Method of Newmark

the method of Newmark allows the numerical resolution of the differential equations second-order whose terms are possibly very complicated.

Principle

The equation is considered

M \ \ ddot X = F (X, \ dowry X, T)

in which the second member can be extremely complicated, the exiting force or the force of damping depend for example on position X. It is thus necessary to recompute the forces for each position and speed, possibly by very heavy calculations.

The principle of this method consists in determining by a development limited the position and speed to the moment $T$ starting from the same sizes at the moment $t-dt$ . This development contains a term of third order error proportional to derived from acceleration. On the basis of the assumption according to which acceleration varies linearly on the interval, one leads to formulas which do not depend any more a derivative third but which introduces the derivative at the moment $T$ in addition to those at the moment $t-dt$ .

It is possible to exploit, up to a certain point, the distribution between the two; we will consider the choice more used. Finally this technique, which functions obviously with linear equations, requires in the nonlinear case of the iterations using of the values with the index $a$ .

Algorithm

With the difference in other methods of numerical integration the algorithm is particularly simple.

Beginning

{The values with the preceding step become the old values}

$x_ {t-dt} = x_t; \ dowry {X} _ {t-dt} = \ dowry {X} _t; \ ddot {X} _ {t-dt} = \ ddot {X} _t$ ;

To repeat

{The new values become the old values}

$x_a = x_t; \ dowry {X} _a = \ dowry {X} _t$ ;

{Calculation of the new acceleration according to the old values}

$\ ddot {X} _t = {1 \ over M} F (x_a, \ dowry {X} _a, T)$ ;

{Calculation of the new speed and the new position according to new acceleration}

$\ dowry {X} _t = \ dowry {X} _ {t-dt} + {1 \ over 2} (\ ddot {X} _ {t-dt} + \ ddot {X} _t) dt$ ;

$x_t = x_ {t-dt} + \ dowry {X} _ {t-dt} dt + {1 \ over 3} (\ ddot {X} _ {t-dt} + {\ ddot {X} _t \ over 2}) dt^2$ ;

Until $|x_t - x_a| < \ varepsilon$ {the new approximation differs rather little from old the}

End.

This algorithm converges reasonably when the step of time is sufficiently small compared to the implied periods (clean periods of the system or periods of excitation) but, unlike its use in the linear case, one cannot impose an unconditional convergence.

See too

Methods of Runge-Kutta

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