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A seminar outline for me 501b engineering analysis focusing on finite elements in two dimensions. It covers topics such as quadratic basis functions, border integral terms, triangular elements, natural coordinates, and linear and higher order basis functions. The document also includes information on boundary terms, selecting basis functions, and computing integrals using gauss quadrature.
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Larry Caretto
Mechanical Engineering 501B
2
Outline
dimensions
3
Review Quadrilateral Equation
∫ ∫ − −
1
1
1
1
k i
i k k
i k k
ξ ξ η ξ η ξ
η η η ξ η ξ
2
2 2
2 2
∫ ∫ ∑∑ − − =^ =
n
k
n
j
k j k j
1 1
1
1
1
1
Review Linear φ i Quadrilateral
4
4
1
1
3 4
1 2
i
(j)
5
Review φ Derivatives
4 4 4
3 3 3
2 2 2
1 1 1
ξ
η
η ϕ
ξ
ξ η ϕ ϕ
ξ
η
η ϕ
ξ
ξ η ϕ ϕ
ξ
η
η ϕ
ξ
ξ η ϕ ϕ
ξ
η
η ϕ
ξ
ξ η ϕ ϕ
6
Review x and y Derivatives
2
2
4 1 3 2
4 1 3 2
2 1 3 4
2 1 3 4
y y y y y
x x x x x
y y y y y
x x x x x
ξ ξ
ξ ξ
η η
η η
η
η
ξ
ξ
η^ = –
ξ η η ξ
ξ η η ξ
xy x y
x y x y J
= −
∂
∂
∂
∂ − ∂
∂
∂
ki
7
4
4
3
3
condition to include in solution
Dirichlet boundary condition
η^ = –
8
(x 4 , y 4 )
3
3
η^ = –
Γ
end
start
k side
k
2 3 2
2 S (^) ξ= 1 = x 3 − x 2 + y − y
9
(x 4 , y 4 )
1
1
η^ = –
Γ
2 3 2
2 1 3 2 S = x − x + y − y ξ=
2 3
−
1
1
1
3
2
1 2
ξ ξ
d S ds S
10
4
4
1
1
η^ = –
Γ
2 1
ξ
=
1
1
1
1 1
2
=
− =
= Γ =
ξ
ξ
ξ ξ
11
2
2
η^ = –
A (^) 21 u 1 + A 22 u 2 + A 23 u 3 + A 24 u 4 = B
A 31 u 1 + A 32 u 2 + A 33 u 3 + A 34 u 4 = B
A 41 u 1 + A 42 u 2 + A 43 u 3 + A 44 u 4 = 0
1
1
=
=
ξ
ξ
2
2
η^ = –
A 21 u 1 + A 22 u 2 + A 23 u 3 + A 24 u 4 = 0
A (^) 31 u 1 + A 32 u 2 + A 33 u 3 + A 34 u 4 = B
A 41 u 1 + A 42 u 2 + A 43 u 3 + A 44 u 4 = B
1
1
=
=
η
η
19
Review April 27 Homework II
1
30 o
1
1
1
1 1 1 1
1
1 1
0
o
1
4 3
2
2
2
1
1
20
Review April 27 Homework III
4
4
1
1
sin ( sin ) 4
(sin ) 4
θ θ=
+ξ θ+
+η
+η
η
ξ
ξ
h x h h
y
h x h h
4 1 3 2
2 1 3 4
2 1 3 4
x x x x x
y y y y y
x x x x x
+ξ − +
+η − +
+η − +
η
ξ
ξ
21
Review April 27 Homework IV
4
4
3
3
cos ( cos) 4
( cos ) 4
4 1 3 2
θ θ=
+ξ θ+
−ξ
+ξ − +
η
h h h
y y y y y
2
cos 0 2
sin
2
cos
2
2 θ ⎟ = ⎠
⎞ ⎜ ⎝
⎛ θ −
θ = (^) ξη− ηξ=
hh h h J xy xy
( )
θ
θ
θ
θ
=
θ
= θ
θ
⎟ ⎠
⎞ ⎜ ⎝
⎛ θ ⎟+ ⎠
⎞ ⎜ ⎝
⎛ θ
=
