Efficient matrix assembly in isogeometric analysis with hierarchical B-splines

Hierarchical B-splines that allow local refinement have become a promising tool for developing adaptive isogeometric methods. Unfortunately, similar to tensor-product B-splines, the computational cost required for assembling the system matrices in isogeometric analysis with hierarchical B-splines is also high, particularly if the spline degree is increased. To address this issue, we propose an efficient matrix assembly approach for bivariate hierarchical B-splines based on the previous work (Pan, Juttler and Giust, 2020). The new algorithm consists of three stages: approximating the integrals by quasi-interpolation, building three compact look-up tables and assembling the matrices via sum-factorization. A detailed analysis shows that the complexity of our method has the order O(Np-3) under a mild assumption about mesh admissibility, where N and p denote the number of degrees of freedom and spline degree respectively. Finally, several experimental results are demonstrated to verify the theoretical results and to show the performance of the proposed method. (C) 2020 Elsevier B.V. All rights reserved.

Automated summary

Hierarchical B-splines with local refinement are a promising tool for adaptive isogeometric methods, but the computational cost for assembling system matrices is high, similar to tensor-product B-splines. To address this issue, an efficient matrix assembly approach for bivariate hierarchical B-splines is proposed based on quasi-interpolation, look-up table construction, and sum-factorization. The method shows a complexity of O(Np-3) under a mild assumption about mesh admissibility.