ABAQUS有限元素法处理原则

[ B] matrix, also called the element strain - displacement matrix:
[
B]
=
d[N(ξ)]
dx
=
d[N(ξ)]
dξ
dξ dx
=
d[N1(ξ), N2(ξ)]
dξ
x2
2 −
x1
∵ξ = 2(x − x1) − 1 x2 − x1
[B]
=
⎡⎣⎢−
∫ ∫ ∫ f δu A dx = Le
Le f [ N ]{δ d} A dx = ={δ d}T ⎛⎝⎜ f Ae
1 [ N]T
−1
e
2
dξ ⎞⎠⎟
{ } = {δ d}T f e
{ } ∫∫ f e
=
Ae e f 2
⎧ ⎪ ⎨ ⎪ ⎩
1
−1 1
N1
dξ
⎫ ⎪ ⎬
=
−1
N
2
dξ
⎪ ⎭
Ae e f 2
in an average (integral) sense ( starting point for finite
element approximation: equilibrium will be satisfied, not at
every point in V, but at some finite number of nodes.
如此内差的表示方式最大的优点就是，不论 shape functions 的形式如何， 所有的函数都可以同一内差形式表示，因此我们可以建立所谓的 element library，library 中含所有的 shape functions，需要时呼叫出来即可。
Quadratic Shape Functions
∫ ∫ ∫ ∑ E du dδu A dx =
L dx dx
f δu A dx + Tδu dx +
L
L
i
Piδui
∀δu such that δu = 0 on Su
Finite Element Discretization
Geometry
Finite Element Mesh (网格)
分割的准则：由使用者判断（通常依准确度的要求以及题目的特性而定，如 本题在有外力作用的点上应置入节点）
e
T δu dx +
Le
i
Piδui
using the same interpolation for δ u : δ u = [ N ]{δ d}, we have
∫ ∫ E du dδu A dx = E[B]{d}[B]{δ d}Adx
Le dx dx
Le
∫= {d}T [B]T E[B]{δ d}Adx Le
Summary
[ ] { } { } ∑ ∑ ∑ ∑ {δ d}T k e {d} = {δ d}T f e + {δ d}T Te + δ di Pi
)
=
1 2
ξ(1
+
ξ)
N3(ξ) = (1 + ξ) (1 − ξ) (1) = (1 + ξ)(1 − ξ)
+
3
∑ Nα (ξ) = 1
1
α =1
Linear and Quadratic Interpolations
f
f2 f=N1 f1+N2 f2 f1
f f=N1 f1+N2 f2+N3 f3
∫ dv u
＝
∫ v u － v du
∫ ∫ ∫ d E du δu dV =
V dx dx
S
nx
⎛⎝⎜
E
du dx
δu⎞⎠⎟
dS
−
E du dδu dV V dx dx
B.C.’s: ( S = Su + ST + SP ), Su 與 ST 或 SP 不应重叠，很容易在 fem 中犯错！
Su: u(0) = 0 ⇒ δu = 0
⎧1⎫ ⎨⎩1⎬⎭
N1
=
1
− 2
ξ
N2
=
1+ξ 2
is called the element body force vector.
∫ { } T δu dx ={δ d}T T e Le
{ }Te
=
Te 2
⎧1⎫ ⎨⎩1⎬⎭
is called the element traction force vector.
知，但在此先假设节点上的位移为已知，在元素中任何
位置的位移则由节点上的位移内差而得。
Note: (1) 使用 index 表示法时，顺序不重要，但使用 matrix 表示法时，
要注意变数的先后顺序。你不一定要精通推导公式，但
一定要看得懂！
(2) 以上的表示方法并不限于两点的线性元素，这是用形状函数内差的好
dx
=
d[ N]{x}
=
d[N]
dξ
dξ{x}
[ ] = {δ d}T k e {d}
[ ] =
−1 2
,
1 2
{x}dξ
=
e
2
dξ
[ ]k e
= Ee Ae
e[B]T [B]
=
Ee Ae
e
⎡1 ⎣⎢− 1
− 1⎤ 1 ⎦⎥
[B] = 1 [− 1,1]
e
is called the (symmetric) element stiffness matrix.
处！
also
x = [ N ]{x}, {x} = [x1, x2 ]T , x1, x2 are nodal coordinates
When same order of shapes functions are used to interpolate the coordinates and displacements, it is called the isoparametric interpolation.
Galerkin 方法： Multiply differential equations by an arbitrary variable: (u
and integrate over the volume
∫V
⎛⎝⎜
d dx
E
du dx
+
f
⎞⎠⎟
δu dV
=
0
Integration by parts (for the 2nd order term)
元素进行处理即可。
Finite Element Interpolation (2-node element)
f(x) = a + b x f1 = f(x1)
= a + b x1
两节点的元素: 仅允许线性的变化
f2 = a + b x2
⎡1 ⎣⎢1
x1 x2
⎤⎧a⎫ ⎦⎥⎨⎩b⎬⎭
=
⎧ ⎨ ⎩
f1 f2
Writing shape N1(ξ) = (1 − ξ) functions
ξ
(−
1 2
)
=
−
1 2
ξ
(1
−
ξ)
Nα (ξβ ) = δαβ
N1(ξ1) = N1(−1) = 1 N1(ξ3) = N1(0) = 0
N1(ξ2 ) = N1(1) = 0
N2 (ξ) = (1 + ξ) ξ
(
1 2
f2 f3 f1
Isoparametric Interpolation (for 2-node elements)
Within each element,
u(x) = [ N ]{d},
[
N
]
=
[
N1,
N2
]
=
⎡1 ⎣⎢
− 2
ξ
,
1
+ 2
ξ
⎤ ⎦⎥
for linear elements
{d} = [d1, d2 ]T ： d1, d2 是节点上的位移。在结果尚未解出前， {d} 是未
1, 2
1⎤ 2 ⎦⎥
x2
2 −
x1
=
x2
1 −
x1
[− 1,1]
[
N
(ξ)]
=
⎡1 ⎣⎢
− 2
ξ
,
1
+ 2
ξ
⎤ ⎦⎥
for linear elements, [B] is a constant matrix.
Stress
σ = Eε = E du = E[ B]{d} = E d2 − d1 constant stress element
If (u is regarded as the virtual displacement, then the Galerkin method becomes the principle of virtual work.
dV → Adx, dS → dx (T is defined as traction force per unit length)
N1 ( x)
=
x2 − x x2 − x1
N2 ( x)
=
x − x1 x2 − x1
these are called the shape functions
They can also be expressed as
N1(ξ)
=
1
− 2
ξ
N
2