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MeshGeneration

MarkFilipiak

EdinburghParallelComputingCentre

TheUniversityofEdinburgh

1Introduction

Continuousphysicalsystems,suchastheairflowaroundanaircraft,thestressconcentrationinadam,theelectricfieldinanintegratedcircuit,ortheconcentrationofreactantsinachemicalreactor,aregenerallymodelledusingpartialdifferentialequations.Toperformsimulationsofthesesystemsonacomputer,thesecontinuumequationsneedtobediscretised,resultinginafinitenumberofpointsinspace(andtime)atwhichvariablessuchasvelocity,density,electricfieldarecalculated.Theusualmethodsofdiscretisation,finitedifferences,finitevolumesandfiniteelements,useneighbouringpointstocalculatederivatives,andsothereistheconceptofameshorgridonwhichthecomputationisperformed.Therearetwomeshtypes,characterisedbytheconnectivityofthepoints.Structuredmesheshavearegularconnectivity,whichmeansthateachpointhasthesamenumberofneighbours(forsomegridsasmallnumberofpointswillhaveadifferentnumberofneighbours).Unstructuredmesheshaveirregularconnectivity:eachpointcanhaveadifferentnumberofneighbours.Figure1givesanexampleofeachtypeofgrid.Insomecasespartofthegridisstructuredandpartunstructured(e.g.,inviscousflowswheretheboundarylayercouldbestructuredandtherestoftheflowunstructured).

Figure1:Structuredmesh(left)andunstructuredmesh(right).

Intherestofthischapter,thevariousdiscretisationmethodsaredescribed,withtheirmeshrequirements.Chapter2describesthemethodsusedtogeneratestructuredmeshesinsimpledomains(algebraicandellipticmethods)andtheextensiontocomplexdomainsusingmultiblock.Chapter3describesmethodsusedtogenerateunstructuredmeshes,concentratingonthetwomainmethodsforproducingtriangular/tetrahedralmeshes:advancingfrontandDelaunaytriangulation.Chapter4givesaverybriefintroductiontoadaptivemeshing.

1.1Discretisationandmeshtype

Themaindiscretisationmethodsarefinitedifferences[8],finitevolumes(whichisequivalenttofinitedifferences)[9]andfiniteelements[10].Toillustratethemethods,weconsidertheconservationformoftheconvectionequation.

Whereisthedensity,isthevelocity,andisasourceterm.Thetermisthefluxof.

Thefinitedifferenceformulationapproximatesthederivativesusingneighbouringpoints,e.g.,foraregular,rectangulargridwithspacinginthedirection.

Althoughirregulargridscanbeusedforfinitedifferences,regulargridsareinvariablyused.Theseleadtosimpledifferenceschemesandefficientsolvers(onvectormachines,forexample).Therecanbeproblemswiththefinitedifferenceschemeatcoordinatesingularitiesforcertainregulargrids(e.g.,sphericalpolarcoordinates)[12]

Inthefinitevolumeformulation,thephysicalspaceissplitupintosmallvolumesandthepartialdifferentialequationsintegratedovereachofthesevolumes:

Thenthevariablesareapproximatedbytheiraveragevaluesineachvolume,andthefluxesthroughthesurfacesofeachvolumeareapproximatedasfunctionofthevariablesinneighbouringvolumes.

Finitevolumediscretisationcanusebothregularandirregularmeshes.Inanirregularmeshthefluxesthroughthesurfacesarestillwelldefined.

Thefiniteelementmethodalsosplitsthespacesupintosmallvolumes,the‘elements’.Ineachelement,thevariablesandfluxesareapproximatedusingweightingfunctions.Thecomputationalvariablesarethecoefficientsoftheseweighting(or‘shape’)functions.

Finiteelementmethodsgenerallyuseirregularmeshes:thereisnospecialadvantageinusingregularmeshes.

1.2Meshcharacteristics

Foralltypesofmeshes,therearecertaincharacteristicsthatwewanttocontrol.

