39 pages

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InterCell Escapade Tutorial

Syaj - N. Fressengeas

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39 pages

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Diusion/Heat Equation Derive a Cellular Algorithm Running on InterCellInterCell/Escapade TutorialN. FressengeasLaboratoire Materiaux Optiques, Photonique et SystemesUnite de Recherche commune a l’Universite Paul Verlaine Metz et a SupelecDowload this document fromhttp://intercell.metz.supelec.frN. Fressengeas LMOPSInterCell/EscapadeDiusion/Heat Equation Derive a Cellular Algorithm Running on InterCellTable of contentsFick Di usion or Heat Transfer laws as an exampleFick/Heat lawsBoundary ConditionsDerive a Cellular AlgorithmInstall the necessary softwareDiscretize your problemLocate your equations on the MeshRunning on InterCellA look at the lesRun the programSee the resultsN. Fressengeas LMOPSInterCell/EscapadeDiusion/Heat Equation Derive a Cellular Algorithm Running on InterCellFick/Heat lawsSteady state Fick/Heat di usion lawsA fairly simple partial di erential problem for didactic purposeFick laws Di usion lawsI IJ = Cgrad () q = kgrad (T )@ @TI I+div (J) = 0 C +div (q) = 0@t @t@ @TI I= C4 C = k4T@t @tSteady state lawSteady state law4 = 0 4T = 0N. Fressengeas LMOPSInterCell/EscapadeDiusion/Heat Equation Derive a Cellular Algorithm Running on InterCellBoundary ConditionsSome boundary conditionsComplexity often lies in the boundary conditionsTJPure diusionqJqBoundary conditionsI Constant ow pipes : J = J = Cgrad ()0@ @I grad (J)kBoxborders : = 0 or = 0@x @yI Dirichlet : constantN. ...

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Publié par	Syaj
Nombre de lectures	12
Langue	English

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LaboratoireMate´riauxOptiques,PhotoniqueetSyst`emes Unit´edeRecherchecommune`al’Universit´ePaulVerlaineMetzeta`Sup´elec

LMOPS

N. Fressengeas

Dowload this document from http://intercell.metz.supelec.fr

InterCell/Escapade Tutorial

sInterCell/

csE/lleCretnI

Running on InterCell A look at the ﬁles Run the program See the results

Derive a Cellular Algorithm Install the necessary software Discretize your problem Locate your equations on the Mesh

Fick Diﬀusion or Heat Transfer laws as an example Fick/Heat laws Boundary Conditions

Table of contents

LMOPS

eadapasengeressN.FﬀiDirevnoeDulalCalen/HeusiouatiatEqllCenIgoerntuRmhninnglArtiro

seesgnae.NrFrCtel/elsIn

Fick laws IJ=−Cgrad(ρ) I∂∂tρ+div(J) = 0 Iρ∂∂t=C4ρ Steady state law 4ρ= 0

Steady state Fick/Heat diﬀusion laws A fairly simple partial diﬀerential problem for didactic purpose

Fick/Heat laws

Diﬀusion laws Iq=−kgrad(T) IC∂∂Tt+div(q) = 0 IC∂∂Tt=k4T Steady state law 4T= 0

LMOPS

acsEedapea/HqutEﬀuDionsionInningellterCoglAralunuRmhtirernDioatllCeeaiv

Boundary conditions IConstant ﬂow pipes :J=J0=−Cgrad(ρ) Igrad(J)kBox borders:∂ρ∂x= 0 or∂ρ∂y= 0 IDirichlet :ρconstant

J q

Pure diﬀusion

J q

ρ T

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LMOPS

SAGE IGNU Free formal computing software IPoint your browser tohttp://www.sagemath.org IDownload, install and learn to useSAGE(Pythonsyntax) IMake sure thesagecommand is available1

GPL License

Install the necessary software

Formal computing software SAGEis what you need

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LMOPS

escabooz.sage IA piece of software written inSAGE IInstall usingmake install ISAGEsyntax to formally describe your diﬀerential problem IWrites aset of ﬁles(*.booz&*.cell) to describe the cellular algorithm

Install the necessary software

InterCellsoftware to derive the Cellular Algorithm Downloadescabooz.sgaefromInterCellweb site

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Running the script IStartSAGE ILoad the script IWrite the ﬁles

Diﬀerential problem speciﬁcation A shortSAGE/Pythonscript

LMOPS

Install the necessary software

acaped

TheSAGEscript ﬁle IDeﬁne a discrete diﬀerential problem IDeﬁne discrete boundary conditions Iset up constants and observers IFeed all this to oneEscapadeobject IThis object can write the ﬁles for you

rCell/EsInteCeerntnIgoinnnRumhtiroglAralulleveaCDeritionEquaeHtaoi/nﬀisuDllsengeasN.Fres

leletCr/HeatEquationDeriDuﬀisnothriunmRngniInonaevilleCraluoglArFse.NaesesgnIntel/EsrCel

Declare your unknowns and constants

LMOPS

Declare the unknownsyou are seeking # Declare the unknown as an unknown function rho=function("rho")

Declare the constants you will be using # Declare the constants you will use formally # It is usually a good idea to have dx and dy declared # We take here dx=dy=dr var("J0 C dr")

Discretize your problem

capa

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Finite Diﬀerences will be used Let us deﬁne some expressions that will be needed several times Center second order derivatives SecondDerivativeX =(rho(-1,0)+rho(+1,0)-2*rho(0,0))/dr^2 SecondDerivativeY =(rho(0,-1)+rho(0,+1)-2*rho(0,0))/dr^2

acap

LMOPS

Discretize your problem

Oﬀ-centered ﬁrst order derivatives UpDerivative=(rho(0,0)-rho(0,-1))/dr DownDerivative=(rho(0,+1)-rho(0,0))/dr LeftDerivative=(rho(+1,0)-rho(0,0))/dr RightDerivative=(rho(0,0)-rho(-1,0))/dr

eCrellognitnInhmitnnRurAlaorlgevCaleulitnoeDirHeatEquaiﬀusion/D

theequation"==0"unllaLlpcaai#nnI#Dcrisizetheetziteqedeitaunolyonasonmenteelere:eoihnsihthtlelyonvehaatqueeonmmoebnaceW#detti]YrivativeSecondDeitav+XevdnocireDanciSe=[uleqlaapgoAlthrillCearulnInoCretnuRmgninsion/HeaDiﬀuDnreviaeEtuqtaoircsidlareneGellN.FressengeasPOMLtnISeCreE/llapscead

Set up your discretized equation

4ρ= 0

Discretization is at your convenience IYou can use 5 (or more) points to improve derivative accuracy IYou can attempt stabilization using Lax method I !Call in all your discretization skills

Discretize your problem

Discretize your problem

LMOPS

And with several coupled equations ?

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General discretized equation4ρ= 0 equlaplacian=[SecondDerivativeX+SecondDerivativeY]

equlaplacianis alist IAdd equations as list elements ILeast Squares will treat them equally

Good normalization practice ISame order of magnitude required for each IIf not : only the highest will be solved. . . mess foreseen

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