Dear Claire,
You are right that this is a total Lagrangian formulation but that doesn't
mean that one is restricted to defining the problem in terms of fully
referential quantities.
One can arrive at the same conclusion from a number of starting points, but
ultimately its because we'd chosen to integrate spatial quantities on the
reference configuration. Along with that, following from the weak form we
need shape functions defined in the spatial configuration in order to
perform the integration correctly. These can be computed using the chain
rule: d/dx_{j} [N^{I}] = d/dX_{K} [N^{i}] . dX_{K}/dX_{j} = d/dx_{j}
[N^{I}] . F^{-1}_{jK}.
Does that help at all? Its a good exercise to derive the variational
problem with a fully referential or two-point description, and then with
some relatively simple (although tedious) manipulations you would end up
with the formulation adopted in the tutorial.
Best,
Jean-Paul
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