ηξ ηξ
η η
cos
sin
4
cos
0 2
cos
2 2
sin
cos
1
4
cos
sin cos 4
4
cos
2
cos 2
sin
2
2
2 2
2
2
2 2 2 2
h
h h h
J
xx yy
h
h
h
h h
J
x y
22
Review April 27 Homework V
ki
ki
k i
k k
i k k
1
1
1
1
2
2 2
2 2
∫ ∫ − −
ξ ξ ηξ η ξ
η η ηξ η ξ
a = 0
23
Review April 27 Homework VI
θ
= θ
⎟+ ⎠
⎞ ⎜ ⎝
⎛
=
ξ+^ ξ
cos
1
4
cos
0 2 2
2
2
2 2
h
h
J
x y
⎥ ⎦
⎤ ⎢ ⎣
⎡
∂ξ
∂ϕ
θ
θ − ∂η
∂ϕ
∂η θ
∂ϕ ⎥+ ⎦
⎤ ⎢ ⎣
⎡
∂η
∂ϕ
θ
θ − ∂ξ
∂ϕ
∂ξ θ
∂ϕ ξ η= i k k i k k f cos
sin
cos
1
cos
sin
cos
1 (, )
⎭
⎬
⎫
⎩
⎨
⎧ ⎥ ⎦
⎤ ⎢ ⎣
⎡
∂ξ
∂ϕ − θ ∂η
∂ϕ
∂η
∂ϕ ⎥+ ⎦
⎤ ⎢ ⎣
⎡
∂η
∂ϕ − θ ∂ξ
∂ϕ
∂ξ
∂ϕ
θ
ξ η= i k k i k k f sin sin cos
1 (,)
⎭
⎬
⎫
⎩
⎨
⎧ ⎥ ⎦
⎤ ⎢ ⎣
⎡
∂ξ
∂ϕ − ∂η
∂ϕ
∂η
∂ϕ ⎥+ ⎦
⎤ ⎢ ⎣
⎡
∂η
∂ϕ − ∂ξ
∂ϕ
∂ξ
∂ϕ ξ η= i k k i k k f 2
1
2
1
3
2 (, )
For q = 30 o , sinθ = ½
and cosθ = 31/2^ /
24
Review April 27 Homework VII
chart for computational ease
25
Review April 27 Homework VIII
i i i
b i a i i
ξ
η
ϕ
η
ξ
ϕ
i i i
i i i
26
Review April 27 Homework IX
⎭
⎬
⎫ ⎥ ⎦
⎤ ⎢ ⎣
⎡ + η −
⎩
⎨
⎧ ⎥ ⎦
⎤ ⎢ ⎣
⎡ + ξ −
4
( 1 )
2
1
4
( 1 )
4
( 1 )
4
( 1 )
2
1
4
( 1 )
4
( 1 )
3
2 (,)
i i k k k k
i i k k k k
b a b a a b
a b a b b a f
⎭
⎬
⎫
⎩
⎨
⎧ ⎥ ⎦
⎤ ⎢ ⎣
⎡
⎤ ⎢ ⎣
⎡ = − 24
1
24 4 4
1
3 4 4
2 ( 0 , 0 )
ai ak bk bi bk ak f
163
2
3 16 32
2 ( 0 , 0 )
ai ak bibk aibk biak aiak bibk aibk biak f
⎞ ⎜ ⎝
⎛ + −
=
27
Review April 27 Homework X
43
2 4 ( 0 , 0 ) ik ik ik ik ki
aa bb ab ba A f
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
−
−
−
=
0 6 0 6
2 0 2 0
0 6 0 6
2 0 2 0
23
1 A
12 44
31 33
22 24
11 13
u u
u u
u u
u u
A 11 = 2, A 13 = -2, A 22 = 6, A 24 = -6, A 31 = -2, A 33 = 2, A 42 = -6, A 44 = 6, A 12 = A 14 = A 21 = A 23 = A 32 = A 34 = A 41 = A 43 = 0
28
Review April 27 Homework XI
i-1,j-1 (^) i,j-1 i+1,j-
i-1,j
i-1,j+
i,j+1 i+1,j+
i+1,j
(UR)
(LL)
(UL)
(LR)
i,j
[ ]
[ ] [ ]
[ ]
[ ] (^1) , 1 0
( ) , (^113)
( ) 23
( ) 14
1 , 1
( ) 1 , 24
( ) 43
( ) 12
,
( ) 44
( ) 33
( ) 22
( ) 1 , 11
( ) 34
( ) 21
1 , 1
( ) , 1 42
( ) 41
( ) 1 , 1 32
( ) 31
− +
−
− − − + −
i j
UR ij
UR UL
i j
UR i j
UR LR
ij
UR UL LL LR i j
UL LL
i j
LR ij
LL LR i j
LL
A A u A u
A A u A u
A A u A A A A u
A u A A u A u
29
Review April 27 Homework XII
i-1,j-1 (^) i,j-1 i+1,j-
i-1,j
i-1,j+ i,j+1 i+1,j+
i+1,j
(UR)
(LL)
(UL)
(LR)
i,j
2 0 0 6 0 0 2 6 2 6
1 , 1 , 1 , 1 1 , 1
1 , 1 , 1 1 , 1 1 , ,
− + + − + + + + + +
