The Newmark-beta method is a 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 analysis to model dynamic systems.
The method is named after Nathan M. Newmark,[1] 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,
{\displaystyle M{\ddot {u}}+C{\dot {u}}+f^{\textrm {int}}(u)=f^{\textrm {ext}}\,}
is the mass matrix,
are internal force per unit displacement and external forces, respectively.
Using the extended mean value theorem, the Newmark-
method states that the first time derivative (velocity in the equation of motion) can be solved as, where therefore 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, where again The discretized structural equation becomes
Explicit central difference scheme is obtained by setting
Average constant acceleration (Middle point rule) is obtained by setting
A time-integration scheme is said to be stable if there exists an integration time-step
, a finite variation of the state vector
induces only a non-increasing variation of the state-vector
calculated at a subsequent time
Assume the time-integration scheme is
The linear stability is equivalent to
is the spectral radius of the update matrix
For the linear structural equation
{\displaystyle {\begin{aligned}H_{1}={\begin{bmatrix}M+\gamma \Delta tC&\gamma \Delta tK\\\beta \Delta t^{2}C&M+\beta \Delta t^{2}K\end{bmatrix}}\qquad H_{0}={\begin{bmatrix}M-(1-\gamma )\Delta tC&-(1-\gamma )\Delta tK\\-({\frac {1}{2}}-\beta )\Delta t^{2}C+\Delta tM&M-({\frac {1}{2}}-\beta )\Delta t^{2}K\end{bmatrix}}\end{aligned}}}
For undamped case (
), the update matrix can be decoupled by introducing the eigenmodes
of the structural system, which are solved by the generalized eigenvalue problem
For each eigenmode, the update matrix becomes
The characteristic equation of the update matrix is
λ + 1 − ( γ −
As for the stability, we have Explicit central difference scheme (
Average constant acceleration (Middle point rule) (
) is unconditionally stable.