ch 3_shape functions and elemental calculations

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1、3ShapeFunctionsandElementalCalculationsLikeitscontinuouscounterpart,thediscontinuousfiniteelementmethodemploysshapefunctionsforlocalapproximations.Theuseof1-DlinearshapefunctionswasdemonstratedinthelastchapterforthediscontinuousGalerkinsolutionofboundaryandinitialvalueproblems.There,itwasalsosh

2、ownthatadiscontinuousfiniteelementformulationhingescriticallyonlocalinterpolationfunctionsandinvolvescomputationsattheelementlevel.Thecalculationsforhigherdimensionsrequireinterpolationfunctionsorshapefunctionsdefinedinmulti-dimensions.Formanyengineeringapplications,elementsbasedonlinearandquad

3、raticinterpolationsmaybesufficient.Becausetheelementsforthediscontinuousfiniteelementformulationsareembeddedinafiniteelementbrokenspace,andbecausetheformulationsarelocalized,higherorderpolynomialinterpolationfunctionscanbeeasilyincorporatedinthesolutionprocedure.Thisflexibilityinlocalapproximat

4、ionhasmotivatedresearcherstodevelopandapplymoreaccurate,localspectralbasisfunctionsforhigherorderanalysis.Inthischapter,theshapefunctions,numericalintegration,andelementalcalculationsusedfordiscontinuousfiniteelementcalculationsarediscussed.Thechapterstartswiththeprocedureandbasiccriteriaforthe

5、constructionofthesimple1-Dfiniteelements.Theideaisthenextendedtoestablishageneralframeworkfordevelopingstandardfiniteelementsinmulti-dimensions.Abriefdiscussionisgivenontheconstructionofspectralelements,whichareknownfortheirhigherorderaccuracyofapproximations.Hierarchicalelementsarewidelyusedin

6、p-typeadaptiveanalysis;andtheconstructionofavarietyofhierarchicalshapefunctionsandtheselectionofhierarchicalpolynomialsforcertainapplicationsarealsodiscussed.Thetechniquesforconstructingspecialelementsandtransitionelementsthroughcoordinatetransformationarealsopresented.Interpolationerrorestimat

7、esforuniformandnon-uniformmeshesarepresentedbyintroducingvariousnormsforerrormeasures.Theseerrorestimatesareofcrucialimportanceindevelopingadaptivealgorithmsfordiscontinuousfiniteelementcomputations.Numericalintegrationisdiscussed