Newmark-beta method

From Wikipedia, the free encyclopedia

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}+\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 Nathan M. Newmark, who introduced it around 1959.


v • d • e
Numerical integration
First order methods
Second order methods
Higher order methods
Languages