− − − + − −
i j i j ij i j
i j ij i j i j ij
u u u u
u u u u u
− 2 ui − 1 , j − 1 − 6 ui + 1 , j − 1 + 16 ui , j − 6 ui − 1 , j + 1 − 2 ui + 1 , j + 1 = 0
30
Higher order Shape Functions
higher order for dependent variable
allow elements with curved sides
37
Triangular Coordinates
x 1 ,y 1 x 2 ,y 2
x3 ,y 3
λ 3
λ 3 = A 3 /A
λ 3 = 1
38
Triangular Coordinates II
3
x 1 ,y 1 x 2 ,y 2
x3 ,y 3
λ 3
λ 3 = A 3 /A
λ 3 = 1
39
Triangular Coordinates III
1
2
3
λ 2
λ 3
λ 1
x 3 , y 3
x 1 , y 1 x 2 , y 2
40
Triangular Coordinates IV
λ 2
λ 3
λ 1
x 3 , y 3
x 2 , y 2 x 1 , y 1
41
Triangular Coordinates V
λ 2
λ 3
λ 1
x 3 , y 3
x 1 , y 1 x 2 , y 2
i
1
2
3
42
Coordinate Transformations
3
2
1
1 2 3
1 2 3
−
1
1 2 3
1 2 3
3
2
1
( )
ji
i j
ij
43
Finding the Inverse
2 1 3 2 1 3
2 3 1 2 31
1 2 3
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
− − − −
− − − − −
− − − −
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
12 12 2 1 2 1
1 3 13 3 1 3 1
2 3 23 3 2 3 2
1
1 2 3
1 2 3 2
1
1 1 1
xy yx y y x x
xy yx y y x x
xy yx y y x x
A y y y
x x x
( )
ji
ij
ij
b 23
44
Getting λ i from x and y
2 3 12 1 2 2 1
13 1 3 3 1 2 3
2 3 2 3 2 3 3 2
3
2
1
45
Triangle Shape Functions (N i ≡ φ i )
i
5, and 6 at midpoints of edge
( )
( )
( ) 3 3 3 6 3 1
2 2 2 5 2 3
1 1 1 4 12
6
5
3 4
1
2
46
Shape Function Derivatives
2
i
47
Shape Function Derivatives II
2
2
1
1
2
2
1
1
λ
λ ψ
λ
ψ λ ψ
λ
λ ψ
λ
ψ λ ψ
∂
x x x y y y
2 3 1 2 1 2 2 1
13 1 3 3 1 2 3
2 3 23 2 3 3 2
3
2
1
23 23 2 3 3 2 1
48
Shape Function Derivatives III
b
x x
A y
a
y y
x
b
x x
A y
a
y y
x
b
x x
A y
a
y y
x
3 1 2 3 3 2 1 3
2 3 1 2 2 1 3 2
1 2 3 1 1 3 2 1
λ λ
λ λ
λ λ
55
Border Terms
1 , 2 , 3
ˆ ˆ ˆ
3
2
1
0
3 23
23 0 () 23
= ∂
∂
∂
= Γ
Nd k n
u Nds L n
u ds n
u A u Nk k k
M
i
kii e
λ
23
23
1
0
2 3
23
23
1
0
3 3 23
23
( 23 ) 3 0
3
n
L u
n
u d L n
u Au B L
M
i
i i ∂
λ λ λ
56
Border Terms II
integrate in a counterclockwise direction
23
23
0
1
2 3
23
23
0
1
2 2 23
23
( 23 ) 2 0
2
ˆ
2 2
ˆ ˆ
n
L u
n
u d L n
u Au B L
M
i
i i ∂
∂ ⎥= ⎦
⎤ ⎢ ⎣
⎡
∂
∂ − =− ∂
∂ = =
=
λ λ λ
equations
57
Border Terms III
( 23 ) 311 32 2 33 3 3
( 23 ) 211 22 2 233 2
11 1 122 133
( 31 ) 311 32 2 33 3 3
211 22 2 23 3
( 31 ) 111 12 2 13 3 1
23
23
( 23 ) 3
( 23 ) 2
ˆ
2 n
L u
31
31
( 31 ) 3
( 31 ) 1
ˆ
2 n
L u
