how to apply follower load

[Aminofa70]

@KnutAM @KristofferC @fredrikekre
I have implemented a code to update the traction with displacement (follower load) for nonlinear problems.
The following is the code, but I am curios if we can do it better or even if mine is incorrect?

function assemble_traction_forces_twoD!(F_ext, dh,  facetsets::Vector,  facetvalues, 
     tractions::Dict{Int, <:AbstractVector}, u::AbstractVector)

    fe_ext = zeros(getnbasefunctions(facetvalues))
    for (idx, facetset) in enumerate(facetsets)
        t_traction = tractions[idx]
        for facetidx in facetset  
            # Get the cell and facet index
            cellid = facetidx[1]      # Cell ID
            facet_idx = facetidx[2]   # Local facet index
            cell = getcells(dh.grid)[cellid]
            
            cell_coords = getcoordinates(dh.grid, cellid)             
 
            updated_coords = copy(cell_coords)
            cell_dofs = celldofs(dh, cellid)  
            for i in eachindex(cell_coords)
                dim = length(cell_coords[1])  
                current_coords = cell_coords[i]
                # Get DOFs for the i-th node (x and y displacements)
                dof_x = cell_dofs[2*i-1]  # x-displacement DOF
                dof_y = cell_dofs[2*i]    # y-displacement DOF
                # Create new coordinates
                new_coords = (current_coords[1] + u[dof_x], current_coords[2] + u[dof_y])
                updated_coords[i] = Ferrite.Vec{2, Float64}(new_coords)
            end
            
            # Reinitialize FacetValues with updated coordinates
            reinit!(facetvalues, cell, updated_coords, facet_idx)
            
            # Compute traction forces
            fill!(fe_ext, 0.0)
            for qp in 1:getnquadpoints(facetvalues)
                dΓ = getdetJdV(facetvalues, qp)
                for i in 1:getnbasefunctions(facetvalues)
                    Nᵢ = shape_value(facetvalues, qp, i)
                    fe_ext[i] += t_traction ⋅ Nᵢ * dΓ
                end
            end
            assemble!(F_ext, cell_dofs, fe_ext)
        end
    end
    return F_ext
end
```

[KnutAM]

This depends on what problem you try to solve? If your problem involves large rotations, you probably want to update the coordinates during assembly of the stiffness as well (i.e. updated lagrange)

The update looks ok in your specific case at a first glance, but has the same limitations as your initial code for the node to dof mapping. I would probably use that code directly instead to update the coordinates in the grid, but haven't implemented updated Lagrange so not 100% sure what the best approach is.

The other option is calculating the deformation gradient and use Nanson's formula to map the normal vector, this avoids having to update the coordinates and if you don't want the incremental linear approximation allows you to get the derivative dfext/du for the Newton iterations.

[Aminofa70]

@KnutAM thanks a lot.
The code is Total Lagrangian for hyperelasticity.

In such a case do we need to update the coordinate based on the displacement for next increment for the traction surface load? or just we need to calculate with the reference?

I believe that the hyperelasticity demo in Ferrite should be updated to include the load increment or something like time step? (both is awesome ) Do you agree?
That's more common in the field.

[KnutAM]

I see, in that case you want to rely on Nanson's formula, and it depends on your specific use case what you want. What physical load do you want to describe? Can you provide the exact weak form of the problem you want to solve, especially being clear with the boundary load on which normal and which area you have the load per? In the Ferrite hyperelasticity example. This is when you end up having to have a deformation dependent load, and you need to include this in the stiffness calculation, i.e. your tangent stiffness becomes K := dr/da = dfint/da - fext/da. So you do not have to update the coordinates.

Regarding load stepping, yes, that is normally done 👍. If you have no buckling and get convergence it won't be needed for accuracy though. However, to not complicate the example one could add a note that this is normally done, and refer to e.g. the plasticity example for how to do load stepping with nonlinear FE.

[Aminofa70]

@KnutAM Super thanks. The problem is simple, a cube under uniaxial tension. The Weak form exactly the same as the hyper-elasticty example.