Isogeometric Parametrization Inspired by Large Elastic Deformation
For researchers in isogeometric analysis, this work provides a novel method for generating volumetric parametrizations, though it is incremental as it builds on existing mesh deformation and Coons patch techniques.
The paper addresses the problem of constructing volumetric parametrizations for isogeometric analysis by modeling the target domain as a deformed configuration of a simple initial geometry, using nonlinear elasticity with a neo-Hookean material law. The method yields bijective parametrizations and is demonstrated on 2D and 3D examples, showing competitive performance compared to established approaches.
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.