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JN Reddy
The Finite Element Method
Read: Chapter 6
1D Eigenvalue and Time-Dependent Problems
• Eigenvalue problems Structural problems Heat transfer-like problems
• Transient problems Semi-discretization Time approximations Mass lumping Stability and accuracy Numerical examples
CONTENTS
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INTRODUCTION
Equations of motion
Static problem (set all derivatives with respect to time
t to zero)
Construct FE model
Construct weak form
Semidiscretization(construct weak form in spatial
coordinates and semi-discrete FEM)
Full discretization (use time
approximations)
Eigenvalue problem (replace the solution with periodic or decaytype solution with respect to time t)
Construct FE model
Construct weak form
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INTRODUCTION (continued)
( )f x
Equilibrium configuration
• • • • • • • •1 2 3
4 56 7 8
3u
( , )f x t
Transient configurations
3 1( )u t
• • • • • • • •
2time,t
1time,t
3time,t
3
3 2( )u t 3 3( )u t
r
0 0
( , )
B.C. : (0, ) 0, ( , ) 0
I.C. : ( , 0) ( ), ( , 0) ( )
u uT f x t
t t x xu t u L t
u x u x u x v x
æ ö æ ö¶ ¶ ¶ ¶ç ÷ ç ÷- =ç ÷ ç ÷ç ÷ ç ÷ç ÷ ç ÷¶ ¶ ¶ ¶è ø è ø= == =
( )
(0) 0, ( ) 0
d duT f x
dx dxu u L
æ öç ÷- =ç ÷ç ÷è ø= =
Eigenvalue and Dynamics Problems : 3
Equilibrium (static) problem
Transient problem
Equation of equilibrium Equation of motion
JN Reddy Eigenvalue and Dynamics Problems : 4
EIGENVALUE PROBLEMS
w
w
r
r w
0
2 00
( , )
For periodic motion, set ( , ) ( )e
e ( ) ( ) 0
i t
i t
u uA EA f x t
t t x x
u x t u x
dudA i u x EA
dx dx
æ ö æ ö¶ ¶ ¶ ¶ç ÷ ç ÷- =ç ÷ ç ÷ç ÷ ç ÷ç ÷ ç ÷¶ ¶ ¶ ¶è ø è ø
=ì üæ öï ïç ÷ï ïç ÷- =í ýç ÷ï ïç ÷è øï ïî þ
2 00
200
( ) 0
( ) 0,
dudA u x EA
dx dxdud
A u x EAdx dx
r w
r l l w
æ öç ÷- - =ç ÷ç ÷è øæ öç ÷- - = =ç ÷ç ÷è ø
Natural vibration of barsGoverning Equation for transient response
Natural vibration
JN Reddy Eigenvalue and Dynamics Problems : 5
t
t
g T x t T x
T TA kAt x x
dTdA T x kA
dx dx
0
00
0, ( , ) ( )e
0
e ( ) ( ) 0
l
l
r
r l
-
-
= =
æ ö¶ ¶ ¶ç ÷- =ç ÷ç ÷è ø¶ ¶ ¶ì üæ öï ïï ïç ÷- - =í ýç ÷ç ÷è øï ïï ïî þ
T TA kA g x tt x x
( , )ræ ö¶ ¶ ¶ç ÷- =ç ÷ç ÷è ø¶ ¶ ¶
Governing equation for transient response
00( ) 0
dTdA T x kA
dx dxr l
æ öç ÷- - =ç ÷ç ÷è ø
Governing equation for eigenvalue analysis: for decay type solution, set
EIGENVALUE PROBLEMS Heat transfer-like problems
Transient configurations
• • • • • • • •
t2time,
t1time,t3time,
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FE MODELS OF EIGENVALUE PROBLEMS
01
( ) ( ) ( )
,b b
a a
n
h j jj
e e e e e
x xje ei
ij ij i jx x
u x u x u x
ddK EA dx M A dx
dx dx
y
l
yyr y y
=
» =
- =
= =
å
ò ò
K u M u Q
FE Model
Weak Form
1 2
1 2
0 ( )
( ) ( ) ( )
,
b
a
b
a
a b
xh
i hx
