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+1−2un+un−1h2
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+1−un)
so that:
(5)
¨un=1hn+12(˙un+12−˙un−12)
(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:
The same formulation is used for rotational velocities.
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.