Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
2.
Numericalproblemanalysis
2.1.Formulationoftheproblem
Fromamathematicalpointofview,tomographyalsobelongstothecategory
ofproblemsoftheinverseelectromagneticfield.Theinverseelectromagneticfield
istheprocessofidentification,optimizationorsynthesisinwhichtheparameters
describingthefieldaredeterminedbasedonthepossessionofsomeinformation
specifictothisfield.Inverseproblemsaredifficulttoanalyse.Asarule,theydonot
haveunambiguoussolutionsandarebadlyconditioned[96].Thereasonforthisis
oftentoolittleortoomuchinformation,whicharesometimescontradictoryorline-
arlydependent.Numericalanalysisoftheproblemwillbepresentedonthebasisof
electricalimpedancetomographyandfiniteelementmethods(FEM).Inverseprob-
lemsarecharacterizedbythefactthattheyoftenhavenounambiguoussolutions
andareusuallypoorlyconditioned.Thisisduetothefactthatonegenerallyhastoo
littleortoomanyinputinformationthataresometimescontradictory.Inthecaseof
datashortages,theauthordescribessub-specifiedtasks,andinthecaseofexcess,
over-specifictasks.Theinverseproblemisformulatedwhencertaininformation
abouttheareaunderconsiderationislacking,forexample;theboundaryconditions
arenotknown.Inordertoobtainanunambiguoussolution,itneedstohaveaddi-
tionalinformation(e.g.potentialdistribution).Thetypeofmissinginformationis
oftenusedasacriterionforthedivisionofinverseproblems.Thisdivisiontakes
intoaccountthephysicalparametersofthetestedobject.
Theinverseproblemisformulatedfor:
•
boundaryconditions,
•
thesourcefunction,
•
geometryofthearea,
•
materialfactors.
2.2.Forwardproblem
Inelectricalimpedancetomography(EIT),thesolutionoftheforwardprob-
lemconsistsofdeterminingthepotentialdistributionwithintheobject,withgiv-
enboundaryconditionsanddataconcerningtheobjectinquestion.Anumerical
analysisoftheproblematthispointispresentedusingtheexampleofelectrical
impedancetomographyandthefiniteelementmethod[304].Itiscurrentlythemost
commonlyusednumericalmethodforanapproximatesolutionofthefieldproblems
18
