Strong and Weak Forms for Multidimensional Scalar Field Problems

Strong and Weak Forms for Multidimensional Scalar Field Problems and multidimensional scalar conservation laws
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6 Strong and Weak Forms for Multidimensional Scalar Field Problems Inthenextthreechapters,wewillretracethesamepaththatwehavejusttraversedforone-dimensional problemsformultidimensionalproblems.WewillagainfollowtheroadmapinFigure3.1,startingwiththe development of the strong form and weak form in this chapter. However, wewill now consider a more narrow class of problems; we have called these scalar problems because the unknowns are scalars like temperatureorapotential.Themethodsthatwillbedevelopedinthesechaptersapplytoproblemssuchas steady-stateheatconduction,idealfluidflow,electricfieldsanddiffusion–advection.Inordertoprovidea physicalsettingforthesedevelopments,wewillfocusonheatconductionintwodimensions,butdetails willbegivenforsomeoftheotherapplications. AscanbeseenfromtheroadmapinFigure3.1,thefirststepindevelopingafiniteelementmethodisto derivethegoverningequationsandboundaryconditions,whicharethestrongform.Wewillseethatintwo dimensions,justasbefore,wewillhaveessentialandnaturalboundaryconditions.Usingaformulasimilar tointegrationbyparts,wewillthendevelopaweakform.Finally,wewillshowthattheweakformimplies thestrongform,sothatwecanusefiniteelementapproximationsfortrialsolutionstoobtainapproximate solutionstothestrongformbysolvingtheweakform. Oneaspectthatwewillstressintheextensiontotwodimensionsisitssimilaritytotheone-dimensional formulation. The major equations in two dimensions are almost identical in structure to those in one dimension,somostofthe learningeffortcanbedevotedtolearningwhattheseexpressionsmeanintwo dimensions.Theexpressionsforthestrongandweakformsintwodimensions,bytheway,areidenticalto thoseforthreedimensions,andattheendofthechapterwewillgiveashortdescriptionofhowtheyareapplied to three dimensions. In engineering practice today, most analyses are done in three dimensions, so it is worthwhile to acquaint yourself with the theory in three dimensions. The extension from two to three dimensionsisalmosttrivial(wehaveusuallyavoidedtheword‘trivial’inthisbookbecauseitisoftenmisused intexts,forwhatoftenseemstrivialtoanauthorcanbequitebaffling,buttheextensionfrom2Dto3Dis indeedtrivial). One complication in extending the methods to two dimensions lies in notation. In two dimensions, variables such as heat flux and displacement arevectors. You have undoubtedly encountered vectors in elementary physics. Vectors are physical quantities that have magnitude and direction, and they can be expressedintermsofcomponentsandbasevectors.Wewilldenotevectorsbysuperposedarrows,suchas q, whichisthefluxmatrix.Lettheunitvectorsinthe xand ydirectionsbe iand j;theseareoftencalledthebase A First Course in Finite Elements J. Fish and T. Belytschko 2007 John Wiley & Sons, Ltd ISBNs: 0 470 85275 5 (cased) 0 470 85276 3 (Pbk)132 STRONG AND WEAK FORMS FOR MULTIDIMENSIONAL SCALAR FIELD PROBLEMS vectorsofthecoordinatesystem.Thenavector qcanbeexpressedintermsofitscomponentsby q¼ q iþ q j; ð6:1Þ x y where q and q arethe xand ycomponentsofthevector,respectively. x y Whenwegettothederivationoffiniteelementequations,itbecomesconvenienttousematrixnotation. Acolumnmatrixcanbeusedtodescribeavector qbylistingthecomponentsofthevectorintheorderas shownbelow:  q x q¼ : ð6:2Þ q y Thoughitisnotcrucialtodeeplyunderstandthedifferencebetweenvectorsandmatricesatthispoint,a vectordiffersfromamatrix:avectorembodiesthedirectionforaphysicalquantity,whereasamatrixis justanarrayofnumbers.Wewillgivemostoftheformulasofthestrongandweakformsinbothvector andmatrixnotations.Inthefiniteelementequations,wewilluseonlymatrixnotation.Youwillseethat thederivationofweakandstrongformsinmatrixnotationisalittleawkwardanddiffersfromtheforms commonly seen in advanced calculus and physics. So if you know vector notation as taught in those courses, you may findit preferable to usevector notation for the material in thischapter. Thetransition tomatrixnotationisquiteeasy.Onthecontrary,somepeopleprefertolearnbothpartsinmatrixnotation for the sake of consistency. An importantoperationinvector