1、Formulation of Isoparametric Finite Element Matrices13.2.2 Lagrange polynomials（拉格朗日多项式）（拉格朗日多项式）Formulation of Isoparametric Finite Element Matrices2Lagrange Multiplier Function in dimensionless form下标下标i表示节点表示节点i，上标，上标n表示表示Lagrange插值多项式的次数插值多项式的次数Formulation of Isoparametric Finite Element Matrice
2、s3Table 3.2 Lagrange polynomials in dimensionless form 011,1 originn=1-1101110111()()()()()()()()()()()niiniiiiiiiinLLLLLFormulation of Isoparametric Finite Element Matrices40121,0,1 Table 3.2 Lagrange polynomials in dimensionless form originn=201110111()()()()()()()()()()()niiniiiiiiiinLLLLLFormula
3、tion of Isoparametric Finite Element Matrices5012311,31,13 Table 3.2 Lagrange polynomials in dimensionless form originn=3Formulation of Isoparametric Finite Element Matrices6Formulation of Isoparametric Finite Element Matrices7n=m=1n=m=4n=m=3n=m=2Fig.3.8 Family of Lagrange elements The nodes exit on
4、 a regular mesh of(n+1)columns,(m+1)lines.Formulation of Isoparametric Finite Element Matrices83.3 Family of Isoparametric Elements（等参元族）（等参元族）Formulation of Isoparametric Finite Element Matrices9Formulation of Isoparametric Finite Element Matrices10Formulation of Isoparametric Finite Element Matric
5、es11110111()(1),()(1)22LLiFormulation of Isoparametric Finite Element Matrices12 Fig.3.9 Variation of shape function over a quadrilateral element with linear interpolation functionN1Formulation of Isoparametric Finite Element Matrices13n =8This is not a Lagrange elementFormulation of Isoparametric F
6、inite Element Matrices14Shape Functions for Mid Side Nodes（边中节点）（边中节点）12101()(1),2LProduct of Lagrange polynomialOr use the Inspective Construction Method（划线法）221()(1)LFormulation of Isoparametric Finite Element Matrices15(2,6)(4,8)Shape Functions for Mid Side NodesFormulation of Isoparametric Finit
7、e Element Matrices16Shape Functions for Corner Nodes（角节点）（角节点）NcN8Formulation of Isoparametric Finite Element Matrices17Shape Functions for Corner NodesNc8Formulation of Isoparametric Finite Element Matrices18Construction of quadratic shape functions for corner node 181Formulation of Isoparametric F
8、inite Element Matrices19Construction of quadratic shape functions for corner node 12210011()()(1)(1)22NLL 1811(1)(1)(1)24cNNN This is not a Lagrange elementFormulation of Isoparametric Finite Element Matrices20Shape Functions for Corner NodesFormulation of Isoparametric Finite Element Matrices21Syst
9、ematic Generation of a Quadratic Shape Function by Starting with a Bilinear Lagrangian Family 821Formulation of Isoparametric Finite Element Matrices22Fig.3.12 Systematic generation of serendipity shape functionsSystematic Generation of a Quadratic Shape Function by Starting with a Bilinear Lagrangi
10、an Family Formulation of Isoparametric Finite Element Matrices23-Bilinear Lagrangian FamilyFormulation of Isoparametric Finite Element Matrices24Systematic Generation of Serendipity Shape Functions Formulation of Isoparametric Finite Element Matrices25Systematic Generation of Serendipity Shape Funct
11、ions Formulation of Isoparametric Finite Element Matrices26Fig.3.13 Shape functions for a transition serendipity element,cubic/linear Cubic/Linear Transition（三次元（三次元/线性元的过渡）线性元的过渡）Formulation of Isoparametric Finite Element Matrices27Fig.3.14 Five node isoparametric quadrilateral element Quadratic/L
12、inear Transition Formulation of Isoparametric Finite Element Matrices28(0,-1)Formulation of Isoparametric Finite Element Matrices29Formulation of Isoparametric Finite Element Matrices30Formulation of Isoparametric Finite Element Matrices31Formulation of Isoparametric Finite Element Matrices32Formula
13、tion of Isoparametric Finite Element Matrices33Construction of Shape Functions By Simple Inspections（划线法）（划线法）1(,)0ikkikikNik Formulation of Isoparametric Finite Element Matrices34Construction of Shape Functions By Simple Inspections Numerator(分子分子)Denominator(分母分母)Formulation of Isoparametric Finit
14、e Element Matrices35Example:Construction of Shape Functions for 8 Node Isoparametric Element Formulation of Isoparametric Finite Element Matrices36(-1,-1)()3()1(,)(,)ijiijjiifNf Formulation of Isoparametric Finite Element Matrices37Similarly 2341(1)(1)(1)(374),(481),(56)41(1)(1)(1)(481),(152),(67)41
