  The construction of volumetric parametrizations for computational domains is
a key step in the pipeline of isogeometric analysis. Here, we investigate a
solution to this problem based on the mesh deformation approach. The desired
domain is modeled as a deformed configuration of an initial simple geometry.
Assuming that the parametrization of the initial domain is bijective and that
it is possible to find a locally invertible displacement field, the method
yields a bijective parametrization of the target domain. We compute the
displacement field by solving the equations of nonlinear elasticity with the
neo-Hookean material law, and we show an efficient variation of the incremental
loading algorithm tuned specifically to this application. In order to construct
the initial domain, we simplify the target domain's boundary by means of an
L2-projection onto a coarse basis and then apply the Coons patch approach. The
proposed methodology is not restricted to a single patch scenario but can be
utilized to construct multi-patch parametrizations with naturally looking
boundaries between neighboring patches. We illustrate its performance and
compare the result to other established parametrization approaches on a range
of two-dimensional and three-dimensional examples.