methodsisthe scalar product.Thescalarproductoftwovectorsin cartesiancoordinatesisthesumoftheproductsofthecomponentsofthevectors;thescalarproductof q withavector risgivenby q r ¼ q r þ q r : x x y y The scalar product is commutative, so the order of the two vectors does not matter. If we consider two matricesqandrthatcontainthecomponentsof qand r,respectively,thenthescalarproductiswrittenas  r T x q r¼½q q  ¼ q r þ q r : x y x x y y r y Sowritingthescalarproductintermsofthematricesrequirestakingthetransposeofthefirstmatrix.Itcan T T easilybeshownthatq r¼ r q.Whenmanipulatingvectorexpressionsinmatrixform,itisimportantto carefullyhandlethetransposeoperation. Anotherimportantoperationinvectormethodsisthe gradient.Thegradientprovidesameasureofthe slope of a field, so it is the two-dimensional counterpart of a derivative. The gradient vector operator is defined by  r¼ i þ j : x y fflfflfflfflfflfflfflfflfflfflzfflfflfflfflfflfflfflfflfflffl r Thegradientofthefunctionðx; yÞisobtainedbyapplyingthegradientoperatortothefunction,which gives   r¼ i þ j : x y Notice that we have simply replaced the bold dot in ( ) byðx; yÞ. The gradient of a function gives the direction of steepest descent. In other words, if you think of the function as describing a ski slope, theDIVERGENCE THEOREM AND GREEN’S FORMULA 133 gradient gives you the direction along which you would go the fastest. This is further illustrated in Example 6.1. Thescalarproductofthegradientoperatorwithavectorfieldgivesthe divergenceofthevectorfield.The termdivergenceprobablyoriginatedinfluidmechanics,whereitreferstotheflowleavingapoint.Wewill seelaterthatthedivergenceoftheheatfluxisequaltotheheatflowingfromapoint(thenegativeofthe sourceinasteady-statesituation).Thedivergenceofavector qisobtainedbytakingascalarproductofthe gradientoperator rand q,whichgives    q q x y r q¼ i þ j  q iþ q j ¼ þ  div q: x y x y x y Noticethatthedivergenceofavectorfieldisascalar.Asindicatedinthelastexpression,thedivergence operatorisoftenwrittenbysimplyprecedingthevectorbytheabbreviation‘div’. Theaboveexpressionscanbewritteninthematrixformasfollows.Thegradientoperatorisdefinedasa column matrix.So 2 3 2 3  6 7 6 7 x x 6 7 6 7 =¼ and =¼ : 4 5 4 5  y y The matrix form of the divergence is written by replacing the dot in the scalar product by a transpose operation,so T div q¼ q¼= q: Itisimportanttonoticethatwhenwewritethegradientoperatorinvectornotation,anarrowisplacedonthe inverteddel;inmatrixnotation,thearrowisomitted. Inthefollowing,thestudentsshouldusewhichevernotationismorenatural.Forthosenotveryfamiliar witheithernotation,theyshouldfirstscanthematerialandseewhichonetheycanunderstandmorereadily. Foradvancedstudents,afamiliaritywithbothnotationsisrecommended. 6.1 DIVERGENCE THEOREM AND GREEN’S FORMULA Thetwo-dimensionalequationswillbedevelopedforabodyofarbitraryshape.Wewilloftenrefertothe pointsinsidethebodyasthedomainoftheproblemwearetreating.Wewillfollowcommonpracticeand drawthisgenericarbitrarybodyasshowninFigure6.1(b);theideaofthisfigureisintendedtoconveythat wearenotplacinganyrestrictionsontheshapeofthebody:Thederivationsthatfollowholdforarbitrary shapes.Thisbodyisoftencalledapotato,thoughheatconductioninpotatoesisseldomofinterest.Itis worthpointingoutthattheshapecanactuallybemuchmorecomplicated:Thebodycanhaveholes,itcan Γ n y n y n x Ω = 0,l n = −1 n =1 x Ω 0 l Γ x (a) (b) Figure 6.1 (a)One-dimensionaldomainand(b)two-dimensionaldomain.134 STRONG AND WEAK FORMS FOR MULTIDIMENSIONAL SCALAR FIELD PROBLEMS havecornersanditcanconsistofdifferentmaterialswithinterfacesbetweenthem.Theboundaryofthe domainisdenotedby.Noticethatournomenclatureisidenticaltothatinthepreviouschapters,butnow thesymbolsrefertomorecomplicatedobjects.Thecorrespondencebetweenthedefinitionsinoneandtwo dimensionsisreadilyapparentbycomparingFigure6.1(a)andFigure6.1(b). Theunitnormalvectortothedomain,denotedby n,isshownatatypicalpointinFigure6.1(b)andis givenby n¼ n iþ n j; ð6:3Þ x y and n and n arethe xand ycomponentsoftheunitnormalvector,respectively;thisvectorisalsocalledthe x y 2 2 normalvectororjustthe normal.As nisaunitvector,itfollowsthat n þ n ¼ 1. x y Theobjectiveofthissectionistodeveloptheformulacorrespondingtotheintegrationbypartsformula (3.16)forascalarfieldðx; yÞ,whereðx; yÞisdefinedonthedomain.Examplesofthescalarfieldsare temperaturefields Tðx; yÞandpotentialfieldsðx; yÞ. Priortodiscussingthedivergencetheorem,itisinstructivetorecallthefundamentaltheoremofcalculus 0 that we developed in Chapter 3: for any C integrable function in a one-dimensional domain, , with boundaries,wehave Z dðxÞ dx¼ðnÞj : ð6:4Þ  dx  Recall that the boundary consists of the two end points of the domain and the unit normals point in the negative x-directionat x¼ 0andpositive x-directionat x¼ l. Thegeneralizationofthisstatementtomultidimensionsisgivenby Green’s theorem,whichstates: 0 If ðx; yÞ2 C and integrable; then Z I Z I ð6:5Þ rd¼ nd or =d¼ nd:     Notethesimilarityof(6.4)and(6.5);theoperatord=dxissimplyreplacedbythegradientr.Infact,d/dx canbeconsideredtheone-dimensionalcounterpartofthegradient.Sotheone-dimensionalform(6.4)is justaspecialcaseof(6.5).Equation(6.5)alsoappliesinthreedimensions.TheproofofGreen’stheoremis giveninAppendixA4. Usingtheabove,wewillnowdevelopatheoremthatrelatestheareaintegralofthedivergenceofavector 0 fieldtothecontourintegralofavectorfield,whichiscalledthe divergence theorem.Itstatesthatif qis C andintegrable,then Z I Z I T T r qd¼ q nd or = qd¼ q nd: ð6:6Þ     Notethat(6.5)intwodimensionsrepresentstwoscalarequations Z Z Z Z   ðaÞ d¼ n d; ðbÞ d¼ n d: ð6:7Þ x y x y    DIVERGENCE THEOREM AND GREEN’S FORMULA 135 Letting¼ q in(6.7a)and¼ q in(6.7b),andaddingthemtogetheryields x y Z I Z I q q x y þ d¼ ðq n þ q n Þd or r qd¼ q nd; ð6:8Þ x x y y x y     whichisthedivergencetheoremgivenin(6.6). Green’s formula,whichisderivednext,isthecounterpartofintegrationbypartsinonedimension.It statesthat Z I Z Z I Z T T T wr qd¼ w q nd rw qd or w= qd¼ wq nd ð=wÞ qd:       TodevelopGreen’sformula,wefirstevaluaterðw qÞbythederivativeofaproductrule: w q w q x y rðw qÞ¼ ðwqÞþ ðwqÞ¼ q þ w þ q þ w x y x y x y x x y y  q q w w x y ð6:9Þ ¼ w þ þ q þ q ¼ wr qþrw q: x y x y x y fflfflfflfflfflfflfflfflfflfflzfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflfflfflfflfflfflzfflfflfflfflfflfflfflfflfflfflfflfflfflffl r q rw q Noticethatwecanimmediatelywritethelaststepoftheaboveifwethinkofthegradientasageneralized derivativeandplacedotsbetweenanytwovectors. Integrating(6.9)overthedomainyields Z Z Z rðw qÞd¼ wr qdþ rw qd: ð6:10Þ    Applying the divergence theorem to the LHS of (6.10) and then rearranging terms yields Green’s formula: Z I Z wr qd¼ w q nd rw qd: ð6:11Þ    It is interesting to observe that for a rectangular domain l1 with one-dimensional heat flow, where q¼ q iand n¼ ni, nð0Þ¼i, nðlÞ¼ i,wehave x Z I Z q w x w d¼ q wnd q d: ð6:12Þ x x x x   136 STRONG AND WEAK FORMS FOR MULTIDIMENSIONAL SCALAR FIELD PROBLEMS Choosing wtobeonlyafunctionof x,i.e. wðxÞ,andintegrating(6.12)in ytheabovereducestotheformula forintegrationbypartsinonedimension(3.16),whichisrepeatedbelow: l l Z Z q w x w dx¼ðÞ q w ðq wÞ  q dx: ð6:13Þ x x x x¼ l x¼0 x x 0 0 Notethesimilarityof(6.11)and(6.13).ForadditionalreadingonGreen’stheorem,Green’sformulaand thedivergencetheorem,werecommendFung(1994)foranintroductoryapproachandMalvern(1969)for amoreadvancedtreatment. Example6.1 2 2 GivenarectangulardomainasshowninFigure6.2.Considerascalarfunction¼ x þ2y .Let qbethe gradientofdefinedas q¼r.Contourlinesarelinesalongwhichafunctionisconstant. (a) Findthenormaltothecontourlineofpassingthroughthepoint x¼ y¼ 0:5. (b) Verifythedivergencetheoremfor q. Thegradientvector qisgivenas   q¼ iþ j¼ 2xiþ4yj: x y Figure6.3depictsthecontourlinesofandthegradientvector q.Itcanbeseenthat qisnormaltothe contourlinesanditsmagnituderepresentstheslopeofatanypoint. Thegradientofat x¼ y¼ 0:5is qð0:5;0:5Þ¼ iþ2j: Atthepoint x¼ y¼ 0:5,thevalueofthescalarfieldisð0:5;0:5Þ¼ 0:75.Theunitnormalvectortothe 2 2 contour line x þ2y 0:75¼ 0 at the point x¼ y¼ 0:5 is obtained by dividing the vector q by its magnitude,whichgives 1 nð0:5;0:5Þ¼pffiffiffiðiþ2jÞ: 5 y (3) n = j 1 D C (2) n = i (4) n = − i −1 1 x −1 A B (1) n = − j Figure 6.2 Domainusedforillustrationofdivergencetheorem.DIVERGENCE THEOREM AND GREEN’S FORMULA 137 2 2 Figure 6.3 Contourlinesofafunction¼ x þ2y anditsgradient. We now verify the divergence theorem. Theunit normal vectors at the four boundaries of the domain