# Quasistatic formulation#

This formulation solves the steady-state equations as described in hyperelasticity 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. The quasistatic example 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 [NW99], 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.