An Adaptive Collocation Method with Weighted Extended PHT-Splines
- PDF / 8,532,189 Bytes
- 21 Pages / 595.276 x 841.89 pts (A4) Page_size
- 19 Downloads / 233 Views
An Adaptive Collocation Method with Weighted Extended PHT-Splines∗ NI Qian · DENG Jiansong · WANG Xuhui
DOI: 10.1007/s11424-020-9390-7 Received: 17 September 2019 / Revised: 19 November 2019 c The Editorial Office of JSSC & Springer-Verlag GmbH Germany 2020 Abstract This paper presents an adaptive collocation method with weighted extended PHT-splines. The authors modify the classification rules for basis functions based on the relation between the basis vertices and the computational domain. The Gaussian points are chosen to be collocation points since PHT-splines are C 1 continuous. The authors also provide relocation techniques to resolve the mismatch problem between the number of basis functions and the number of interpolation conditions. Compared to the traditional Greville collocation method, the new approach has improved accuracy with fewer oscillations. Several numerical examples are also provided to test our the proposed approach. Keywords
1
Collocation method, Gaussian points, PHT-splines, weighted extended PHT-splines.
Introduction
B-splines play an important role in the field of computer aided design (CAD). However, the regular structure of tensor product meshes impedes the further applications of B-splines in finite element method (FEM). To make B-splines more applicable to solving partial differential equations, isogeometric analysis (IGA) was introduced by Hughes, et al.[1] . Isogeometric analysis is mainly based on defining the geometry and the solution using the same basis functions. Since the standard tensor-product mechanism of NURBS prevents the localized editing of the mesh, several locally refinable splines were introduced in IGA. For example, these include 1) T-splines[2, 3] , AST-Splines and AS++ T-splines[4–7] ; 2) PHT-splines[8–12] ; 3) HB-splines[13–15] and THBsplines[16, 17] ; 4) LR-splines[18] . It should also be noted that the framework of isogeometric analysis based on generalized B-splines was also introduced in [19]. Using this IGA framework, NI Qian · DENG Jiansong School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China. WANG Xuhui (Corresponding author) School of Mathematics, Hefei University of Technology, Hefei 230009, China. Email: [email protected]. ∗ This research was supported by the National Natural Science Fondation of China under Grant Nos. 11601114, 11771420, 61772167. This paper was recommended for publication by Editor-in-Chief GAO Xiao-Shan.
2
NI QIAN · DENG JIANSONG · WANG XUHUI
the problem on computational domain bounded by transcendental curves or surfaces can be dealt with more efficiently. The parametrization of the computational domain is a key step in isogeometric analysis. Cohen, et al.[20] found that the results of parametrization not only affect the accuracy and convergence speed of solutions but also affect the total running time of improvement. However, when conducting IGA with complex geometric forms (e.g., problem over the annular domain in Figure 1(a)), the parametrization process is not easy to accomplish (for m
Data Loading...