Thelocaldensityofpoints.Highdensitygivesmoreaccuracy,butcomputationtakeslonger.Thisleadstoadaptivemeshingmethods,treatedverybrieflyinChapter4.

Thesmoothnessofthepointdistribution.Largevariationsingriddensityorshapecancausenumericaldiffusionoranti-diffusionanddispersionorrefractionofwaves.Thiscanleadtoinaccurateresultsorinstability[12].

Theshapeofthegridvolumes.Forinstance,boundarylayersinfluidflowrequireagridthatisverycompressednormaltotheflowdirection.Inthefiniteelementmethodusingtriangularelementsthemaximumanglemustbeboundedstrictlybelowtoproveconvergenceofthemethodastheelementsizeisreduced[11].

Forsimpledomains,thechoicebetweenregularorirregularmeshesisgovernedmainlybythediscretisationmethod.However,forcomplexdomains(e.g.,theinsideofahydro-electricturbine)irregularmeshesarepreferredtoregularmeshes.Irregularmeshgeneration(atleastfortriangularortetrahedralelements)canbefullyautomaticandfast.Regularmeshgenerationrequiresthedomaintobesplitupintosimpleblockswhicharethenmeshedautomatically.Thisblockdecompositionisatbestsemi-automaticandcanrequireman-monthsofusereffort.

2Structuredmeshes

Thischapterbeginswithadiscussionofboundary-fittedgridsandthediscretisationofPDEsonthem,thendealswiththemainmethodsforgridgenerationofsimpledomains(usingalgebraicordifferentialequationmethods)andthenexplainsthemultiblockconceptusedformorecomplicateddomains.References[12]and[13]aredetailedexpositionsofstructuredmeshgeneration.

2.1Boundary-fittedmeshes

Structuredmeshesarecharacterisedbyregularconnectivity,i.e.,thepointsofthegridcanbeindexed(by2indicesin2D,3indicesin3D)andtheneighboursofeachpointcancalculatedratherthanlookedup(e.g.,theneighboursofthepointareat,,etc.).Meshesonarectangulardomainaretrivialtogenerate(thoughcomecareneedstobetakeninthediscretisationatconvexcorners)andstructuredmeshgenerationtechniquesconcentrateonmeshingdomainswithirregularboundaries,e.g.,theflowinbloodvessels,orthedeformation,flowandheattransferinmetalbeingformedindies.Generallythemeshesaregeneratedsothattheyfittheboundaries,withonecoordinatesurfaceforming(partof)theboundary.Thisgivesaccuratesolutionsneartheboundaryandenablestheuseoffastandaccuratesolvers.Forfluidflowthesegridsallowtheeasyapplicationofturbulencemodels,whichusuallyrequirethegridtobealignedwiththeboundary.Thealternativeistousearectangulargridwhichisclippedattheboundary,withlocalgridrefinementnearsharpfeaturesontheboundary(‘Cartesiangrids’).Thiswillreducethetruncationorderattheboundaryandwillrequirethemeshcellstobeclippedattheboundary,increasingthecomplexityofthesolver(evenforafinitevolumecode).CartesiangridgenerationisveryfastandhasbeenusedforEuleraerodynamics[14];itdoesnotappeartobeapplicabletoNavier-Stokesaerodynamics.

Themostcommonmethodofgeneratingboundary-fittinggridsistohaveonecontinuousgridthatfitstoalltheboundaries.Theeffectistofitacontiguoussetofrectangularcomputationaldomainstoaphysicaldomainwithcurvedboundaries(Figure2).

Figure2:Two-blockboundary-fittedgrid

Itisdifficulttofitcomplexdomainswithonemappingfromarectangularcomputationaldomainwithoutgeneratingexcessivelyskewedgrids.Togetroundthisproblemthedomainissplitupintoblocksandeachblockisgridded,withsomecontinuityrequirementsattheblockinterfaces;thisismultiblock.ThedecompositionofthedomainintoblocksisusuallydonemanuallyusingCADtechniquesandisslow.

