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1、Chapter8Finiteelementmethod8.1Basicsofthefiniteelementmethod&1・1IntroductionVirtuallyeveryphenomenoninnature,whetherbiological,geological,ormechanical,canbedescribedwiththeaidofthelawsofphysics,intermsofalgebraic,differential,orintegralequationsrelatingvariousquantitiesofinterest.Determini
2、ngthestressdistributioninapressurevesselwithoddlyshapedholesandnumerousstiffenersandsubjectedtomechanical,thermal,and/oraerodynamicloads,findingtheconcentrationofpollutantsinseawaterorintheatmosphere,andsimulatingweatherinanattempttounderstandandpredictthemechanicsofformationoftornadoesand
3、thunderstormsareafewexamplesofmanyimportantpracticalproblems.Mostengineersandscientiststudyingphysicalphenomenaareinvolvedwithtwomajortasks:1.Mathematicalformulationofthephysicalprocess2.NumericalanalysisofthemathematicalmodelWhilethederivationofthegoverningequationsformostproblemsisnotund
4、ulydifficult,theirsolutionbyexactmethodsofanalysisisaformidabletask.Insuchcases,approximatemethodsofanalysisprovidealternativemeansoffindingsolutions.Amongthese,thefinitedifferencemethodandthefiniteelementmethodaremostfrequentlyusedintheliterature.Thefiniteelementmethodisderivedbythevariat
5、ionalmethodssuchastheRayleigh-RitzandGalerkinmethods.Inthefinitedifferenceapproximationofadifferentialequation,thederivativesinthelatterarereplacedbydifferencequotients(orthefunctionisexpandedinaTaylorseries)thatinvolvethevaluesofthesolutionatdiscretemeshpointsofthedomain.Theresultingalgeb
6、raicequationsaresolved,afterimposingtheboundaryconditions,forthevaluesofthesolutionatthemeshpoints.Inthesolutionofadifferentialequationbyavariationalmethod,theequationisputintoanequivalentweighted-integralformandthentheapproximatesolutionoverthedomainisassumedtobealinearcombination⑺心①)ofap
7、propriatelychosenapproximationfunctions①jandundeterminedcoefficients,Cj.Thecoefficientscfaredeterminedsuchthattheintegralstatementequivalenttotheoriginaldifferentialequationissatisfied.Variousvariationalmethods,e.g・,theRayleigh-Ritz,Galerkin,andleast-squaresme
