工程有限元分析英文课件：Lagrange-polynomials-(拉格朗日多项式)

13.2.2Lagrangepolynomials（拉格朗日多项式）

13.2.2Lagrangepolynomials（拉2LagrangeMultiplierFunctionindimensionlessform下标i表示节点i，上标n表示Lagrange插值多项式的次数2LagrangeMultiplierFunction3Table3.2Lagrangepolynomialsindimensionlessform

originn

=1-113Table3.2Lagrangepolynomial4Table3.2Lagrangepolynomialsindimensionlessform

originn

=24Table3.2Lagrangepolynomial5Table3.2Lagrangepolynomialsindimensionlessform

originn

=35Table3.2Lagrangepolynomial6InterpolationFunctionsfor2DProblems6InterpolationFunctionsfor27n=m

=1n=m

=4n=m

=3n=m

=2Fig.3.8FamilyofLagrangeelements

Thenodesexitonaregularmeshof(n+1)columns,(m+1)lines.7n=m=1n=m=4n=m=3n=m83.3FamilyofIsoparametricElements（等参元族）

83.3FamilyofIsoparametricE9910LinearLagrangePolynomials

10LinearLagrangePolynomials11i11i12Fig.3.9VariationofshapefunctionoveraquadrilateralelementwithlinearinterpolationfunctionN112Fig.3.9Variationofshapef13n

=8ThisisnotaLagrangeelement13n=8ThisisnotaLagrange14ShapeFunctionsforMid–SideNodes（边中节点）12ProductofLagrangepolynomialOrusetheInspectiveConstructionMethod（划线法）14ShapeFunctionsforMid–Si15(2,6)(4,8)ShapeFunctionsforMid–SideNodes15(2,6)(4,8)ShapeFunctions16ShapeFunctionsforCornerNodes

（角节点）NcN816ShapeFunctionsforCornerN17ShapeFunctionsforCornerNodesNc817ShapeFunctionsforCornerN18Constructionofquadraticshapefunctionsforcornernode18118Constructionofquadraticsh19Constructionofquadraticshapefunctionsforcornernode1ThisisnotaLagrangeelement19Constructionofquadraticsh20ShapeFunctionsforCornerNodes20ShapeFunctionsforCornerN21SystematicGenerationofaQuadraticShapeFunction

byStartingwithaBilinearLagrangianFamily

82121SystematicGenerationofaQ22Fig.3.12Systematicgenerationof‘serendipity’shapefunctionsSystematicGenerationofaQuadraticShapeFunction

byStartingwithaBilinearLagrangianFamily

22Fig.3.12Systematicgenerati23-BilinearLagrangianFamily23-BilinearLagrangianFamily24SystematicGenerationof‘Serendipity’ShapeFunctions

CapMethod24SystematicGenerationof‘Se25SystematicGenerationof‘Serendipity’ShapeFunctions

25SystematicGenerationof‘Se26Fig.3.13Shapefunctionsforatransition‘serendipity’element,cubic/linear

Cubic/LinearTransition（三次元/线性元的过渡）

26Fig.3.13Shapefunctionsfor27Fig.3.14Five–nodeisoparametricquadrilateralelement

Quadratic/LinearTransition

27Fig.3.14Five–nodeisopara28(0,-1)28(0,-1)292930303131323233ConstructionofShapeFunctionsBySimpleInspections

（划线法）33ConstructionofShapeFuncti34ConstructionofShapeFunctionsBySimpleInspectionsNumerator(分子)Denominator(分母)34ConstructionofShapeFuncti35Example:ConstructionofShapeFunctionsfor8–NodeIsoparametricElement

35Example:36(-1,-1)36(-1,-1)373738(0,-1)38(0,-1)393940BasicRequirementsforShapeFunctions40BasicRequirementsforShape41ShapeFunctions41ShapeFunctions423.4.1Interpolation

Functions

3.4FormulationofIsoparametricFiniteElementMatricesforPlaneElasticity(平面弹性问题)423.4.1InterpolationFunctio43InterpolationFunctionsforGeometry43InterpolationFunctionsfor44Fig.3.16Fourtoninevariable-number-nodes2Delement

Variable-Number-Nodes2DElement44Fig.3.16FourtonineVaria45IsoparametricFiniteElementsTable3.3Shapefunctionsoffourtoninevariable–number–nodestwodimensionalelement45IsoparametricFiniteElement464647InterpolationFunctionsforGeometry（形状的插值函数）47InterpolationFunctionsfor48InterpolationFunctionsforDisplacements（位移的插值函数）48InterpolationFunctionsfor49Four-NodeIsoparametricElement（四节点等参元）49Four-NodeIsoparametricElem50Four-NodeIsoparametricElement50Four-NodeIsoparametricElem51Four-NodeIsoparametricElement51Four-NodeIsoparametricElem52Eight-NodeIsoparametricElement52Eight-NodeIsoparametricEle53CoordinatesMappingbetweenCoordinateSystemsEight-NodeIsoparametricElement53CoordinatesMappingbetween54uvEight-NodeIsoparametricElement54uvEight-NodeIsoparametricE553.4.2Strain–DisplacementTransformationMatrix

553.4.2Strain–Displacement56Strain–DisplacementRelationsuv56Strain–DisplacementRelat57Strain–DisplacementTransformationMatrix57Strain–DisplacementTransf5858595960JacobianOperator（雅克比矩阵）60JacobianOperator616162Inverseof