Analternativetocontinuousboundary-fittedgridswithmultipleblocksistouseaboundaryfittinggridneareachboundary,andsimplerectangulargridintheinterior,andinterpolatebetweenthem.Thesearecalledoversetorchimeragrids(Figure3)[15].

Figure3:Oversetgrids

Thistypeofgridiseasiertogeneratethanamultiblockgridsinceeachgridislocalanddoesn’tneedtomatchtoothers.Theindividualgridswillgenerallybeofhighquality(lowdistortion).However,theinterpolationcanbedifficult,especiallywithmorethantwogridsoverlapping,anditaddstothesolvertime(increasingitby10%-30%).Theoverlappinggridscannotbetoodifferentinresolutionandthiscancauseproblemswiththegridsrequiredforboundarylayersinviscousflowsimulations.

Chimeragridsareveryusefulformovingboundaries(e.g.,helicopterblades)ormultipleboundaries(e.g.,particlesinafluid).Mostofthegridremainsfixedbuttheinterpolationchangesasthegridsmovewiththeboundaries.

Chimeragridsdohavecertainadvantages,andtherecentwork[16]onconservativeinterpolationmethodshaveincreasedtheirusefulness.However,thebulkofstructuredgridgenerationisbasedonmultiblocktypegrids,andwewillconcentrateonthesefortherestofthischapter.

2.2Problemsolutiononcurvedgrids

Oncethegridisgenerated,thephysicalproblemhastobediscretisedandsolvedonthisgrid.Itispossibletocalculatefinitedifferenceorfinitevolumeformulasusingthephysicalgriddirectly,butthiswillreducethetruncationorderoftheFD/FVschemesincethegridsaregenerallynon-uniform.Thepreferredmethodistotransformtheequationsusedtomodeltheproblemintocomputationalspace.Sincethephysicalgridisdefinedassometransformationfromtherectangularcomputationalgrid,thisprocessisstraightforward.Theresultingequationsinthecomputationalspacewillincludetheeffectsofthecoordinatetransformation,e.g.,timederivativesofmovinggrids.Thefinitedifferencingcanbedoneonthetransformedequations,onthecomputationalgrid.Sincethecomputationalgridisusuallycompletelyregular,highordermethodscanbeused,andthetruncationorderispreservediftheresultingFD/FVequationsaretransformedbacktothephysicalgrid.(Flowsolverscanbedevelopedtousechimeragrids,withtheextrabook-keepingrequiredtointerpolatebetweeneachgrid.)

AlthoughtheorderofFD/FVmethodscanbepreserved,thereareothereffectsofcurvilineargridsontheaccuracyofthesolution.

Gridsizevariationgivesnumericaldiffusion(oranti-diffusionleadingtoinstability).Theeffectisworstwithsmallgridsizeandlargesolutiongradients.

Non-orthogonality,orskewness,ofthe(physical)gridincreasesthecoefficientofthetruncationerror,butdoesn’tchangetheorder.Skewnessofuptoareacceptableinthecentreofthegrid,butone-sideddifferencesareusedatboundaries,whereorthogonalityshouldbemaintained.Theselimitsonskewnessarethemainreasonthatmultiblockmethodsareused.

Inamultiblockscheme,thecornerjunctionsbetweenblocksareusuallypointswithnon-standardconnectivity(e.g.,in2D,5gridlinesfromapointratherthan4)andtheseneedspecialtreatment(e.g.,forfinitevolumediscretisationthesepointsneedtobediscretisedinphysicalspace).

