Time integrators

The required top-level time integrator section selects the time-stepping scheme and its parameters.

time integrator:
  type: quasi static
  initial time: 0.0
  final time:   1.0
  time step:    0.1

Keys common to all integrators

Key	Required	Default	Description
`type`	no	`quasi static`	Scheme — see below.
`initial time`	no	`0.0`	Start time `t₀`.
`final time`	yes	—	End time `t_f`. Missing key raises a `KeyError`.
`time step`	yes	—	Nominal step `Δt`. Missing key raises a `KeyError`.

Unlike most sections, final time and time step are indexed directly rather than fetched with a default, so omitting either produces a bare KeyError rather than a friendly message.

`type` values

Value	Aliases	Kind
`quasi static`	`quasistatic`, `static`	Rate-independent equilibrium; no inertia
`newmark`	`newmark-beta`	Implicit dynamics (Newmark-β / HHT-α)
`central difference`	`centraldifference`, `cd`	Explicit dynamics

The value is lowercased before matching. An unrecognised type is a hard error.

Adaptive time stepping

Available to quasi static and newmark (parsed for central difference as well, though an explicit run is governed by stability rather than convergence).

All four keys must appear together, or none of them:

  minimum time step: 0.001
  maximum time step: 0.1
  decrease factor:   0.5
  increase factor:   1.5

Key	Constraint	Description
`minimum time step`	≤ `maximum time step`	Floor on Δt.
`maximum time step`	—	Ceiling on Δt.
`decrease factor`	must be < 1.0	Δt multiplier after a failed solve.
`increase factor`	must be > 1.0	Δt multiplier after a successful solve.

Supplying some but not all four is an error:

Adaptive time stepping requires all four: minimum time step, maximum time step,
decrease factor, increase factor.

Each constraint is checked individually, so decrease factor: 1.5 fails with decrease factor must be < 1.0. When the keys are omitted, Δt is fixed: min_dt = max_dt = time step and both factors are 1.0.

Quasi-static

For problems with no inertia. Requires a solver section.

time integrator:
  type: quasi static
  initial time: 0.0
  final time:   1.0
  time step:    0.1
  initial equilibrium: false

Key	Default	Description
`initial equilibrium`	`false`	Solve `R = 0` at `t₀` before advancing.

If initial equilibrium: true and that first solve does not converge, the run aborts with Failed to establish initial equilibrium.

Velocity and traveling-wave initial conditions are ignored by this integrator (with a warning) — see Initial conditions.

Newmark (implicit dynamics)

Second-order implicit integration. Requires a solver section.

time integrator:
  type: newmark
  initial time: 0.0
  final time:   0.01
  time step:    1.0e-4
  beta:  0.25
  gamma: 0.5

Key	Aliases	Default	Description
`beta`	`β`	`0.25`	Displacement-update parameter.
`gamma`	`γ`	`0.5`	Velocity-update parameter. `0.5` = no numerical damping.
`alpha`	—	`0.0`	HHT-α parameter. See the warning below.

beta and gamma accept either the ASCII or the Unicode spelling; the ASCII key is checked first and wins if both are present.

`alpha` overrides `beta` and `gamma`

When alpha is nonzero, β and γ are derived from it and any user-supplied beta/gamma are silently discarded:

β = (1 − α)² / 4        γ = (1 − 2α) / 2

So alpha: -0.1 always yields β = 0.3025, γ = 0.6 regardless of what you wrote. To set β and γ independently, leave alpha at its default of 0.0.

Common parameter choices:

Scheme	`beta`	`gamma`	`alpha`	Damping
Trapezoidal rule (average acceleration)	0.25	0.5	0.0	none
HHT-α, mild	—	—	−0.05	light
HHT-α, moderate	—	—	−0.1	moderate

Central difference (explicit dynamics)

Conditionally stable explicit integration. Uses no nonlinear solver — the solver section is not consulted. This is the matrix-free path: Carina opts the assembler into matrix-free mode, skipping sparse preallocation entirely.

time integrator:
  type: central difference
  initial time: 0.0
  final time:   0.001
  time step:    1.0e-7
  gamma: 0.5
  CFL:   0.9
  stable time step interval: 0

Key	Aliases	Default	Description
`gamma`	`γ`	`0.5`	Velocity-update parameter. `0.5` = standard central difference.
`CFL`	`cfl`	`0.0`	If > 0, compute a stable Δt and cap `time step` with it.
`stable time step interval`	—	`0`	Steps between recomputations of the stable Δt. `0` = compute once at startup only.

The CFL cap

When CFL > 0, Carina computes a stable step from the element characteristic length and the material dilatational wave speed,

Δt_stable = CFL · min(h) / c_p,     c_p = √(M / ρ)

with M the p-wave modulus, and then takes Δt = min(Δt_stable, time step). The estimate is logged at startup:

[SETUP]   Stable Δt = 2.05e-06 (CFL = 0.90)

If your requested time step exceeds the stable step, Carina warns and uses the stable one — it will not run an unstable explicit simulation just because you asked:

[WARNING] Δt = 5.00e-06 exceeds stable Δt = 2.05e-06 — using stable step.

Setting stable time step interval: N with N > 0 recomputes the estimate every N steps, which matters when large deformation changes element geometry enough to shrink the stable step mid-run. Leaving it at 0 computes the estimate once, from the undeformed configuration.

Note that the stable-step computation needs a nonzero density; see Materials.