SteadySteady HeatHeat ConductionConduction inin CartesianCartesian CoordinatesCoordinates andand aa LibraryLibrary ofof Green'sGreen's FunctionsFunctions KevinKevin D.D. ColeCole Dept.Dept. ofof MechanicalMechanical EngineeringEngineering PaulPaul E.E. CrittendenCrittenden Dept.Dept. ofof MathematicsMathematics UniversityUniversity ofof NebraskaNebraska----LincolnLincoln 1 MotivationMotivation VerificationVerification ofof fullyfully--numericnumeric codescodes Sponsor:Sponsor: SandiaSandia NationalNational LaboratoryLaboratory Personnel:Personnel: K.K. DowdingDowding,, D.D. AmosAmos ((SandiaSandia);); J.J. BeckBeck ,, D.D. Yen,Yen, R.R. McMastersMcMasters,, (MI(MI State);State); K.K. D.D. Cole,Cole, P.P. E.E. CrittendenCrittenden (Nebraska)(Nebraska) Geometry:Geometry: ParallelepipedParallelepiped Steady Heat Conduction and a Library of Green’s Functions 2 OutlineOutline • Temperature problem, Cartesian domains • Green’s function solution • Green’s function in 1D, 2D and 3D • Web-based Library of Green’s Functions •Summary Steady Heat Conduction and a Library of Green’s Functions 3 Temperature Problem Domain R includes the slab, rectangle, and parallelepiped. The boundary condition represents one of three types : Type 1. ki=0, hi=1, and fi a specified temperature; 2 Type 2. ki=k, hi=0, and fi a specified heat flux [W/m ]; 2 o Type 3. ki=k and hi a heat transfer coefficient [W/m / K]. 4 WhatWhat isis aa Green'sGreen's Function?Function? Green's function (GF) is the response of a body (with homogeneous boundary conditions) to a concentrated energy source. The GF depends on the differential equation, the body shape, and the type of boundary conditions present. Given the GF for a geometry, any temperature problem can be solved by integration. Green's functions are named in honor of English mathematician George Green (1793-1841). Steady Heat Conduction and a Library of Green’s Functions 5 Green’sGreen’s functionfunction solutionsolution T ( r ) = Green’s function G is the response at location r to an infinitessimal heat source located at coordinate r’. 6 Green’sGreen’s functionfunction forfor 1D1D SlabSlab Boundary conditions are homogeneous, and of the same type (1, 2, or 3) as the temperature problem. There are 32 = 9 combinations of boundary types for the 1D slab. Steady Heat Conduction and a Library of Green’s Functions 7 1D1D ExampleExample G=0G=0 atat y=0y=0 andand atat y=W.y=W. Y11Y11 case.case. TwoTwo forms:forms: Series.Series. Polynomial.Polynomial. Steady Heat Conduction and a Library of Green’s Functions 8 Y11Y11 case,case, continuedcontinued Plot of G(y,y’) versus y Y11 Geometry. for several y’ values. Steady Heat Conduction and a Library of Green’s Functions 9 GFGF forfor thethe 2D2D RectangleRectangle •Here G is dimensionless. •There are 34 = 81 different combinations of boundary conditions (different GF) in the rectangle. Steady Heat Conduction and a Library of Green’s Functions 10 2D2D ExampleExample Case X21Y11. G=0 at edges, except insulated at x=0. Double sum form: where Steady Heat Conduction and a Library of Green’s Functions 11 2D2D Example,Example, casecase X21Y11X21Y11 Single sum form: where kernel function Pn for this case is: Steady Heat Conduction and a Library of Green’s Functions 12 CaseCase X21Y11X21Y11 heatedheated atat (0.4,0.4)(0.4,0.4) Steady Heat Conduction and a Library of Green’s Functions 13 GFGF forfor thethe 3D3D ParallelepipedParallelepiped There are 36=729 combinations of boundary types. Steady Heat Conduction and a Library of Green’s Functions 14 3D3D ExampleExample CaseCase X21Y11Z12X21Y11Z12 TripleTriple sumsum form:form: Steady Heat Conduction and a Library of Green’s Functions 15 3D3D Example,Example, X21Y11Z12X21Y11Z12 AlternateAlternate doubledouble--sumsum forms:forms: Steady Heat Conduction and a Library of Green’s Functions 16 WebWeb Publication:Publication: PromisePromise •• MaterialMaterial cancan bebe presentedpresented inin multiplemultiple digitaldigital formats,formats, maymay bebe cutcut andand pastedpasted intointo otherother digitaldigital documents.documents. •• ImmediateImmediate worldworld--widewide distribution.distribution. •• RetainRetain controlcontrol ofof content,content, easilyeasily updated.updated. •• HyperlinksHyperlinks toto relatedrelated sites.sites. Steady Heat Conduction and a Library of Green’s Functions 17 WebWeb Publication:Publication: PitfallsPitfalls •• NoNo editorialeditorial support,support, nono royalties.royalties. •• UnclearUnclear copyrightcopyright protection.protection. •• ContinuedContinued operatingoperating costscosts (service(service provider,provider, computercomputer maintenance,maintenance, etc.)etc.) •• LittleLittle academicacademic reward;reward; doesn’tdoesn’t “count”“count” asas aa publication.publication. Steady Heat Conduction and a Library of Green’s Functions 18 NISTNIST DigitalDigital LibraryLibrary ofof MathematicalMathematical FunctionsFunctions •• WebWeb--basedbased revisionrevision ofof handbookhandbook byby AbramowitzAbramowitz andand StegunStegun (1964).