15、(1)(1)(1)(152),(263),(78)4NNN The general equation for shape functions at all corner nodes is 1(1)(1)(1)(1,2,3,4)4iiiiiNi Formulation of Isoparametric Finite Element Matrices38(0,-1)Formulation of Isoparametric Finite Element Matrices39Formulation of Isoparametric Finite Element Matrices40Basic Requ
16、irements for Shape FunctionsFormulation of Isoparametric Finite Element Matrices41Shape FunctionsFormulation of Isoparametric Finite Element Matrices423.4.1 Interpolation Functions 3.4 Formulation of Isoparametric Finite Element Matrices for Plane Elasticity(平面弹性问题平面弹性问题)Formulation of Isoparametric
17、 Finite Element Matrices43Interpolation Functions for GeometryFormulation of Isoparametric Finite Element Matrices44Fig.3.16 Four to nine variable-number -nodes 2D element Variable-Number-Nodes 2D ElementFormulation of Isoparametric Finite Element Matrices45Isoparametric Finite ElementsTable 3.3 Sha
18、pe functions of four to nine variable number nodes two dimensional elementFormulation of Isoparametric Finite Element Matrices46Formulation of Isoparametric Finite Element Matrices47Interpolation Functions for Geometry（形状的插值函数）（形状的插值函数）Formulation of Isoparametric Finite Element Matrices48Interpolat
19、ion Functions for Displacements（位移的插值函数）（位移的插值函数）Formulation of Isoparametric Finite Element Matrices49123411(1)(1),(1)(1)4411(1)(1),(1)(1)44NNNNFour-Node Isoparametric Element（四节点等参元）Formulation of Isoparametric Finite Element Matrices50Four-Node Isoparametric ElementFormulation of Isoparametric Fi
20、nite Element Matrices51For four node isoparametric element,the coordinate interpolation is 123412341111(1)(1)(1)(1)(1)(1)(1)(1)44441111(1)(1)(1)(1)(1)(1)(1)(1)4444xxxxxyyyyy(3.34)and the displacement interpolation is 123412341111(1)(1)(1)(1)(1)(1)(1)(1)44441111(1)(1)(1)(1)(1)(1)(1)(1)4444uuuuuvvvvv(
21、3.35)Four-Node Isoparametric ElementFormulation of Isoparametric Finite Element Matrices52 Considering an eight node two-dimensional element,the coordinate interpolations are 81(,)iiiixxNyy (3.36)221(1)(1)(1)1,2,3,441(1)(1)5,721(1)(1)6,82iiiiiiiiiNiNiNiEight-Node Isoparametric ElementFormulation of
22、Isoparametric Finite Element Matrices53Coordinates Mapping between Coordinate SystemsEight-Node Isoparametric ElementFormulation of Isoparametric Finite Element Matrices54uvEight-Node Isoparametric ElementFormulation of Isoparametric Finite Element Matrices553.4.2 Strain Displacement Transformation
23、Matrix Formulation of Isoparametric Finite Element Matrices56Strain Displacement RelationsuvFormulation of Isoparametric Finite Element Matrices57Strain Displacement Transformation Matrix 12inBBBBBFormulation of Isoparametric Finite Element Matrices58Formulation of Isoparametric Finite Element Matri
24、ces5934(,)(,)fx yfx yFormulation of Isoparametric Finite Element Matrices60Jacobian Operator（雅克比矩阵）（雅克比矩阵）Formulation of Isoparametric Finite Element Matrices611(,)niiiixxNyy xyJxyFormulation of Isoparametric Finite Element Matrices6234(,)(,)fx yfx y12(,)(,)xfyf Inverse of Jacobian Operator Formulat
25、ion of Isoparametric Finite Element Matrices63Element with possible Singular(奇异的奇异的)JacobianFormulation of Isoparametric Finite Element Matrices64Fig.3.17 Elements with possible singular JacobianFor J nonsingular,all interior angles must be smallerthan 180 degreesx and y are undefined(folding back u
26、pon itself)Formulation of Isoparametric Finite Element Matrices65 xyJxyFormulation of Isoparametric Finite Element Matrices66Inverse of Jacobian Operator at a Specific Point xyJxyFormulation of Isoparametric Finite Element Matrices67DiscussionFormulation of Isoparametric Finite Element Matrices68Fig
27、.1Fig.2 Formulation of Isoparametric Finite Element Matrices69 Formulation of Isoparametric Finite Element Matrices70 Formulation of Isoparametric Finite Element Matrices71Formulation of Isoparametric Finite Element Matrices72Formulation of Isoparametric Finite Element Matrices73Formulation of Isoparametric Finite Element Matrices74Formulation of Isoparametric Finite Element Matrices75