ABCDareshowninFigure6.2.Toverifythedivergencetheorem(6.6),wefirstevaluatetheintegrandon theLHSof(6.6): q q x y r q¼ þ ¼ 2þ4¼ 6: x y Integratingtheaboveovertheproblemdomaingives 0 1 1 1 Z Z Z A r qd¼ 6dy dx¼ 24:  1 1 Evaluatingtheboundaryintegralcounterclockwisegives I Z Z Z Z q nd¼ ð4yÞ d þ 2x d þ 4y d þ ð2xÞ d z z z z  AB dx BC dy CD dx DA dy 1 1 1 1 Z Z Z Z ¼ 4dxþ 2dyþ 4dxþ 2dx¼ 24: 1 1 1 1 Thus,wehaveverifiedthedivergencetheoremforthisexample.138 STRONG AND WEAK FORMS FOR MULTIDIMENSIONAL SCALAR FIELD PROBLEMS y C (0,1) (2) n (3) n B x A (2,0) (1) n Figure 6.4 Triangularproblemdomainusedforillustrationofdivergencetheorem. Example6.2 2 3 3 Given a vector field q ¼ 3x yþ y , q ¼ 3xþ y on the domain shown in Figure 6.4, verify the x y divergencetheorem. TheintegrandontheLHSof(6.6)isgivenas q q x y 2 r q¼ þ ¼ 6xyþ3y : x y Integratingtheaboveyields Z Z Z Z 2 10:5x 2h i 2 3 2 r qd¼ ð6xyþ3y Þdy dx¼ 3xð10:5xÞ þð10:5xÞ dx¼ 1:5: 0 0 0  ThecounterclockwisecomputedboundaryintegralonABis 2 Z Z Z ð1Þ q n d¼ q ð1Þd¼ 3xdx¼6; x AB AB 0 ð1Þ where n ¼j,d¼ dxand y¼ 0onAB. ForthecounterclockwisecomputedboundaryintegralonBC,notethatequationofthelineBCisgiven pffiffiffi pffiffiffi ð2Þ by y¼ 10:5xand n ¼ 5=5ðiþ2jÞ,d¼ 5=2dxonBC.TheboundaryintegralonBCisthen givenby pffiffiffi 0 Z Z Z h i 5 1 ð2Þ 2 3 3 q n d¼ ðq iþ q jÞ ð iþ2jÞd¼  ð3x þ yÞþ2ð3xþ y Þ dx¼ 7:75: x y 5 2 BC AB 2 Finally,thecounterclockwiseboundaryintegralonCAis 0 Z Z Z ð3Þ 3 q n d¼ ðq iþ q jÞðiÞd¼ y dy¼0:25; x y BC AB 1 ð3Þ where n ¼i,d¼dyand x¼ 0onCA.STRONG FORM 139 Addingthecontributionsofthethreesegmentsgives I Z Z Z q nd¼ q ndþ q ndþ q nd¼6þ7:750:25¼ 1:5;  AB BC CA whichcompletesthedemonstrationthatthedivergencetheoremholdsforthiscase. 1 6.2 STRONG FORM To derive the strong form we will apply energy balance to a control volume. The strong form will be completedbyaddingtheFourierlaw,whichrelatesheatfluxtothetemperaturegradientandtheboundary conditions.Finally,theweakformwillbeformulatedbyintegratingtheproductofthegoverningequation and the natural boundary condition with the weight function over the domains where they hold. A symmetric form is obtained by applying Green’s formula (equivalent to the integration by parts in one dimension).Weconsideronlysteady-stateproblemswherethetemperatureisnotafunctionoftime. ConsideraplateofunitthicknessshowninFigure6.5(a);theplatecontainsaheatsource sðx; yÞ(energy perunitareaandtime).ThecontrolvolumeisshowninFigure6.5(b).Thebalanceofheatenergyinthe controlvolumerequiresthattheheatflux q flowing outthroughtheboundariesofthecontrolvolumeequals theheatgenerated s.ThisisthesameenergybalanceweusedinChapter3:asthebodyisinsteadystate,the heatenergyinanycontrolvolumemuststayconstant,whichmeansthattheflowouthastoequaltheheat energygeneratedbythesource. The flux vector q can be expressed in terms of two components: the component tangential to the boundary q andthecomponentnormaltotheboundary q .Thetangentialcomponent q doesnotcontribute t n t totheheatenteringorexitingthecontrolvolume.Recallthat 2 2 q¼ q iþ q j; n¼ n iþ n j; n þ n ¼ 1: x y x y x y Thenormalcomponent q isgivenbythescalarproductoftheheatfluxwiththenormaltothebody: n T q ¼ q n¼ q n¼ q n þ q n : ð6:14Þ n x x y y On AD, where n¼i, the heat inflow isq ¼ q ðiÞ¼ q whereas on BC,where n¼ i, the heat n x inflowisq ¼ q i¼q . n x ∆ y q y qx (,y + ) y 2 DC n sx (,y) Ω ∆ y Ox (,y) q n ∆ y ∆ x ∆ x ∆ x qx(, + y) x qx(, − y) x 2 2 ∆ x AB ∆ y x qx (,y − ) y 2 (a) (b) Figure 6.5 Problemdefinition:(a)domainofaplatewithacontrolvolumeshadedand(b)heatfluxesinandoutofthe controlvolume. 1 Recommended for Science and Engineering Track.140 STRONG AND WEAK FORMS FOR MULTIDIMENSIONAL SCALAR FIELD PROBLEMS InFigure6.5(b),onlythenormalcomponentsofthefluxareshown,asthesearetheonlyonesthatcontribute totheenergyflowintothecontrolvolume.Theenergybalanceinthecontrolvolumeisgivenby   x x q x ; y y q xþ ; y y x x 2 2   y y þ q x; y x q x; yþ xþ sðx; yÞxy¼ 0: y y 2 2 wherethefirstfourtermsarethenetheatinflow.Dividetheabovebyxyandrecallthedefinitionofa partialderivative:  x x q xþ ; y  q x ; y x x 2 2 q x lim ¼ ; x0 x x  y y q x; yþ  q x; y y y q 2 2 y lim ¼ : y0 y y Theaboveenergybalanceequation(afterachangeofsign)canthenbewrittenas q q x y þ  s¼ 0; x y