A Refineable Space of Smooth Spline Surfaces of Arbitrary Topological Genus

Abstract A PDE surface is generated by solving partial differential equations subject to boundary conditions. To obtain an approximation of the PDE surface in the form of a B-spline surface the finite element method, with the basis formed from B-spline basis functions, can be used to solve the equations. The procedure is simplest when uniform B-splines are used, but it is also feasible, and in some cases desirable, to use non-uniform B-splines. It will also be shown that it is possible, if required, to modify the non-uniform B-spline approximation in a variety of ways, using the properties of B-spline surfaces.

Download Full-text

A Local Approach for Computing Smooth B-spline Surfaces for Arbitrary Quadrilateral Base Meshes

Journal of Computing and Information Science in Engineering ◽

10.1115/1.4051121 ◽

2021 ◽

pp. 1-28

Author(s):

Dennis Mosbach ◽

Katja Schladitz ◽

Bernd Hamann ◽

Hans Hagen

Keyword(s):

Degrees Of Freedom ◽

Tangent Plane ◽

Local Approach ◽

Approximation Process ◽

Surface Approximation ◽

B Spline ◽

Spline Surfaces ◽

Continuous Surface ◽

Normal Vectors ◽

The Individual

Abstract We present a method for approximating surface data of arbitrary topology by a model of smoothly connected B-spline surfaces. Most of the existing solutions for this problem use constructions with limited degrees of freedom or they address smoothness between surfaces in a post-processing step, often leading to undesirable surface behavior in proximity of the boundaries. Our contribution is the design of a local method for the approximation process. We compute a smooth B-spline surface approximation without imposing restrictions on the topology of a quadrilateral base mesh defining the individual B-spline surfaces, the used B-spline knot vectors, or the number of B-spline control points. Exact tangent plane continuity can generally not be achieved for a set of B-spline surfaces for an arbitrary underlying quadrilateral base mesh. Our method generates a set of B-spline surfaces that lead to a nearly tangent plane continuous surface approximation and is watertight, i.e., continuous. The presented examples demonstrate that we can generate B-spline approximations with differences of normal vectors along shared boundary curves of less than one degree. Our approach can also be adapted to locally utilize other approximation methods leading to higher orders of continuity.

Download Full-text

Sequential B-Spline Surface Construction Using Multiresolution Data Clouds

Journal of Computing and Information Science in Engineering ◽

10.1115/1.4006000 ◽

2012 ◽

Vol 12 (2) ◽

Author(s):

Yuan Yuan ◽

Shiyu Zhou

Keyword(s):

Control Points ◽

Finite Sample ◽

B Spline ◽

Spline Surface ◽

Spline Surfaces ◽

Filtering Technique ◽

Engineering Practices ◽

Asymptotical Convergence ◽

Surface Construction ◽

Fitting Error

B-spline surfaces are widely used in engineering practices as a flexible and efficient mathematical model for product design, analysis, and assessment. In this paper, we propose a new sequential B-spline surface construction procedure using multiresolution measurements. At each iterative step of the proposed procedure, we first update knots vectors based on bias and variance decomposition of the fitting error and then incorporate new data into the current surface approximation to fit the control points using Kalman filtering technique. The asymptotical convergence property of the proposed procedure is proved under the framework of sieves method. Using numerical case studies, the effectiveness of the method under finite sample is tested and demonstrated.

Download Full-text