3.2.6.5. Central Difference

integrator CentralDifference

Note

• The calculation of \(U_{t+\Delta t}\), as shown below, is based on using the equilibrium equation at time t. For this reason the method is called an explicit integration method.

• If there is no rayleigh damping and the C matrix is 0, for a diagonal mass matrix a diagonal solver may and should be used.

• For stability, \(\frac{\Delta t}{T_n} < \frac{1}{\pi}\)

3.2.6.5.1. THEORY:

The Central difference approximations for velocity and acceleration:

:math:` v_n = frac{d_{n+1} - d_{n-1}}{2 Delta t}`

:math:` a_n = frac{d_{n+1} - 2 d_n + d_{n-1}}{Delta t^2}`

In the Central Difference method we determine the displacement solution at time \(t+\delta t\) by considering the the eqilibrium equation for the finite element system in motion at time t:

\(M \ddot U_t + C \dot U_t + K U_t = R_t\)

which when using the above two expressions of becomes:

:math:` left ( frac{1}{Delta t^2} M + frac{1}{2 Delta t} C right ) U_{t+Delta t} = R_t - left (K - frac{2}{Delta t^2}M right )U_t - left (frac{1}{Delta t^2}M - frac{1}{2 Delta t} C right) U_{t-Delta t} `