I'm with the background of computer engineering and generally use FEM for graphics simulation. As far as I know, FEM formulation is usually expressed with respect to the reference configuration, i.e., the volume integration is over the initial domain and the stress-based force $f$ is computed using the first Piola-Kirchhoff stress $P$ integrated over the initial element $\Omega_0$ as: $$ f = \int_{\Omega_0}P\nabla_X Nd\Omega $$
Is there any good reason why initial configuration is preferred over current configuration? Theoretically the force can be expressed with the cauchy stress $\sigma$ integrated over current volume.
The integration is usually approximated with Gauss quadrature. One reason I assume is that accuracy and robustness of the numerically approximated volume integration maybe at stake if the shape of current domain is close to degenerated (with nearly not invertible jacobian). Integrating over initial volume is better as we assume the initial mesh quality is good.