Week - 2 - Explicit and Implicit Analysis

Aim:

To understand the difference between Implicit and Explicit method with application of forward and backward euler method.

Explanation:

Implicit Analysis:

• The time taken for each step of analysis is more.
• The analysis is stable even for larger time steps.

Explicit analysis:

• The time taken for each step is less.
• The analysis is unstable for larger time step.

Calculation:

Using forward euler method the working of Implicit and explicit analysis can be understood.

The formula for force with respect to displacement is given as 

`F(u) = u^3 + 9u^2 + 4u`

The formula for stiffness can be derived as 

`K(u) = 3u^2 + 18u+4`

The exact solution of force for particular displacement u can be obtained from formular F(u).

But, for our understanding of explicit and implicit method the `deltaF` for each `delta u` is obtained using the formula `deltaF = K deltau`

In explicit method the Force is calculated directly from the above formula. For implicit method the 'detal u' is further refined using Newton Raphson iteration for acurate results.

The force applied and the correspoinding displacement for explicit analysis is tabulated below.

Step	'delta'F	u	Fext	fint	R = fint - Fext
1	1	0.25	1.578	1	0.5781
2	1	0.3651	2.182	2	0.7088
3	1	0.4563	3.109	3	0.7935
4	1	0.5342	4.078	4	0.8568
5	1	0.6033	5.0609	5	0.9078

 The force applied and the correspoinding displacement for Implicit analysis is tabulated below.

Step	u	Fext	fint	R = fint - Fext
1	0.1777	1.0003	1	0.0003
2	0.2962	2.0002	2	0.0002
3	0.3910	3.0001	3	0.0001
4	0.4721	4.0001	4	0.0001
5	0.5440	5.0001	5	0.0001

The below graphs shows the value obtained from the explicit and implicit analysis in comparision with exact solution.

Frome the above graph we can observe that the Implicit method gives solution close to exact solution. The Value from explicit method, even though is faster gives out erroneous solutions (unstable).

The accuracy of the explicit method can be increased by reducing the step size. Below graph shows the result from explicit analysis with 0.1 step w.r.t to exact solution.

Conclusion

From the above analysis the following observation can be made

• Implicit method is more accurate (stable) in terms of results but takes more time for solving.
• Explicit method is much faster, but with larger step size the results are less accurate (unstable).

Hence, It is efficient to use Implicit method for simpler problems by using larger time step and  Explicit analysis to use complex problems with smaller time step.

Appendix:

Find the attachment for MATLAB code used for this assignment.