Computer Methods in Applied Mechanics and Engineering(4)

一些ME专业提升的论文。

T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–21462133

time.ThebodyBisassumedtohaveaprescribedsurfacemotion.Followingthematrix–vectornotationof[4],wecanwriteEqs.(1)–(3)semi-discretelyasM

dq

dt

þGpÀHf¼NðqÞþLqþbc1;ð4ÞDq¼0þbc2;

ð5ÞEq¼unþ1

B;

ð6Þ

whereq,p,andfarethediscretevelocity uxvector,pres-sure,andboundaryforce.Thediscretevelocity,u,canberelatedtoqbymultiplyingthecellfaceareanormaltothevector,i.e.,q=(qu,i,qv,i)=(uiDyi,viDxi).Theabove rst,second,andthirdequationsrepresentthediscretizedmomentumequation,continuityequation,andno-slipcon-ditionalongoB.Discretizednon-linearconvectivetermÀuÆ$uisdenotedbyNðqÞandoperatorsMandLarethe(diagonal)massmatrixanddiscreteLaplacian,respectively.

Wenotethatallofthematricesintheabove(andallthatfollow)aresparseandaremoste cientlycodedaspoint-operators-subroutinesreturnthematrix–vectormul-tiplysuchthatthematricesareneverexplicitlyformed.Forconvenience,point-operatorrepresentations(forthecaseofauniformgrid)aregiveninAppendixA.

OperatorsGandDarethediscretegradientanddiver-genceoperatorsTandcanbeformulatedsuchthatG=ÀD[27,4].TheremainingoperatorsofEandHaretheinterpolationandregularizationoperatorsresultingfromtheregularizationoftheDiracdeltafunctionsinEqs.(1)and(3).Theno-slipconstraintisenforcedbyequatingtheboundaryvelocity,uB,tothevelocityvaluealongoBinterpolatedbyEfromtheneighboringcells.Ontheotherhand,theregularizationoperatorsmearsthee ectofthesingularboundaryforcealongoBtotheCartesiangrid.Topreservesymmetryinthe nalalgo-

rithm,weconstructtheseoperatorstosatisfyE=ÀHT;see[36]forfurtherdiscussion.WementionthatmatricesG,D,E,andHarenotsquare.Consequently,Eqs.(4)–(6)2canbewritten30asasystemofalgebraicequations:nþ110n16AGÀHr0bc11

4D0

07Bq

CBCBC

E005@pfA¼@0unAþ@bc2A:ð7ÞB

þ10SubmatrixA¼1

mentofthevelocityMÀaLLresultsfromtheimplicittreat-term.Hereweapplytheimplicittrap-ezoidruleontheviscoustermwithaL¼Thetermisdiscretizedwiththesecond-order2

convectiveAdam–Bashforth(AB2)r¼Âmethod.ÃInthiscasetheright-handsidevectorn11qn

DtMþ2Lþ3NðqnÞÀ1NðqnÀ1Þ.TheAB2meth-odisnotself-starting2andwereplace2

itwithbackwardEu-lerforthe rsttimestep.Theinhomogeneoustermsbc1andbc2dependontheparticularboundaryconditionsandarediscussedin[36].Boundaryconditionsaredis-cussedingreaterdetailinSections3and4.

WiththeuseofstaggeredCartesiangrid,weareabletogloballyconservemass,momentum,kineticenergy,andcirculation[17,23,26].Detaileddiscussiononspatialdis-cretizationsofvariousformsofthenon-linearconvectiveterm(rotational,divergence,skew-symmetric,andadvec-tiveforms)areprovidedin[23,26].Theexplicitright-handsidetermingeneralalsoincludesinhomogeneousterms,bc1andbc2,generatedbytheboundaryconditionsfromthediscreteLaplacianLandthedivergenceDoperators,respectively.

Byapplyingthepropertiesofthesub-matrices,Eq.(7)can2berestated3as

0nþ1106AGET

qrn

þbc14G

T0075B

@pCA¼B1@Àbc2CAð8ÞE00~funB

þ1;where~f

istheboundaryforcewithanincorporatedscalingfactor.ThisformoftheequationisknownKahn–Tucker(KKT)systemwhereðp;~astheKarush–

f

ÞTappearasasetofLagrangemultipliertosatisfyasetofkinematiccon-straints.Inthediscretizedsetofequations,theconstraintsarepurelynumericalanditisnolongernecessarytodistin-guishthepressureandboundaryforce. neacombinedvariablek¼ðp;~Insteadwecande-f

ÞTfortheLagrangemultipliersandgroupthesubmatricesasQ=[G,ET].Notethatbyremovingtheboundaryforceandno-slipconditionalongoB,thetraditionaldiscretizationoftheincompress-ibleNavier–Stokesequationscanberetrieved.

SincewenowhaveformulatedtheimmersedboundaryformulationoftheNavier–Stokesequationsinanalgebra-icallyidenticalmannertothetraditionaldiscretizationoftheincompressibleNavier–Stokesequations,standardsolutiontechniquescanbeutilized.Hereweapplythepro-jection(fractional-step)algorithmtoEq.(8),whichcanbeexpressedasanapproximateLUdecompositionoftheleft-handsidematrix[27],toproducetheimmersedboundaryprojectionmethod[36]

: