Newmark beta method

Newmark beta methods are methods for the implicit numerical integration of differential equations . The methods belong to the one-step method , since only the values ​​of the previous time step are currently required to calculate the values . Two parameters and are introduced with which the stability and the accuracy of the method are controlled. The method class is widely used in the numerical analysis of the dynamics of solids as in the finite element method . It is named after Nathan M. Newmark , who developed it in 1959 for use in structural dynamics . ${\displaystyle t_{n+1}}$ ${\displaystyle t_{n}}$ ${\displaystyle \gamma }$ ${\displaystyle \beta }$

Derivation

Assumption of linear or constant acceleration

In the time interval in which a solution of a differential equation of the second order in time is sought, a strictly monotonically increasing sequence of points in time is given, at which the solution is to be calculated. The value of the variable , its rate and acceleration are currently known. The acceleration is interpolated linearly in the interval , see picture: ${\displaystyle [t_{0,}t_{e}]}$ ${\displaystyle x(t)}$ ${\displaystyle \{t_{0},t_{1},\dotsc ,t_{e}\}}$ ${\displaystyle x(t)}$ ${\displaystyle x_{n}}$ ${\displaystyle {\dot {x}}_{n}}$ ${\displaystyle {\ddot {x}}_{n}}$ ${\displaystyle t_{n}}$ ${\displaystyle t\in [t_{n},t_{n+1}]}$

(I)     ${\displaystyle {\ddot {x}}^{h}(t)={\frac {t_{n+1}-t}{t_{n+1}-t_{n}}}{\ddot {x}}_{n}+{\frac {t-t_{n}}{t_{n+1}-t_{n}}}{\ddot {x}}_{n+1}}$

where denotes an approximate solution of the function sought . Integration over time provides : ${\displaystyle x^{h}}$ ${\displaystyle x}$ ${\displaystyle \Delta t=t-t_{n}}$

(II)     ${\displaystyle {\dot {x}}^{h}(t)={\dot {x}}_{n}+\int _{t_{n}}^{t}{\ddot {x}}^{h}(\tau )\mathrm {d} \tau ={\dot {x}}_{n}+\Delta t[(1-\gamma ){\ddot {x}}_{n}+\gamma {\ddot {x}}^{h}(t)]}$

(III)     ${\displaystyle x^{h}(t)=x_{n}+\int _{t_{n}}^{t}{\dot {x}}^{h}(\tau )\mathrm {d} \tau =x_{n}+\Delta t{\dot {x}}_{n}+\Delta t^{2}\left[\left({\frac {1}{2}}-\beta \right){\ddot {x}}_{n}+\beta {\ddot {x}}^{h}(t)\right]}$

With

${\displaystyle \gamma ={\frac {1}{2}}}$    and    ${\displaystyle \beta ={\frac {1}{6}}}$

these formulas are exact for linear systems and provide the linear acceleration method . The values ​​originally reported by Newmark

${\displaystyle \gamma ={\frac {1}{2}}}$    and    ${\displaystyle \beta ={\frac {1}{4}}}$

correspond to the #constant average acceleration method with

${\displaystyle {\ddot {x}}^{h}(t)={\frac {1}{2}}({\ddot {x}}_{n}+{\ddot {x}}_{n+1})}$.

Assuming that the extreme values ​​of the acceleration in the interval occur at the limits of the interval, the integrals in equations (II) and (III) represent a terminated Taylor series with a remainder, with and other approximations being found. In this way, other values ​​for the constants and can also be motivated. ${\displaystyle [t_{n},t_{n+1}]}$ ${\displaystyle 0<\gamma <1}$ ${\displaystyle 0<\beta <1}$ ${\displaystyle x^{h}}$ ${\displaystyle \gamma }$ ${\displaystyle \beta }$

Start the calculation

The Newmark algorithm starts at the time with . It is mostly assumed that the accelerations vanish. With this assumption, given the initial values and initial speed , the algorithm is self- starting , i.e. H. the initial accelerations do not need to be calculated in a first step. ${\displaystyle t=t_{0}}$ ${\displaystyle n=0}$ ${\displaystyle t<t_{0}}$ ${\displaystyle x_{0}}$ ${\displaystyle {\dot {x}}_{0}}$