2.3Boundaryfittinggridsonasingleblock

Forsimpledomains,asinglegridcanbeusedwithoutleadingtoexcessiveskewness.Thecomputationalspaceisarectangleorcuboid,oratleasthasarectangularboundary,witharegular,rectangulargrid.Wethenneedtodefinea1-1mappingfromthecomputationalspacetothephysicalspace.Themethodscurrentlyusedare

algebraicgridgeneration,wherethegridisinterpolatedfromthephysicalboundaries,

methodswhereaPDEforthegridpointcoordinatesissolved,

variationalgridgeneration,wheresomefunctional,e.g.,smoothness,ismaximised/minimised(variationalgridgenerationisnottreatedinthisreport,see[13]fordetails),

othermethods,(nottreatedinthisreport),e.g.,conformalmapping[12],[13].

2.4Algebraicgridgeneration:interpolation

Themethodusedinthealgebraicmethodofgridgenerationistransfiniteinterpolation(TFI).

2.4.1Transfiniteinterpolation(TFI)

Workingin2D(theextensionto3Disstraightforward),takethecomputationaldomainastheunitsquarewithcoordinatesand,andthephysicaldomainassomeregionwithcoordinatesand.Togeneratethegridinthephysicalspace,wewouldcreateagridintheunitsquareandmapthisintothephysicaldomain.Therearetworequirementsforthismapping(Figure4):

Itmustbe1-1,and

theboundariesofthecomputationalspacemustmapintotheboundariesofthephysicalspace,butotherwisethemappingcanbearbitrary,althoughitmaynotproduceagoodgrid.

Figure4:Mappingofboundaries

TFIisonekindofmapping,wherethephysicalcoordinates,treatedasafunctionofthecomputationalcoordinates,areinterpolatedfromtheirvaluesontheboundariesofthecomputationaldomain.Sincethedataisgivenatanon-denumerablenumberofpoints,theinterpolationiscalledtransfinite.

The2Dinterpolationisconstructedasalinearcombinationof21Dinterpolations(alsocalledprojections)andtheirproduct.Firstdefinetheblendingfunctionsandthatwillbeusedtointerpolateineacherpolatesvaluesfromtheboundaryintothedomain,interpolatesvaluesfromtheboundaryintothedomain,andsimilarlyforinthedirection.Therequirementsonandare

andsimilarlyforinthedirection.Thesimplestblendingfunctionislinear,givinglinearinterpolation

The1Dtransfiniteinterpolations(projections)areformedasfollows(forthecoordinate)

andtheproductprojectionis

whichisthefiniteinterpolantforthevaluesofatthefourcornersofthecomputationaldomain.

The2Dtransfiniteinterpolationisthen

Figure5:Transfiniteinterpolation

whichinterpolatestotheentireboundary.Toformthegridinthephysicalspace,thisinterpolantisusedtomapthepointsofaregulargridinthecomputationalspace(Figure5).

ItispossibletoextendtheTFItosothatitinterpolatestoseveralcoordinatelines,notjusttheboundaries.Thisisusefultocontrolthegriddensityandalsomayberequiredtoensurea1-1mapping(i.e,avalidgrid).Theblendingfunctionswouldthenbe,forexample,cubicsplines.Itisalsopossibletomatchnormalderivatives,e.g.,attheboundaries.Thisallowsgridslopecontinuitybetweentheblocksofamultiblockgrid.

Figure6:Overspill

TherearesomeproblemswithTFI.Themappingwillpropagateboundarysingularities(corners)intotheinteriorofthedomain,whichisnotidealforfluidflowsimulations.Amoreseriousproblemisthatifthemappingisnot1-1thenoverspilloccurs(Figure6).

Thiscanbecorrectedbere-parameterisingtheboundaries,oraddingconstraintlines(surfacesin3D)insidethedomain.

However,TFIgridgenerationisveryfastandistheonlycompetitivemethodfor3D.ItispossibletouseanPDEbasedgeneratortorefineagridproducedusingTFI.

2.5PDEgridgeneration

TheideabehindthesemethodsistodefinePDEsforthephysicalspacecoordinatesintermsofthecomputationalspacecoordinates,andthensolvetheseequationsonagridincomputationalspacetocreateagridinphysicalspace.ThemaintypeofPDEusediselliptic,whichissuitablefordomainswithclosedboundaries(forunboundeddomains,afictitiousboundaryatlargedistancesisused).Thealternativesarehyperbolicandparabolicequations,whichareusedforunboundeddomains.

