Central Difference Algorithm

The central difference algorithm corresponds to the Newmark algorithm with

γ = \frac{1}{2}

and

β = 0

so that Newarks Method, 式 7 and 式 8 become:(1)

{\dot{u}}_{n + 1} = {\dot{u}}_{n} + \frac{1}{2} h_{n + 1} ({\ddot{u}}_{n} + {\ddot{u}}_{n + 1})

(2)

u_{n + 1} = u_{n} + h_{n + 1} {\dot{u}}_{n} + \frac{1}{2} h_{n + 1}^{2} {\ddot{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)

{\ddot{u}}_{n} = \frac{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)

{\dot{u}}_{n + \frac{1}{2}} = \dot{u} (t_{n + \frac{1}{2}}) = \frac{1}{h_{n + 1}} (u_{n + 1} - u_{n})

so that:(5)

{\ddot{u}}_{n} = \frac{1}{h_{n + \frac{1}{2}}} ({\dot{u}}_{n + \frac{1}{2}} - {\dot{u}}_{n - \frac{1}{2}})

(6)

h_{n + \frac{1}{2}} = (h_{n} + h_{n + 1}) / 2

Time integration is explicit, in that if acceleration

{\ddot{u}}_{n}

is known (Combine Modal Reduction), the future velocities and displacements are calculated from past (known) values in time:

${\dot{u}}_{n + \frac{1}{2}}$ is obtained from 式 5: (7)
${\dot{u}}_{n + \frac{1}{2}} = {\dot{u}}_{n - \frac{1}{2}} + h_{n + \frac{1}{2}} {\ddot{u}}_{n}$

The same formulation is used for rotational velocities.

$u_{n + 1}$ is obtained from 式 4: (8)
$u_{n + 1} = u_{n} + h_{n + 1} {\dot{u}}_{n + \frac{1}{2}}$

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 $h$ 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.