Editing Newmark-beta method (section)

{{Short description|Concept in differential equation mathematics}}
The '''Newmark-beta method''' is a [[Time integration method|method]] of [[numerical integration]] used to solve certain [[differential equations]].  It is widely used in numerical evaluation of the dynamic response of structures and solids such as in [[Finite element method in structural mechanics | finite element analysis]] to model dynamic systems. The method is named after [[Nathan M. Newmark]],<ref>{{citation|last=Newmark|first= Nathan M. |authorlink=Nathan M. Newmark|year=1959|title= A method of computation for structural dynamics|journal= Journal of the Engineering Mechanics Division |volume= 85 (EM3)|issue= 3 |pages= 67–94|doi= 10.1061/JMCEA3.0000098 }}</ref> former Professor of Civil Engineering at the [[University of Illinois at Urbana–Champaign]], who developed it in 1959 for use in [[structural dynamics]]. The semi-discretized structural equation is a second order ordinary differential equation system,

<math>M\ddot{u} + C\dot{u} + f^{\textrm{int}}(u) = f^{\textrm{ext}} \,</math>

here <math>M</math> is the [[mass matrix]], <math>C</math> is the [[damping matrix]], <math>f^{\textrm{int}}</math> and <math>f^{\textrm{ext}}</math> are internal force per unit displacement and external forces, respectively.

Using the [[extended mean value theorem]], the Newmark-<math>\beta</math> 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>

The discretized structural equation becomes

<math>\begin{aligned}
&\dot{u}_{n+1}=\dot{u}_n + (1 - \gamma) \Delta t~\ddot{u}_n + \gamma \Delta t~\ddot{u}_{n+1}\\
&u_{n+1}=u_n + \Delta t~\dot{u}_n + \frac{\Delta t^2}{2}\left((1 - 2\beta)\ddot{u}_n + 2\beta\ddot{u}_{n+1}\right)\\
&M\ddot{u}_{n+1} + C\dot{u}_{n+1} + f^{\textrm{int}}(u_{n+1}) = f_{n+1}^{\textrm{ext}} \,
\end{aligned}</math>

'''Explicit central difference scheme''' is obtained by setting <math>\gamma=0.5 </math> and <math>\beta=0 </math>

'''Average constant acceleration (Middle point rule)''' is obtained by setting <math>\gamma=0.5 </math> and <math>\beta=0.25 </math>