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2005 February, 2 Page 1
Finite Element AnalysisBasics – Part 2/2
Johannes Steinschaden
2005 February, 2 Page 2
Finite Element Analysis Procedure
1. Preliminary analysis of the system:Perform an approximate calculation to gain some insights about the system
2. Preparation of the finite element model:a Geometric and material information of the systemb Prescribe how is the system supportedc Determine how the loads are applied to the system
3. Perform the calculation:Solve the system equations and compute
displacements, strains and stresses
4. Post-processing of the results:Viewing the stresses and displacementsInterpret the results
2005 February, 2 Page 3
Direct Stiffness MethodTwo-dimensional Truss Elements
⎭⎬⎫
⎩⎨⎧
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡+−−+
=⎭⎬⎫
⎭⎬⎫
⎩⎨⎧
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡+−−+
=⎭⎬⎫
2
1
2
1
2
1
2
1
1111xx
LEA
FF
xx
KKKK
FF
F1 K=EA/L F2
N1 N2
x2x1
2005 February, 2 Page 4
ad Two-dimensional Truss Elements
X
YF2
F1K
N1
N2X1
Y2 X2Y1
local stiffness matrix
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎪⎪⎩
⎪⎪⎨
⎧
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−
−+
=
⎪⎪⎭
⎪⎪⎬
⎫
2
2
1
1
2
2
1
1
0000010100000101
YXYX
LEA
FFFF
Y
X
Y
X
2005 February, 2 Page 5
ad Two-dimensional Truss Elements
Coordinate transformation equation
( ) ( )( ) ( )
( ) ( )( ) ( )( ) ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( ) ⎪
⎪⎩
⎪⎪
⎨
⎧
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−=
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−=
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎩
⎪⎨⎧
⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡−
=⎭⎬⎫
==
Yg
Xg
Yg
Xg
Y
X
Y
X
g
g
g
g
g
g
FFFF
cssc
cssc
FFFF
YXYX
cssc
cssc
YXYX
YX
cssc
YX
cs
2
2
1
1
2
2
1
1
2
2
1
1
2
2
1
1
1
1
1
1
0000
0000
0000
0000
cossin
θθθθ
θθθθ
θθθθ
θθθθ
θθθθ
θθθθ
2005 February, 2 Page 6
ad Two-dimensional Truss Elements
} [ ]{ }{[ ] }{ [ ][ ] }{[ ] [ ] }{ [ ] [ ][ ] }{}{ [ ] [ ][ ] }{
[ ] [ ]}{ [ ] [ ][ ] }{}{ [ ] }{
[ ] [ ] [ ][ ]lKlK
XKFXlKlF
ll
XlKlF
XlKlFll
XlKFlXKF
tg
gg
gt
t
g
g
g
g
=
=
=
=
=
=
=
=
−
−
−−
1
1
11
2005 February, 2 Page 7
ad Two-dimensional Truss Elements
[ ]( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−−−−
−−−
=
θθθθθθθθθθθθ
θθθθθθθθθθθθ
22
22
22
22
sscsscsccscc
sscsscsccscc
LEAKg
2005 February, 2 Page 8
ad Two-dimensional Truss ElementsExample
Truss ATruss B
Node 1 Node 3
Node 2
F
2005 February, 2 Page 9
ad Two-dimensional Truss ElementsExample
Element A:
local stiffness matrix
global stiffness matrix
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ⎪
