# Finite element

## Introductiontocalculation

English:FiniteElement

Thefiniteelementmethodisamoderncalculationmethodthathasdevelopedrapidlywiththedevelopmentofelectroniccomputers.Itisaneffectivenumericalanalysismethodfirstappliedinthefieldofcontinuummechanics-thestaticanddynamiccharacteristicsanalysisofaircraftstructuresinthe1950s,andthenitwasquicklyandwidelyusedtosolvecontinuityproblemssuchasheatconduction,electromagneticfields,andfluidmechanics.

## Methods

Thethinkingandmethodsoffiniteelementmethodanalysisandcalculationcanbesummarizedasfollows:

### Selectthedisplacementmode

Inthefiniteelementmethod,whenthenodaldisplacementisselectedasthebasicunknownquantity,itiscalledthedisplacementmethod;whenthenodalforceisselectedasthebasicunknownquantity,itiscalledtheforcemethod;Whenapartofthenodalforceandapartofthenodaldisplacementareregardedasthebasicunknownquantity,itiscalledthehybridmethod.Thedisplacementmethodiseasytorealizethecalculationautomation,sothedisplacementmethodisthemostwidelyusedinthefiniteelementmethod.

Whenthedisplacementmethodisused,aftertheobjectorstructureisdiscretized,thetotalphysicalquantitiesoftheelement,suchasdisplacement,strainandstress,canberepresentedbythenodedisplacement.Atthistime,someapproximatefunctionsthatcanapproximatetheoriginalfunctioncanbeusedtodescribethedisplacementdistributionintheelement.Usually,inthefiniteelementmethod,weexpressthedisplacementasasimplefunctionofcoordinatevariables.Thisfunctioniscalledthedisplacementmodeordisplacementfunction.

### Analyzethemechanicalproperties

Accordingtotheelement’smaterialproperties,shape,size,numberofnodes,positionandmeaning,etc.,findouttherelationshipbetweentheelementnodeforceandthenodedisplacement.ThisisAkeystepinunitanalysis.Atthistime,itisnecessarytoapplythegeometricequationsandphysicalequationsinelasticmechanicstoestablishtheequationsofforceanddisplacement,soastoderivetheelementstiffnessmatrix,whichisoneofthebasicstepsofthefiniteelementmethod.

### Equivalentnodalforce

Aftertheobjectisdiscretized,itisassumedthattheforceistransferredfromoneelementtoanotherthroughthenodes.However,fortheactualcontinuum,theforceistransferredfromthecommonedgeoftheunittoanotherunit.Therefore,thesurfaceforce,volumeforceandconcentratedforceactingontheelementboundaryneedtobeequivalentlymovedtothenode,thatis,theequivalentnodalforceisusedtoreplacealltheforcesactingontheelement.

## Elementgroupset

### Definition

Usingtheequilibriumconditionsandboundaryconditionsofstructuralmechanicstoreconnecteachelementaccordingtotheoriginalstructuretoformanoverallfiniteelementequation.

## Solve

Solvethefiniteelementequationtogetthedisplacement.Here,theappropriatecalculationmethodcanbeselectedaccordingtothespecificcharacteristicsoftheequations.

Throughtheaboveanalysis,itcanbeseenthatthebasicideaof​​thefiniteelementmethodis"onedivisionandoneintegration",thedivisionisforunitanalysis,andthecombinationisforthecomprehensiveanalysisoftheoverallstructure.

### Developmentoverview

In1943,couranttookapiecewisecontinuousfunctiondefinedonatriangulardomaininhispaper,andusedtheprincipleofminimumpotentialenergytostudySt.Venant’storsionproblem.

In1960,cloughusedthename"FiniteElementMethod"inhispaperonplaneelasticity.

In1970,withthedevelopmentofcomputersandsoftware,finiteelementsdeveloped.

### Contentsinvolved

Thetheorybehindthefiniteelement,theprincipleofunitdivision,theselectionandcoordinationofshapefunctions.

### Thefiniteelementmethodinvolves

numericalcalculationmethodsandtheirerrors,convergenceandstability.

### Scopeofapplication

Solidmechanics,fluidmechanics,heatconduction,electromagnetics,acoustics,biomechanics

## Development

Overview:

Withtherapiddevelopmentofcomputertechnology,intheengineeringfield,finiteelementanalysis(FEA)Itisincreasinglyusedforsimulationtosolverealengineeringproblems.Overtheyears,moreandmoreengineers,appliedmathematiciansandphysicistshaveprovedthatthismethodofsolvingpartialdifferentialequations(PDE)cansolvemanyphysicalphenomena.Thesepartialdifferentialequationscanbeusedtodescribetheflow,electromagneticfieldandstructure.Mechanicsandsoon.Thefiniteelementmethodisusedtoconvertthesewell-knownmathematicalequationsintoapproximatedigitalimages.

Beforethe1990s,duetothelackofcomputerresources,multi-physicssimulationonlystayedatthetheoreticalstage,andfiniteelementmodelingwasalsolimitedtothesimulationofasinglephysicalfield,themostcommonbeingSimulationofmechanics,heattransfer,fluidsandelectromagneticfields.Itseemsthatthefateoffiniteelementsimulationisthesimulationofasinglephysicalfield.