orinthevectorandmatrixforms: T ðaÞ r q s¼ 0 or div q s¼0or ðbÞ = q s¼ 0: ð6:15Þ Ifwerecallthedefinitionofthedivergenceoperator,wecanseethatthisequationcanbeobtainedjustby reasoning: the first term is the divergence of the flux, i.e. the heat flowing out from the point. The heat flowing out from the pointr q must equal the heat generated s to maintain a constant amount of heat energy,i.e.temperature,atapoint,whichgivesequation(6.15). RecallFourier’slawinonedimension: dT q¼k ¼krT: dx Intwodimensions,wehavetwofluxcomponentsandtwotemperaturegradientcomponents.Forisotropic materialsintwodimensions,Fourier’slawisgivenby q¼krT or q¼k=T; ð6:16Þ where k 0. As in one dimension, the minus sign in (6.16) reflects the fact that heat flows in the directionoppositetothegradient,i.e.fromhightemperaturetolowtemperature.Iftheconductivity kis constant,theenergybalanceequationexpressedintermsoftemperatureisobtainedbysubstituting(6.16) into(6.15): 2 kr Tþ s¼ 0; ð6:17Þ where 2 2 2 T ð6:18Þ r ¼rr¼= =¼ þ : 2 2 x ySTRONG FORM 141 2 Equation(6.17)iscalledthePoissonequationandr iscalledtheLaplacianoperator. ThefluxandthetemperaturegradientvectorsarerelatedbyageneralizedFourier’slaw: 2 3 T   6 7 x q k k x xx xy 6 7 ¼ ; 4 5 q k k T y yx yy fflfflfflfflfflfflfflzfflfflfflfflfflfflffl y D orinthematrixform: q¼D=T; ð6:19Þ whereDistheconductivitymatrix.Wewritethisequationonlyinthematrixformbecausethevectorform cannotbewrittenwithoutsecond-ordertensors,whicharenotcoveredhere. SubstitutingthegeneralizedFourierlaw(6.19)intotheenergybalanceequation(6.15)yields T = ðD=TÞþ s¼ 0: ð6:20Þ ThematrixDmustbepositivedefiniteasheatmustflowinthedirectionofdecreasingtemperature. Forisotropicmaterials,  k 0 D¼ ¼ kI: ð6:21Þ 0 k Intwodimensions,thesymmetryofthematerialisanimportantfactorintheformoftheFourierlaw.A material is said to have isotropic symmetry if the properties are the same in any coordinate system. For example,mostmetals,concreteandasiliconcrystalareisotropic.Theformoftherelationbetweenheat flux and temperature gradient in an isotropic material is independent of how the coordinate system is placed.Inanisotropicmaterials,Ddependsonthecoordinatesystem.Examplesofanisotropicmaterials areradialtires,fibercompositesandrolledaluminumalloys.Forexample,inaradialtire,heatflowsmuch morerapidlyalongthedirectionofthesteelwiresthanintheotherdirections. To solve the partial differential equation (6.20), boundary conditions must be prescribed. In multi- dimensions,thesamecomplementarityconditionsthatwelearnedinonedimensionhold.Atanypointof theboundary(seeFigure6.6),eitherthetemperatureorthenormalfluxmustbeprescribed,buttheyboth cannotbeprescribed.Therefore,ifwedenotetheboundarywherethetemperatureisprescribedby and T theboundarywherethefluxisprescribedby ,thenwehave q   ¼;  \ ¼ 0: ð6:22Þ q T q T y Γ= Γ Γ q T Ω T = T on Γ T x Figure 6.6 Problemdomainandboundaryconditions. ⊂142 STRONG AND WEAK FORMS FOR MULTIDIMENSIONAL SCALAR FIELD PROBLEMS Wewritetheprescribedtemperatureboundaryconditionas Tðx; yÞ¼ Tðx; yÞ on  ; ð6:23Þ T where Tðx; yÞ is the prescribed temperature; these are the essential boundary conditions; these are also calledDirichletconditions.Asindicated,theprescribedtemperaturealongtheboundarycanbeafunction ofthespatialcoordinates. On a prescribed flux boundary, only the normal flux be prescribed. We can write the prescribed flux conditionas  q ¼ q n¼ q on  : ð6:24Þ n q These arealsocalledNeumannconditions.Foranisotropicmaterial,thenormalfluxisproportionaltothe gradient of the temperature in the normal direction, i.e. it follows from (6.19) and (6.21) that T q ¼kn rT. It can be seen that the flux depends on the derivatives of the temperature, so this is the n naturalboundarycondition. Theresultingstrongformfortheheatconductionproblemintwodimensionsisgiveninthevectorform forisotropicmaterialsinBox6.1andinthematrixformforgeneralanisotropicmaterialsinBox6.2.These forms differ from what we used in one dimension in that the energy balance and Fourier’s law are not combined.Thissimplifiesthedevelopmentoftheweakformandextendstheapplicabilityoftheweakform tononlinearheatconduction. Box6.1.Strongform(vectornotation)forheatconduction ðaÞ energy balance : r q s¼0on ; 0 ðbÞ Fouriers law : q¼krT on ; ð6:25Þ ðcÞ natural BC : q ¼ q n¼ q on  ; n q ðdÞ essential BC : T ¼ T on  : T Box6.2.Strongform(matrixnotation)forheatconduction T ðaÞ energy balance : = q s¼0on ; 0 ðbÞ Fouriers law : q¼D=T on ; ð6:26Þ T ðcÞ natural BC : q ¼ q n¼ q on  ; n q ðdÞ essential BC : T ¼ T on  ; T Thevariables s,D, T and qarethedatafortheproblem.These,alongwiththegeometryofthedomain, mustbegiven. 