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Newmark-beta method
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<p>The Newmark-beta method is a <a href="/facts/Time_integration_method/rbrYBco3">method</a> of <a href="/facts/Numerical_integration/MwJnvcDV">numerical integration</a> used to solve certain <a href="/facts/Differential_equations/9MGbE3Ca">differential equations</a>. It is widely used in numerical evaluation of the dynamic response of structures and solids such as in <a href="/facts/Finite_element_method_in_structural_mechanics/J5gSNPEE"> finite element analysis</a> to model dynamic systems. The method is named after <a href="/facts/Nathan_M._Newmark/4ak9zwtV">Nathan M. Newmark</a>, former Professor of Civil Engineering at the <a href="/facts/University_of_Illinois_at_Urbana%E2%80%93Champaign/gusw7HoS">University of Illinois at Urbana–Champaign</a>, who developed it in 1959 for use in <a href="/facts/Structural_dynamics/xWUPczL6">structural dynamics</a>. The semi-discretized structural equation is a second order ordinary differential equation system, </p><p> M u ¨ + C u ˙ + f int ( u ) = f ext {\displaystyle M{\ddot {u}}+C{\dot {u}}+f^{\textrm {int}}(u)=f^{\textrm {ext}}\,} </p><p>here M {\displaystyle M} is the mass matrix, C {\displaystyle C} is the damping matrix, f int {\displaystyle f^{\textrm {int}}} and f ext {\displaystyle f^{\textrm {ext}}} are internal force per unit displacement and external forces, respectively. </p><p>Using the <a href="/facts/Extended_mean_value_theorem/JnjkzQb4">extended mean value theorem</a>, the Newmark- β {\displaystyle \beta } method states that the first time derivative (velocity in the <a href="/facts/Equation_of_motion/25msr1vE">equation of motion</a>) can be solved as, </p> u ˙ n + 1 = u ˙ n + Δ t u ¨ γ {\displaystyle {\dot {u}}_{n+1}={\dot {u}}_{n}+\Delta t~{\ddot {u}}_{\gamma }\,} <p>where </p> u ¨ γ = ( 1 − γ ) u ¨ n + γ u ¨ n + 1 0 ≤ γ ≤ 1 {\displaystyle {\ddot {u}}_{\gamma }=(1-\gamma ){\ddot {u}}_{n}+\gamma {\ddot {u}}_{n+1}~~~~0\leq \gamma \leq 1} <p>therefore </p> u ˙ n + 1 = u ˙ n + ( 1 − γ ) Δ t u ¨ n + γ Δ t u ¨ n + 1 . {\displaystyle {\dot {u}}_{n+1}={\dot {u}}_{n}+(1-\gamma )\Delta t~{\ddot {u}}_{n}+\gamma \Delta t~{\ddot {u}}_{n+1}.} <p>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, </p> u n + 1 = u n + Δ t u ˙ n + 1 2 Δ t 2 u ¨ β {\displaystyle u_{n+1}=u_{n}+\Delta t~{\dot {u}}_{n}+{\begin{matrix}{\frac {1}{2}}\end{matrix}}\Delta t^{2}~{\ddot {u}}_{\beta }} <p>where again </p> u ¨ β = ( 1 − 2 β ) u ¨ n + 2 β u ¨ n + 1 0 ≤ 2 β ≤ 1 {\displaystyle {\ddot {u}}_{\beta }=(1-2\beta ){\ddot {u}}_{n}+2\beta {\ddot {u}}_{n+1}~~~~0\leq 2\beta \leq 1} <p>The discretized structural equation becomes </p><p> u ˙ n + 1 = u ˙ n + ( 1 − γ ) Δ t u ¨ n + γ Δ t u ¨ n + 1 u n + 1 = u n + Δ t u ˙ n + Δ t 2 2 ( ( 1 − 2 β ) u ¨ n + 2 β u ¨ n + 1 ) M u ¨ n + 1 + C u ˙ n + 1 + f int ( u n + 1 ) = f n + 1 ext {\displaystyle {\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}}} </p><p>Explicit central difference scheme is obtained by setting γ = 0.5 {\displaystyle \gamma =0.5} and β = 0 {\displaystyle \beta =0} </p><p>Average constant acceleration (Middle point rule) is obtained by setting γ = 0.5 {\displaystyle \gamma =0.5} and β = 0.25 {\displaystyle \beta =0.25} </p>