafundamentalunderstandingofthese phenomena. Ifwehaveprovidedideasandinformationthatcauseothersto Preface vii testthemexperimentallyorintellectually,agreewiththemordisputethem vigorously,anddevelopthemfurther,wewillconsiderthatwehaveachieved ouraim. Colchester April,2008 JohnDowden Contents 1MathematicsinLaserProcessing JohnDowden. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 1 MathematicsanditsApplication. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 2 FormulationinTermsofPartialDi?erentialEquations. . . . . . . . . 3 1. 2. 1 LengthScales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. 2. 2 ConservationEquationsandtheirGeneralisations. . . . . . 4 1. 2. 3 GoverningEquationsofGeneralised ConservationType. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1. 2. 4 Gauss’sLaw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1. 3 BoundaryandInterfaceConditions. . . . . . . . . . . . . . . . . . . . . . . . . . 11 1. 3. 1 GeneralisedConservationConditions. . . . . . . . . . . . . . . . . 11 1. 3. 2 TheKinematicConditioninFluidDynamics. . . . . . . . . . 13 1. 4 Fick’sLaws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1. 5 Electromagnetism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1. 5. 1 Maxwell’sEquations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1. 5. 2 Ohm’sLaw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2SimulationofLaserCutting WolfgangSchulz,MarkusNießen,UrsEppelt,KerstinKowalick. . . . . . . . 21 2. 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2. 1. 1 PhysicalPhenomenaandExperimentalObservation. . . . 23 2. 2 MathematicalFormulationandAnalysis. . . . . . . . . . . . . . . . . . . . . . 26 2. 2. 1 TheOne-PhaseProblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2. 2. 2 TheTwo-PhaseProblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2. 2. 3 Three-PhaseProblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2. 3 Outlook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2. 4 Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 x Contents 3KeyholeWelding:TheSolidandLiquidPhases AlexanderKaplan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3. 1 HeatGenerationandHeatTransfer. . . . . . . . . . . . . . . . . . . . . . . . . . 71 3. 1. 1 Absorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .