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2.Numericalproblemanalysis
whichthevaluesofthepotentialuareknown(intwonodesadjacenttothecurrent
electrodes).
Themethodoftheconjugatevariableconsistsofsolvingtheadjointequationwith
thecoupledvectorandusingthisvectortodeterminethevector
(2.52)
Theconstructionofthecoupledequationisnottime-consumingintermsofthe
numberofnumericaloperations.Theformoftheleftsideofthisequationisthe
sameastheequationofstate.Hence,inthedeterminationoftheconjugatevector,
areducedstatematrixYjdeterminedinsolvingaforwardproblemisused.
Todeterminetherightsideoftheadjointequation,thetargetfunctionforthej-pair
ofelectrodesinrelationtothevectorofnodalvoltageuj(statevector)shouldbe
differentiated:
(2.53)
Byanalogywiththestateequationitmaybeconcludedthattheadjointequation
resultsfromtheapproximationbythefiniteelementmethodofthePoissonequation
foranadjointvariablepoftheform
(2.54)
Inequation(2.54),thesourcefunctionµisdifferentfromzeroonlyinspecific
sub-areasofthetestedobject,inwhichthephysicalmagnitudesoftheelectromag-
neticfieldareset(theelectromagneticfieldisaimedatthesemagnitudesinthe
optimizationprocess).Intheproblemunderconsideration,thesetmagnitudesof
theelectromagneticfieldarethepotentialsofpointsadjacenttotheelectrodes(more
precisely,inter-electrodevoltagesarerequired).Therefore,thesourceµwilloccur
inplacesadjacenttothevoltageelectrodes.
Theobjectivefunctionisameasureofthedistanceofinter-electrodevoltages
determinedonthebasisofthevectorofdecisionvariables(electricconductivity
distributionvector)determinedatagivenstageofthecalculationandthemeasured
inter-electrodevoltages.Thecalculatedinter-electrodevoltagesareequaltothe
measuredvoltages,incasethedecisionvariablesreachtherealvalue.IntheFEM
approximation,thesourcefunctionµisrepresentedbyavector
.Itmaybecon-
cludedthatthevaluesofthefunctionofthesourceoftheadjointequationexpress
themismatchofthepotentialvalues(andthustheinter-electrodevoltages)seton
thebasisofmeasurementstotheappropriatepotentialsobtainedinagiveniteration
oftheoptimizationprocess.Inthefinalstageofthecalculations,thevaluesofthe
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