⎪⎩
⎪⎪⎨
⎧
⎪⎪⎭
⎪⎪⎬
⎫
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−−−−
−−−
=
⎪⎪⎩
⎪⎪⎨
⎧
⎪⎪⎭
⎪⎪⎬
⎫
2
2
1
1
22
22
22
22
2
2
1
1
YXYX
sscsscsccscc
sscsscsccscc
LEA
FFFF
g
g
g
g
A
A
YAg
XAg
Yg
Xg
θθθθθθθθθθθθ
θθθθθθθθθθθθ
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎪⎪⎩
⎪⎪⎨
⎧
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−
−+
=
⎪⎪⎭
⎪⎪⎬
⎫
2
2
1
1
2
2
1
1
0000010100000101
YXYX
LEA
FFFF
A
A
Y
X
Y
X
2005 February, 2 Page 10
ad Two-dimensional Truss ElementsExample
Element B:
local stiffness matrix
global stiffness matrix
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ⎪
⎪
⎩
⎪⎪
⎨
⎧
⎪⎪⎭
⎪⎪⎬
⎫
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−−−−
−−−
=
⎪⎪
⎩
⎪⎪
⎨
⎧
⎪⎪⎭
⎪⎪⎬
⎫
3
3
2
2
22
22
22
22
3
3
2
2
YXYX
sscsscsccscc
sscsscsccscc
LEA
FFFF
g
g
g
g
B
B
Yg
Xg
YBg
XBg
φφφφφφφφφφφφ
φφφφφφφφφφφφ
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎪⎪⎩
⎪⎪⎨
⎧
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−
−+
=
⎪⎪⎭
⎪⎪⎬
⎫
3
3
2
2
3
3
2
2
0000010100000101
YXYX
LEA
FFFF
B
B
Y
X
Y
X
2005 February, 2 Page 11
ad Two-dimensional Truss ElementsExample
Summing the two seta of global force-displacement equations:
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
−−
−++−−
−−++−−
−
−−
=
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
3
3
2
2
1
1
22
22
2222
2222
22
22
3
3
2
2
1
1
00
00
00
00
YXYXYX
sLAsc
LAs
LAsc
LA
scLAc
LAsc
LAc
LA
sLAsc
LAs
LAs
LAsc
LAsc
LAs
LAsc
LA
scLAc
LAsc
LAsc
LAc
LAc
LAsc
LAc
LA
sLAsc
LAs
LAsc
LA
scLAc
LAsc
LAc
LA
E
FFFFFF
g
g
g
g
g
g
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
A
A
B
B
A
A
A
A
A
A
B
B
B
B
B
B
A
A
B
B
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
Yg
Xg
Yg
Xg
Yg
Xg
φφφφφφ
φφφφφφ
φφφφθφφθθθθθ
φφφφφθθφθθθθ
θθθθθθ
θθθθθθ
2005 February, 2 Page 12
ad Two-dimensional Truss ElementsExample
Nodes 1 and 3 are fixed and only load on node 2 in global X direction
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
−−
−++−−
−−++−−
−
−−
=
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
00
00
00
00
00
00
2
2
22
22
2222
2222
22
22
3
3
1
1
YX
sLAsc
LAs
LAsc
LA
scLAc
LAsc
LAc
LA
sLAsc
LAs
LAs
LAsc
LAsc
LAs
LAsc
LA
scLAc
LAsc
LAsc
LAc
LAc
LAsc
LAc
LA
sLAsc
LAs
LAsc
LA
scLAc
LAsc
LAc
LA
E
FF
FF
g
g
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
A
A
B
B
A
A
A
A
A
A
B
B
B
B
B
B
A
A
B
B
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
Yg
Xg
Yg
Xg
φφφφφφ
φφφφφφ
φφφφθφφθθθθθ
φφφφφθθφθθθθ
θθθθθθ
θθθθθθ
0F
2005 February, 2 Page 13
ad Two-dimensional Truss ElementsExample
Nodes 1 and 3 are fixed and only load on node 2 in global X direction
Solve for nodal displacements:
