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

Lagrange Multiplier Function in dimensionless form
Lni ( )
n k 0
( (i
k ) k )
k i
( 0 )( 1) ( i1)( i1) ( n ) (i 0 )(i 1) (i i1)(i i1) (i n )
where i denotes the natural coordinate of node i.
Table 3.2 Lagrange polynomials in dimensionless form
-1
0 1, 1 1
origin
1
n=1
L10 ( )
1
1 1
1 2
(1 )
L11( )
(1)
1 (1)
1 2
(1 )
Lni
(
)
( 0 )( (i 0 )(i
1) 1)
( i1)( i1) ( n ) (i i1)(i i1) (i n )
7
Formulation of Isoparametric Finite Element Matrices
3.3 Family of Isoparametric Elements（等参元族）
A whole family of two – dimensional isoparametric elements can be formulated but the quadratic element （二次单元） is considered to be the best. Elements of an order higher than three are not really practical since it is time consuming to generate stiffness matrices.
nm
f (,)
Lni ( )Lmj () fij
i0 j0
(3.14)
where n and m stand for the number of subdivisions in the
and directions respectively.
6
Formulation of Isoparametric Finite Element Matrices
3
Formulation of Isoparametric Finite Element Matrices
Table 3.2 Lagrange polynomials in dimensionless form
origin
0 1, 1 0, 2 1
L20 (
)
0
1 0
1
1 1
1 2
Table 3.2 Lagrange polynomials in dimensionless form
origin
0 1,
1
1 3
,
n=3
2
1 3
,
3 1
L30 (
)
1 3 1 1 3
1 3
11 3
1
1 1
1 16
(3
1)(3
1)(1
)
L13
(
)
1
1 3 1
1
1 3
3 1
3
1
Formulation of Isoparametric Finite Element Matrices
3.2.2 Lagrange polynomials （拉格朗日多项式）
The Lagrange interpolation polynomial in a single coordinate system (-1≤ ≤1) with (n 1) nodes is
1
3
1
9 16
(1
2
)(1
3
)
L32
(
)
1 1 3 1
1 3 1 3 1 3
1
1 31
9 16
(1
2
)(1
3
)
L33 (
)
1 1 1
1 1 1
3 3
1
11
3 3
1 16
(1
)(3
1)(3
1)
5
Formulation of Isoparametric Finite Element Matrices
n
f ( ) Lni ( ) fi i0
(3.11)
where fi - the function f ( ) evaluated at node i Lni ( ) - the Lagrange multiplier（乘子） function
1
Formulation of Isoparametric Finite Element Matrices
nm
f (,)
Lni ( )Lmj () fij
i0 j0
n = m =1
n = m =2
n = m =3
n = m =4
Fig.3.8 Family of Lagrange elements The nodes exit on a regular mesh of (n+1) columns, (m+1) lines.
8
Formulation of Isoparametric Finite Element Matrices
(
1)
L12
(
)
0
(1) (1)
0
1 1
(1
2
)
L22
(
)
(1)
1 (1)
0
1 0
1 2
(1
)
n=2
Lni
(
)
( 0 )( (i 0 )(i
1) 1)
( i1)( i1) ( n ) (i i1)(i i1) (i n )
4
Formulation of Isoparametric Finite Element Matrices
The shape functions can be constructed directly as the products
（ 乘 积 ） of Lagrange Polynomials. Thus the interpolation
functions for a two – dimensional problem would be
It can be seen that Lni ( ) is an nth degree polynomial as it is
given by the product of n linear factors.
下标i表示节点i，上标n表示Lagrange插值多项式的次数
2
Formulation of Isoparametric Finite Element Matrices