IGA Using Offset-based Overlapping Domain Parameterizations

Isogeometric analysis (IGA) is a numerical method that connects computer-aided design (CAD) with finite element analysis (FEA). In CAD the computational domain is usually represented by B-spline or NURBS patches. Given a NURBS parameterization of the domain, an isogeometric discretization is defined on the domain using the same NURBS basis as for the domain parameterization. Ideally, such an isogeometric discretization allows an exact representation of the underlying CAD model. CAD models usually represent only the boundary of the object, thus, for planar domains, it is given as a collection of curves. Finding a suitable parameterization of the interior is one of the major issues in IGA, similar to the mesh generation process in FEA. The objective of this parameterization problem is to obtain a set of patches, which exactly represent the boundary of the domain and which are parameterized regularly and without self-intersections. This can be achieved by segmenting the domain into patches which are matching along interfaces, or by covering the domain with overlapping patches. In this paper we follow the second approach. To construct from a given boundary a planar parameterization suitable for IGA, we propose an offset-based domain parameterization algorithm. Given a boundary curve, we obtain an inner curve by generalized offsetting. Those two curves define a ring-shaped patch, which has a hole that can be covered by a multi-cell domain. Consequently, the domain is represented as a union of two overlapping subdomains which are both regularly parameterized. On such a configuration, one can employ the overlapping multi-patch method introduced in (Kargaran, J\"uttler, Kleiss, Mantzaflaris, Takacs; CMAME, 2019), to solve PDEs on the given domain. The performance of the proposed method is reported in several numerical examples, considering different shapes of the domain.