Central Difference Algorithm

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

with h n + 1 the time step between t n and t n + 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)
u ¨ n = u n + 1 2 u n + u n 1 h 2
From the algorithmic point of view, it is, however, more efficient to use velocities at half of the time step:(4)
u ˙ n + 1 2 = u ˙ ( t n + 1 2 ) = 1 h n + 1 ( u n + 1 u n )
so that:(5)
u ¨ n = 1 h n + 1 2 ( u ˙ n + 1 2 u ˙ n 1 2 )
(6)
h n + 1 2 = ( h n + h n + 1 ) / 2
Time integration is explicit, in that if acceleration u ¨ n is known (Combine Modal Reduction), the future velocities and displacements are calculated from past (known) values in time:
  • u ˙ n + 1 2 is obtained from Equation 5: (7)
    u ˙ n + 1 2 = u ˙ n 1 2 + h n + 1 2 u ¨ n
The same formulation is used for rotational velocities.
  • u n + 1 is obtained from Equation 4: (8)
    u n + 1 = u n + h n + 1 u ˙ n + 1 2

The accuracy of the scheme is of h 2 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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaaaa@36E3@ 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.