Week - 2 - Explicit and Implicit Analysis

Implicit and Explicit analysis:

Both analysis are methods used for obtaining numerical solutions with respect to time.

If the matrix equation of the system was Mx'' +Cx' + Kx=f, then

(a) Implicit: x is solved through matrix inversion. and Newton-Raphson iterations are used. Mathematicaly if u(n) is the state of the system then implicit analysis uses equation u(n+1)=u(n) + dt.f(u(n+1)) where dt is the time step. This facilitates use of larger time steps but requires more computattional time. It is unconditionally stable.

(b) Explicit: Here x'' (accelaration) which is a diagonal matrix is solved. Thereby requiring no iterations. Here equation
u(n+1)=u(n) + dt.f(u(n)) is solved. The time step is governed by Courant–Friedrichs–Lewy equation and is very small. This is useful when the change of state is very fast and occurs in a very short period and is impractical if it is a steady and slow proccess. It is not unconditionally stable.

Here we use the equation f(u)= u^3 +9u^2 + 4u.

Stiffness matrix k(u) = df/du , that implies k(u)= 3u^2 + 18u + 4.

Calculations.

Kindly refer the attached excel sheet to find Explicit calculations and results and implicit methodology

The calculations of implicit are done using the python code used below,developed using the abouve equations and criterion.

import math
def K(i):
    k=4+(18*i)+(9*math.pow(i,2));
    return k;
def fi(j):
    f=4*int(j) +(9*math.pow(j,2))+(math.pow(j,3));
    return f;
def newton(p,r):
    x=-r/(K(p));
    return x;
u=0;
df=1;
for F in range(1,4):
    k=K(u);
    du=df/k;
    u=u+du;
    f=fi(u);
    R=abs(f-F);
    while R>0.01:
        du=newton(u,R);
        u=u+du;
        f=fi(u);
        R=abs(f-F);
    print("F=",F," ui=",u," fi=", f," R=",R);

3 load steps (F) each of value 1 is used . R is the residual means f(intenal) -F(external)

Conclusion

As we can see implicit provides a more accurate results but the time required for it is too much if it comes to a very fast sudden change in state. Hence it is preferred if the change if state is slow.
Explicit is used when there is a sudden change in state of system, like drop test, crash analysis.