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This paper presents a method for isogeometric analysis using rational Triangular Bézier Splines (rTBS) where optimal convergence rates are achieved. In this method, both the geometry and the physical field are represented by bivariate splines in Bernstein Bézier form over the triangulation of a domain. From a given physical domain bounded by NURBS curves, a parametric domain and its triangulation are constructed. By imposing continuity constraints on Bézier ordinates, we obtain a set of global C r smooth basis functions. Convergence analysis shows that isogeometric analysis with such C r rTBS basis can deliver the optimal rate of convergence provided that the C r geometric map remains unchanged during the refinement process. This condition can be satisfied by constructing a pre-refinement geometric map that is sufficiently smooth. Numerical experiments verify that optimal rates of convergence are achieved for Poisson and linear elasticity problems.

This paper presents an approach for isogeometric analysis of 3D objects using rational Bézier tetrahedral elements. In this approach, both the geometry and the physical field are represented by trivariate splines in Bernstein Bézier form over the tetrahedrangulation of a 3D geometry. Given a NURBS represented geometry, either untrimmed or trimmed, we first convert it to a watertight geometry represented by rational triangular Bézier splines (rTBS). For trimmed geometries, a compatible subdivision scheme is developed to guarantee the watertightness. The rTBS geometry preserves exactly the original NURBS surfaces except for an interface layer between trimmed surfaces where controlled approximation occurs. From the watertight rTBS geometry, a Bézier tetrahedral partition is generated automatically. By imposing continuity constraints on Bézier ordinates of the elements, we obtain a set of global C r smooth basis functions and use it as the basis for analysis. Numerical examples demonstrate that our method achieve optimal convergence in C r spaces and can handle complicated geometries.

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Spline paths in NC machining are advantageous over linear and circular paths due to their smoothness and compact representation, thus are highly desirable in high-speed machining (HSM) where frequent change of tool position and orientation may lead to inefficient machining, tool wear, and chatter. This paper presents an approach for calculating spline NC paths directly from discrete points with controlled accuracy. Part geometry is represented by discrete points via an implicit point set surface (PSS). Cutter location (CL) points are generated directly from implicit part surfaces and interpolated by B-spline curves. A computing procedure for calculating maximum scallop height is given. The procedure is general and suitable for part surfaces in various surface representations provided that the closest distance from a point to the part surface can be calculated. Our results affirm that the proposed approach can produce high-quality B-spline NC paths directly from discrete points. The resulting spline paths make it possible for directly importing discrete points into Computer Numerical Control (CNC) machines for high-speed machining.