Solvers

The solver block selects the algorithm that solves each time step's system. The type is required and case-sensitive, and must match the time integrator:

`type` value	Solver	Pair with
`Hessian minimizer`	Newton's method (Hessian minimizer)	`quasi static`, `Newmark`
`explicit solver`	Explicit lumped-mass update	`central difference`
`steepest descent`	Matrix-free steepest descent	implicit schemes

`Hessian minimizer`

Newton's method with a configurable step and optional backtracking line search.

solver:
  type: Hessian minimizer
  step: full Newton
  minimum iterations: 1
  maximum iterations: 16
  relative tolerance: 1.0e-10
  absolute tolerance: 1.0e-06

Key	Required	Default	Meaning
`step`	yes	—	Newton step type (see Steps)
`minimum iterations`	yes	—	minimum nonlinear iterations per step
`maximum iterations`	yes	—	maximum nonlinear iterations per step
`absolute tolerance`	yes	—	absolute residual-norm convergence tolerance
`relative tolerance`	yes	—	relative residual-norm convergence tolerance
`linear solver absolute tolerance`	no	`0.0`	absolute tolerance for the inner CG linear solve
`linear solver relative tolerance`	no	`sqrt(eps)`	relative tolerance for the inner CG linear solve
`use line search`	no	`false`	enable backtracking line search
`line search backtrack factor`	no	`0.5`	step-reduction factor per backtrack
`line search decrease factor`	no	`1.0e-04`	Armijo sufficient-decrease coefficient
`line search maximum iterations`	no	`16`	maximum backtracking iterations

`explicit solver`

Single-shot explicit update for central-difference dynamics; no nonlinear iteration or tolerances. Requires only a step, which is explicit.

solver:
  type: explicit solver
  step: explicit

`steepest descent`

Matrix-free steepest descent with a mandatory line search.

solver:
  type: steepest descent
  step: steepest descent
  minimum iterations: 1
  maximum iterations: 32
  relative tolerance: 1.0e-10
  absolute tolerance: 1.0e-06

Key	Required	Default	Meaning
`step`	yes	—	`steepest descent`
`minimum iterations`	yes	—	minimum iterations per step
`maximum iterations`	yes	—	maximum iterations per step
`absolute tolerance`	yes	—	absolute convergence tolerance
`relative tolerance`	yes	—	relative convergence tolerance
`line search backtrack factor`	no	`0.5`	step-reduction factor per backtrack
`line search decrease factor`	no	`1.0e-04`	Armijo sufficient-decrease coefficient
`line search maximum iterations`	no	`16`	maximum backtracking iterations
`energy stagnation window`	no	`0` (disabled)	mesh smoothing only: window length for the energy stagnation exit
`energy stagnation tolerance`	no	`1.0e-06`	mesh smoothing only: relative energy decrease below which a window counts as stalled

Energy stagnation exit (mesh smoothing)

A mesh being smoothed can reach the energy floor its topology permits long before the residual tolerances are met: near that floor the solver keeps reducing the gradient without reducing the energy, since the remedy is topology modification with an external tool, not more iterations. With energy stagnation window set to $W > 0$, the solve stops once the energy decrease over the last $W$ iterations, relative to the current energy, stays below energy stagnation tolerance for $W$ consecutive iterations. The energy still recoverable past that point is of the order of the tolerance times the energy itself. On trigger the step is accepted as converged, a message advising topology modification is logged, and the solver's stagnated flag is raised. The criterion applies to mesh smoothing models with the steepest descent solver only; setting the keys anywhere else aborts. See examples/ems/awful-cube/awful-cube-lbfgs.yaml.

Steps

The step key inside solver names the update direction. It is required and case-sensitive:

`step` value	Step	Notes
`full Newton`	Newton step from the tangent (Hessian)	for `Hessian minimizer`
`explicit`	Explicit lumped-mass update	for `explicit solver`
`steepest descent`	Negative-gradient step	for `steepest descent`
`lbfgs`	Limited-memory BFGS quasi-Newton step	for `Hessian minimizer`

Key	Required	Default	Applies to	Meaning
`step length`	no	`1.0`	all steps	scaling applied to the computed step
`memory`	no	`10`	`lbfgs`	number of stored curvature pairs

Canonical examples

Newton (implicit): examples/single/implicit-dynamic-solid
Explicit: examples/single/explicit-dynamic-solid
Quasi-static Newton: examples/single/static-solid