A weak condition for the convexity of tensor-product Bézier and B-spline surfaces

1994 ◽  
Vol 2 (1) ◽  
pp. 67-80 ◽  
Author(s):  
Michael S. Floater
Keyword(s):  
Tensor Product ◽  
Weak Condition ◽  
B Spline ◽  
Spline Surfaces

scholarly journals Innovative tooth contact analysis with non-uniform rational b-spline surfaces

2021 ◽  
Author(s):  
Felix Müller ◽  
Stefan Schumann ◽  
Berthold Schlecht
Keyword(s):  
Tensor Product ◽  
Measurement Data ◽  
Software Tool ◽  
Contact Analysis ◽  
Tooth Root ◽  
Surface Representation ◽  
Tooth Contact Analysis ◽  
Tooth Contact ◽  
B Spline ◽  
Spline Surfaces

AbstractMore and more simulation tools are being used in the development of gears in order to save development time and costs while improving the gears. BECAL is a comprehensive software tool for the tooth contact analysis (TCA) of bevel, hypoid, beveloid and spur gears. The gear geometry is provided by a manufacturing simulation or a geometry import. To determine the exact contact conditions in the TCA, the discrete flank points are converted into a continuous and differentiable surface representation. At present, it is an approximation by means of Bézier tensor product surfaces. With this surface representation, significant deviations to the target points can occur depending on the tooth geometry. In particular tip, root and end relief, strongly curved tooth root geometries or discontinuous topological measurement data due to e.g. micro-pitting can only be considered insufficiently.Hence, a new method for surface approximation with non-uniform rational b‑spline surfaces (NURBS) is presented. Its application can significantly improve the surface representation compared to the target geometry, leading to more realistic results regarding contact stress, tooth root stress and transmission error. To illustrate the advantages, NURBS-based surfaces are compared with the Bézier tensor product surfaces. Finally, the potential of the new approach regarding the prediction of lifetime and acoustics is demonstrated by application to different gear geometries.

Knot line refinement algorithms for tensor product B-spline surfaces

1985 ◽  
Vol 2 (1-3) ◽  
pp. 133-139 ◽  
Author(s):  
T. Lyche ◽  
E. Cohen ◽  
K. Mørken
Keyword(s):  
Tensor Product ◽  
B Spline ◽  
Spline Surfaces

scholarly journals Progressive Iterative Approximation for Extended B-Spline Interpolation Surfaces

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Yeqing Yi ◽  
Zixuan Tang ◽  
Chengzhi Liu
Keyword(s):  
Convergence Rate ◽  
Tensor Product ◽  
Spline Interpolation ◽  
Optimal Shape ◽  
Data Interpolation ◽  
Shape Parameters ◽  
Iterative Approximation ◽  
B Spline ◽  
Numerical Examples ◽  
Spline Surfaces

In order to improve the computational efficiency of data interpolation, we study the progressive iterative approximation (PIA) for tensor product extended cubic uniform B-spline surfaces. By solving the optimal shape parameters, we can minimize the spectral radius of PIA’s iteration matrix, and hence the convergence rate of PIA is accelerated. Stated numerical examples show that the optimal shape parameters make the PIA have the fastest convergence rate.

Generation and Modification of Non-Uniform B-Spline Surface Approximations to PDE Surfaces Using the Finite Element Method

1990 ◽  
Author(s):  
Joanna M. Brown ◽  
Malcolm I. G. Bloor ◽  
M. Susan Bloor ◽  
Michael J. Wilson
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Spline Approximation ◽  
The Finite Element Method ◽  
B Spline ◽  
B Splines ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Spline Basis ◽  
Element Method

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.

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

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.

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