Update of the variables

With the Newmark algorithm, the corresponding values ​​are calculated from given values and at the time . The values in the interval can be interpolated with equations (I) to (III). With and one gets from equations (II) and (III): ${\displaystyle x_{n},{\dot {x}}_{n}}$ ${\displaystyle {\ddot {x}}_{n}}$ ${\displaystyle t_{n}}$ ${\displaystyle t_{n+1}}$ ${\displaystyle [t_{n},t_{n+1}]}$ ${\displaystyle t=t_{n+1}}$ ${\displaystyle {\ddot {x}}^{h}(t_{n+1})={\ddot {x}}_{n+1}}$

(IV)     , ${\displaystyle {\dot {x}}^{h}(t_{n+1})={\dot {x}}_{n+1}={\dot {x}}_{n}+\Delta t[(1-\gamma ){\ddot {x}}_{n}+\gamma {\ddot {x}}_{n+1}]}$

(V)     . ${\displaystyle x^{h}(t_{n+1})=x_{n+1}=x_{n}+\Delta t{\dot {x}}_{n}+\Delta t^{2}\left[\left({\frac {1}{2}}-\beta \right){\ddot {x}}_{n}+\beta {\ddot {x}}_{n+1}\right]}$

The two equations (IV) and (V) contain three unknowns and . The third equation you need to complete provides the differential equation to be solved. ${\displaystyle x_{n+1},{\dot {x}}_{n+1}}$ ${\displaystyle {\ddot {x}}_{n+1}}$

At can also be selected as the primary unknown: ${\displaystyle \beta \neq 0}$ ${\displaystyle x_{n+1}}$

${\displaystyle {\ddot {x}}_{n+1}={\frac {1}{\beta \Delta t^{2}}}(x_{n+1}-x_{n})-{\frac {1}{\beta \Delta t}}{\dot {x}}_{n}-{\frac {{\frac {1}{2}}-\beta }{\beta }}{\ddot {x}}_{n}}$

${\displaystyle {\dot {x}}_{n+1}={\frac {\gamma }{\beta \Delta t}}(x_{n+1}-x_{n})+\left(1-{\frac {\gamma }{\beta }}\right){\dot {x}}_{n}+\Delta t{\frac {\beta -{\frac {1}{2}}\gamma }{\beta }}{\ddot {x}}_{n}}$.

Once the values and have been calculated, the counter is incremented and the calculation continues until the end of the time interval of interest is reached. ${\displaystyle x_{n+1},{\dot {x}}_{n+1}}$ ${\displaystyle {\ddot {x}}_{n+1}}$ ${\displaystyle n}$

Special cases

Constant average acceleration method

The original form of the Newmark method corresponds to a constant mean acceleration

${\displaystyle {\ddot {x}}^{h}(t)={\frac {1}{2}}({\ddot {x}}_{n+1}+{\ddot {x}}_{n})}$

with which one in the above formulas (IV) and (V)

${\displaystyle \gamma ={\frac {1}{2}}}$ and ${\displaystyle \beta ={\frac {1}{4}}}$

gets.

equation	Inference
${\displaystyle {\dot {x}}^{h}(t)={\dot {x}}_{n}+\int _{t_{n}}^{t}{\ddot {x}}^{h}(\tau )\mathrm {d} \tau ={\dot {x}}_{n}+{\frac {\Delta t}{2}}({\ddot {x}}_{n}+{\ddot {x}}_{n+1})}$	${\displaystyle \gamma ={\frac {1}{2}}}$
${\displaystyle x_{n}+\int _{t_{n}}^{t}{\dot {x}}^{h}\mathrm {d} \tau =x_{n}+\int _{t_{n}}^{t}\left({\dot {x}}_{n}+{\frac {\tau -t_{n}}{2}}({\ddot {x}}_{n}+{\ddot {x}}_{n+1})\right)\mathrm {d} \tau =x_{n}+\Delta t{\dot {x}}_{n}+{\frac {\Delta t^{2}}{4}}({\ddot {x}}_{n}+{\ddot {x}}_{n+1})}$	${\displaystyle \beta ={\frac {1}{4}}}$

