Splines Suitable For Analysis And Modeling

Computer Aided Geometric Design (CAGD) has developed rapidly in the later part of the twentieth century as the NURBS (Non-Uniform Rational B-Splines) be-came a standard in computer-aided design/manufacture (CAD/CAM) for its math-ematical elegance and geometrical flexibility. However, geometric models are becom-ing more and more complicated as the advent of3D digital acquisition technology, and the traditional NURBS representations face with new challenges. The main reason is that the tensor-product structure makes the control points must lie on a rectangular grid topologically, leading to large redundant control points. T-splines was introduced by Sederberg in2003to overcome the disadvantage of NURBS, which permits T-junctions in the control nets. T-splines are a generalization of NURBS and are capable of significantly reducing the number of superfluous control points. While T-splines are suitable for local refinement in geometric modelling, they face issues in analysis, since the blending functions of T-splines may linearly dependent. Thus T-splines are not well suited for iso-geometric analysis (IGA)-a new concept recently introduced by Huges et al. The rapid development of CAGD and IGA technology calls for a new kind of locally refinable splines that are suitable for both geometric modeling and analysis.The main focus of the current thesis is to design local refinable splines which are suitable for both geometric modeling and iso-geometric analysis. Particularly two kinds of local refinable splines are defined-Modified T-splines and hierarchical splines on triangular partitions. In chapter two, a new type of splines called Modi-fied T-splines is defined. The main idea is to construct a set of basis functions which have some good properties such as nonnegativity, partition unity, local support and linear independence. Applications in surface fitting are provided to demonstrate the usefulness of the new splines in adaptive geometric modeling. For splines over triangular partitions, although the general theory is not far from mature, the theory of splines on some regular triangulation, such as type-I triangular partition, type-II triangular partition is well developed. These splines also have some nice properties, and have been applied in many scientific computing problems. Like NURBS, splines over regular triangular partitions do not hold local refinement property. In chapter three and chapter four, the nature refinement paradigm for tensor product hierar-chical splines is extended to splines defined on type-I and type-II triangulation, and box splines. Applications in surface fitting and solving elliptic PDEs are provided and the results seem promising.Curve fitting with splines is a fundamental problem in computer-aided design and engineering. However, how to place the knots appropriately in spline fitting re-mains a difficult issue. Treating spline knots as free parameters has been discussed in earlier literature, however the number of knots has to be fixed in advance with-out optimization. Until recently, with the advent of compressed sensing, sparsity theory has been applied in splines fitting. For such methods, the number of knots and positions are found automatically. However, there are still redundant knots which cannot be removed. Several strategies like iteration and deletion one by one, were proposed to settle this problem. Unfortunately they can’t solve the problem essentially. In chapter five, a two-stage framework for computing knots (including the number and positions) in curve fitting is proposed based on a sparse optimiza-tion model. The resulting interior knots will be deleted or shifted locally according to certain rules by the new approach, and thus are not necessarily a subset of ini-tial given knots, which is a main difference between our methods and other related methods. The new method produce less number of knots, and particularly when the data points are sampled enough from a spline, our algorithm can recover the ground truth knots within a given tolerance.