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
⎪⎩
⎪⎨⎧
⎭⎬⎫
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
−−
++
++
−
−−
=
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
2
2
2
2
22
22
2
2
3
3
1
1
YX
sLAsc
LA
scLAc
LA
sLAs
LAsc
LAsc
LA
scLAsc
LAc
LAc
LA
sLAsc
LA
scLAc
LA
E
FF
FF
g
g
B
B
B
B
B
B
B
B
B
B
A
A
B
B
A
A
B
B
A
A
B
B
A
A
A
A
A
A
A
A
A
A
Yg
Xg
Yg
Xg
φφφ
φφφ
φθφφθθ
φφθθφθ
θθθ
θθθ
0F
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ⎪⎩
⎪⎨⎧
⎭⎬⎫
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
++
++=
⎩⎨⎧
⎭⎬⎫
2
2
22
22
YX
sLAs
LAsc
LAsc
LA
scLAsc
LAc
LAc
LA
E g
g
B
B
A
A
B
B
A
A
B
B
A
A
B
B
A
A
φθφφθθ
φφθθφθ
0F
2005 February, 2 Page 14
ad Two-dimensional Truss ElementsExample
Substitute the known displacements and solve for the reaction forces:
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
⎪⎩
⎪⎨⎧
⎭⎬⎫
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
−−
−
−−
=
⎪⎪
⎩
⎪⎪
⎨
⎧
⎪⎪⎭
⎪⎪⎬
⎫
2g
2g
YX
φφφ
φφφ
θθθ
θθθ
2
2
2
2
3
3
1
1
sLAsc
LA
scLAc
LA
sLAsc
LA
scLAc
LA
E
FFFF
B
B
B
B
B
B
B
B
A
A
A
A
A
A
A
A
Yg
Xg
Yg
Xg
2005 February, 2 Page 15
ad Two-dimensional Truss Elements
Truss element A:
( ) ( )( ) ( )
( ) ( )( ) ( )
⎪⎪⎩
⎪⎪⎨
⎧
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−
−+
=
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−=
⎪⎪⎭
⎪⎪⎬
⎫
2
2
1
1
2g
2g
YXYX
YX00
0000010100000101
0000
0000
2
2
1
1
2
2
1
1
A
A
Y
X
Y
X
LEA
FFFF
cssc
cssc
YXYX
θθθθ
θθθθ
2005 February, 2 Page 16
Stress and Momentum Balance
15 unknown variables
– 3 displacements – 6 strains– 6 stresses
15 equations
– 6 displacement-strain equations– 6 strain-stress equations– 3 equilibrium equations
[ ][ ][ ]zxyzxyzzyyxx
t
zxyzxyzzyyxxt
t wvuu
τττσσσσ
γγγεεεε
,,,,,
,,,,,
,,
=
=
=
0=+⋅
⋅=⋅=
pD
EuD
t σ
εσε
2005 February, 2 Page 17
Strains
xw
zu
yzv
yw
yu
xv
zw
yv
xu
zxyzxy
zzyyxx
∂∂
+∂∂
=∂∂
+∂∂
=∂∂
+∂∂
=
∂∂
=∂∂
=∂∂
=
γγγ
εεε
;;
;;
2005 February, 2 Page 18
ad Strains
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
∂∂
∂∂
∂∂
∂∂∂∂
∂∂
∂∂
∂∂
∂∂
=
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
wvu
xz
yz
xy
z
y
x
γγγεεε
zx
yz
xy
zz
yy
xx
0
0
0
00
00
00
uD ⋅=ε
2005 February, 2 Page 19
Material law
( )( ) ( ) ( )[ ]
( )( ) ( ) ( )[ ]
( )( ) ( ) ( )[ ]
( )
( )
( ) zxzx
yzyz
xyxy
yyxxzzzz
xxzzyyyy
zzyyxxxx
E
E
E
E
E
E
γν
τ
γν
τ
γν
τ
εενεννν
σ
εενεννν
σ
εενεννν
σ
+=
+=
+=
++−−+
=
++−−+
=
++−−+
=
12
12
12
1211
1211
1211
2005 February, 2 Page 20
ad Material law
( )( )
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
⋅
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−
−−
−−
−+=
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
zx
yz
xy
zz
