IGA-suitable planar parameterization with patch structure simplification of closed-form polysquare

A primary challenge for isogeometric analysis (IGA)-suitable parameterization is to efficiently decompose a complex computational domain into a small number of high-quality IGA-suitable patches without internal singularities. Aiming at domains with arbitrary topology, we propose integrating frame field and polysquare structure, resulting in a structure associated with a frame field that is locally integrable everywhere, i.e. a closed differential form or closed-form. Such a structure is more general than a common polysquare but has no internal singularity as well, which makes it suitable for IGA because it usually has fewer undesired boundary corners and low distortion. Since IGA prefers a small number of patches in the parameterization structure, we further propose a patch simplification method with distortion control. A key challenge in simplification is to prevent the resulting patches from degenerating. Previous methods rely on a mesh of the domain to characterize the non-degenerate condition. In sharp contrast, our approach simply uses the boundary of the domain, which obviously improves efficiency because the number of variables is much lower. The patch degeneration issue in such a boundary-only strategy is addressed by solving a constrained optimization problem with a set of automatically constructed inequality constraints. We prove that such constraints are sufficient to make the common polysquare structure homeomorphic to the input shape. We also demonstrate that it prevents the patches from degenerating for the structure of closed-form polysquares, and it can be extendedly applied to 3D closed-form polycubes. Numerical experiments using bi-quadratic B-spline surfaces with C1-constraints are also provided to justify the accuracy of applying the resulting planar parameterization for IGA application.