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The '''Newmark-beta method''' is a [[Time integration method|method]] of [[numerical integration]] used to solve [[differential equations]]. It is widely used in numerical evaluation of the dynamic response of structures and solids such as in [[finite element analysis]] to model dynamic systems. The method is named after [[Nathan M. Newmark]],<ref>Newmark, N. M. (1959) A method of computation for structural dynamics. Journal of Engineering Mechanics, ASCE, 85 (EM3) 67-94.</ref> former Professor of Civil Engineering at the University of Illinois, who developed it in 1959 for use in [[Structural dynamics]].
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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,
 
:<math>\dot{u}_{n+1}=\dot{u}_{n}+{\Delta}t~\ddot{u}_\gamma \,</math>
 
where
 
:<math>\ddot{u}_{\gamma} = (1 - \gamma)\ddot{u}_n + \gamma \ddot{u}_{n+1}~~~~0\leq \gamma \leq 1</math>
 
therefore
 
:<math>\dot{u}_{n+1}=\dot{u}_{n}+(1 - \gamma){\Delta}t~\ddot{u}_n + \gamma {\Delta}t~\ddot{u}_{n+1}.</math>
 
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,
 
:<math>{u}_{n+1}=u_n + {\Delta}t~\dot{u}_{n}+\begin{matrix} \frac{1}{2} \end{matrix}{\Delta}t^{2}~\ddot{u}_\beta </math>
 
where again
 
:<math>\ddot{u}_{\beta} = (1 - 2\beta)\ddot{u}_n + 2\beta\ddot{u}_{n+1}~~~~0\leq 2\beta\leq 1</math>
 
Newmark showed that a reasonable value of <math>\gamma</math> is&nbsp;0.5, therefore the update rules are,
 
:<math>\dot{u}_{n+1}=\dot{u}_{n}+ \begin{matrix}\frac{{\Delta}t}{2}\end{matrix}~(\ddot{u}_n + \ddot{u}_{n+1})</math>
 
:<math>{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}</math>
 
Setting ''β'' to various values between 0 and 1 can give a wide range of results. Typically ''β''&nbsp;=&nbsp;1/4, which yields the constant average acceleration method, is used.
 
==References==
 
{{Reflist}}
 
{{Numerical integrators}}
 
{{DEFAULTSORT:Newmark-Beta Method}}
[[Category:Numerical differential equations]]

Latest revision as of 09:30, 27 December 2014

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