2.5.1Ellipticgridgeneration

ThesemethodsstartwithanPDEforthecomputationalspacecoordinatesandintermsofthephysicalspacecoordinatesand(theextensionto3Disstraightforward).ThesimplestexampleistheLaplaceequation

Partsofthephysicalboundary,correspondingtocoordinatelines,arethen‘equipotentialsurfaces’(Figure7)

Figure7:Ellipticgridgenerationboundaries

In2D,theextremumprincipleguaranteesthatthemappingis1-1,andtheLaplaceoperatorissmoothingsothatboundaryslopediscontinuitiesarenotpropagatedintotheinterior.

Inordertosolvetheequationinitscurrentform,youwouldalreadyneedagridinthephysicaldomain,sotheequationistransformedsothatarethedependentvariables,andaretheindependentvariables

Usingtheequationforintherelation

gives

and(themetrictensor)canbeexpressedintermsof(see[12]fordetails).

Thisquasi-linearequationcanthenbesolvedintherectangulardomain,withthelocationofthepointsonthephysicalboundariesastheboundarydataatthepointsonthecomputationalboundaries.Anystandardsolvercanbeused,andanalgebraicallygeneratedgridcanbeusedastheinitialguesstoensure/increaseconvergence.

GridsbasedontheLaplaceequationarenotveryflexible.Thecoordinatelinestendtobeequallyspaced,exceptnearconvexorconcaveboundaries,anditisnotpossibletocontrolthegridlineslopeattheboundaries(sincethisisa2ndorderequation).

Tocontrolthegridspacingandslope,wecanaddasourceterm(controlfunction),givingPoisson’sequation

TheeffectofthecontrolfunctionisshowninFigure8

Figure8:Effectofcontrolfunctions

Themetrictensorisfactoredoutofthesourcetermstogive,ingeneral

Thetransformedequationsnowbecome

Usuallyonlyone-dimensionalstretchingineachdirectionisrequired;thenthesourcetermis(nosum).Intheory,thesourcetermsaredefinedintermsofthephysicalcoordinates,butinpracticetheyaredefinedintermsofthecomputationalcoordinates,sotheywillcauseattractionto/repulsionfromgridpointsratherthanfixedphysicalpoints.Thismakesthedirect(manual)useofcontrolfunctionsratherdifficultandtheytendtobeusedindirectlytoachieveavarietyofeffects

Becauseofthestrongsmoothing,thecoordinatelineswilltendtobeequallyspacedintheinteriorofthedomain.Togettheboundarypointspacingtopropagateintotheinterior,sourcetermscanbederivedfromboundarydatatogivethedesiredspacing,andtheninterpolated(usingtransfiniteinterpolation)

Thepointspacingisfixedontheboundary,andsincethegridgenerationequationsare2ndorder,itisnotpossibletocontrolthecoordinatelineslopeattheboundaryaswell.OnepossiblesolutionistousethebiharmonicequationratherthanLaplace’sequation,butthepreferredmethodistousecontrolfunctions.InStegerandSorenson’smethod[12]thecontrolfunctionsareiterativelyadjustedastheellipticequationissolved,andthegridlineslopeattheboundary,thegridpointspacingalongtheboundary,andthespacingofthefirstparallelgridlinefromtheboundary,canallbecontrolled.Thiscanbeusedtogiveagridthatisorthogonalattheboundary(e.g.,forboundarylayers).Itcanalsobeusedtogiveslopeandspacingcontinuityforpatchedgrids(forembeddedgridsinmultiblock,butseelaterforabetterbutslowermethodofmatchingatblockboundaries).

