دانلود رایگان مقاله ادغام توابع بی اسپلاین به تعمیم در سطوح کتمول - کلارک در نقاط تکین

Abstract

Subdivision surfaces are a common tool in geometric modelling, especially in computer graphics and computer animation. Nowadays, this concept has become established in engineering too. The focus here is on quadrilateral control grids and generalized B-spline surfaces of Catmull–Clark subdivision type. In the classical theory, a subdivision surface is defined as the limit of the repetitive application of subdivision rules to the control grid. Based on Stam’s idea, the labour-intensive process can be avoided by using a natural parameterization of the limit surface. However, the simplification is not free of defects. At singularities, the smoothness of the classically defined limit surface has been lost. This paper describes how to rescue the parameterization by using a subdivision basis function that is consistent with the classical definition, but is expensive to compute. Based on this, we introduce a characteristic subdivision finite element and use it to discretize integrals on subdivision surfaces. We show that in the integral representation the complicated parameterization reduces to a decisive factor. We compare the natural and the characteristic subdivision finite element approach solving PDEs on surfaces. As model problem we consider the mean curvature flow, whereby the computation is done on the step-by-step changing geometry.

5. Conclusion and future work

The paper introduces a new subdivision finite element approach that is consistent with the classical idea of subdivision sur-faces. The aim of this paper is to derive the mathematical principles of this approach to use it for PDEs on surfaces. In our approach, we make use of the labour-intensive characteristic parameterization of the limit surface. However, in the presented concepts the inversion of the characteristic map is done only implicitly, i.e. in the integral representation, it reduces to a valence dependent scaling factor. That makes our approach practicable for PDE applications. Due to the complexity of the to be integrated functions, the use of an appropriate numerical integration is still an open problem. On the other hand, the computational effort that accompanies the integration on irregular elements also indicates that more work needs to be done in this area. Moreover, a more valid explanation for the choice of the number of subdivision levels have to be found. In the future, we will investigate the dependence of the valence of the extraordinary vertex on the number of subdivision levels to be integrated.