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-$\beta$ 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,

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

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

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