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Title

Real time boundary element node location optimization
Authors

Samuel Smith
Robert Baxter
Joshua Menges
T. V. Hromadka II
Steven B. Horton
Journal / Anthology

Engineering Analysis with Boundary Elements
Year: 2012
Volume: 36
Issue: 6
Page range: 1049-1054
Description

BoundaryElementMethod(BEM)computermodelstypicallyinvolveuseofnodalpointsthatarethe locationsofsingularpotentialfunctionssuchasthelogarithmorreciprocaloftheEuclideandistance function.Thesesingularfunctionsaretypicallyassociatedwiththenodesthemselvesasfaras identification.TheComplexVariableBoundaryElementMethod(CVBEM)isanotherapplicationof similartypesofsingularpotentialfunctionsandincludesotherfunctionsthatarenotsingularbutare fundamentalsolutionsofthegoverningpartialdifferentialequation(PDE).Thesevarioussingular potentialfunctionsformabasiswhosespanoflinearcombinations(eitherrealorcomplexspace,as appropriate)isavectorspace.Aspartoftheapproximationapproach,onedeterminesthatelementin the vectorspacethatisclosest(usuallyinaleastsquaresresidualmeasure)totheexactsolutionofthe PDE andrelatedboundaryconditions.RecentresearchonthetypesofbasisfunctionsusedinaBEMor CVBEMapproximationhasshownthatconsiderableimprovementincomputationalaccuracyand efficiencycanbeachievedbyoptimizingthelocationofthesingularbasisfunctionswithrespectto possiblelocationsontheproblemboundaryandalsolocationsexterioroftheproblemboundary(in general,exterioroftheproblemdomain).Todevelopsuchoptimumlocationsforthemodelingnodes (and associatedsingularbasisfunctions),theapproachpresentedinthispaperistodevelopaRealTime BoundaryElementNodeLocationmodulethatenablestheprogramusertoclickanddragnodes(oneat a time)throughouttheexterioroftheproblemdomain(thatis,nodesareallowedtobepositionedon or arbitrarilyclosetotheproblemboundary,andalsotobepositionedexterioroftheproblemdomain union boundary).TheprovidedmoduleinterfaceswiththeCVBEMprogram,builtwithincomputer programMathematica,sothatvarioustypesofinformationflowstothedisplaymoduleasthenodeis moved,inrealtime.Theinformationdisplayedincludesagraphicoftheproblemboundaryand domain,theexteriorofthedomainunionboundary,evaluationpointsusedtorepresentproblem boundaryconditions,nodallocations,modelingerrorin L2 and also LN norms,andaplotofproblem boundaryconditionsversusmodelingestimatesontheproblemboundarytoenableavisualizationof closenessoffitofthemodeltotheproblemboundaryconditions.Asthetargetnodeismovedonthe screen,thesevariousinformationformschangeandaredisplayedtotheprogramuser,enablingthe usertoquicklynavigatethetargetnodetowardsapreferredlocation.Onceanodeisestablishedat someoptimizedlocation,anothernodecanthenbeclickeduponanddraggedtonewlocations,while reducingmodelingerrorintheprocess.
Subject

*Unclassified