JacobianOperator

62InverseofJacobianOperato63ElementwithpossibleSingular(奇异的)Jacobian63Elementwithpossible64Fig.3.17ElementswithpossiblesingularJacobianForJnonsingular,allinterioranglesmustbesmallerthan180degreesxandyareundefined(foldingbackuponitself)64Fig.3.17Elementswithpossi656566InverseofJacobianOperatorataSpecificPoint66InverseofJacobianOperator67Discussion67Discussion68Fig.1Fig.2

68Fig.1Fig.269

6970

7071717272737374747575763.2.2Lagrangepolynomials（拉格朗日多项式）

13.2.2Lagrangepolynomials（拉77LagrangeMultiplierFunctionindimensionlessform下标i表示节点i，上标n表示Lagrange插值多项式的次数2LagrangeMultiplierFunction78Table3.2Lagrangepolynomialsindimensionlessform

originn

=1-113Table3.2Lagrangepolynomial79Table3.2Lagrangepolynomialsindimensionlessform

originn

=24Table3.2Lagrangepolynomial80Table3.2Lagrangepolynomialsindimensionlessform

originn

=35Table3.2Lagrangepolynomial81InterpolationFunctionsfor2DProblems6InterpolationFunctionsfor282n=m

=1n=m

=4n=m

=3n=m

=2Fig.3.8FamilyofLagrangeelements

Thenodesexitonaregularmeshof(n+1)columns,(m+1)lines.7n=m=1n=m=4n=m=3n=m833.3FamilyofIsoparametricElements（等参元族）

83.3FamilyofIsoparametricE84985LinearLagrangePolynomials

10LinearLagrangePolynomials86i11i87Fig.3.9VariationofshapefunctionoveraquadrilateralelementwithlinearinterpolationfunctionN112Fig.3.9Variationofshapef88n

=8ThisisnotaLagrangeelement13n=8ThisisnotaLagrange89ShapeFunctionsforMid–SideNodes（边中节点）12ProductofLagrangepolynomialOrusetheInspectiveConstructionMethod（划线法）14ShapeFunctionsforMid–Si90(2,6)(4,8)ShapeFunctionsforMid–SideNodes15(2,6)(4,8)ShapeFunctions91ShapeFunctionsforCornerNodes

（角节点）NcN816ShapeFunctionsforCornerN92ShapeFunctionsforCornerNodesNc817ShapeFunctionsforCornerN93Constructionofquadraticshapefunctionsforcornernode18118Constructionofquadraticsh94Constructionofquadraticshapefunctionsforcornernode1ThisisnotaLagrangeelement19Constructionofquadraticsh95ShapeFunctionsforCornerNodes20ShapeFunctionsforCornerN96SystematicGenerationofaQuadraticShapeFunction

byStartingwithaBilinearLagrangianFamily

82121SystematicGenerationofaQ97Fig.3.12Systematicgenerationof‘serendipity’shapefunctionsSystematicGenerationofaQuadraticShapeFunction

byStartingwithaBilinearLagrangianFamily

22Fig.3.12Systematicgenerati98-BilinearLagrangianFamily23-BilinearLagrangianFamily99SystematicGenerationof‘Serendipity’ShapeFunctions

CapMethod24SystematicGenerationof‘Se100SystematicGenerationof‘Serendipity’ShapeFunctions

25SystematicGenerationof‘Se101Fig.3.13Shapefunctionsforatransition‘serendipity’element,cubic/linear

Cubic/LinearTransition（三次元/线性元的过渡）

26Fig.3.13Shapefunctionsfor102Fig.3.14Five–nodeisoparametricquadrilateralelement

Quadratic/LinearTransition

27Fig.3.14Five–nodeisopara103(0,-1)28(0,-1)10429105301063110732108ConstructionofShapeFunctionsBySimpleInspections

（划线法）33ConstructionofShapeFuncti109ConstructionofShapeFunctionsBySimpleInspectionsNumerator(分子)Denominator(分母)34ConstructionofShapeFuncti110Example:ConstructionofShapeFunctionsfor8–NodeIsoparametricElement

35Example:111(-1,-1)36(-1,-1)11237113(0,-1)38(0,-1)11439115BasicRequirementsforShapeFunctions40BasicRequirementsforShape116ShapeFunctions41ShapeFunctions1173.4.1Interpolation

Functions

3.4FormulationofIsoparametricFiniteElementMatricesforPlaneElasticity(平面弹性问题)423.4.1InterpolationFunctio118InterpolationFunctionsforGeometry43InterpolationFunctionsfor119Fig.3.16Fourtoninevariable-number-nodes2Delement

Variable-Number-Nodes2DElement44Fig.3.16FourtonineVaria120IsoparametricFiniteElementsTable3.3Shapefunctionsoffourtoninevariable–number–nodestwodimensionalelement45IsoparametricFiniteElement12146122InterpolationFunctionsforGeometry（形状的插值函数）47InterpolationFunctionsfor123InterpolationFunctionsforDisplacements（位移的插值函数）48InterpolationFunctionsfor124Four-NodeIsoparametricElement（四节点等参元）49Four-NodeIsoparametricElem125Four-NodeIsoparametricElement50Four-NodeIsoparametricElem126Four-NodeIsoparametricElement51Four-NodeIsoparametricElem127Eight-NodeIsoparametricElement52Eight-NodeIsoparametri