math lessons - Newmark-beta method

The Newmark-beta method is a method of numerical integration used to solve differential equations. It is used in finite element analysis to model dynamic systems.

Recalling the continuous time equation of motion,

$u = \dot{u}t + \begin{matrix} \frac{1}{2} \end{matrix} \ddot{u}t^2$

Using the extended mean value theorem, The Newmark- $β$ method states that the first time derivative (velocity in the equation of motion) can be solved as,

$\dot{u}_{n+1}=\dot{u}_{n}+{\Delta}t~\ddot{u}_{\gamma}$

where

$\ddot{u}_{\gamma} = (1 - \gamma)\ddot{u}_n + \gamma \ddot{u}_{n+1}~~~~0\leq \gamma \leq 1$

therefore

$\dot{u}_{n+1}=\dot{u}_{n}+(1 - \gamma){\Delta}t~\ddot{u}_n + \gamma {\Delta}t~\ddot{u}_{n+1}$ .

Because acceleration also varies with time, however, the extended mean value theorem must also be extended to the second time derivative to obtain the correct displacement. Thus,

${u}_{n+1}=u_n + {\Delta}t~\dot{u}_{n+1}+\begin{matrix} \frac{1}{2} \end{matrix}{\Delta}t^{2}~\ddot{u}_{\beta}$

where again

$\ddot{u}_{\beta} = (1 - 2\beta)\ddot{u}_n + 2\beta\ddot{u}_{n+1}~~~~0\leq \beta\leq 1$

Newmark showed that a reasonable value of $γ$ is 0.5, therefore the update rules are,

$\dot{u}_{n+1}=\dot{u}_{n}+ \begin{matrix}\frac{{\Delta}t}{2}\end{matrix}~(\ddot{u}_n + \ddot{u}_{n+1})$

${u}_{n+1}=u_n + {\Delta}t~\dot{u}_{n} + \begin{matrix}\frac{1-2\beta}{2}\end{matrix} {\Delta} t^2 \ddot{u}_{n} + \beta {\Delta} t^2 \ddot{u}_{n+1}$

Setting $β$ to various values between 0 and 1 can give a wide range of results. Typically $β = 1 / 4$ , which yields the constant average acceleration method, is used.

The method is named for N. M. Newmark, who introduced it around 1959.

Categories: Numerical analysis | Differential equations | Mathematical methods

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