xi h
i h i a i bx
h h
x x
dudw A u x EA dx
dx dx
dw duA w u x EA dx w x Q w x Q
dx dxdu du
Q EA Q EAdx dx
r l
r l
ì üæ öï ïï ïç ÷=- +í ýç ÷ç ÷è øï ïï ïî þæ öç ÷= - + - -ç ÷ç ÷è ø
æ ö æ öç ÷ ç ÷=- =ç ÷ ç ÷ç ÷ ç ÷è ø è ø
ò
ò
Eigenvalue and Dynamics Problems : 6
Eigenvalue problem
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Linear element
[Ke] =aehe
1 −1−1 1
[Me] =ce0he6
2 11 2
Quadratic element
[Ke] =ae3he
⎡⎣ 7 −8 1−8 16 −81 −8 7
⎤⎦+[Me] =
ce0he30
⎡⎣ 4 2 −12 16 2−1 2 4
⎤⎦
Explicit form of the matrices involved
12
3x he
(b)
3e
bx x=ex2
1e
ax x=x
1 2x he
(a)
3e
bx x=
1e
ax x=x
Eigenvalue and Dynamics Problems : 7
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EXAMPLES: Axial vibrations of a bar (fundamental frequency)
Deformable bar
Problem 1
x
E, A k
L Deformable bar
Linear springProblem 2
1 1
2 2
1
2
21 1 2 161 1 1 2
u uEA ALu uL
wrì ü ì üé ù é ù- ï ï ï ïï ï ï ïê ú ê ú-í ý í ýê ú ê úï ï ï ï-ë û ë ûï ï ï ïî þ î þ
ì üï ïï ï= í ýï ïï ïî þ
0 0
0
22 2
2 306
orEA AL EuL L
rw w
ræ ö÷ç - = =÷ç ÷çè ø
1 1
2 2
1
2
21 1 2 161 1 1 2
u uEA ALu uL
wrì ü ì üé ù é ù- ï ï ï ïï ï ï ïê ú ê ú-í ý í ýê ú ê úï ï ï ï-ë û ë ûï ï ï ïî þ î þ
ì üï ïï ï= í ýï ïï ïî þ
0 0
2ku-
22
2
03
3( / A)or
EA ALk uL
E kLL
rw
wr
æ ö÷ç + - =÷ç ÷çè ø
+=
Eigenvalue and Dynamics Problems : 8
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TRANSIENT ANALYSIS(steps involved)
2 2 2
1 2 02 2 2 ( , )u u u uc c a b c u f x tt t x x x x
æ öæ ö¶ ¶ ¶ ¶ ¶ ¶ ÷ç÷ç ÷+ - + + =÷ çç ÷÷ç ç ÷çè ø¶ ¶ ¶ ¶ ¶ ¶è ø
Model Equation
Approximate solution
1
( , ) ( , ) ( ) ( )n
h j jj
u x t u x t t xϕΔ=
» =å1. Spatial approximation (semidiscretization)
2. Time approximation (full discretization)
Δ Δ Δ+ + =C M K F
1 , 1K̂ Fs s sΔ + +=
Eigenvalue and Dynamics Problems : 9
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SPATIAL APPROXIMATION
2 2 2
1 2 02 2 2 ( , )u u u uc c a b c u f x tt t x x x x
æ öæ ö¶ ¶ ¶ ¶ ¶ ¶ ÷ç÷ç ÷+ - + + =÷ çç ÷÷ç ç ÷çè ø¶ ¶ ¶ ¶ ¶ ¶è ø
2 2
1 2 02 2 2
2 2 2
1 2 02 2 2
0 ( ) ( , )
( , )
b
a
b
a
xh h h h
i hx
xh h i h i h
i i i h ix
i
u u u uw x c c a b c u f x t dxt t x x x x
u u dw u d w uc w c w a b c w u w f x t dxt t dx x dx x
w
é ùæ ö æ ö¶ ¶ ¶ ¶¶ ¶÷ ÷ç çê ú= + - + + -÷ ÷ç ç÷ ÷ç çê úè ø è ø¶ ¶ ¶ ¶ ¶ ¶ë û
é ùæ öæ ö¶ ¶ ¶ ¶ ÷ç÷ê úç ÷= + + + + -÷ çç ÷÷ê úçç ÷çè ø¶ ¶ ¶ ¶è øë û
-
ò
ò
1 3 2 4( ) ( )a b
i ia i b
x x
dw dwx Q w x Q Q Qdx dx
æ ö æ ö÷ ÷ç ç- - - - -÷ ÷ç ç÷ ÷ç çè ø è ø
Model Equation
Weak Form for semi-discretization
Eigenvalue and Dynamics Problems : 10
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Approximation
Finite element model
1 2
22
02 2
1 3 2 4
,
( ) ( )
b b
a a
b
a
b
aa b
x xe eij i j ij i jx x
x j je i iij i jx
xe i ii i i a i bx
x x
C c dx M c dx
d dd dK b a c dxdx dx dx dx
d dF f dx x Q x Q Q Qdx dx
jj jj
