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framework/doc/content/source/timeintegrators/CentralDifference.md
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# CentralDifference | ||
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!syntax description /Executioner/TimeIntegrator/CentralDifference | ||
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## Description | ||
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Central difference integration is a common explicit integration scheme, typically used for structural dynamics problems. For a variable, $u$, at time $t$ with a time step, $\Delta t$, the central difference approximations for the first and second time derivatives, $\dot{u}$ and $\ddot{u}$, are given as, | ||
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\begin{equation} | ||
\begin{aligned} | ||
\mathbf{\dot{u}}(t) &=& \frac{\mathbf{u}(t+\Delta t)-\mathbf{u}(t-\Delta t)}{2 \Delta t} \\ | ||
\mathbf{\ddot{u}}(t) &=& \frac{\mathbf{u}(t+\Delta t)-2\mathbf{u}(t)+\mathbf{u}(t-\Delta t)}{\Delta t^2} | ||
\end{aligned} | ||
\end{equation} | ||
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The central difference time integrator derives from the [`ActuallyExplicitEuler`](ActuallyExplicitEuler.md) class and therefore circumvents the nonlinear solver. It can be used with `consistent`, `lumped` or `lump_preconditioned`, `solve_type` options. | ||
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## Assumptions | ||
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The implementation of the central difference timeintegrator assumes that corresponding kernels involving the first and second derivatives directly use the derivatives calculates by the timeintegrator. Currently, only the `InertialForce` kernel and the `NodalKernels` corresponding to nodal inertias are modified to work with the central difference timeintegrator. Therefore, the central difference time integrator can currently only be used for mechanics problems with solid elements. | ||
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!syntax parameters /Executioner/TimeIntegrator/CentralDifference | ||
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!syntax inputs /Executioner/TimeIntegrator/CentralDifference | ||
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!syntax children /Executioner/TimeIntegrator/CentralDifference |