IGA Using Offset-based Overlapping Domain Parameterizations. (October 2021) - Kargaran, Somayeh; Jüttler, Bert; Takacs, Thomas
Abstract: Isogeometric analysis (IGA) is a numerical method, proposed in Hughes et al. (2005), 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 B-spline or NURBS parameterization of the domain, an isogeometric discretization is defined on the domain using the same B-spline or 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. For planar domains, the CAD model is given as a collection of curves representing the boundary. Finding a suitable parameterization of the interior is one of the major issues for IGA, similar to the mesh generation process in the FEA setting. The objective of this isogeometric 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 a planar parameterization suitable for IGA from a given boundary curve, we propose an offset-based domain parameterization algorithm. Given a boundary curve, we obtain an inner curve by generalized offsetting. The inner curve, together with the boundary curve, naturally defines a ring-shaped patch with an associated parameterization. By definition, the ring-shaped patch has a hole, which can be covered by a multi-cell domain. Consequently, the domain is represented as a union of two overlapping subdomains which are regularly parameterized. On such a configuration, one can employ the overlapping multi-patch (OMP) method, as introduced in Kargaran et al. (2019), to solve PDEs on the given domain. The performance of the proposed method is reported in several numerical examples, considering different shape properties of the given boundary curve. Graphical abstract: Highlights: We propose an offset-based domain parameterization (OODP) algorithm suitable for IGA. The algorithm produces two overlapping subdomains for a given boundary curve. The subdomains are a ring-shaped patch and a multi-cell domain. To solve PDEs we employ the overlapping multi-patch (OMP) method.