j jj jjj
j jj j j
Δ Δ Δ+ + == =
æ ö÷ç ÷ç= + + ÷ç ÷ç ÷çè øæ ö æ ö÷ ÷ç ç÷ ÷= + + + - + -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
ò ò
ò
ò
C M K F
SPATIAL DISCRETIZATIONFinite Element Model
1
( , ) ( , ) ( ) ( )n
h j jj
u x t u x t t xjΔ=
» =å
Eigenvalue and Dynamics Problems : 11
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PARABOLIC EQUATION (heat transfer, fluid mechanics, and like problems
1 0
1 2
,
( ) ( )
yyyy yy
y y y
+ =
æ ö÷ç ÷= = +ç ÷ç ÷çè ø
= + +
ò ò
ò
Cu Ku F
b b
a a
b
a
x x je e iij i j ij i jx x
xe
i i i a i bx
ddC c dx K a c dxdx dx
F f dx x Q x Q
TIME APPROXIMATIONS
HYPERBOLIC EQUATION (structural mechanics problems)
C M K FΔ Δ Δ+ + =
Eigenvalue and Dynamics Problems : 12
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TIME APPROXIMATIONS OF PARABOLIC EQUATIONS
D
D
1
1
11
1
, forward difference
, backward difference
s sj js
js
s sj js
js
u uu
t
u uu
t
+
++
+
+
-»
-»
Approximation of thefirst derivative
a a a
a a
D
D
11
1
1 11
(1 ) , 0 1
(1 )
s sj js s
j js
s s s sj j s j j
u uu u
t
u u t u u
++
+
+ ++
-+ - » £ £
é ù= + + -ê úë û
Alfa (α)-family of approximation
Eigenvalue and Dynamics Problems : 13
11( )s
j j su u t--=
( )sj j su u t=
11( )s
j j su u t++=
stD
1sju+
0s =1s =
2s = 1s - s 1s +
( )ju t
t
•
• •
sju
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TIME APPROXIMATIONS (Parabolic)
Cu Ku F Cu K u F Cu K u F
u u u u
Cu Cu Cu Cu
s s s ss s s s
s s s ss
s s s ss
t T
t
t
a a
a a
D
D
1 11 1
1 11
1 11
, 0 ,
(1 )
(1 )
+ ++ +
+ ++
+ ++
+ = < < + = + =
é ù= + + -ê úë ûé ù= + + -ê úë û
Alfa-family of approximation (Parabolic equation)
Cu F K us s ss
1 1 11 + + +
+= -
( ) ( ) C K u C K u
F F
s ss s s s
s ss
t t
t
a a
a a
D D
D
11 1
11
(1 )
(1 )
++ +
++
+ = - -
é ù+ + -ê úë û1 1
1
1 1 1
1 11 1 1
ˆ ˆ
ˆ ,
ˆ (1 ) (1 )
s ss
s s s
s s s ss s s
where
t
t t
a
a a a
D
D D
+ ++
+ + +
+ ++ + +
=
= +
é ù é ù= - + + + -ê ú ê úë û ë û
K u F
K K C
F K C u F F
Cu F K us s ss = -
Eigenvalue and Dynamics Problems : 14
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C u M u K u Fe e e e e e e+ + =
( )u u u u
u u u u u u
s s s s
s s s s s s
t t
t
g
a q q q
D D
D
21 ,12
1 , , 1, (1 )
+
+ +
= + +
= + º - +
Semidiscrete FE model
Newmark scheme (hyperbolic equations)
( ) ( )
K u F , K K M C
F F M u u u C u u u
s ss s s s s
s s s s s s s ss s
a a
a a a a a a
1 11 1 1 3 1 5 1
1 11 3 4 5 1 5 6 7
ˆ ˆ ˆ
ˆ
+ ++ + + + +
+ ++ +
= = + +
= + + + + + +
Fully discretized model
TIME APPROXIMATIONS (Hyperbolic)
Eigenvalue and Dynamics Problems : 15
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General form of the time-marching scheme
The scheme is called explicit if the coefficient matrix is diagonal (and hence, no inversion of equations is necessary); otherwise, the scheme is said to be implicit.