2.6Implementationin3D

In3Dwehaveacuboidasthecomputationaldomainandthegenerationprocessproceedsintheorder:edges,surfaces,volume.Firstwechoosethecomputationalgrid.Thegriddingoftheedgesiscalculatedusinga1DPoissonequation.Thentheboundarysurfacegridsaregenerated.Thisissimilarto2Dgridgeneration,butthesurfacecurvatureneedstobetakenintoaccount(surfacesareusuallydefinedintheCADsystemasafunctionof2Dparameterspace,e.g.,contiguousbicubicsplinespatches,andsimple2Dgridgenerationcanbeperformedinthisspace).Finally,the3Dgridgenerationscanbedone,usingthesurfacegridsasdata.

Therearesomeimplementationdifficultieswithellipticgridgeneration

Thetransformedequationsarenon-linear,sousingagoodinitialguess(fromanalgebraicgridgenerator)willspeedupconvergence.

Therecanbeslowconvergence(orfailure)forlargecontrolfunctions,sharpcorners,orhighlystretchedgrids.

Themethodisreallytooslowfor3Dgrids,wherealgebraictechniquesaregenerallyused.

2.7Othermethods

2.7.1Hyperbolicgridgeneration

InsteadofusinganellipticPDEtogeneratethegrid,hyperbolicequationshavebeenused,e.g.in2D

wherethefirstequationenforcesorthogonalityandthesecondcontrolsthecellarea.

Thesemethodsarealsoavailablein3D.Theyhavecertaincharacteristics

theyproduceorthogonalgrids,

theydon’tsmoothboundarydiscontinuities,

theyareapplicabletoexteriordomains(e.g.,theflowaroundanaerofoil).Thesolutionforthegridcoordinatesismarchedoutfromtheboundary,

theycanfailforconcaveboundarysurfaces(asthesolutionproceeds,gridlinesoverlapandcellvolumesbecomenegative),

theyarefast,takingaboutthesametimeasoneiterationofanellipticgridgenerator.

2.7.2Parabolicgridgeneration

Parabolicmethodshavebeenusedforgridgeneration,combiningthespeedofthehyperbolicgenerator(marching)withincreasedsmoothing,butthesemethodsdonotseemtobewidelyused.

2.8Multiblock

Intheory,complexgoemetriescanbemappedtoonerectangular(heretakerectangulartoapplyto2Dand3D)region(perhapsnotsimply-connected)butthiswillleadtounacceptabledistortionofthegridcells.Inpractice,thephysicalregionisbrokenupintopiecesthateachhavea(relatively)simplemappingfromarectangulargrid.Thesepieces(blocks)arefittedtogether(multiblock)withsomedegreeofgridcontinuityattheirinterfaces,rangingfromnone(eventhegridpointcoordinatesdiffer)tocomplete(lookslikeasinglegridwithnoslopeorspacingdiscontinuities).Sothegridgenerationprocesssplitsintotwoparts,thedecompositionofthephysicaldomainintoblocksandthegriddingofeachblock.Thedecompositionprocesshasnotyetbeenfullyautomated,andrequiresconsiderableuserinteraction(e.g.,choosingblockedgestoalignwithobjectedges)toproducegoodmeshes.Themeshingoftheblockscanproceedautomatically,usingoneofthemethodsdiscussedabove.

Beforelookingatmultipleblocks,welookatsomeextensionstoasimplerectangular(physical)blockthatcanbeusedtofitobjectsandthencanbeinsertedintothemultiblockschemeasasingleblock.

2.8.1C,O,Hgrids

Figure9:Degeneratepoints

C,O,andHgrids,sonamedbecauseoftheirapproximateshape,areslightextensionsofthesimple‘rectangular’block.Thecomputationalblocksareusuallysimplerectangles,butcornersinthecomputationalspaceneednotcorrespondtocornersinthephysicalspaceandviceversa(Figure9).Atthesedegeneratepointscareneedstobetakenwiththedifferenceformulation.

Figure10:Multiply-connectedblock

Thecomputationalblockcanbemultiplyconnected(Figure10),butcutscanbemadetotransformthemtosimply-connectedblocks,mappabletoarectangle,andthisleadstotheC,OandHgrids(infacttheHgridtypeincludesthismultiply-connectedblock).

