Newmark Method

The Newmark method is an implicit time-stepping method for solving the transient problem:

<math> M \ddot U_{t+\Delta t} + C \dot U_{t+\Delta t} + R(U,\dot U)_{t+\Delta t} = F_{t+\Delta t}</math}

Using the Taylor series approximation of <math>U_{t+\Delta t}</math>

and <math>\dot U_{t+\Delta t}</math>:

<math> u_{t+\Delta t} = u_t + \Delta t \dot u_t + \tfrac{\Delta t^2}{2} \ddot u_t + \tfrac{\Delta t^3}{6} \dot \ddot u_t + \cdots </math>

<math> \dot u_{t+\Delta t} = \dot u_t + \Delta t \ddot u_t + \tfrac{\Delta t^2}{2} \dot \ddot u_t + \cdots </math>

Newton truncated these using the following:

<math> u_{t+\Delta t} = u_t + \Delta t \dot u_t + \tfrac{\Delta t^2}{2} \ddot u + \beta {\Delta t^3} \dot \ddot u </math>

<math> \dot u_{t + \Delta t} = \dot u_t + \Delta t \ddot u_t + \gamma \Delta t^2 \dot \ddot u </math>

and he further assumed linear acceleration within a time step, i.e.

<math> \dot \ddot u = \frac{{\ddot u_{t+\Delta t}} - \ddot u_t}{\Delta t} </math>

which results in the following expressions:

<math> u_{t+\Delta t} = u_t + \Delta t \dot u_t + [(0.5 - \beta) \Delta t^2] \ddot u_t + [\beta \Delta t^2] \ddot u_{t+\Delta t}</math>

<math> \dot u_{t+\Delta t} = \dot u_t + [(1-\gamma)\Delta t] \ddot u_t + [\gamma \Delta t ] \ddot u_{t+\Delta t} </math>