Central difference quotients

→ Main article : difference quotient

The central difference quotients

(VI)    ${\displaystyle {\dot {x}}_{n}={\frac {1}{2\Delta t}}(x_{n+1}-x_{n-1})}$

(VII)     ${\displaystyle {\ddot {x}}_{n}={\frac {1}{\Delta t^{2}}}(x_{n+1}-2x_{n}+x_{n-1})}$

correspond to the above formulas (IV) and (V) with

${\displaystyle \gamma ={\frac {1}{2}}}$and . ${\displaystyle \beta =0}$

equation	Inference
${\displaystyle {\begin{array}{lcl}{\ddot {x}}_{n}&=&{\frac {1}{\Delta t^{2}}}(x_{n+1}-2x_{n}+x_{n-1})\\[1ex]\rightarrow x_{n-1}&=&\Delta t^{2}{\ddot {x}}_{n}-x_{n+1}+2x_{n}\\[1ex]{\dot {x}}_{n}&=&{\frac {1}{2\Delta t}}(x_{n+1}-x_{n-1})\\[1ex]\rightarrow \Delta t{\dot {x}}_{n}&=&{\frac {1}{2}}(x_{n+1}-x_{n-1})=x_{n+1}-{\frac {\Delta t^{2}}{2}}{\ddot {x}}_{n}-x_{n}\\[1ex]\rightarrow x_{n+1}&=&x_{n}+\Delta t{\dot {x}}_{n}+{\frac {\Delta t^{2}}{2}}{\ddot {x}}_{n}\end{array}}}$	${\displaystyle \beta =0}$
${\displaystyle {\begin{array}{lcl}{\frac {\Delta t}{2}}{\ddot {x}}_{n}&=&{\frac {1}{2\Delta t}}(x_{n+1}-2x_{n}+x_{n-1})\\[1ex]{\frac {\Delta t}{2}}{\ddot {x}}_{n+1}&=&{\frac {1}{2\Delta t}}(x_{n+2}-2x_{n+1}+x_{n})\\[2ex]{\dot {x}}_{n}&=&{\frac {1}{2\Delta t}}(x_{n+1}-x_{n-1})\\[1ex]{\dot {x}}_{n+1}&=&{\frac {1}{2\Delta t}}(x_{n+2}-x_{n})\\[1ex]\rightarrow {\frac {\Delta t}{2}}({\ddot {x}}_{n}+{\ddot {x}}_{n+1})&=&{\frac {1}{2\Delta t}}(x_{n+2}-x_{n}-x_{n+1}+x_{n-1})={\dot {x}}_{n+1}-{\dot {x}}_{n}\\[1ex]\rightarrow {\dot {x}}_{n+1}&=&{\dot {x}}_{n}+{\frac {\Delta t}{2}}({\ddot {x}}_{n}+{\ddot {x}}_{n+1})\end{array}}}$	${\displaystyle \gamma ={\frac {1}{2}}}$

Explicit time integration

→ Main article : Explicit Euler method

The explicit time integration method does not belong to the family of (implicit!) Newmark-beta algorithms and is only given here for comparison purposes. The above formulas (VI) and (VII) for the central differences are equivalent to

${\displaystyle {\dot {x}}_{n+1/2}={\frac {1}{\Delta t}}(x_{n+1}-x_{n})}$

${\displaystyle {\ddot {x}}_{n}={\frac {1}{\Delta t}}({\dot {x}}_{n+1/2}-{\dot {x}}_{n-1/2})}$.

It is noticeable here that the speeds are always calculated in the middle of the time intervals. With the assumption

${\displaystyle {\begin{array}{lcl}{\dot {x}}_{n+1}&\approx &{\dot {x}}_{n+1/2}={\dot {x}}_{n-1/2}+\Delta t{\ddot {x}}_{n}\\x_{n+1}&=&x_{n}+\Delta t{\dot {x}}_{n+1/2}=x_{n}+\Delta t({\dot {x}}_{n-1/2}+\Delta t{\ddot {x}}_{n})\end{array}}}$

the values and the speeds at the point in time can be traced back to known results and the differential equation supplies the determining equation for the now only unknown . ${\displaystyle x_{n+1}}$ ${\displaystyle {\dot {x}}_{n+1}}$ ${\displaystyle t_{n+1}}$ ${\displaystyle x_{n},{\dot {x}}_{n-1/2},{\ddot {x}}_{n}}$ ${\displaystyle {\ddot {x}}_{n+1}}$

example

Time integration with algorithms from the Newmark family

A vibration obeys the homogeneous differential equation in the absence of excitation

${\displaystyle {\ddot {x}}(t)+x(t)=0}$.