6.3 WEAK FORM To obtain the weak form we will follow the same basic procedure as for the one-dimensional problem in Chapter3.However,aswehavealreadymentioned,wewilldeveloptheweakformofthebalanceequation (6.15a). Then wewill express the heat flux in terms of the temperaturegradient by the Fourierlaw. We start with the energy balance equation (6.15a) and the natural boundary condition (6.25c). We premultiplythetwoequationsbyaweightfunction wandintegrateovertheproblemdomainandtheWEAK FORM 143 naturalboundary ,respectively: q Z Z ðaÞ wðr q sÞd¼ 0 8w; ðbÞ wðq q nÞd¼ 0 8w: ð6:27Þ   q Fortheequivalenceofthestrongandweakforms,itiscrucialthattheweakformholdforallfunctions w.As inonedimension,wewillfindthatsomerestrictionsmustbeimposedontheweightfunction,butwewill developtheseasweneedthem.ApplyingGreen’sformulatothefirsttermin(6.27a)yields Z I Z wr qd¼ w q nd rw qd 8w: ð6:28Þ    Inserting(6.28)into(6.27a)yields Z I Z Z Z Z rw qd¼ w q nd wsd¼ w q ndþ w q nd wsd: ð6:29Þ       q T where we have subdivided the first integral on the RHS of (6.29) into the prescribed temperature and prescribedfluxboundaries,whichispermissiblebecauseof(6.22).Substituting(6.27b)intotheintegralon  (6.29)yields q Z Z Z Z rw qd¼ wqdþ w q nd wsd:     q T WenowfollowthesamereasoningasinChapter3.Itiseasytoconstructweightfunctionsthatvanishona portion of the boundary, so we set w¼ 0 on the prescribed temperature boundary, i.e. the essential boundary.Thereforetheintegralon vanishesandtheweakformisgivenby T Z Z Z rw qd¼ w qd wsd 8w2 U ; ð6:30Þ 0    q where U isthesetofsufficientlysmoothfunctionsthatvanishontheessentialboundary,itisthespaceof 0 functions defined in (3.48). The space of admissible trial solutions U satisfies the essential boundary conditionsandissufficientlysmoothasdefinedin(3.47).Recallthataccordingtothedefinitionofthese 0 spaces,thetrialsolutionsandweightfunctionshavetobe C continuous. Expressing(6.30)inmatrixformgives Z Z Z T ð=wÞ qd¼ w qd wsd 8w2 U : 0    q The above is the weak form for any material, linear or nonlinear. To obtain the weak form for linear materials,wesubstituteFourier’slawintothefirsttermoftheabove,whichyields Box6.3.Weakform(matrixnotation)forheatconduction find T 2 Usuch that: Z Z Z T ð=wÞ D=Td¼ wqdþ wsd 8w2 U : ð6:31Þ 0    q144 STRONG AND WEAK FORMS FOR MULTIDIMENSIONAL SCALAR FIELD PROBLEMS 2 6.4 THE EQUIVALENCE BETWEEN WEAK AND STRONG FORMS Todemonstratetheequivalenceofthestrongformandtheweakform,itmustbeshownthattheweakform implies the strong form. This demonstration is similar to the one used in Chapter 3 for showing the equivalenceforone-dimensionalproblems:wereversethestepsthatwehavefollowedingoingfromthe strongformtotheweakformandtheninvokethearbitrarinessoftheweightfunctionstoextractthestrong formfromtheintegralequations.Wewilldothisfortheweakformforarbitrarymaterials. Westartwith(6.30),rewrittenbelow: Z Z Z rw qd¼ wqd wsd:    q NowweapplyGreen’sformula(6.11)tothefirstterm,whichgives Z Z Z wðr q sÞdþ wð q q nÞd w q nd¼ 0 8w2 U : ð6:32Þ 0    q T WefollowthesamestrategyasinChapter3.Sincetheweightfunction wðxÞisarbitrary,itcanbeassumedto beanyfunctionthatvanisheson . T Wetakeadvantageofthearbitrarinessoftheweightfunctionandmakeitequaltotheintegrandthatis,we let  0on  w¼ ðxÞðr q sÞ; where ðxÞ¼ : ð6:33Þ 0on  Inserting(6.33)into(6.32)yields Z 2 ðr q sÞ d¼ 0: ð6:34Þ  Theboundarytermshavevanishedbecauseourchoiceof wðxÞ,(6.33),vanishesontheboundaries.Since ðxÞ0in,theintegrandin(6.34)ispositiveateverypointinthedomain.Fortheintegralin(6.34)to vanish,theintegrandhastovanishaswell.Hence,since ðxÞ 0, r q s¼0in ; ð6:35Þ which is the energy balance equation (6.15). After substituting (6.35) into (6.32) we select a weight function that is nonzero on the natural boundary, but vanishes on the essential boundary (it does not matterwhatitsvalueisinsidethedomain,asby(6.35)weknowthatthefirsttermin(6.32)willvanish). So we let  0on  T w¼’ðq q nÞ; where ’¼ : ð6:36Þ 0on  q Substituting(6.36)into(6.32)yields Z 2 ’ð q q nÞ d¼ 0: ð6:37Þ  q 2 Recommended for Advanced Track.