Newmark beta/gamma (Newmark) — 1 scheme

Newmark and generalized-alpha advance position and velocity together in second-order mechanics. Average-acceleration Newmark preserves linear modal energy, while generalized-alpha can deliberately damp unresolved fast modes. Check the force, mass and accepted-acceleration assumptions before choosing the damping parameters.

The classical Newmark time integrator for second-order dynamics $x'' = f(t, x, v)$, driven by drivers.mechanics.newmark — a repo-native version of the standard form:

$$x_{n+1} = x_n + \mathrm{dt}\, v_n + \mathrm{dt}^2\big( (1/2 - \beta)\, a_n + \beta\, a_{n+1} \big),$$ $$v_{n+1} = v_n + \mathrm{dt}\big( (1 - \gamma)\, a_n + \gamma\, a_{n+1} \big),$$

with the implicit unknown $a_{n+1}$ closed by the same nonlinear-solve machinery as the other implicit families (JFNKSolver by default; a fixed-point solver is accepted through the usual solver= override).

Parameters and variants

$\beta, \gamma \in [0, 1]$ (validated when stepping); gamma=1/2 is the second-order condition for ordinary Newmark. The registered default and callable linear-acceleration variant are:

Scheme	$\beta$	$\gamma$	Note
Newmark (default = newmark_average_acceleration)	$1/4$	$1/2$	constant average acceleration — the standard structural-dynamics choice; unconditionally stable for the linear oscillator, no numerical damping
newmark_linear_acceleration	$1/6$	$1/2$	linear acceleration — conditionally stable, with no algorithmic damping on the stable undamped linear oscillator

The default newmark(state, dt, f, ..., beta=0.25, gamma=0.5) accepts any admissible pair. The RHS must expose a velocity derivative — an update with a dudt field, or fields tagged velocity_derivative / acceleration — which is what supplies $a_n$ and the implicit $a_{n+1}$ guess.

Properties

Order 2. For the linear oscillator the average-acceleration variant is unconditionally stable (its frequency response is bounded for all $\mathrm{dt}\,\omega$); linear acceleration is stable only for $\mathrm{dt}\,\omega\le\sqrt{12}$ and does not add numerical damping there. Phase error remains even when linear modal energy is preserved. Newmark is not symplectic in general (measured dissipative in the Hamiltonian test suite — it is absent from the symplectic sets in tests/test_hamiltonian.py; the geometric alternatives for conservative dynamics are the symplectic family and Gauss-Legendre 2, coupled-rk).

The step's cost is one nonlinear solve (plus the endpoint evaluation recorded for the history) and the standard SolveDiagnostics entry. priorStep is refused; history= is threaded like the other one-step implicit schemes.

When to use

• Structural/second-order dynamics where the position and velocity both need the implicit endpoint (contact, constrained systems): Newmark gives both in one solve, with the damping behaviour of the $\gamma$ choice.
• Not for conservative Hamiltonian dynamics where energy fidelity over long runs is the point — use a symplectic scheme instead.

References

The two-mass structural demo compares the actual registered driver and the parameterized package GeneralizedAlpha against an exact modal solution and an independent dense generalized-alpha reference. It shows why preserving an underresolved fast mode's energy is not the same as representing its motion accurately.

• Newmark, N. M., Equations of motion for structural dynamics, Journal of Engineering Mechanics, ASCE 77 (1959) 679-709 — the $\beta$-$\gamma$ family.
• Hughes, T. J. R., The Finite Element Method (1987), Ch. 15 — the unconditional-stability and damping analysis.

Tests: tests/test_newmark.py (order 2, the linear-oscillator stability and damping of the two variants, the gradient through the solve).