Implicit Vs Explicit

hi sneha raj,

implicit algorithms are unconditionally stable but explicit algorithms are conditionally stable.i.e explicit algorithms are stable under only certain conditions.

Rajnikanth Reddy

IIT Kanpur. India

rajreddy@iitk.ac.in

arkayreddy@gmail.com

<img src=http://physweb.bgu.ac.il/cgi-bin/mimetex.cgi?\left[{\bf M}\right] \left\{ \ddot{u}\right\}+\left[\bf K\right] \left\{u\right\}=0 (1)alt="" border=0 align=middle>

http://www.aero.iisc.ernet.in/cdfm

Mu''+Ku=0

is the governing discretized FEM equations one wants to solve to obtain the time dependent solution for the displacement u.

u'' is the acceleration vector

M is the mass matrix (consistent or lumped)

K is the stiffness matrix

u is the displacement vector

u'', u' and u can be related by a backward difference as

u''(t+dt)= (u'(t+dt)-u'(t))/dt

u' (t+dt)=(u(t+dt)-u(t))/dt

in the above approximation both the left and right hand side have the unknown u'(t+dt) and u(t+dt) and hence calls for modification and inversion of the stiffness matrix at every time step and hence called implicit.

If once uses a forward difference approximation for the derivatives wrt time we get

u''(t)=(u'(t+dt)-u(t))/dt

u'(t)=(u(t+dt)-u(t))/dt

due to the above approximation inversion of the K matrix at all time steps can be overcome.

Coding an implicit scheme is harder than an explicit scheme. However, the implicit scheme can be more accurate even when a large time-step (if the physics of the problem permits) is adopted. The explicit scheme can also be accurate at the cost of taking smaller time steps (will be useful when the physics demands a small time-step). In either case one needs to use the time-step as a fraction of that obtained by the CFL (Courant) condition.

R. Chennamsetti, Scientist, India

The difference -

If the semi-discretised equation ma+cv+kx=f(t), is written at 'i+1' time step to get 'x' at i+1 => Implicit

If it is written at 'i' time step to get 'x' at 'i+1' => Explicit.

[where a = acceleration, v=velocity and x=deflection/deformation, and v are expressed in terms of time derivatives of 'x'.]

In general 'x' at i+1 is a function of x at i, v at i etc in explicit, but, in implicit schemes 'x' at i+1 is a function of x at i, v at i+1 etc. => parameters in i+1 step are also appearing in the function.

In general, explicit schemes are conditionally stable and implicit schemes are unconditionally stable. But, there are some implicit schemes, which are conditionally stable.

Explicit schemes are used for shot duration phenomenon like shock loads, blast, impact etc. (high frequencies). Implicit schemes => long durarion   phenomenon (low frequencies).

You may refer 'Newmark's' time integration technique available in Structural dynamics books.

hi

i siggest FEM book by Bathe for the excellent explanation of this.

ragards, 

Rajnikanth Reddy

IIT Kanpur. India

rajreddy@iitk.ac.in

arkayreddy@gmail.com