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2.Numericalproblemanalysis
ofstatetoindividualdecisionvariables.Forthispurpose,theequationofthestate
equationwithrespecttomaterialcoefficientsshouldbemadeonbothsidesγ
(2.45)
where:ũj–vectoruj,whoseelementsaretreatedasconstantexpressionsinthe
partialdifferentiationprocess.
ũj–isavectorofstatefromthepreviousiterationofcalculations.
BecausethematrixYjisnon-singular(matrixYj[Yj]formedbythereductionofthe
matrixHaftertheintroductionofDirichletboundaryconditions(inq=2nodes)),
sothefollowingtransformationsaretrue:
Aftersubstituting(2.42)into(2.47):
Letacertainvectorpjbedefinedas:
(2.46)
(2.47)
(2.48)
(2.49)
DuetothefactthatthereducedstatematrixYj,obtainedasaresultofthefinite
elementmethodforapproximatingtheflowfieldinthestudiedarea,isamatrix
symmetricaltothemaindiagonal,theexpression(2.49)mayberepresentedas:
AftermultiplyingbothsidesoftheequationbythematrixYj[Yj]:
(2.50)
(2.51)
Itisanequationcoupledtoastateequation,andthevectorpjisavectorcoupledto
astatevectoruj.Fortheconjugateequation,zeroboundaryconditionsoftheDir-
ichlettypefortheadjointvariablepwereadoptedforthoseFEMmeshnodesfor
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