# Time integration
(static-quasistatic)=
## Static and Quasistatic

The above formulations solve the steady-state equations but with updates to the time parameter in the boundary conditions.
Ratel also offers a static solve option via a PETSc SNES solve; this is covered in the [static elasticity example](example-static).
The [quasistatic example](example-quasistatic) uses the PETSc Timestepper (TS) object to manage pseudo-timestepping.

Large deformation solid mechanics exhibits both geometric and material nonlinearities, leading to path dependence by which there are be multiple static solutions for a specified set of boundary conditions.
To disambiguate the multiple solutions, we solve the hyperelastic problem as a non-autonomous differential algebraic equation of index 1, with boundary conditions/loading a function of time $t \in [0, 1]$.
The current quasistatic example uses applied load (rather than displacement) and use backward Euler from PETSc's TS with extrapolation-based hot starts disabled for simplicity.

Each pseudo time step requires a nonlinear solve, which is implemented using PETSc's Scalable Nonlinear Equations Solver (SNES).
By default, Ratel uses Newton-CG in which a multigrid V-cycle is used as a preconditioner for conjugate gradients.
Additionally, we use a "critical point" line search, which supposes that the residual is the functional gradient of a latent objective function, $\bm F(\bm u) = \nabla_{\bm u}\Psi(\bm u)$ and uses one step of a secant method to find $\alpha$ for which $F(\bm u + \alpha \delta \bm u)^{T} \delta \bm u = 0$ where $\delta \bm u$ is the search direction found by Newton.
This line search is inspired by the strong Wolfe conditions in optimization {cite}`nocedal99`, but without explicit evaluation of the objective $\Psi$, which may not be available or may not exist (e.g., for non-conservative models).

More information on the PETSc TS and SNES objects can be found in the [PETSc documentation](https://petsc.org/release).


(elastodynamics)=
## Elastodynamics

In {eq}`residual-matrix-free`, we represented the static problem i.e. $\ddot{\bm u} = 0$. However, in the [dynamic example](example-dynamic) acceleration is **not zero** and we use the PETSc Timestepper (TS) object to manage timestepping. Specifically, Ratel uses the Generalized-Alpha method for second order systems (`TSALPHA2`) to solve the second order system of equations given by the momentum balance. More information on the PETSc `TSALPHA2` time stepper can be found in the [PETSc documentation TS](https://petsc.org/release/docs/manualpages/TS/index.html).

For elastodynamics we need to add

$$
  \int_{\Omega_0} \bm v \cdot \left(\rho_0 \, \ddot{\bm u} \right) \, dV,
$$ (elastodynamics-residual)

to the residual equation {eq}`residual-matrix-free` (or adding $\rho_0 \ddot{\bm u}$ to $\bm f_0$ term) and its jacobian

$$
  \mathrm{shift_a} \, \int_{\Omega_0} \bm v \cdot \left(\rho_0 \, \diff \bm u \right) \, dV,
$$ (elastodynamics-jacobian)

Note that in general for a time dependent residual function

$$
  F (t, \bm u, \dot{\bm u}, \ddot{\bm u}) = 0,
$$ (time-dependent-residual)

we can write the Jacobian as

$$
  \diff F = \frac{\partial F}{\partial \bm u} \, \diff \bm u + \mathrm{shift_v} \, \frac{\partial F}{\partial \dot{\bm u}} \, \diff \bm u + \mathrm{shift_a} \, \frac{\partial F}{\partial \ddot{\bm u}} \, \diff \bm u,
$$ (time-dependent-jacobian)

where $\mathrm{shift_v}, \mathrm{shift_a}$ is related to the inverse of time step $\Delta t$ and its square.