Transcript Lecture 10 Notes

MECH300H Introduction to Finite
Element Methods
Lecture 10
Time-Dependent Problems
Time-Dependent Problems
In general,
ux, t 
Key question: How to choose approximate functions?
Two approaches:
ux, t   u j j x, t 
ux, t   u j t  j x
Model Problem I – Transient Heat Conduction
u   u 
c
  a   f x, t 
t x  x 
Weak form:
u
 w u
0  a
 cw
 wf
x x
t
x1 
x2
 du 
Q1  a  ;
 dx  x1

dx  Q1w( x1 )  Q2 w( x2 )

 du 
Q2  a 
 dx  x2
Transient Heat Conduction
n
u x, t    u j t  j x 
let:
and
w  i x 
j 1
u
 w u

0  a
 cw
 wf dx  Q1w( x1 )  Q2 w( x2 )
x x
t

x1 
x2
K u M u  F
i  j
K ij   a
dx
x x
x1
ODE!
x2
x2
M ij   ci j dx
x2
Fi   i fdx  Qi
x1
x1
Time Approximation – First Order ODE
du
a
 bu  f t 
dt
0t T
u0  u0
Forward difference approximation - explicit
t
uk 1  uk   f k  buk 
a
Backward difference approximation - implicit
t
uk 1  uk 
 f k  buk 
a  bt
Time Approximation – First Order ODE
du
a
 bu  f t 
dt
0t T
u0  u0
a - family formula:
uk 1  uk  t 1  a  uk  a uk 1 
Equation
uk 1 
a  1  a  tbuk  t a f k 1  1  a a f k 
a  atb
Time Approximation – First Order ODE
du
a
 bu  f t 
dt
0t T
u0  u0
Finite Element Approximation
2tb 
tb 
 f k 2 f k 1 


a 
 uk 1   a 
 uk  t  
3 
3 
3 


 3
Stability of a – Family Approximation
Example
Stability
A 1
a  1  a  tb
a  atb
1
FEA of Transient Heat Conduction
K u M u  F
a - family formula for vector:
uk 1  uk   t 1  a uk   a uk 1
uk 1   M   a  K  t   M   1  a   K  t uk   t 1  a  f k   at  f k 1
1
Stability Requirment
t  tcri 
where
2
1  2a max
K   M u  Q
Note: One must use the same discretization for solving
the eigenvalue problem.
Transient Heat Conduction - Example
u  2u
 2 0
t x
u0, t   0
ux,0  1.0
0  x 1
u
1, t   0
t
t0
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 EulerBernoulli Beam
 2   2u 
 2u
 EI 2   A 2  f x, t 
2 
x  x 
t
Weak form:
Where:
  2 w  2u
 2u
0    EI 2 2  Aw 2  wf
x x
t
x1 
x2

dx

 w 
 w 
 Q1w( x1 )  Q2  
  Q3 w( x2 )  Q4  

 x  x1
 x  x2
    2u 
  2u 
Q1    EI 2  Q2   EI 2 
 x  x
 x  x  x
1
1
    2u  
  2u 
Q3     EI 2  Q4    EI 2 
 x  x2
 x  x  x2
Transverse Motion of Euler-Bernoulli Beam
n
let:
u x, t    u j t  j x 
and
w  i x 
j 1
  2 w  2u
 2u
0    EI 2 2  Aw 2  wf
x x
t
x1 
x2

dx

 w 
 w 
 Q1w( x1 )  Q2  
  Q3 w( x2 )  Q4  

 x  x1
 x  x2
K u M u  F
Transverse Motion of Euler-Bernoulli Beam
K u M u  F
 i   j
K ij   EI 2
dx
2
x x
x1
x2
2
2
x2
M ij   Ai j dx
Fi   i fdx  Qi
x1
x2
x1
ODE Solver – Newmark’s Scheme
1 2

us 1  us  tus  t us 
2
us 1  us  us a t
where
us  1   us   us1
Stability requirement:
t  tcri
where
1 2

  max a   
2

K   M u  F
2

1
2
ODE Solver – Newmark’s Scheme
1
1
2
2
1
1
a  ,   2 
2
3
a  ,   2 
Constant-average acceleration method (stable)
Linear acceleration method (conditional stable)
1
a  ,   2  0
2
Central difference method (conditional stable)
3
8
a  ,   2 
2
5
Galerkin method (stable)
3
a  ,   2  2
2
Backward difference method (stable)
Fully Discretized Finite Element Equations
Transverse Motion of Euler-Bernoulli Beam
2w 4w
 4 0
2
t
x
w
0, t   0
w0, t   0
t
0  x 1
w1, t   0
wx,0  sin x  x1  x 
w
1, t   0
t
w
 x ,0   0
t
Transverse Motion of Euler-Bernoulli Beam
Transverse Motion of Euler-Bernoulli Beam
Transverse Motion of Euler-Bernoulli Beam