TheNewmark-beta method is amethod ofnumerical integration used to solve certaindifferential 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 afterNathan M. Newmark,^[1] former Professor of Civil Engineering at theUniversity of Illinois at Urbana–Champaign, who developed it in 1959 for use instructural dynamics. The semi-discretized structural equation is a second order ordinary differential equation system,

${\displaystyle M\ddot{u}+C\dot{u}+f^{\text{int}}(u)=f^{\text{ext}}}$

here${\displaystyle M}$ is themass matrix,${\displaystyle C}$ is thedamping matrix,${\displaystyle f^{\text{int}}}$ and${\displaystyle f^{\text{ext}}}$ are internal force per unit displacement and external forces, respectively.

Using theextended mean value theorem, the Newmark-${\displaystyle \beta }$ method states that the firsttime derivative (velocity in theequation of motion) can be solved as,

${\displaystyle {\dot{u}}_{n+1}={\dot{u}}_n+\mathrm{\Delta }t{\ddot{u}}_{\gamma }}$

where

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

therefore

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

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,

${\displaystyle u_{n+1}=u_n+\mathrm{\Delta }t{\dot{u}}_n+\begin{array}{c}\frac{1}{2}\end{array}\mathrm{\Delta }{t}^2{\ddot{u}}_{\beta }}$

where again

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

The discretized structural equation becomes

Explicit central difference scheme is obtained by setting${\displaystyle \gamma =0.5}$ and${\displaystyle \beta =0}$

Average constant acceleration (Middle point rule) is obtained by setting${\displaystyle \gamma =0.5}$ and${\displaystyle \beta =0.25}$

A time-integration scheme is said to be stable if there exists an integration time-step${\displaystyle \mathrm{\Delta }{t}_0>0}$ so that for any${\displaystyle \mathrm{\Delta }t\in (0,\mathrm{\Delta }{t}_0]}$, a finite variation of the state vector${\displaystyle q_n}$ at time${\displaystyle t_n}$ induces only a non-increasing variation of the state-vector${\displaystyle q_{n+1}}$ calculated at a subsequent time${\displaystyle t_{n+1}}$. Assume the time-integration scheme is

${\displaystyle q_{n+1}=A(\mathrm{\Delta }t)q_n+g_{n+1}(\mathrm{\Delta }t)}$

The linear stability is equivalent to${\displaystyle \rho (A(\mathrm{\Delta }t))\le 1}$, here${\displaystyle \rho (A(\mathrm{\Delta }t))}$ is thespectral radius of the update matrix${\displaystyle A(\mathrm{\Delta }t)}$.

For the linear structural equation

${\displaystyle M\ddot{u}+C\dot{u}+Ku=f^{\text{ext}}}$

here${\displaystyle K}$ is the stiffness matrix. Let${\displaystyle q_n=[{\dot{u}}_n,u_n]}$, the update matrix is${\displaystyle A=H_1^{-1}{H}_0}$, and

For undamped case (${\displaystyle C=0}$), the update matrix can be decoupled by introducing the eigenmodes${\displaystyle u=e^{i{\omega }_{i}t}{x}_i}$ of the structural system, which are solved by the generalized eigenvalue problem

${\displaystyle {\omega }^{2}Mx=Kx}$

For each eigenmode, the update matrix becomes

The characteristic equation of the update matrix is

${\displaystyle {\lambda }^2-\left(2-(\gamma +\frac{1}{2}){\eta }_i^2\right)\lambda +1-(\gamma -\frac{1}{2}){\eta }_i^2=0{\eta }_i^2=\frac{{\omega }_i^2\mathrm{\Delta }{t}^2}{1+\beta {\omega }_i^2\mathrm{\Delta }{t}^2}}$

As for the stability, we have

Explicit central difference scheme (${\displaystyle \gamma =0.5}$ and${\displaystyle \beta =0}$) is stable when${\displaystyle \omega \mathrm{\Delta }t\le 2}$.

Average constant acceleration (Middle point rule) (${\displaystyle \gamma =0.5}$ and${\displaystyle \beta =0.25}$) is unconditionally stable.

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