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.

翻译：等几何分析是一种连接计算机辅助设计与有限元分析的计算方法。在CAD中，计算域通常由B样条或NURBS曲面片表示。给定域的NURBS参数化，等几何离散化采用与域参数化相同的NURBS基函数定义于该域上。理想情况下，这种等几何离散化能精确表示底层CAD模型。CAD模型通常仅表示物体边界，因此对平面域而言，其以曲线集合形式给出。寻找合适的内部参数化是等几何分析的主要难题之一，类似于有限元分析中的网格生成过程。此参数化问题的目标是获得一组既能精确表示域边界、又具有规则参数化且无自交的曲面片。这可通过将域分割成沿界面匹配的曲面片，或通过覆盖重叠曲面片来实现。本文采用第二种方法。为从给定边界构造适用于等几何分析的平面参数化，我们提出一种基于偏移的域参数化算法。给定边界曲线，通过广义偏移获取内部曲线。这两条曲线定义环形曲面片，其内部孔洞可由多单元域覆盖。因此，该域表示为两个规则参数化的重叠子区域之并集。在此配置下，可采用(Kargaran, Jüttler, Kleiss, Mantzaflaris, Takacs; CMAME, 2019)引入的重叠多曲面片方法求解给定域上的偏微分方程。通过多个数值算例（考虑不同域形状）验证了所提方法的性能。