Figure11:Otypegrid

TheOgridjustapolargrid(Figure11).TheCgrid(Figure12)wasdevelopedforaerofoils,withroundedleadingedgesandsharptrailingedges.Itgivesamuchbettergrid(andmoreaccuratesolutions)thananHtypegridaboutanaerofoil.

Figure12:Ctypegrid

TheHgridisaCartesian-typegrid,butallowingamultiply-connectedregionasshownbefore,orwiththeinteriorslabreducedtoaslit(Figure13).

Figure13:Htypegrid

Acrossthecut,thecoordinatedirectionsorspeciescanchange(seeFigure12).Thiswillmeanthatspecialcaseswillhavetobeprocessedinthedifferenceformulationoftheproblem(andforthegridgenerationifaPDEmethodisused).Thiscomplicationcanbeavoidedbyusingaextralayerofpoints,ahalo,toholdcopiesoftheconnectingpointsfromtheothersideofthecut(Figure14).Thisisalmostthesameasusinghaloesindomaindecompositionforparallelprocessing,butinFigure14the‘boundary’isalongaline/surfaceofgridpoints.Inafinitevolumeformulationthedatapointsarecenteredonthegridcells,andinthiscasethehaloesareexactlythesameasindomaindecomposition.

Figure14:Halopoints

Oncethehaloesaresetup,thesamedifferenceformulascanbeusedintheinteriorandonthecuts.

2.8.2Multiblock

UsingC,OandHgridsinthecomponentblocks,wecannowcreatecompositegrids:multiblock.

Thefirststepistosegmentthephysicalregionintosub-regionseachboundedby4(curved)sidesin2Dor6surfacesin3D.Thissteprequiresconsiderableuserinteractiontogenerateagooddecompositionofthedomain(e.g.,withblockedgesalignedwithsharpfeatures).

Thesecond,automatic,stepistogenerategridsineachsub-region.Theoverallgridisformedbyjoiningthesesub-gridstogether.Thedegreeofcontinuityofthegridlinesattheboundariesbetweenthesub-regionscanbeanythingfromcompletecontinuityofallderivativestonocontinuity,noteventhegridpointsontheboundary.Evidently,thelesscontinuity,themoreinterpolation/approximationwillhavetobedoneinthesolverwhenprocessingsub-regionboundaries,andtherewillbesomelossofaccuracyinthedifferenceformulations.Ifalgebraicmethodsareusedineachblock,gridlinecontinuityisensuredbygeneratingthecommonedgesandthensurfaces(in3D)andthenvolumes,eachstepusingtheboundarypointsgeneratedbythepreviousone.ItisalsopossibletoapplyTFItogridlineslopes,andthisallowsslope/spacingcontinuitytobeensured.IfPDE(elliptic)methodsareused,theequationscanbesolvedinthewholedomain,usingthehaloestotransferphysicalcoordinateinformationfromoneblocktoitsneighbours.Thisgivescompletecontinuityofthegridcoordinatesandtheregionboundarieswillalsoadjust.Solvingoverthewholedomainisverytimeconsuming,sothealternativeistosolveineachblockanduseStegerandSorensontypecontrolfunctionstoensureslope/spacingcontinuity.

Figure15showsasimplemultiblockexample,withthecomputationalgridpointsthatmaptoasinglephysicalgridpointidentified.

Figure15:Exampleofmultiblock:3C-typegrids

Therearesomedisadvantageswithfullyboundary-fittedmultiblockgrids:

Theblockingrequiresagreatdealofusereffort,thiscanbe1man-monthforcomplexconfigurations.Severaldifferentphysicalconfigurationsmaybeinvestigatedinthedesignphaseofaproduct,andeachmayneedadifferentblockstructure

Changesofgeometryinoneblockcancausechangestomanyotherblocks

Changeinthegri