Thousandsofwordsareendless.Thefollowingcanonlyshowsomepotentialapplicationsofmultiphysicsfiniteelementanalysisinthefuturethroughafewexamples.

Piezoacoustictransducercanconvertelectriccurrentintoacousticpressurefield,orviceversa,convertsoundfieldintoelectriccurrentfield.Thiskindofdeviceisgenerallyusedinsoundsourcedevicesinairorliquid,suchasphasedarraymicrophones,ultrasoundbioimaging,sonarsensors,acousticbiotherapy,etc.Itcanalsobeusedinsomemechanicaldevicessuchasinkjetsandpiezoelectrics.Motoretc.

Piezoelectricloudspeakersinvolvethreedifferentphysicalfields:structurefield,electricfieldandsoundfieldinfluid.Onlysoftwarewithmultiphysicsanalysiscapabilitiescansolvethismodel.

Thekeyissueishowtodesignthenozzleshapeoftheductsothatitcanoptimizethefluidvelocityandreducestructuraldeformation.Kalafut'sresearchteamusesmultiphysicsmodelingmethodstocoupletheforcesgeneratedbylaminarflowtostress-strainanalysis,andthenconductfluid-structurecouplinganalysisonvariousnozzleshapesandlayouts."Oneofourtraineesestablisheddifferentnozzlelayoutsfordifferentfluidareasandanalyzedthem,"saidDr.Kalafut."Weusetheseanalysisresultstoevaluatethefeasibilityofthesenewideas,therebyreducingthenumberofphysicalmodelmanufacturing."

Thefirstresearchresultisamathematicalmodeloffrictionstirwelding,whichallowsAirbusengineersto"seethrough"theweldtochecktemperaturedistributionandmicrostructurechanges.Dr.ColegroveandhisresearchteamalsowroteasimulationtoolwithagraphicalinterfacesothatAirbusengineerscandirectlyextractthethermalpropertiesofthematerialandtheultimatestrengthoftheweld.

Inthisfrictionstirweldingsimulationprocess,three-dimensionalheattransferanalysisandtwo-dimensionalaxisymmetriceddycurrentsimulationarecoupled.Theheattransferanalysiscalculatestheheatdistributionofthestructureaftertheheatfluxisappliedtothesurfaceofthetool.Thedisplacementofthetool,thethermalboundaryconditions,andthethermalpropertiesoftheweldingmaterialcanbeextracted.Next,thethree-dimensionalheatdistributionatthetoolsurfaceismappedtothetwo-dimensionalmodel.Thecoupledmodelcancalculatetheinteractionbetweenheatandfluidduringprocessing.

Couplingtheelectromagnetic,electricalresistance,andheattransferbehaviorofthesubstraterequiresatruemultiphysicsanalysistool.Atypicalapplicationisintheprocessofsemiconductorprocessingandannealing.Thereisahotwallfurnaceusinginductionheating,whichisusedtogrowsemiconductorwafers,whichisakeytechnologyintheelectronicsindustry.

Incircuitdesign,animportantaspectthataffectsmaterialselectionisthedurabilityandservicelifeofthematerial.Thetrendofminiaturizationofelectricalapplianceshasledtotherapiddevelopmentofelectroniccomponentsthatcanbemountedoncircuitboards.Asweallknow,theresistorsandothercomponentsmountedonthecircuitboardwillgeneratealotofheat,whichmaycausecracksinthesolderfeetofthecomponents,andfinallycausetheentirecircuitboardtobescrapped.

Multi-physicsanalysistoolscananalyzetheheattransferoftheentirecircuitboard,structuralstresschangesanddeformationduetotemperaturerise.Thiscanbeusedtoimprovetherationalityofthecircuitboarddesignandtherationalityofmaterialselection.

## Domesticdevelopmentmileage

Inthemid-1970s,DalianUniversityofTechnologydevelopedDDJ,JIGFEXfiniteelementanalysissoftwareandDDDUstructuraloptimizationsoftware;ProfessorLiMingruiofBeijingAgriculturalUniversitydevelopedFEMsoftware;mid-1980s,PekingUniversityProfessorYuanMingwuThroughthetransplantationandmajortransformationofforeignSAPsoftware,SAP-84wasdeveloped;duetotheneedsoftheaviationindustry,theMinistryofAviationIndustrybegantoorganizethedevelopmentofHAJIF(I,II,III),YIDOYU,COMPASS,Andwonmanynationalawardsandsoon.Comparedwithsimilarforeignproducts,thesedomesticCAEsoftwarehasmanycharacteristicsincorealgorithmsandseveralfunctions,reflectingtheachievementsofChinesescholarsincomputationalmechanicsresearch,andfullyconsideringtheactualconditionsofcomputerhardwareinmycountry.Ithasplayedanimportantroleinconstructionandengineeringstructuredesignandhasawiderangeofapplications.