Week - 2 - Explicit and Implicit Analysis
AIM:
To solve a non- linear dynamic analysis using both implicit and explicit methods.
To solve the equation F(u) = u3+9u2+4u, using both implicit and explicit method.
Implicit and Explicit analysis are used to solve dynamics problems.
EXPLICIT ANALYSIS :-
Explicit analysis are the analysis where there is dynamic equilibrium, the explicit method should be used where the strain rates are higher, eg in automotive crash, meteor impact.
IMPLICIT ANALYSIS :-
Implicit analysis should be used where the events are much slower and the effects of strain rates are minimal. Implicit analysis involves matrix inversions that are extremely complicated and do not directly scale with the number of processors.
Explicit method:
F(u) = u3+9u2+4u
Stiffness of the bar k(u) is :
dF(u)/du = K(u) = d\du{u3+9u2+4u}
dF(u)/du = K(u) = 3u2+18u+4
Step 1 :
u = 0
k(u0) = 4 , delta F = 1
delta u1 = delta F/k(u0)
= ¼ = 0.25
F_int = f(u1) = (0.25)3 +9(0.25)2 + 4*0.25
= 0.015625+ 0.5625 + 1 = 1.578125
Residual = f_int – f_ext = 1.578125 – 1
= 0.578125
Step 2:
F(u1) = 1.578125
U1 = 0.25
K(u1) = 3*(0.25)2 + 18*0.25 + 4 = 0.1875 + 4.5 + 4 = 8.6875
delta u2 = delta F/k(u1) = 1/8.6875 = 0.1151
u2 = u1 + delta u2 = 0.25+0.1151 = 0.3651
f_int = f(u2) = (0.3651)3 + 9*(0.3651)2 + 4*0.3651 = 2.70806
f_ext = 1+1 = 2
Residual = f_int = f_ext = 2.70806-2 = 0.70806
Step 3:
F(u2) = 2.70806
K(u2) = 3*(0.3651)2 + 18 *0.3651 + 4 = 0.4 + 6.5718 + 4 = 10.9718
u2 = 0.3651
delta u3 = 1/10.9718 = 0.0911
u3 = u2+delta u3 = 0.3651+0.0911 = 0.4562
f_int = f(u3) = (0.4562)3 + 9(0.4562)2 + 4*0.4562 = 0.095+1.873 + 1.8248
= 3.7928
F_ext = 1+1+1 = 3
Residual = f_int-f_ext = 3.7928-3 = 0.7928
|
U (exp) |
F(u) (exp) |
|
0 |
0 |
|
0.250 |
1.578125 |
|
0.3651 |
2.7081 |
|
0.4562 |
3.7928 |
Implicit Method :
Implicit analysis is the same as explicit, except that at the end of each step a newton-raphson iterations are used to enforce equilibrium.
Step 1:-
Consider , U(0) = 0
K(u0) = 9u2 +18u+4 = 4
Delta u = deltaf/k(0) = ¼ = 0.25
U1= u(0)+delta u = 0.25
F_ext = 1
F _int = (u1)3 + 9(u1)2 + 4u1 = (0.25)3 + 9(0.25)2 + 4*0.25
= 1.518
Residual = R0 = f_int – f_ext = 1.518 – 1
= 0.518
Which is greater than 0.01 thus newton raphson model has to be followed,
U1 = U10
Delta U1 =[-k(u1)0]-1 *R0 = -0.06
Then the value of u1 will be
U11 = u10 +delta u1 = 0.25 -0.06 = 0.19
R1 = f_int – f_ext = 1.0469-1 = 0.0469 > 0.01
U1 = U10
Delta u2 = delta u1- [ku11]-1 *r1 = -0.06 - 0.005 = -0.0713
U12 = u10 + delta u2 = 0.25 – 0.0713 =0.178
F_int = (u13)+9(u2)+4u
= (0.178)3 + 9*(0.178)2 + 4*0.178
=1.007
R2 = 1.007-1 = 0.007
The R2 value is less than 0.01, thus the value of u1 = 0.178
Step 2:
U = u1 = 0.178
K(u1) = 9*(0.178)2 + 18*0.178 + 4 = 7.489
Delta u2 = delta f / k(u1) = 1/7.489 = 0.133
u2 = u1 + delta u2 = 0.178+0.133 = 0.311
f_ext = 1+1 = 2
f_int = (0.311)3 + 9*(0.311)2 + 4*0.311 = 2.144
Residula R0 = f_int – f_ext = 2.144-2 = 0.144 > 0.01
Thus solving for newton raphson method
U2 = 0.296
Step 3 :
u=u2 = 0.296
K(u2) = 9(0.296)2 + 18*0.296 + 4 = 10.116
`delta` u3 = delta F/k(u2) = 1/10.116 = 0.098
U3 = u2+delta u3 = 0.296 + 0.098 = 0.394
F_ext = 1+1+1 = 3
F_int = (u3)3 + 9(u3)2 + 4 u3
= (0.394)3 + 9(0.394)2 + 4*0.394 = 3.034
Residual R0 = f_int - f_ext = 3.034-3 = 0.034 >0.01
Thus solving using newton raphson method :
U3 = 0.391
|
K(u) |
F_ext |
F_int |
u |
|
4 |
1 |
1.077 |
0.178 |
|
7.489 |
2 |
2.144 |
0.296 |
|
10.116 |
3 |
3.034 |
0.391 |
CONCLUSION :-
Implicit method requires and extra computation and can be hard sometimes to calculate. On the other hand explicit method can be used for problems where a short duration analysis has to be done, such as an automotive crash, etc .
For implicit analysis it takes much less computational time because of its larger time step, which in case of explicit analysis takes greater computational time because of smaller time step in explicit.
The choice of analysis depends upon the results to be obtained and also the problem which need to be solved. For some cases implicit analysis suits good, whereas for some cases explicit analysis suits best.