AbstractPHT-splines are a type of polynomial splines over hierarchical T-meshes which posses perfect local refinement property. This property makes PHT-splines useful in geometric modeling and iso-geometric analysis. Current implementation of PHT-splines stores the basis functions in Bézier forms, which saves some computational costs but consumes a lot of memories. In this paper, we propose a de Boor like algorithm to evaluate PHT-splines provided that only the information about the control coefficients and the hierarchical mesh structure is given. The basic idea is to represent a PHT-spline locally in a tensor product B-spline, and then apply the de-Boor algorithm to evaluate the PHT-spline at a certain parameter pair. We perform analysis about computational complexity and memory costs. The results show that our algorithm takes about the same order of computational costs while requires much less amount of memory compared with the Bézier representations. We give an example to illustrate the effectiveness of our algorithm.
Space fractional advection dispersion equation includes non-local differential operators, which leads to calculating numerical integrals with weakly singular kernel on every subintervals. Oscillations are visible in computed solutions when fractional order tends to 1. Studies show that the stiffness matrix of time semi-discretization can be calculated directly by formulas established from a special variational formulation. Oscillations are eliminated by using adaptive moving mesh and De Boor algorithm, while the number of nodes remains unchanged.
This paper realizes direct interpolation of NURBS with use of solving the non-linear equation and de Boor algorithm, which improves the efficiency greatly by avoiding computing the derivatives and basic functions; meanwhile, look-ahead adaptive interpolation algorithm (LAIA) adjusts the feedrate according to the curvature to guarantee the contour accuracy, and the new backtracking and re-interpolation strategy makes it more efficient. Reverse interpolation with use of NURBS symmetry is proposed to predict the deceleration point precisely. The algorithm performs very well in experiments, which prove its viability and reliability.
This paper studies geometric design of uniform developable B-spline surfaces from two boundary curves. The developability constraints are geometrically derived from the de Boor algorithm and expressed as a set of equations that must be fulfilled by the B-spline control points. These equations help characterize the number of degrees of freedom (DOF’s) for the surface design. For a cubic B-spline surface with a first boundary curve freely chosen, five more DOF’s are available for a second boundary curve when both curves contain four control points. There remain (7-2m) DOF’s for a cubic surface consisting of m consecutive patches with C2 continuity. The results are in accordance with previous findings for equivalent composite Be´zier surfaces. Test examples are illustrated to demonstrate design methods that fully utilize the DOF’s without leading to over-constrained systems in the solution process. Providing a foundation for systematic implementation of a CAGD system for developable B-spline surfaces, this work has substantial improvements over past studies.
Efficient Computation of Isosurface Curvatures on GPUs Based on the de Boor Algorithm