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2.Numericalproblemanalysis
Theproblemofanalysingthefieldunderexaminationmeanssolvingtheequation
describingthisfieldinagivenenvironment.Theforwardproblemconsistsofsolv-
ingtheLaplaceequationforthedescribedareaundergivenboundaryconditions,
whichcorrespondstofindingtheminimumofthefollowingfunctional:
(2.9)
Solvingtheproblembyusingthefiniteelementconsistsofcalculatingthevaluesof
nodepotentialsthatcorrespondtothestationarystateofthefunctionalpointI(fl).
Thispointisachievedbythefunctionfornodalvaluesu,forwhichthefunctional
varianceis:
where:nnumberoffiniteelementmeshnodes.
Theequation(2.10)isonlytrueif
(2.10)
(2.11)
Afterdiscretizingtheareausingthefiniteelementmethod,thefunctionI(fl)may
bepresentedasthesumofindividualfunctionalitiesofeachfiniteelementdefined
asfollows:
(2.12)
Therefore,therealdependenceis:
(2.13)
ThevarianceIe)isdeterminedonlyonthebasisofthenodalvaluesassociatedwith
thee-element.Theequation(2.13)showsthat:
Thefullsetofequationsofthefiniteelementmethodgivestheappropriateassembly
offunctionalI(u)derivativesforallelements
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(2.14)