With the initial conditions

${\displaystyle x(t=0)=x_{0}=0}$

${\displaystyle {\dot {x}}(t=0)={\dot {x}}_{0}=1}$

the differential equation has the analytical solution

${\displaystyle x(t)=\sin(t)}$

to which the initial acceleration

${\displaystyle {\ddot {x}}(t=0)=-\sin(0)=0}$

heard. The differential equation gives the equation for the primary unknown  : ${\displaystyle {\ddot {x}}_{n+1}}$

${\displaystyle {\ddot {x}}_{n+1}+x_{n+1}=0\rightarrow {\ddot {x}}_{n+1}=-x_{n+1}}$

Time integration using the Newmark method gives the equations for the values and rates from the table ${\displaystyle x_{n+1}}$ ${\displaystyle {\dot {x}}_{n+1}}$

parameter	Update rule
${\displaystyle \gamma ={\frac {1}{2}},\;\beta ={\frac {1}{6}}}$	${\displaystyle {\begin{array}{lcl}x_{n+1}&=&x_{n}+\Delta t{\dot {x}}_{n}+{\frac {\Delta t^{2}}{6}}(2{\ddot {x}}_{n}-x_{n+1})\rightarrow x_{n+1}={\frac {6x_{n}+6\Delta t{\dot {x}}_{n}+2\Delta t^{2}{\ddot {x}}_{n}}{6+\Delta t^{2}}}\\[1ex]{\dot {x}}_{n+1}&=&{\dot {x}}_{n}+{\frac {\Delta t}{2}}({\ddot {x}}_{n}-x_{n+1})\end{array}}}$
${\displaystyle \gamma ={\frac {1}{2}},\;\beta ={\frac {1}{4}}}$	${\displaystyle {\begin{array}{lcl}x_{n+1}&=&x_{n}+\Delta t{\dot {x}}_{n}+{\frac {\Delta t^{2}}{4}}({\ddot {x}}_{n}-x_{n+1})\rightarrow x_{n+1}={\frac {4x_{n}+4\Delta t{\dot {x}}_{n}+\Delta t^{2}{\ddot {x}}_{n}}{4+\Delta t^{2}}}\\[1ex]{\dot {x}}_{n+1}&=&{\dot {x}}_{n}+{\frac {\Delta t}{2}}({\ddot {x}}_{n}-x_{n+1})\end{array}}}$
${\displaystyle \gamma ={\frac {1}{2}},\;\beta =0}$	${\displaystyle {\begin{array}{lcl}x_{n+1}&=&x_{n}+\Delta t{\dot {x}}_{n}+{\frac {\Delta t^{2}}{2}}{\ddot {x}}_{n}\\[1ex]{\dot {x}}_{n+1}&=&{\dot {x}}_{n}+{\frac {\Delta t}{2}}({\ddot {x}}_{n}-x_{n+1})\end{array}}}$
explicit	${\displaystyle {\begin{array}{lcl}{\dot {x}}_{n+1}&=&{\dot {x}}_{n+1/2}={\dot {x}}_{n-1/2}+\Delta t{\ddot {x}}_{n}\\[1ex]x_{n+1}&=&x_{n}+\Delta t{\dot {x}}_{n-1/2}+\Delta t^{2}{\ddot {x}}_{n}\end{array}}}$

The solutions in the interval and have the course in the picture. The mean deviation ${\displaystyle t\in [0,10\pi ]}$ ${\displaystyle \Delta t=0,2}$

${\displaystyle e={\sqrt {\frac {\sum _{n=1}^{159}{(x_{n}-\sin(t_{n}))}^{2}}{159}}}}$

gives the table:

Procedure	Mean deviation ${\displaystyle e}$
Linear acceleration method	0.021778594324355638
Central differences procedure	0.022202937295615111
Constant mean acceleration	0.043283257071468406
Explicit procedure	0.022202937295615576

literature

• Robert Gasch, Klaus Knothe, Robert Liebich: Structural Dynamics , Springer Verlag 2012, ISBN 978-3-540-88977-9
• T. Belytschko, TJR Hughes (Ed.): Computational methods for transient analysis. North-Holland 1986. ISBN 9780444864796

Individual evidence

1. ^ Newmark, Nathan M .: A method of computation for structural dynamics . In: Journal of Engineering Mechanics . 85 (EM3). ASCE, 1959, pp. 67-94 .