FESolvers

Package to easily solve nonlinear problem, in particularily tailored to Ferrite.jl. Requires julia v1.9 or later.

Your problem - your way

You define your own problem type, FESolvers just needs to be able to request updates and access to variables.

For a standard case, using Newton iterations to solve the problem, $$\mathbf{r}(\mathbf{x}(t),t) = \mathbf{0}$$ the following get* functions

x = getunknowns(problem)
r = getresidual(problem)
K = getjacobian(problem)    # ∂r/∂x (or an approximation thereof)

must be defined.

Define the solver and timestepper, e.g.

solver = QuasiStaticSolver(;nlsolver=NewtonSolver(), timestepper=FixedTimeStepper(collect(0:0.1:1.0)))

The problem can then be solved by calling

solve_problem!(problem, solver)

For each new time, t, in timestepper, the user-defined function

update_to_next_step!(problem, t)

is called to update boundary conditions and other time-dependent items to t. Thereafter, the user-defined function

update_problem!(problem, nothing, update_spec)

is called. If requested by update_spec, the residual and/or jacobian should be updated. Then starts the iterations to find the new value of the unknowns, x.

Iterations

First, the residual is checked by calling the user-defined function

calculate_convergence_measure(problem, Δx, iter) < gettolerance(nlsolver)

where the user defines calculate_convergence_measure. If false, then continue by calculating the update, Δx=-[∂r/∂x]\r using the linear solver given to nlsolver.

Then,

update_problem!(problem, Δx, update_spec)

is called once more, but this time with the update Δx such that the unknowns, x, the residual, and the jacobian can be updated. Keep looping over the iterations until calculate_convergence_measure returns a sufficiently small value.

After convergence

The first thing that happens after convergence (also directly after calling solve_problem) is that the user-defined function

postprocess!(problem, step, [solver])   # Optional to define with or without solver

is called. This can be used to save any data required for the given step. Directly after,

handle_converged!(problem)

is called. This should be used to update old values, such as old unknowns, old time, old state variables etc. to the current values.

End of simulation

At the end of the simulation, even if it fails due to an error, the function

close_problem(problem)

is called. This allows closing any open files etc.

Summary of functions to define in standard cases

x = getunknowns(problem)
r = getresidual(problem)
K = getjacobian(problem)                  # K = ∂r/∂x (or an approximation thereof)
update_to_next_step!(problem, t)          # Update boundary conditions etc. for a new time step
update_problem!(problem, Δx, update_spec) # Assemble stiffness and residual for x+=Δx 
calculate_convergence_criterion(problem)  # Get a scalar value to compare with the iteration tolerance
postprocess!(problem, step)               # Do all postprocessing for current step (after convergence)
handle_converged!(problem, t)             # Do stuff if required after the current time step has converged. 
close_problem(problem)                    # Close any open file streams etc. Called in a `finally` block.