Study On Spline Curves And Surfaces In CAD/CAE

Geometric modeling is one of the kernel technologies of CAD system. For a long time, parametric curves and surfaces have been the primary tools in geometric modeling. Curves and surfaces parameterization and the surface blending are the foundation of geometric modeling in CAD. In this dissertation, we develop a set of functions, called representation set, that represent a given Bezier curve. Then, the optimal-parameterization curve can be selected in the representation set. For a polynomial Bezier curve, its representation set is deduced by polynomial and rational reparameterization, respectively. For a rational Bezier curve, the representation set is calculated by a general degree elevation. Moreover, we develop a geometric method to generate the representation set for a given Bezier curve.Surface blending is a useful operation in geometric design for rounding sharp edge or corner. Meanwhile, NURBS has already become the de facto industrial standard in existing CAD/CAM systems. Therefore, it is required to study how to blend two B-spline surfaces. However, two arbitrary B-spline surfaces (called base surfaces) are hard to be blended with a B-spline surface (called blending surface) because the knot vectors of the two base surfaces are usually mismatched. In this dissertation, we propose a curve-based spline representation, i.e., the semi-structured B-spline surface, which is generated by skinning a series of B-spline curves with different knot vectors. By assigning suitable knot vectors to the head and tail skinned curves, the semi-structured B-spline surface can blend two B-spline surfaces smoothly without disturbing them at all. We formulate the B-spline surface blending problem as an optimization problem with continuity constraints, and the continuity between the base and blending surfaces can reach G2or C2.CAE systems receive CAD models for analysis. To avoid the mesh transformation and to advance the seamless integration of CAD and CAE, isogeometric analysis(IGA) is proposed. Isogeometric analysis approximates the unknown solution of a boundary value problem (or initial value problem) by a NURBS function with non-linear NURBS basis functions. If the order of the NURBS function is high enough, the boundary value problem can be solved by isogeometric collocation method (IGA-C) applied to its strong form. Isogeometric collocation (IGA-C) method has shown its superior behavior over Galerkin method in terms of accuracy-to-computational-time ratio and other aspects. However, relatively little has been published about numerical analysis of the IGA-C method. This dissertation develops theoretical results on consistency and convergence of the IGA-C method to a generic boundary (initial) problem. It shows that the IGA-C method is convergent when differential operator of the boundary (initial) problem is stable or strongly monotone. We also show some concrete examples whose differential operators are strongly monotone, and the IGA-C method is convergent. However, in IGA-C method, the selection of the collocation points has great influence on the numerical solution. And the selection of desirable collocation points still remains as an open problem. To reduce the influence of the selection of the collocation points to the solution, in this dissertation, we develop the isogeometric least-square collocation method (IGA-L). IGA-L method determines the unknown solution by making the approximate differential operator of the numerical solution fit the real differential operator in the least-square meaning. Therefore, while IGA-C method requires the number of the collocation points equal to that of the coefficients of the numerical solution, the number of the collocation points employed in IGA-L method is larger than that of the unknown coefficients. This brings great flexibility in selecting collocation points in IGA-L method, and makes IGA-L method more robust and efficient than IGC method.In a word, we proposed new ideas and new algorithms of B-spline curves and surfaces in CAD/CAE, and presented new means and new techniques for CAD/CAE in this dissertation. The validity of all the theoretical results has been validated by lots of examples in this dissertation. It indicates that they have wide applications in computer aided design and computer aided engineering.