面向等几何分析的高效矩阵装配算法及其应用

等几何分析是一种旨在无缝集成计算机辅助设计与计算机辅助工程的新型数值计算方法，目前已成为几何设计、分析及计算领域新的研究热点。具有局部细分能力的样条克服了NURBS无法实现局部加细这一弊端，已被成功地应用于等几何分析并展现出很好的性能。然而，与NURBS类似，基于可局部加细样条的等几何分析方法中的矩阵装配过程会很耗时（特别是对于高次的样条基函数以及大规模高维问题），目前鲜有处理此问题的研究工作。本项目拟采用拟插值、改进的和分解以及低秩逼近等技术，发展面向两类可局部加细样条（层次B样条和截断层次B样条）的高效矩阵装配算法，重点研究如何构造高效的拟插值算子、改进经典的和分解方法以及使用低秩逼近技术。此外，我们将探索所提出的算法在无矩阵方法、线性弹性问题与热传导问题中的应用。本项目的研究成果有助于提高等几何分析方法在工程中的实用性，为真正实现设计与分析的无缝融合做出积极的贡献。

Isogeometric analysis, introduced to seamlessly integrate computer-aided design and computer-aided engineering, is a new approach for numerical simulation. It has become a research hotspot in the field of geometric design, analysis and computation. Local refinable splines, which overcome the drawback of NURBS, have been successfully applied in isogeometric analysis and have shown good performance. However, similar as for NURBS, assembling the system matrices in isogeometric analysis with local refinable splines can be very expensive from the computational point of view, especially for large-scale problems with high spatial dimensions and spline degrees. To the best of our knowledge, few works have been proposed to address the above issue. In this project, we plan to investigate fast matrix assembly for isogeometric analysis with hierarchical B-splines and truncated hierarchical B-splines based on the quasi-interpolation, improved sum-factorization and low-rank approximation techniques. The effort will be devoted to constructing efficient quasi-interpolation operators, improving the classical sum-factorization approach and applying the low-rank approximation technique. In addition, we will explore the applications of the proposed assembly algorithms in matrix-free methods, linear elastic and heat conduction problems. The research results of this project can improve the practicability of isogeometric analysis in engineering, and make a contribution to the real seamless integration of design and analysis.