yy
xx
zx
yz
xy
zz
yy
xx
E
γγγεεε
ν
ν
νννν
νννννν
νν
τττσσσ
22100000
02210000
00221000
000100010001
211
εσ ⋅= E
2005 February, 2 Page 21
Equilibrium Equations
2005 February, 2 Page 22
ad Equilibrium Equations
0=⋅⋅⋅
+⋅⋅⎟⎠⎞
⎜⎝⎛
∂∂
++⋅⋅−
−⋅⋅⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂++⋅⋅−
−⋅⋅⎟⎠⎞
⎜⎝⎛
∂∂
++⋅⋅−
dzdydxp
dydxdzz
zdydx
dzdxdyy
dzdx
dzdydxx
dzdy
x
zxzx
yxyxyx
xxxxxx
τττ
τττ
σσσ
0
0
0
=+∂∂
+∂
∂+
∂∂
=+∂
∂+
∂
∂+
∂
∂
=+∂∂
+∂
∂+
∂∂
zzzyzxz
yzyyyxy
xzxyxxx
pzyx
pzyx
pzyx
σττ
τστ
ττσ
2005 February, 2 Page 23
ad Equilibrium Equations
0
000
000
000
000
=+⋅
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
=
=
=
pD
ppp
xyz
zxy
zyx
t
z
y
x
zx
yz
xy
zz
yy
xx
xzzx
zyyz
yxxy
σ
τττσσσ
ττ
ττ
ττ
2005 February, 2 Page 24
Plain Stress
( )yyxxzz
xy
yy
xx
xy
yy
xx
zz
yzzxzz
E
εεννε
γεε
νν
ν
ντσσ
ε
ττσ
+−−
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⋅
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
≠
===
1
2100
0101
1
0
0;0;0
2
2005 February, 2 Page 25
Plain Strain
( )( )
( )yyxxzz
xy
yy
xx
xy
yy
xx
zz
zxyzzz
E
σσνσ
γεε
ννν
νν
νντσσ
σ
γγε
+=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⋅
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−
−+=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
≠
===
22100
0101
211
0
0;0;0
2005 February, 2 Page 26
Principle of Virtual WorkPrinciple of Virtual Displacements
dVdOqudVpuFu
dVW
dOqudVpuFuW
WW
V
t
O
t
V
tt
V
ti
O
t
V
tto
io
∫∫∫
∫
∫∫
⋅=⋅+⋅+⋅
⋅=
⋅+⋅+⋅=
=
σεδδδδ
σεδδ
δδδδ
δδ
2005 February, 2 Page 27
ad Principle of Virtual Work
( )
pdk
dOqGdVpGFGddVGDEGD
dOqGddVpGdFGdddVGDEDGd
Gdu
dGu
dOqudVpuFDuudVDEDu
uDEEDu
O
t
V
tt
V
t
O
tt
V
tttt
V
ttt
ttt
O
t
V
tttt
V
ttt
ˆ=⋅
⋅+⋅+⋅=⋅⋅⋅⋅⋅
⋅⋅+⋅⋅+⋅⋅=⋅⋅⋅⋅⋅⋅
⋅=
⋅=
⋅+⋅+⋅⋅=⋅⋅⋅⋅
⋅⋅=⋅=⋅=
∫∫∫
∫∫∫
∫∫∫
δδδδ
δδ
δδδδ
εσδεδ
2005 February, 2 Page 28
Basis Function
Example: two-dimensional beam element
Basis function to approximate displacement inside element
( ) ( )
[ ]
62
2
0
3
3
2
210
2
321
32
3
2211
4
1
xcxcxccw
xcxccw
xccwcw
wwd
dxgxw
w
t
ii
i
IV
+++=
++=′
+=′′=′′′=
=
=
∑=
ψψ
2005 February, 2 Page 29
ad Basis Function
( )
⎟⎠⎞
⎜⎝⎛ −−−=
+++−=
−==
=′
−=′
=
=
=
=
=
=
2212
2332
212133
212122
11
10
2
10
2
10
LLwwL
c
LLwwL
c
cwc
w
w
ww
ww
Lx
x
Lx
x
ψψ
ψψ
ψ
ψ
ψ
2005 February, 2 Page 30
ad Basis Function
( )
LLx
Lxg
Lx
Lxg
LLx
Lx
Lxg
Lx
Lxg
LLx
Lxw
Lx
LxL
Lx
Lx
Lxw
Lx
Lxxw
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−+−=
+−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−+−+⎟⎟
⎠
⎞⎜⎜⎝
⎛+−=
3
3
2
2
4
3
3
2
2
3
3
3
2
2
2
3
3
2
2
1
23
3
2
2
23
3
2
2
13
3
2
2
13
3
2
2
23
2
231
232231 ψψ
2005 February, 2 Page 31
ad Basis Function
2005 February, 2 Page 32
Thank you!
