mech593 introduction to finite element methods eigenvalue problems and time-dependent problems
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MECH593 Introduction to Finite Element Methods
Eigenvalue Problems and Time-dependent Problems
Eigenvalue Problems
A Bu uDefinition:
Examples:
Eigenvalue problems refer to the problems with the following type:
where A, B are operators.
1 2 1 1
3 4 2 2
a a u u
a a u u
2
2
d uu
dx
,G x y u x dx u y2 2 2
2 2 20
d d w d wEI N
dx dx dx
Eigenvalue Problems – Engineering Applications
Example: Vibration of an axial bar
2
2,
u uA EA f x tt x x
General solution scheme: h pu u u
, i tu x t U x e let
2 0d dU
EA AUdx dx
Eigenvalue Problems – A Model Problem
Model problem: 0
d dUa x c x U x c x U x
dx dx
Weak form: 0 1
1 ,
b
a
a b
x e ea n bx
e en
x x
dw dUa cwU c wU dx Q w x Q w xdx dx
dU dUQ a Q a
dx dx
1eQ
enQ
ax bx
Assume: 1
ne e e
j jj
U x u x
e e e e eK u M u Q
0 b b
a a
eex xje e e e e eiij i j ij i jx x
ddK a c dx M c dx
dx dx
Assembling: K M u Q
Eigenvalue Problems – Heat Conduction2
02
T Tc
x t
Non-dimensional variables/parameters:
20 0
T Tx t
x t uc L L T T
0 L
x
Initial condition: 00,T x T
Boundary conditions: 0,T t T
0x L
TT T
x
2
20
u u
x t
1
0, 1 0, 0 0 x
u Lu x u t Hu H
x
Eigenvalue Problems – Heat Conduction
Let ,u x t U x T t
0 L
x
Initial and boundary conditions:
2
21
0 0 0 0x
d U dUU U HU
dx dx
U T
U T
Element equation for linear element:
1 1
2 2
1 1 2 11
1 1 1 26
U Qh
U Qh
2
20
u u
x t
1
0, 1 0, 0 0 x
u Lu x u t Hu H
x
Eigenvalue Problems – Heat Conduction
2
21
0 0 0 0x
d U dUU U HU
dx dx
Element equation for linear element:
1 1
2 2
1 1 2 11
1 1 1 26
U Qh
U Qh
1 2
h h
Assembled system: 1 1
2 2
3 3
1 1 0 2 1 01
1 2 1 1 4 16
0 1 1 0 1 2
U Qh
U Qh
U Q
Boundary conditions:1
2 2
3 3
1 1 0 2 1 0 01
1 2 1 1 4 1 06
0 1 1 0 1 2
Qh
U Qh
U HU
Eigenvalue Problems – Heat Conduction
2
21
0 0 0 0x
d U dUU U HU
dx dx
1 2
h h
Condensed system:
2
3
4 2 4 1 0
2 2 1 2 012
U
UH
Eigenvalues:
Eigenvectors:
Mode shapes:
Time-Dependent Problems
In general,
Key question: How to choose approximate functions?
Two approaches:
txutxu jj ,,
txu ,
xtutxu jj ,
Model Problem I – Transient Heat Conduction
Weak form:
txfx
ua
xt
uc ,
)()(0 2211
2
1
xwQxwQdxwft
ucw
x
u
x
wa
x
x
;21
21xx dx
duaQ
dx
duaQ
Transient Heat Conduction
let:
)()(0 2211
2
1
xwQxwQdxwft
ucw
x
u
x
wa
x
x
n
jjj xtutxu
1
, and xw i
FuMuK
2
1
x
x
jiij dx
xxaK
2
1
x
x
jiij dxcM
i
x
x
ii QfdxF 2
1
ODE!
Time Approximation – First Order ODE
tfbudt
dua Tt 0 00 uu
Forward difference approximation - explicit
Backward difference approximation - implicit
1k k k k
tu u f bu
a
1 1k k k k
tu u f bu
a b t
𝑑𝑢𝑑𝑡 |𝑡=𝑡 𝑘≈
𝑢𝑘+1−𝑢𝑘
∆ 𝑡
𝑑𝑢𝑑𝑡 |𝑡=𝑡 𝑘+1
≈𝑢𝑘+1−𝑢𝑘
∆ 𝑡
Time Approximation – First Order ODE
tfbudt
dua Tt 0 00 uu
a - family formula:
1 11k k k ku u t u u
Equation
𝑢𝑘+1=[𝑎− (1−𝛼 )∆ 𝑡𝑏 ]𝑢𝑘+∆ 𝑡 [𝛼 𝑓 𝑘+1+(1−𝛼 ) 𝑓 𝑘 ]
𝑎+𝛼∆ 𝑡𝑏
,
Time Approximation – First Order ODE
tfbudt
dua Tt 0 00 uu
Finite Element Approximation
11
22
3 3 3 3k k
k k
f ftb tba u a u t
𝑢 (𝑡 )=∑𝑚
𝑢𝑚𝜙𝑚 (𝑡 ) 𝑡𝑘≤ 𝑡≤ 𝑡𝑘+1
𝑚=2 , linear element ,
Weak form : 𝑡 𝑘
𝑡 𝑘+1
𝑤 (𝑡 ) [𝑎 𝑑𝑢𝑑𝑡 +𝑏𝑢− 𝑓 ]𝑑𝑡=0
{𝑎2 [−1 1−1 1]+𝑏∆ 𝑡6 [2 1
1 2]}{ 𝑢𝑘
𝑢𝑘+1}= ∆𝑡
6 { 𝑓 𝑘
2 𝑓 𝑘+1}
Stability of – Family Approximation
Stability
a
Example
11 1
a tbA
a tb
𝑢𝑘+1=A ∙𝑢𝑘+𝐹 𝑘 ,𝑘+1
𝑢𝑘+1=𝐴𝑘+1 ∙𝑢0+𝐴𝑘∙𝐹 0 ,1+⋯+𝐹 𝑘 ,𝑘+1
FEA of Transient Heat Conduction
FuMuK
a - family formula for vector:
1 11k k k ku u t u u
1
1 11 1k k k ku M K t M K t u t f t f
Stability Requirement
max21
2
critt
QuMK where
Note: One must use the same discretization for solvingthe eigenvalue problem.