⋅= Γ GENERALIZATION TO THREE-DIMENSIONAL PROBLEMS 145 z q n TT = on Γ qn q on q T k r i y j x Figure 6.7 Problemdomainandboundaryconditionsinthreedimensions. Astheintegrandin(6.37)ispositiveon ,thequantityinsidetheparenthesesmustvanishoneverypointof q thenaturalboundary,sothenaturalboundarycondition(6.25c)follows. 3 6.5 GENERALIZATION TO THREE-DIMENSIONAL PROBLEMS Theextensionfromtwotothreedimensionsisalmosttrivial.Thedifferenceisnotinthestructureofthe strong and weak form equations, which are identical, but in the definitions of the vectors, gradient, divergenceandLaplacianoperators. Inthreedimensions,thebase(unit)vectorsare i, jand kasshowninFigure6.7.Avector qexpressedin termsofitscomponentsis 2 3 q x 4 5 q¼ q iþ q jþ q k; q¼ q ; ð6:38Þ x y z y q z where thematrixformisshown ontheright-handside.In threedimensions, theproblem domainisa volume (which looks like the potato in Figure 6.7) and its boundary is a surface. The progression of dimensionality of the problem domain and its boundary from one-dimensional to three-dimensional problemsissummarizedinTable6.1. Theboundary,whichisthesurfaceencompassingthethree-dimensionaldomain,consistsofthe complementaryessentialandnaturalboundaries,asshowninFigure6.7. Table 6.1 Dimensionality of the problem domain and its boundary. Entity Domain Boundary Onedimension(1D) Linesegment Twoendpoints Twodimensions(2D) Two-dimensionalarea Curve Threedimensions(3D) Volume Surface 3 Recommended for Advanced Track.146 STRONG AND WEAK FORMS FOR MULTIDIMENSIONAL SCALAR FIELD PROBLEMS Thegradientoperatorinthreedimensionsinvectorandmatrixnotationsisdefinedas    r¼ i þ j þ k ; r¼ i þ j þ k ; x y z x y z 2 3 2 3  6 7 6 7 x x 6 7 6 7 6 7 6 7  6 7 6 7 =¼ ; =¼ : 6 7 6 7 y y 6 7 6 7 4 5 4 5  z z Withtheabovedefinitionsofvectorsandthegradientvectoroperator,thedivergenceofthevectorfieldand theLaplacianare q q q x y z div q¼ þ þ ; x y z 2 2 2 2 T r ¼rr¼= =¼ þ þ 2 2 2 x y z : ThestrongforminvectorandmatrixnotationsisidenticaltothatgiveninEquations(6.25)and(6.26).Note thattheFourierlawrelatingthethreecomponentsoftemperaturegradienttothethreefluxcomponentsis definedintermsofa33symmetricpositivedefinitematrixD: 2 3 k k k xx xy xz 4 5 D¼ k k k : yx yy yz k k k zx zy zz Theweakformisalsoidenticaltothatfortwo-dimensionalproblemsasgivenin(6.31). 6.6 STRONG AND WEAK FORMS OF SCALAR STEADY-STATE 4 ADVECTION–DIFFUSION IN TWO DIMENSIONS The advection-diffusion equations are obtained from a conservation principle (often called a balance principle), just like heat conduction. The conservation principle states that the species (be it a material, anenergyorastate)areconservedineachcontrolvolumeofareaxyandunitthicknessshownin Figure 6.8. The amount of species entering minus the amount of species leaving equals the amount produced(anegativevolumewhenthespeciesdecays).Therearetwomechanismsforinflowandoutflow, advection(orconvection),whichisgivenby v,anddiffusion,whichisgivenby q. Dy Dy vx q(,y + ) qx (,y + ) y y 2 2 Dx x D vx(, − y) q vx q(, + y) x x Ox (,y) 2 2 Dy Dx Dx qx(, + y) qx(, − y) x x 2 2 D x y Dy D vx q(,y − ) qx (,y − ) y y 2 2 Figure 6.8 Controlvolumeforadvection–diffusionproblem. 4 Recommended for Advanced Track.STRONG AND WEAK FORMS OF SCALAR STEADY-STATE ADVECTION–DIFFUSION 147 Inaddition,advectiononeachsurfaceresultsinaninflowofv n.Theconservationprinciplecanthen bedevelopedasinsection6.2:     x x y y v  x ; y yþ q x ; y yþ v  x; y xþ q x; y x x x y y 2 2 2 2     x x y y  v  xþ ; y y q xþ ; y y v  x; yþ x q x; yþ x x x y y 2 2 2 2 þxysðx; yÞ¼ 0: Dividingtheabovebyxyandtakingthelimitx 0,y 0,weobtain ðv Þ ðv Þ ðq Þ ðq Þ x y x y þ þ þ  s¼ 0: x y x y Theabovecanbewritteninthevectorformas rð vÞþr q s¼ 0: ð6:39Þ Thisisthegeneralformoftheadvection–diffusionequation.Thefirsttermaccountsfortheadvectionor transportofthematerialandthesecondtermaccountsforthediffusion. In many cases, the material carrying the species is incompressible. For steady-state problems and incompressible materials, the rate of material volume entering control volume is equal to the rate of