Central Difference Algorithm

The central difference algorithm corresponds to the Newmark algorithm with γ=12 and β=0 so that Newarks Method, Equation 7 and Equation 8 become:(1)
˙un+1=˙un+12hn+1(¨un+¨un+1)
(2)
un+1=un+hn+1˙un+12h2n+1¨un

with hn+1 the time step between tn and tn+1 .

It is easy to show that the central difference algorithm 1 can be changed to an equivalent form with 3 time steps, if the time step is constant.(3)
¨un=un+12un+un1h2
From the algorithmic point of view, it is, however, more efficient to use velocities at half of the time step:(4)
˙un+12=˙u(tn+12)=1hn+1(un+1un)
so that:(5)
¨un=1hn+12(˙un+12˙un12)
(6)
hn+12=(hn+hn+1)/2
Time integration is explicit, in that if acceleration ¨un is known (Combine Modal Reduction), the future velocities and displacements are calculated from past (known) values in time:
  • ˙un+12 is obtained from Equation 5: (7)
    ˙un+12=˙un12+hn+12¨un
The same formulation is used for rotational velocities.
  • un+1 is obtained from Equation 4: (8)
    un+1=un+hn+1˙un+12

The accuracy of the scheme is of h2 order, that is, if the time step is halved, the amount of error in the calculation is one quarter of the original. The time step h may be variable from one cycle to another. It is recalculated after internal forces have been computed.

1
Ahmad S., Irons B.M., and Zienkiewicz O.C., “Analysis of thick and thin shell structures by curved finite elements”, Computer Methods in Applied Mechanics and Engineering, 2:419-451, 1970.