Central Difference

This command is used to construct a Central Difference integrator object.

integrator CentralDifference

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Examples

integrator CentralDifference

{"integrator": ["CentralDifference"]}

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NOTES:

1. 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.
2. 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.
3. For stability, \(\frac{\Delta t}{T_n} < \frac{1}{\pi}\)

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Theory

The Central difference approximations for velocity and acceleration:

\[v_n = \frac{d_{n+1} - d_{n-1}}{2 \Delta t}\]

\[a_n = \frac{d_{n+1} - 2 d_n + d_{n-1}}{\Delta t^2}\]

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:

\[\left ( \frac{1}{\Delta t^2} M + \frac{1}{2 \Delta t} C \right ) U_{t+\Delta t} = R_t - \left (K - \frac{2}{\Delta t^2}M \right )U_t - \left (\frac{1}{\Delta t^2}M - \frac{1}{2 \Delta t} C \right) U_{t-\Delta t} \]

References

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Code Developed by: fmk