sysnoise——IndirectBEM

IndirectBoundaryElementMethod,-Definitions-BoundaryIntegralEquation(BIE)-BoundaryConditions-SpecialConditions,Objective:Analysisoftheinteriorandexteriorfieldatthesametime.Definetwonewvariables:Jumpofpressure(doublelayerpotential)Jumpofpressurenormalderivatives(singlelayerpotential),Definitions:IndirectBEM,Foranyclosedsurface,thedirectBIEisgivenby:,BoundaryIntegralEquation(BIE)(1),ForathinstructuretheintegraloverSbecomes,S-,Y+,X,Y-,BoundaryIntegralEquation(BIE)(2),Letthethicknessofbodygotozero,thenweobtain,BoundaryIntegralEquation(BIE)(3),Thustheintegralequationbecomes,S,Y,X,BoundaryIntegralEquation(BIE)(4),Recall:,BoundaryConditions(1),Thepressureandvelocityboundaryconditionsdiscussedbefore,canbeexpressedasfollows:xonS1:xonS2:,BoundaryConditions(2),TakingintoaccounttheprescribedandvaluesonS1andS2,respectively,onecanderivethefollowingequationsforXonS1:XonS2:,SpecialConditions(1),Treatmentoffreeedges:Becauseofp+=p-,thefollowingboundaryconditionhastobeprescribed:=0,p+,p-,Directconnectionbetweenpositiveandnegativeside,TreatmentofJunctionsNote:Thenodeshavetobeduplicated.(Doneautomaticallybythesoftware!),p1+,p1-,p2-,p2+,p3+,p3-,SpecialConditions(2),Ifcoincideswithaninnerresonancefrequency,PinViiskeptsmallbymodelinganadditionalelementwhereimpedanceboundaryconditionsareprescribed.,Va,Vi,n,S,ExteriorProblemwithclosedboundarysurface.Non-uniquesolutionforresonancefrequenciesofinnerproblemmaycauseproblems.Help:,IrregularFrequencies(indirectBEM),Applicabletoarbitrarysurfaces(open,closed,withjunctions,etc.).Inthecaseofclosedsurfaces,theinteriorandtheexteriorproblemaresolvedsimultaneously.Inafirststep,jumpofpressureandjumpofvelocityarecomputedonthesurface.Inasecondstep,pressure,velocity,intensityetc.canbecomputedatanypointnotonthesurface.Formulationleadstofullypopulatedbutsymmetricsystemmatrices=fasterthandirectBEMforbiggerproblems!,WhatYouShouldKeepinMind.(SummaryindirectBEM),AStandardBECalculation(Summary),Discretizationofthestructuralsystem=FE-model.Definitionofthestructuralboundaryconditions(+loads).Calculationofthestructuraldeformations.ModificationoftheFE-discretization=BE-model.Structuraldisplacementsasfluidboundaryconditions.Calculationoftheprimarysurfaceresults.Postprocessingatthefieldpointsinordertoobtainacousticpressure,velocities,intensities,coupledmodes,radiatedpower,etc.,Rayleigh-Method(1),Simplifications:Thesurfaceofthevibratingsystemisreplacedbyanumberofradiatingpistons(elements).Eachpistonradiatesindependentlyofitsenvironment(andislocatedinaninfiniterigidplane).Reflectionanddiffractionarenottakenintoaccount.ResultingEquation:,Rayleigh-Method(2),Pistoninaninfiniteplane:,Step-by-Step:ModelingofthesystemwithFiniteElements.Determinationofthevelocitiesofeachelement(withFEM).ConsiderationofeachelementasaradiatingpistonanddeterminationofthesoundpressureatagivenpointP.Summationofthesoundpressurevaluesofall(visible)elementstocalculatethetotalpressureatP.,Rayleigh-Method(3),Rayleigh-Method(4),Soundpressuredistributioninthevicinityofacatalyst(BEM),Rayleigh-Method(5),System,Comparison,Th