s+1K u F
K K C F K C u F FD D D
1 1
1 1 11 1 1 1 1
ˆ
ˆ ˆ, (1 ) (1 )
s s
s s s s ss s s s st t ta a a a
+ +
+ + ++ + + + +
=
é ù é ù= + = - + + + -ê ú ê úë û ë û
The alfa-family scheme is explicit if and only ifC(1) 0 and (2) is diagonal .a =
Ku Bu F1 , 1ˆ s s s s+ += +
Eigenvalue and Dynamics Problems : 16
K̂
EXPLICIT AND IMPLICIT FORMULATIONS
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ROW-SUM MASS LUMPING
Eigenvalue and Dynamics Problems : 17
For the Lagrange linear and quadratic elements we have
[Me]C =ρhe6
2 11 2
, [Me]C =ρhe30
⎡⎣ 4 2 −12 16 2−1 2 4
⎤⎦
[Me]L =ρhe2
1 00 1
, [Me]L =ρhe6
⎡⎣ 1 0 00 4 00 0 1
⎤⎦
6e eA hr
2e eA hr
6e eA hr
30e eA hr
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STABILITY OF APPROXIMATIONS
K u Ku F
u Au B
1 1 , 1
1 , 1
ˆ s s s s s
s s s s
+ + +
+ +
= +
= +
The scheme is called stable if the repeated solution of the above equation does not result in unbounded solution us+1. The necessary and sufficient condition for the above scheme to be stable is that the maximum eigenvalue of the coefficient matrix A is less than unity:
max 1Al £
Eigenvalue and Dynamics Problems : 18
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STABILITY OF APPROXIMATIONS (continued)
( )
( )
D D
12
12
critmax
, the scheme is
, the scheme is
2Stability condition:
(1 2 )
0.0, Forward difference Euler scheme (conditionally stable)
0.5, Crank-Nicolson's scheme (stabl
stable
conditionally stable
t t
a
a
a l
a
a
³
<
£ =-
=
= e)
2, Galerkin's scheme (stable)
31.0, Backward difference scheme (stable)
a
a
=
=
Alfa-family of approximation scheme
Eigenvalue and Dynamics Problems : 19
( )- C K u 0+ =l
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STABILITY OF APPROXIMATIONS (continued)
( )D Dcrit
max
13
2Stability condition:
( )
0.5, 2 0.5, Constant-average acceleration scheme (stable)
0.5, 2 , Linear acceleration scheme (conditionally stable)
1.5, 2 1.6, Galerkin's scheme (stab
t ta g l
a g b
a g b
a g b
£ =-
= = =
= = =
= = = le)
1.5, 2 2.0, Backward difference scheme (stable)a g b= = =
Newmark’s scheme for Structural Dynamics
gets smaller as the mesh is refined.( )crittD
Eigenvalue and Dynamics Problems : 20
( )- M K u 0+ =l
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0.0 1.0 2.0 3.0 4.0 5.0 6.0
Time, t
-2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
Tem
pera
ture
,u(
1,t)
1 linear element (α = 0.0, Δt = 0.675)
2 linear elements (α = 0.0, Δt = 0.065)
2 linear elements (α = 0.5, Δt = 0.065)
NUMERICAL EXAMPLES
Heat transfer in a rod
2
2 0, 0 1
(0, ) 0, (1, ) 0
( , 0) 1
u ux
t xu
u t tx
u x
¶ ¶- = < <¶ ¶
¶= =¶=
2 / 3 0.66667
2 / 31.689 0.063
one-element model:
two-element model: crit
crit
t
t
Δ = =
Δ = =
Eigenvalue and Dynamics Problems : 21
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NUMERICAL EXAMPLES (continued)
Bending of a clamped beam
p p
2 4
2 4 0, 0 1
(0, ) 0, (0, ) 0,
(1, ) 0, (1, ) 0
( , 0) sin (1 )
w wx
t xw
w t txw
w t tx
w x x x x
¶ ¶+ = < <
¶ ¶¶= =¶¶= =¶
= - -
12 / 516.930.023214
0.00897
one-element model:
two-element model:
Δ ==
Δ =
crit
crit
t
tEigenvalue and Dynamics Problems : 22
w(0
.5,t)
0.0 0.5 1.0 1.5 2.0
Time, t
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
Def
lect
ion,
175.0=Δt15.0=Δt
175.0=Δt
α = 0.5
(γ = 0.5)
(γ = 1/3)05.0=Δt
C.S.
