finite element methods in engineering-1 036015 pinhas z. bar-yoseph computational mechanics lab....
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FINITE ELEMENT METHODS IN ENGINEERING-1
036015
Pinhas Z. Bar-Yoseph
Computational Mechanics Lab.
Mechanical Engineering, Technion
Winter Semester
Tutorial # 4Copyright by PZ Bar-Yoseph ©
0 1
( ) , 0 1
0 , 1 ; ( ) 0
d dux a x u f x x
dx dx
u u u u a x
2 1 2x 3 1x x1 0x
1
1 1h
Finite Element mesh- 1 quadratic element
Quadratic Lagrangian element
11
7 8 1
8 16 83
1 8 7
K1
1
4 2 1
2 16 230
1 2 4
a a
K
11
1
461
f
F
1
7 8 11
8 16 831 8 7
K K 1
120
431
F F
2
240 , 0 1
0 1 , 1 3 ;
d ux
dxu u
2( ) 20 22 1u x x x
1
2
3
7 8 1 1
8 16 8 20 4
1 8 7 1
u
u
u
Imposing B.C.’s
1 0 3 10 1 , 1 3 ;u u u u u u
2 7u
e
0
1
1
2 ( )N
1
1( )N 3( )N
1 2 3
1u 2u 3u
2heu2 1
heu 2 1
heu
hi iu N u
Postprocessing
1 eni n
3en
For quadratic element
n
1
22
3
1 11 1 1
2 2h
u
u u
u
2
11 1
1 1 1 72 2
3
hu
2 1 2 122 1e e
e e
x xx x x
h h
2( ) 20 22 1 ( )hu x x x u x
3x 5 1x 1 0x
1 2
1h 2h
Finite element mesh- 2 quadratic elements
x1 2 3 4 5
2 2
1 1
e e
e e K K F F
1 1 111 12 131 1 121 22 231 1 1 2 2 231 32 33 11 12 13
2 2 221 22 232 2 231 32 33
0 0
0 0
0 0
0 0
K K K
K K K
K K K K K K
K K K
K K K
1112
1 23 1
2223
F
F
F F
F
F
F
The higher the element order
the higher is the bandwidth
How the problem size
can be significantly reduced?
Static condensation
x1 2 3 4 enn1enn 2enn
2hu 3
hu 4hu 2en
hnu 1en
hnu
iU
1hu
en
hnu
oU
Given a higher order element:
oo oi o o
io ii i i
K K U F
K K U F
-1i ii i io o U K F K U
oo
-1 -1oo oi ii io o o oi ii i
FK
K K K K U F K K F
o o oK U F
oo o oi i o K U K U F
-1oo o oi ii i io o oi
K U K K F K U F
oo
-1 -1oo oi ii io o o oi ii i
FK
K K K K U F K K F
o o oK U F
2 2 2 1
2oK
3oK
4oK
K
1oK
4oF
3oF
2oF
1oF
F
At this point, element matrix assembly is proceeded as if the element is a linear element:
0
( ) , 0 1
0 , 0@ 1; ( ) 0
d dux a x u f x x
dx dx
duu u hu x a x
dx
3x 7 1x
1 2
1 2
5h 2 3
5h
1 0x 2x 4
2
5x 5x 6x
Finite Element mesh- 2 cubic elements
Higher order Lagrangian elements
1
1
1
( )
( )
en
en
en
n
jjj in
i n
i jjj i
l
Lagrange polynomials:
1enni ijl
1enni iN l
1
1 1
4 ( )N
1
1( )N 2 ( )N 3( )N
2 3 4
cubic element
2
1
3 3
1
3 4 1
cubic element
2 3 431 1
1 2 1 3 1 4
1 3 432 2
2 1 2 3 2 4
1 11
9 1 13 31
2 4 16 3 323 3
11 1
27 131
2 2 4 16 33 3 3
N l
N l
1 2 433 3
3 1 3 2 3 4
2 3 434 4
3 2 3 3 3 4
1
11 1
27 131 1
4 2 2 16 33 3 3
1 11
9 13 31
4 2 16 323 3
N l
N l
1
3
cubic element
3 1
3 2
1
1
2 2
2
148 -189 54 -13
-189 432 -297 541
54 -297 432 -18940
-13 54 -189 148
e
e
x eT e eT e ee e
e ex
e
e
dN dN dN dN hdx d
dx dx h d h d
h
K
cubic element
3 1
3 2
1
1 2
128 99 36 19
99 648 81 361
36 81 648 991680
19 36 99 128
e
e
x ea e eT e e eT e
x
e e
hN aN dx a N N d
a h
K
cubic element
3 1
3 2
1
1
( )2
1
31
38
1
: ( )
e
e
x ee eT e eT
x
e e
e e
hF N f x dx f N d
h f
other option f N f
3x 7 1x
1 2
1 2
5h 2 3
5h
1 0x 2x 4
2
5x 5x 6x
Finite Element mesh- 2 cubic elements
2
0
( ) 1
( )
( ) 0
1, 2
x x
f x x
a x
u h
1 1 1
11121314
-189 54 3
54 -189 3
-189 54
54 -18
148 -1
432 -297
-297 432
432 -297
-2
-18 3 1
-13 14
9 54
3 1
40 500
54 -189
-189 54
17
240
54 -
97
8 1
1
9
48 -13
-
4
18 1 8
3
1
2
93 4
F
u
u
u
u
K K
2 2 2
21222324
147
4
1
000
1
3
3
F
u
u
u
u
K K
11 1 1
11 1 1
2
1114
1213
148 -13 1
-13 148 1
14
432 -297
-189 54
54 -
-297 432
189
3 1
4 -1
8 -
0 500
17
240
89 54 3
54 -189
13
13 148
3
-
ooo oi o
iio ii i
oo
UK K F
FK K U
K
u
u
u
u
22 2
22 2 2
1114
1213
432 -297
-297 43
-189 54
54 -189
147
4-189 54 3
54 -189 2
00
3
1
1
0ooi o
iio ii i
UK F
FK K U
u
u
u
u
1 1 1 1 1 1 1 1 1 1 1 1oo oi ii io oi ii
2 2 2 2 1 2 2 2 2 2 1 2oo oi ii io oi ii
1 1 113 ,
1 1 1125
1 1 117
13 3
317 17
6 617 17
6 6
147,
1 1 16 1000
0
,
0
3
o o o i
o o o i
K F
K K K K K F F K K F
K K K K K F F K K F
147
1000147
100
1
0
251
125
1 0
1
04 4. .
7 7
1
4
7
13 3 0 125
35 17 15535 17 1556 6 10006 6 100017 17 14717 17 147
0 6 6 10006 6 10
3
00
3b c
u uu
uu u
u u
u
u
h
u
2 1 1 -1 1 1 1i ii i io o
3
5 2 2-1 2 2 2i ii i io o
6
1
0.7768
0.4858
0.9262
0.8518
0.6913
0.5943
u
u
u
u
U K F K U
U K F K U