Transient Heat Conduction - Example
02
2
x
u
t
u 10 x 0,0 tu 0,1
tx
u
0.10, xu
0t
Element equation for linear element
1 1 1
2 2 2
2 1 1 11
1 2 1 16
u u Qh
u u Qh
1
1 11 1k k k ku M K t M K t u t Q t Q
a - family formula:
Initial condition: 1 1 2 20 0 1u x u x
Boundary conditions: 1 20 0k k
u Q for 0k
One element mesh:
[ 13h+𝛼 ∆ 𝑡
h16h−𝛼 ∆ 𝑡
h16h−𝛼
∆ 𝑡h
13h+𝛼
∆𝑡h
]{𝑢1
𝑢2}𝑘+1
=[ 13 h− (1−𝛼 ) ∆ 𝑡h
16h+(1−𝛼 ) ∆ 𝑡
h16h+(1−𝛼 ) ∆ 𝑡
h13h− (1−𝛼 ) ∆ 𝑡
h]{𝑢1
𝑢2}𝑘
+∆ 𝑡 {𝑄1
𝑄2}
𝑢1 (0 )=1 ,𝑢2 (0 )=1
and𝜕𝑢𝜕 𝑥|𝑡=0
=0 𝑄2 (0 )=0
Transient Heat Conduction - Example
Element equation:
2 1
322 6
3
h
uh t t
h
2 211
3 3k k
h t h tu u
h h
1k for
Stability requirement: max21
2
critt QuMK
1 1
2 2
1 1 2 11
1 1 1 26
u Qh
u Qh
2
3crit t
Transient Heat Conduction - Example
Transient Heat Conduction - Example
Transient Heat Conduction - Example
Transient Heat Conduction - Example
Transient Heat Conduction - Example
Model Problem II – Transverse Motion of Euler-Bernoulli Beam
Weak form:
txft
uA
x
uEI
x,
2
2
2
2
2
2
21
2
1
423211
2
2
2
2
2
2
)()(
0
xx
x
x
x
wQxwQ
x
wQxwQ
dxwft
uAw
x
u
x
wEI
22
11
2
2
42
2
3
2
2
22
2
1
xx
xx
x
uEIQ
x
uEI
xQ
x
uEIQ
x
uEI
xQ
Where:
Transverse Motion of Euler-Bernoulli Beam
let:
n
jjj xtutxu
1
, and xw i
FuMuK
21
2
1
423211
2
2
2
2
2
2
)()(
0
xx
x
x
x
wQxwQ
x
wQxwQ
dxwft
uAw
x
u
x
wEI
Transverse Motion of Euler-Bernoulli Beam
FuMuK
2
1
2
2
2
2x
x
jiij dx
xxEIK
2
1
x
x
jiij dxAM
i
x
x
ii QfdxF 2
1
ODE Solver – Newmark’s Scheme
tuuu
ututuu
sss
ssss
1
21 2
1
11 sss uuu where
Stability requirement:
2
1
2max2
1
critt
FuMK 2where
ODE Solver – Newmark’s Scheme
2
12 ,
2
1 Constant-average acceleration method (stable)
3
12 ,
2
1
02 ,2
1
5
82 ,
2
3
22 ,2
3
Linear acceleration method (conditional stable)
Central difference method (conditional stable)
Galerkin method (stable)
Backward difference method (stable)
Fully Discretized Finite Element Equations
2
1
1
2s s s su u t u t u
1 11 1 1s ss s sK u M u F
2 2
1 1 11 1s s ss s sM t K u M b t F
211
2s s s sb u t u t u
2 21 1 11 1
1 1s s ss s s
M K u M b Ft t
One needs: 0 0 0, , u u u
1
0 0 0u M F K u
Element equation
, , s s s
u u u 1 1 1, ,
s s su u u
Transverse Motion of Euler-Bernoulli Beam
04
4
2
2
x
w
t
w10 x
0,0 tw 0,1
tt
w
xxxxw 1sin0,
0,1 tw 0,0
tt
w
00,
xt
w
Element equation of one element:
1 1 12 2 2 2
2 2 2
33 3 3
2 2 2 24 4 4
156 22 54 13 6 3 6 3
22 4 13 3 3 2 32
54 13 156 22 6 3 6 3420
13 3 22 4 3 3 2
u u Qh h h h
u u Qh h h h h h h hhu u Qh h h hh
u u Qh h h h h h h h
Transverse Motion of Euler-Bernoulli BeamSymmetry consider only half the beam 0 0.5x
Boundary conditions: 1 2 4 30, =0, =0, Q =0u u u
Initial conditions: xxxxw 1sin0, 1 2 3 40, =0, =0.2146, =0u u u u
1 2 3 40, =0, =0, =0u u u u
33 3 33 3 33 3 3 4 3 5 31
3 4 3 52
1 1, , 1
s sK a M u M a u a u a u
a a a t at
Imposing bc and ic:
33 3 03 0
33
K uu
M
𝜃 (𝑥 ,0 )=𝜋 cos𝜋 𝑥−𝜋 (1−2 𝑥 )
Transverse Motion of Euler-Bernoulli Beam
Transverse Motion of Euler-Bernoulli Beam