0 0
1 j (1 i )
0 0
e
y ati j
v
(3.57)
x
v
1 4
Jij
1
0 0
1 j 1 i
0 0
(1 j ) 1 i
0 0
(1 j ) (1 i )
0 0
1 j (1 i
)
e
y ati j
with
T e
u1
v1
u2
v2
u3
v3
u4
v4
- the vector listing the element nodal point displacements8
1
B 2
B i
B
n
(3.42)
N
i
x
[B]i 0
Ni y
0
Ni y
Ni x
x
J
1
y
x y
J
x y
3
Formulation of Isoparametric Finite Element Matrices
Inverse of Jacobian Operator at a Specific Point
Formulation of Isoparametric Finite Element Matrices 3.4 Formulation of Isoparametric Finite Element Matrices
for Plane Elasticity (平面弹性问题)
The interpolation of the element coordinates and element displacements using the same interpolation functions, which are defined in a natural coordinate system, is the basis of the isoparametric finite element formulation.
(1
)
y2
1 4
(1
)
y3
1 4
(1
)
y4
(3.55)
y
1 4
(1
)
y1
1 4
(1
)
y2
1 4
(1
)
y3
1 4
(1
)
y4
By evaluating above x , x and y , y at a
specific point
i ,
j,
1
Jij is obtained.
6
Formulation of Isoparametric Finite Element Matrices
(1
)u2
1 4
(1
)u3
1 4
(1
)u4
Fin ite
x
Element Jij
M1 atrices
Four-Node Isoparametric Element
u
y ati j
ati j
x
u
1 4
Jij
1
1 j 1 i
0 0
(1 j ) 1 i
0 0
(1 j ) (1 i )
3.4.2 Strain – Displacement Transformation Matrix
For a two - dimensional n - node element, the strain – displacement
relations are given by
Be
(3.41)
B
B
where
y
1
Jij
1 J ij
x
y
x
at i
j
x y y x
J ij
(
)at
i
j
(3.52) (3.53)
- determinant（行列式） of Jij
4
Formulation of Isoparametric Finite Element Matrices
x
y
n i 1
Ni
(
,
)
xi yi
Four-Node Isoparametric Element
For a 4-node isoparametric element, to evaluate the
displacement derivatives, we need to evaluate
x
y
4 i1
Ni
xi
4 i 1
Ni
yi
N3
1 4
(1
)(1 ),
N4
1 4
(1
)(1 )
It is noted that
x
1 4
(1
)
x1
1 4
(1
)
x2
1 4
(1
)
x3
1 4
(1
)
x4
x
1 4
(1
)
x1
1 4
(1
)
x2
1 4
(1
)
x3
1 4
(1
)
x4
y
1 4
(1
)
y1
1 4
J
x
y
4 i 1
Ni
xi
y
4 i1
Ni
xi
4
i 1
Ni
yi
4
i 1
Ni
yi
(3.54)
5
x
J
x
y
Foi4r1mNui lxai tioi4n1
oNf i
IysiopaNr1am14e(t1ric
)F(1inite),
ENle2me14n(t1
Ma)(t1rice)s
1
Formulation of Isoparametric Finite Element Matrices
3.4.1 Interpolation Functions
Considering a general two – dimensional n - node
isoparametric element, the coordinate interpolations are
1 4
(1 )u3
1 4
(1
)u4
v
1 4
(1
)v1
1 4
(1
)v2
1 4
(1
)v3
1 4
(1
)v4
v
1 4(1 )v1来自1 4(1 )v21 4
(1
)v3
1 4
(1 )v4
(3.56)
7
u u
1 4 1 4
(1 )u1 (1 )u1
1 4 1 4
(F1or)mu2 u14la(1tion)u3of14I(1sop)au4rametric
x
y
n i 1
Ni
(
,
)
xi yi
(3.27)
The displacement interpolations have the form
u
v
n i 1
Ni
(
,
)
uvii
(3.29)
2
Formulation of Isoparametric Finite Element Matrices
Four-Node Isoparametric Element
u
v
n i 1
Ni
(
,
)
ui vi
To evaluate the element strains we use
u
1 4
(1 )u1
1 4
(1
)u2
1 4
(1 )u3
1 4
(1 )u4
u
1 4
(1 )u1
1 4
(1
)u2