Time integrators

The time integrator block selects the time-stepping scheme and the time window. The type is required and case-sensitive:

`type` value	Scheme	Use
`quasi static`	Quasi-static (no inertia)	static/incremental loading
`Newmark`	Newmark-β implicit dynamics	implicit transient
`central difference`	Central-difference explicit dynamics	explicit transient

Time window (all schemes)

Key	Required	Default	Meaning
`initial time`	yes	—	start time
`final time`	yes	—	end time
`time step`	yes	—	base (target) time increment Δt

In a multidomain simulation the time window is set on the top-level controller and injected into each subdomain, so subdomain input files may omit initial time and final time.

Adaptive time stepping (all schemes)

Adaptive stepping is all-or-nothing: either supply none of the four keys below (fixed step) or supply all four. The step is grown after successful steps and shrunk after failed ones, staying within the min/max bounds.

Key	Required	Default	Constraint
`minimum time step`	with the group	`time step`	≤ `maximum time step`
`maximum time step`	with the group	`time step`	—
`decrease factor`	with the group	`1.0`	≤ 1.0
`increase factor`	with the group	`1.0`	≥ 1.0

See examples/adaptive-time-stepping/.

`quasi static`

Drops inertia; solves a sequence of static equilibria over the time window. Pair with the Hessian minimizer solver.

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

Key	Required	Default	Meaning
`initial equilibrium`	no	`false`	solve for static equilibrium before the first step

`Newmark`

Implicit dynamics with the Newmark-β family. Pair with the Hessian minimizer solver.

time integrator:
  type: Newmark
  initial time: 0.0
  final time: 3.0e-6
  time step: 1.0e-6
  β: 0.25
  γ: 0.5

Key	Required	Default	Meaning
`β`	yes	—	Newmark β (use the Unicode key, not `beta`)
`γ`	yes	—	Newmark γ (use the Unicode key, not `gamma`)
`HHT α` (or `HHT alpha`)	no	`0.0`	Hilber–Hughes–Taylor α, in `[0, 1/3]`; when positive it overrides γ = 0.5 + α and β = 0.25(1 + α)² for controlled numerical damping

The common choice β = 0.25, γ = 0.5 is the unconditionally stable, non-dissipative average-acceleration (trapezoidal) rule.

`central difference`

Explicit dynamics. Pair with the explicit solver. The stable time step is estimated from the mesh and wave speed scaled by the Courant number CFL.

time integrator:
  type: central difference
  initial time: 0.0
  final time: 3.0e-6
  CFL: 0.2
  γ: 0.5

Key	Required	Default	Meaning
`CFL`	yes	—	Courant number used to bound the stable step from the element size and wave speed
`γ`	yes	—	central-difference γ (0.5 is the standard second-order value)

The CFL value scales an element-size-and-wave-speed estimate of the stable step; it is a safety factor rather than the exact operating Courant ratio. The element-size estimate is a characteristic diameter, so a value near 1.0 can exceed the true stability limit on hexahedral meshes. Values of about 0.5 or below are a conservative choice.

Canonical examples

Quasi-static: examples/single/static-solid
Newmark: examples/single/implicit-dynamic-solid
Central difference: examples/single/explicit-dynamic-solid
Adaptive stepping: examples/adaptive-time-stepping/