materialvolumeexitingcontrolvolume.Mathematically,thisisgivenby     x y x y v x ; y yþ v x; y x v xþ ; y y v x; yþ x¼ 0: x y x y 2 2 2 2 Dividingtheabovebyxyandtakingthelimitx 0,y 0gives ðv Þ ðv Þ x y þ ¼ 0: x y Theaboveinmatrixandvectornotationsis T r v¼0or = v¼ 0: ð6:40Þ Equation (6.40) is known as the continuity equation for steady-state problems of incompressible materials. Substitutingthecontinuityequation(6.40)into(6.39)yieldstheconservationequationforaspeciesina movingincompressiblefluid,whichcanbewrittenas T T vrþr q s¼0or v =þ= q s¼ 0: ð6:41Þ AssumingthatthegeneralizedFourier’slaw(6.19)holds,theconservationofspeciesequationinthematrix formbecomes T T v == ðD=Þ s¼ 0: ð6:42Þ148 STRONG AND WEAK FORMS FOR MULTIDIMENSIONAL SCALAR FIELD PROBLEMS ForisotropicmaterialsD¼ kIandtheconservationequationreducesto 2 T 2 vr kr  s¼0or v = kr  s¼ 0; ð6:43Þ 2 where r is the Laplacian defined in (6.18). We consider the usual essential and natural boundary conditions  ¼ on  ;  ð6:44Þ q n¼ q on  ; q where and arecomplementary.  q To obtain the weak form of (6.43) we multiply the conservation equation (6.41) and the natural boundary condition by an arbitrary weight function w and integrate over the corresponding domains: Z Z ðaÞ wð vrþr q sÞd¼ 0; ðbÞ wð q q nÞd¼0on 8w: ð6:45Þ   q Integrationbypartsofthesecondterm(thediffusionterm)in(6.45a)gives Z Z Z Z w vrd rw qdþ w qd wsdðaÞ8w2 U ; ð6:46Þ 0    q  wherewehaveexploited(6.45b)andthat w¼0on .  Finally,theweakformiscompletedbysubstitutingthegeneralizedFourierlawinto(6.46),whichgives find the trial solution ðx; yÞ2 U such that Z Z Z Z T T ð6:47Þ wv =dþ ð=wÞ D=dþ w qd wsd 8w2 U : 0     q Theaboveistheweakformfortheadvection–diffusionequation.Notethatthefirsttermisunsymmetricin theweightfunction wandthesolution.Thiswillresultinunsymmetricdiscretesystemequationsandhas importantramificationsonthenatureofthesolutions,becauseasin1D,thesolutionscanbeunstableifthe velocityislargeenough. REFERENCES Fung, Y.C. (1994) A First Course in Continuum Mechanics, 3rd edn, Prentice Hall, Englewood Cliffs, NJ. Malvern, L.E. (1969) Introduction to the Mechanics of a Continuous Medium, Prentice Hall, Englewood Cliffs, NJ. Problems Problem 6.1 2 Given a vector field q ¼y , q ¼2xy on the domain shown in Figure 6.2. Verify the divergence x y theorem. Problem 6.2 2 3 3 Givenavectorfieldq ¼ 3x yþ y ,q ¼ 3xþ y onthedomainshowninFigure6.9.Verifythedivergence x y theorem.Thecurvedboundaryofthedomainisaparabola.REFERENCES 149 y (0,4) 2 n x (−2,0) (2,0) 1 n Figure 6.9 ParabolicdomainofProblem6.2usedforillustrationofdivergencetheorem. Problem 6.3 Usingthedivergencetheoremprove I nd¼ 0:  Problem 6.4 Startingwiththestrongform dq  s¼ 0; qð0Þ¼ q; TðlÞ¼ T; dx developaweakform.Notethattheflux qisrelatedtothetemperaturethroughFourier’slaw,butdevelopthe weakformfirstintermsoftheflux. Problem 6.5 Consider the governing equation for the heat conduction problem in two dimensions with surface convection: T = ðD=TÞþ s¼ 2hðT  T Þ on ; 1 T q ¼ q n¼ q on  ; n q T ¼ T on  : T Derivetheweakform. Problem 6.6 Derivethestrongformforaplatewithavariablethickness tðx; yÞ. Hint:ConsidercontrolvolumeinFigure 6.5(b),andaccountforthevariablethickness.Forexampletheheatinflowatðxx=2; yÞis  x x q x ; y yt x ; y : x 2 2 Derivetheweakformfortheplatewithvariablethickness.150 STRONG AND WEAK FORMS FOR MULTIDIMENSIONAL SCALAR FIELD PROBLEMS y qq = on Γ n q U Γ U Γ Γ= Γ q h T Ω TT=on Γ T qh=– (T T ) on Γ ∞ n h x Figure 6.10 Problemdomainandboundaryconditionsforheatconductionwithboundaryconvection. Problem 6.7 Consideraheatconductionproblemin2Dwithboundaryconvection(Figure6.10). Constructtheweakformforheatconductionin2Dwithboundaryconvection. Problem 6.8 Consideratime-dependentheattransfer.Theenergybalanceinacontrolvolume(seeFigure6.5)isgiven by    x x y q x ; y y q xþ ; y yþ q x; y x x x y 2 2 2  y T  q x; yþ xþ sðx; yÞxy¼ cr xy y 2 t; where Tðx; y; tÞ, candrdenotethetemperature,materialspecificheatanddensity,respectively,and tisthe time. The above equation states that thechange in internal energyis not zero, but is rathergovernedby density,specificheatandrateofchangeoftemperature. Derivetheweakandstrongformsforthetime-dependentheattransferproblem.

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