58
Border Terms III
311 32 2 333
( 12 ) 211 22 2 233 2
( 12 ) 111 12 2 13 3 1
i
i
(jk)
12
12
( 12 ) 2
( 12 ) 1
ˆ
2 n
L u
311 32 2 33 3 3
211 22 2 23 3 2
111 12 2 133 1
59
Assembly
one equation (from three) from each element
where node appears
60
Mesh
61
Assembly for Six Triangles
system is node 3 for all
triangles in local system
a
b
d c
e
f
(α)
(β)
(γ)
(δ)
(ε)
(ζ)
g
( ) 3
() 33
() 32
() 31
() 3
() 33
() 32
() 31
() 3
() 33
() 32
() 31
() 3
() 33
() 32
() 31
() 3
() 33
() 32
() 31
() 3
() 33
() 32
() 31
γ γ γ γ ς ς ς ς
β β β β ε ε ε ε
α α α α δ δ δ δ
A u A u A u R A u A u A u R
A u A u A u R A u A u A u R
A u A u A u R A u A u A u R
b c g e f g
a b g d e g
f a g c d g
62
Assembly for Six Triangles II
have all node g
coefficients as A 33 a
b
d c
e
f
(α)
(β)
(γ)
(δ)
(ε)
(ζ)
g
( ) ( ) ( )
( ) ( ) ( )
( )
( ) 3
() 3
() 3
() 3
() 3
() 3
() 33
() 33
() 33
() 33
() 33
() 33
() 31
() 32
() 31
() 32
() 31
() 32
() 31
() 32
() 32
() 31
() 31
() 32
α β γ δ ε ς
α β γ δ ε ς
δ ε ε ς ς α
α β γ β γ δ
g
d e f
a b c
63
Triangle Quality
2
2 2
64
65
May 4 Homework
week using gradient boundaries
1
30 o
1
1
1 1 1 1
0
1 1
Zero gradient on these elements
boundary node
0
66
May 4 Homework II
i i i
i ai bi
4 cos
sin 3
ik ik
ikik ik ik
ki
ab ba
aabb aa bb
A
73
Jacobian Details
( )( ) ( )( )
3 3 32 13 12
33 31 2 3 2 1
3 2 3 1 3 1 3 2
2
1 2 1 2
74
Integrating Triangle Area
λ 2 λ 1
x 3 , y 3
x1 , y 1 x2 , y 2
Length = λ 1 = 1 – λ 2
∫ ∫
∫ ∫
−
Ω Ω
1
0
1
0
1 2 3 1 2
1 2 3 1 2 3 1 2
2
λ
λ 3 = 0
75
Confirm Integration Limits
A dxdy Ad d A d d A d
= ⎥ ⎦
⎤ ⎢ ⎣
⎡ ⎥ = − ⎦
⎤ ⎢ ⎣
⎡ = − = −
= = = =
∫
∫ ∫ ∫ ∫ ∫
− − Ω Ω
2
1 2 1 2
2 1 2
2 2 2
1
0
2 2 2 2
1
0
2
1
0
1
0
2
1
0
1 (^121210)
2 2
λ λ λ λ
λ λ λλ λ λ
λ λ
i
76
Linear Shape Functions
NaN dxdy y
y
x
x
A (^) k i i k i k ki (^) ∫ Ω
2
∫ ∫ ∫ ∫
∫ ∫ ∫ ∫
− −
− −
1
0
1
0
1
0
1
0
1 2
2 1 2
1
0
1
0
1
0
1
0
1 2
2 1 2
2 2
2 2
λ λ
λ λ
λ λ λλ λ λ
λ λ λλ λ λ
λ λ λ λ
d d a A d d A
b
A
b
A
a
A
a A
d d a A d d x x y y
k i
i k i k
k i
i k i k ki
77
Linear Shape Functions II
1
2
3
ik ik ik ik ik ik
i k i k
1
0
1
0
1 2
1
0
1
0
1 2
2
2
∫ ∫
∫ ∫
−
−
λ
λ
( 2 )!
1 2 3
1 2 3 1 2 3
1 2 3
∫ Ω
m m m
78
Linear Shape Functions III
ki
i
i
ik
k i k i
Aa
i k
Aa a A
i k
Aa a A
NaNdxdy a dxdy
∫ ∫ Ω Ω
2 2
2 2
2 2