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NUMERICAL EXAMPLES (continued)
Eigenvalue and Dynamics Problems : 23
0.0 0.1 0.2 0.3 0.4 0.5
Time, t
-0.20
-0.10
0.00
0.10
0.20
Def
lect
ion,
2 elements1 element
Δt = 0.005
w(0
.5, t
)w(0.5, t)●
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STRUCTURAL STABILITY (Buckling)
L x
P
z
Beam-column
P
L
Governing Equation2 2
2 2
2
20d d w d wEI P
dx dx dx
+ =
Weak Form over an Element2 2
2 2
2 2
2 2
1 3 2 4
0
( ) ( )
b
a
b
a
a b
x
x
x
x
a b
x x
d d d dwv EI dxdx dx dx dxd v d dv
P
dwEI dxdx dx dx dx
dv dvv x Q v x Q Q Qdx dx
w
w P
= +
= −
− − − − − −
Finite Element Model
( )e e e eP− =K G Δ Q
JN Reddy 2-D Problems: 25
STRUCTURAL STABILITY (continued)
( )e e e eP− =K G Δ Q2 2
2 2,b
a
b
a
x i jeij x
x i jeij x
d dK EI dxdx dx
d dG dxdx dx
j j
j j
=
=
1 2ee21 Δ≡θ ee
42 Δ≡θ
eew 11 Δ≡ eew 32 Δ≡
eh
ee q,Q 22 ee q,Q 44
ee q,Q 11ee q,Q 33
1 2ee21 Δ≡θ
eh
For element-wise constant values of EeIe and qe:
[Ke] =2EeIeh3e
⎡⎢⎣6 −3he −6 −3he−3he 2h2e 3he h2e−6 3he 6 3he−3he h2e 3he 2h2e
⎤⎥⎦2 2
2 2
36 3 36 33 4 336 3 36 303 3 4
13
e e
e e e e
e
e e
e
e
e e e
h hh h h h
h hh h h h
h
− − − −
− −
=
−
G
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EXAMPLE: Buckling of a clamped-clamped beam
2 2 2 2
2 2 2
3
2
6 3 6 3 36 3 36 33 2 3 3 4 326 3 6 3 36 3 36 3
3 4 33
3 40
3
e e e e
e e e e e e e e
e e e e
e e e e e e e e
e e
h h h hh h h h h h h hEI
h h h hh h h h h h
hh
P
hh
− − − − − − − − − − − − − −
1
2
3
4
UUUU
(1)1
(1)2
(1)3
(1)4
QQQQ
=
1
2
3 3
4
00
0
UUU UU
=
(1) (1)1 1
(1) (1)2 2
(1)3
(1) (1)4 4
0
Q QQ QQQ Q
=
33 3 2
12 36 12 300 1030 36
EI EI L EIU PL L L L
P =
− = =
One-element (EBT) in half beam
Boundary conditions
Solution x
z
P
●
●
1
2
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SUMMARY
In this lecture the following topics were covered:
Transient and eigenvalue problems and their finite element formulations
Time-approximations of first-order andsecond-order equations
Explicit and implicit schemes, mass lumping, as well as stability of the schemes
Numerical examples to illustrate tostability and accuracy.
Buckling of beams
Eigenvalue and Dynamics Problems : 27