桥梁结构计算分析chapter6.pdf

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1、Chapter6SolutionofLinearEquations1.IntroductionMatrixformulationofstructuralproblemsorfiniteelementidealizationofacontinuumstructureusuallyproducesalargesystemofsimultaneousequations.Efficientwaysofsolvingtheseequationsarethemajorconcernofnumericalanalystsinthepast.Basically,therearetwodifferents

2、olutionapproaches:directanditerativemethods.Thedirectmethodsobtainthesolutionsbysuccessivelydecouplingthesimultaneousequationssothattheunknownscanbereadilycalculated.MostofthedirectmethodsaresomesortofvariationsoftheGaussianeliminationmethod.Theseinclude:•Gaussianelimination•CholeskysquarerootT•C

3、holeskynon-squareroot(LDLtransformation)•Gauss-Jordan•Directmatrixinversion•IterativemethodsIterativemethodsgiveapproximatesolutionswhichcanbeimprovedbysuccessiveiterations.Thenumberofiterationsneededforagivenproblemisgenerallynotknownanditvarieswiththeaccuracyrequired,thenumericalconditionofthep

4、roblem,andconvergencecriterionchosen.Thesemethodsusuallydonotrequirelargeamountsofcentralmemoryasthedirectmethods.However,itssolutionconvergenceandaccuracyaredifficulttocontrol.Therefore,thedirectmethodsaremostlypreferred.Insolvingthelinearsystemofsimultaneousequationsarisinginstructuralanalysis,

5、thefollowingspecialcharacteristicsareutilizedincoding:•Symmetryconditionisfullyutilizedincalculations.•Sparsityofthestiffnessmatrixispreservedinstorageandavoidingmultiplicationsby0’sand1’s.1•Inthecaseofmultipleloadvectors,matrixdecompositionisdoneonlyonce.2.GaussEliminationWeconsiderasystemofline

6、arequationsoftheformAx=b(1)whereAisannbynnon-singularmatrixwithconstantcoefficients,xandbarenbyeqeqeq1vectors,andxisunknown.ThebasicideaofGausseliminationistocombinetherowsofEq.(1)insuchawaythatthecoefficientmatrixistransformedintouppertriangularform.Thisiscalledaforwardreductionprocess.Then,ther

7、esultingequationsaresufficientlyuncoupledsothatalltheunknownsofxcanbedeterminedbyaback-substitutionprocess.Toillustratetheaboveprocedures,weconsidera4×4matrixequationoftheform⎡A11A12A13A14⎤⎡x1⎤⎡b1⎤⎢⎥⎢⎥⎢⎥AAAAxb⎢21222324