mech300h introduction to finite element methods lecture 10 time-dependent problems
Post on 21-Dec-2015
241 views
TRANSCRIPT
MECH300H Introduction to Finite Element Methods
Lecture 10
Time-Dependent Problems
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
1k k k k
tu u f bu
a b t
Time Approximation – First Order ODE
tfbudt
dua Tt 0 00 uu
- family formula:
1 11k k k ku u t u u
Equation
1
1
1 1k k k
k
a tbu t f fu
a 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
Stability of – Family Approximation
Stability
Example
11 1
a tbA
a tb
FEA of Transient Heat Conduction
FuMuK
- 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 Requirment
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
tt
u
0.10, xu
0t
Transient Heat Conduction - Example
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
Transverse Motion of Euler-Bernoulli Beam
04
4
2
2
x
w
t
w 10 x
0,0 tw 0,1
tt
w
xxxxw 1sin0,
0,1 tw 0,0
tt
w
00,
xt
w
Transverse Motion of Euler-Bernoulli Beam
Transverse Motion of Euler-Bernoulli Beam
Transverse Motion of Euler-Bernoulli Beam