4. The Newmark schema [bib1], [bib2]#

4.1. Presentation of the diagram#

Newmark has introduced two parameters \(\gamma\) and \(\beta\) for calculating positions and speeds at step \(t+\Delta t\):

\({\dot{X}}_{t+\Delta t}={\dot{X}}_{t}+\Delta t\text{.}\left[(1-\gamma )\text{.}{\ddot{X}}_{t}+\gamma \text{.}{\ddot{X}}_{t+\Delta t}\right]\)

\({X}_{t+\Delta t}=\text{}{X}_{t}+\Delta t\text{.}{\dot{X}}_{t}+\Delta {t}^{2}\text{.}\left[(\frac{1}{2}-\beta )\text{.}{\ddot{X}}_{t}+\beta \text{.}{\ddot{X}}_{t+\Delta t}\right]\)

Let’s consider the equilibrium equations at time \(t+\Delta t\):

\(M\text{.}{\ddot{X}}_{t+\Delta t}+C\text{.}{\dot{X}}_{t+\Delta t}+\text{}K\text{.}{X}_{t+\Delta t}={R}_{t+\Delta t}\)

Let’s postpone the previous relationships by eliminating \({\dot{X}}_{t+\Delta t }\) and \({\ddot{X}}_{t+\Delta t }\), it comes in « D-form » parametrization:

\(\tilde{K}\text{.}{X}_{t+\Delta t}=\tilde{R}\text{où :}\tilde{K}=K+{a}_{0}\text{.}M+{a}_{1}\text{.}C\)

\(\tilde{R}={R}_{t+\Delta t}+C\text{.}\left\{{a}_{1}\text{.}{X}_{t}+{a}_{4}\text{.}{\dot{X}}_{t}+{a}_{5}\text{.}{\ddot{X}}_{t}\right\}+M\text{.}({a}_{0}\text{.}{X}_{t}+{a}_{2}\text{.}{\dot{X}}_{t}+{a}_{3}\text{.}{\ddot{X}}_{t})\)

with: \(\begin{array}{cccc}{a}_{0}=\frac{1}{(\beta \text{.}\Delta {t}^{2})}& {a}_{1}=\frac{\gamma }{(\beta \text{.}\Delta t)}& {a}_{2}=\frac{1}{(\beta \text{.}\Delta t)}& {a}_{3}=\frac{1}{2\beta }-1\\ {a}_{4}=\frac{\gamma }{\beta }-1& {a}_{5}=\frac{\Delta t}{2}(\frac{\gamma }{\beta }-2)& {a}_{6}=\Delta t\text{.}(1-\gamma )& {a}_{\mathrm{²7}}=\gamma \text{.}\Delta t\end{array}\)

4.2. Full Newmark schema algorithm#

a) initialization:

  1. initial conditions \({X}_{0}\), \({\dot{X}}_{0}\) and \({\ddot{X}}_{0}\)

  2. choice of \(\Delta t\) and calculation of the coefficients \({a}_{1}\),… \({a}_{8}\) (see above)

  3. assemble the stiffness \(K\), mass \(M\) and damping \(\mathrm{C}\) matrices

  4. form the effective stiffness matrix \(\tilde{K}=K+{a}_{0}\text{.}M+{a}_{1}\text{.}C\)

  5. Factorize \(\tilde{K}\)

b) at each time step:

++-+————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————-+ | |Calculate the effective load :math:`\tilde{R}` | ++-+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ ||2|\(\tilde{R}={R}_{t+\Delta t}+M\text{.}({a}_{0}\text{.}{X}_{t}+{a}_{2}\text{.}{\dot{X}}_{t}+{a}_{3}\text{.}{\ddot{X}}_{t})+C\text{.}\left\{{a}_{1}\text{.}{X}_{t}+{a}_{4}\text{.}{\dot{X}}_{t}+{a}_{5}\text{.}{\ddot{X}}_{t}\right\}\) | ++-+————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————-+ || |solve :math:`\tilde{K}\text{.}{X}_{t+\theta \text{.}\Delta t}=\tilde{R}` | ++-+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ || |Calculate velocities and accelerations at time :math:`t+\Delta t` :math:`\begin{array}{c}{\ddot{X}}_{t\text{+}\Delta t}={a}_{0}\text{.}({X}_{t\text{+}\Delta t}-{X}_{t})-{a}_{2}\text{.}{\dot{X}}_{t}-{a}_{3}\text{.}{\ddot{X}}_{t}\\ {\dot{X}}_{t\text{+}\Delta t}={\dot{X}}_{t}+{a}_{6}\text{.}{\ddot{X}}_{t}+{a}_{7}\text{.}{\ddot{X}}_{t\text{+}\Delta t}\end{array}`| ++-+————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————-+ ||5|Calculate the next time step: back to the beginning | ++-+————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————-+

4.3. Stability conditions of the Newmark schema#

The Newmark method is unconditionally stable if:

\(\gamma >0\text{.}5\) and \(\beta >\frac{{(2\gamma +1)}^{2}}{4}\)

Positive numerical damping is introduced if \(\gamma >0\text{.}5\) and negative if \(\gamma <0\text{.}5\). When \(\gamma =0\text{.}5\) and \(\beta =0\), the Newmark formula is reduced to the pattern of centered differences. A very often used combination is \(\gamma =0\text{.}5\) and \(\beta =\frac{1}{4}\), as it leads to a second-order, unconditionally stable pattern without numerical damping.

This integration diagram is used quite widely in the field of mechanics, because it makes it possible to choose the order of integration, to introduce numerical damping or not, and has very good precision. It is integrated into the operator DYNA_VIBRA/BASE_CALCUL =” PHYS “. The keyword factor NEWMARK allows you to specify the use of this integration scheme and the choice of the value of \(\beta\) and \(\gamma\). By default, the value for \(\beta\) is set to \(0.25\) and the value for \(\gamma\) is set to \(0.5\).