(1964). •• EmphasisEmphasis onon text,text, graphicsgraphics withwith fewfew colors,colors, photosphotos usedused sparingly.sparingly. •• NavigationalNavigational toolstools onon everyevery page.page. •• NoNo proprietaryproprietary filefile formatsformats (HTML(HTML only).only). •• SourceSource codecode developeddeveloped inin AMSAMS--TeX.TeX. Steady Heat Conduction and a Library of Green’s Functions 19 Green’sGreen’s FunctionFunction LibraryLibrary •• SourceSource codecode isis LateX,LateX, convertedconverted toto HTMLHTML withwith sharewareshareware codecode latex2htmllatex2html runrun onon aa LinuxLinux PCPC •• GFGF areare organizedorganized byby equation,equation, coordinatecoordinate system,system, bodybody shape,shape, andand typetype ofof boundaryboundary conditionsconditions •• EachEach GFGF alsoalso hashas anan identifyingidentifying numbernumber Steady Heat Conduction and a Library of Green’s Functions 20 ContentsContents ofof thethe GFGF LibraryLibrary • Heat Equation. Transient Heat Conduction Rectangular Coordinates. Transient 1-D Cylindrical Coordinates. Transient 1-D Radial-Spherical Coordinates.Transient 1-D • Laplace Equation. Steady Heat Conduction Rectangular Coordinates. Steady 1-D Rectangular Coordinates. Finite Bodies, Steady. Cylindrical Coordinates. Steady 1-D Radial-Spherical Coordinates.Steady 1-D • Helmholtz Equation. Steady with Side Losses Rectangular Coordinates. Steady 1-D Steady Heat Conduction and a Library of Green’s Functions 21 Steady Heat Conduction and a Library of Green’s Functions 22 Steady Heat Conduction and a Library of Green’s Functions 23 Steady Heat Conduction and a Library of Green’s Functions 24 SummarySummary •• GFGF inin slabs,slabs, rectangle,rectangle, andand parallelepipedparallelepiped forfor 33 typestypes ofof boundaryboundary conditionsconditions •• TheseThese GFGF havehave componentscomponents inin common:common: 99 eigenfunctionseigenfunctions andand 1818 kernelkernel functionsfunctions •• AlternateAlternate formsforms ofof eacheach GFGF allowallow efficientefficient numericalnumerical evaluationevaluation Steady Heat Conduction and a Library of Green’s Functions 25 Summary,Summary, continued.continued. WebWeb Publishing:Publishing: widewide dissemination,dissemination, locallocal control,control, updatable;updatable; continuingcontinuing expense,expense, littlelittle academicacademic reward.reward. Green’sGreen’s FunctionFunction Library:Library: sourcesource codecode developeddeveloped inin LateXLateX (runs(runs onon anyany computer)computer) andand convertedconverted toto HTMLHTML withwith latex2htmllatex2html (runs(runs onon Linux).Linux). Steady Heat Conduction and a Library of Green’s Functions 26 WorkWork inin progress:progress: DynamicDynamic MathMath •• CurrentlyCurrently GFGF webweb pagepage isis static,static, bookbook--likelike •• TemperatureTemperature solutionssolutions areare tootoo numerousnumerous forfor prepre--determineddetermined displaydisplay •• WorkingWorking toto createcreate andand displaydisplay temperaturetemperature solutionssolutions onon demand,demand, inin responseresponse toto useruser input.input. •• CodeCode withwith openopen standardsstandards PerlPerl,, latex2htmllatex2html Steady Heat Conduction and a Library of Green’s Functions 27 AcknowledgmentsAcknowledgments •• Green’sGreen’s FunctionFunction workwork supportedsupported byby SandiaSandia NationalNational Laboratories,Laboratories, UniversityUniversity ofof NebraskaNebraska--Lincoln,Lincoln, andand byby J.J. V.V. BeckBeck •• WebWeb--pagepage developmentdevelopment assistedassisted byby undergraduateundergraduate studentstudent researchersresearchers ChristineChristine Lam,Lam, LloydLloyd Lim,Lim, SeanSean Dugan,Dugan, andand ChootepChootep TeppratuangtipTeppratuangtip.. Steady Heat Conduction and a Library of Green’s Functions 28.