Issue link: https://resources.system-analysis.cadence.com/i/1472078
Preparation of Geometry Models for Mesh Generation and CFD 3 www.cadence.com Boundary Representation Boundary representation is the method of describing a solid object implicitly from its boundary and generally consists of two sub-representations: geometry and topology. Geometry describes individual shapes such as points, curves, and surfaces, typically parametric mappings from a lower-dimensional space into 3D. Topology is the term used to describe the entities that limit the portion of each geometric shape and the interconnections between those defined shapes. Analytic Geometry Geometry models produced in mechanical computer-aided design (Mechanical CAD or MCAD) software and typically shared via data exchange standards like STEP [4], and IGES [5] are referred to as analytic boundary representations. The surfaces can be explicit and piecewise (e.g., NURBS) or implicit (e.g., spheres, cylinders, and planes). Likewise, the curves can be explicit and piecewise (e.g., NURBS) or implicit (e.g., circles, conics, and lines). B-Splines and NURBS Modern CAD software predominantly uses an analytic geometry representation in which parametric equations describe the shape of curves and surfaces in 3D. The predominant form of parametric spline used in modern CAD software is NURBS (Non-Uniform Rational B-Spline, where B-Spline is an abbreviation of Basis Spline). See Farin [6] and Townsend [7] for brief histories of the evolution of splines in computer-aided geometric design (CAGD). Figure 1: Example of a NURBS surface (gray), its network of isoparametric lines (red and green), and its array of control points (black). Image from Issa [8]. Image used with the permission of the author. A parametric spline surface such as a NURBS maps a 2D rectangular parametric space of control points (u,v) to 3D (x,y,z) (Figure 1). A rectangular parametric space makes it difficult or virtually impossible to model a complex shape with a single NURBS surface due to the difficulty of deforming a rectangle into an arbitrary shape. Because of this, analytic boundary representation models from CAD software consist of a patchwork of NURBS surfaces. [At the core of MCAD software, one finds the mathematics of geometry and topology. These fundamental computations are performed by software known as a geometric modeling kernel. There are many kernels available today including open source tools like Open CASCADE [9] and EGADS [10] and commercial tools such as ACIS [11], C3D [12], CGM [13], Dyndrite [14], Geode [15], Granite [16], and Parasolid [17].]