多重因素作用下等几何配点方法设计及其理论分析和加速

The isogeometric collocation method is a new method in the field of isogeometric analysis that arises in recent years. It is easy to construct, simple in form and relatively fast in computation speed. However, its low computational accuracy, slow convergence rate, and lack of theoretical analysis, limit the isogeometric collocation method widely used. This project starts from the research of multiple factors that affect the isogeometric collocation method, develops the computational domain parameter optimization algorithms, designs the effective collocation formats and the collocation selection methods, expounds the compatibility and convergence conditions of the isogeometric collocation method, and deduces the formula of the theoretical convergence rate. Then, we set up the theoretical analysis system of the isogeometric collocation method, and significantly improve its accuracy and convergence rate, making the isogeometric collocation method become the mainstream calculation method in the field of isogeometric analysis. In addition, this project will study the acceleration algorithms of the isogeometric collocation method from two aspects of algorithm design and hardware performance, greatly improving its computing speed. Moreover, the results of this project will be employed to implement the aircraft wing profile curve optimization system based on the isogeometric collocation method, and we will verify the validity of generating smooth curves and surfaces with the physical simulation target as the objective function. In this way, new theory and tools are provided for the field of geometric design, and the integration of CAD and CAE is promoted.

等几何配点法是近几年兴起的等几何分析计算新方法，它容易构造，形式简单，运算速度相对较快。但是，它的计算精度不高，收敛率较低，缺乏理论分析，这些缺点限制了等几何配点法的广泛应用。本项目从影响等几何配点法的多重因素着手研究，发展计算域参数化优化算法，设计有效配点格式和配点选取方法，阐明等几何配点法的相容性和收敛性条件，推导理论收敛率公式，初步建立等几何配点法的理论分析体系，显著提高等几何配点法的计算精度和收敛率，促使等几何配点法成为主流等几何分析计算方法。另外，本项目将从算法设计和硬件性能两方面研究等几何配点法的加速算法，大幅度提升它的计算速度。进而，利用本项目相关成果实现以等几何配点法为计算引擎的飞机机翼翼型曲线优化系统，验证以物理仿真目标为目标函数生成光顺曲线曲面的有效性，为几何设计提供新理论、新技术、新工具，促进CAD和CAE的融合发展。

等几何配点法是一种新兴的等几何分析计算方法，它构造简单、计算速度快；但是，与等几何伽辽金方法相比，它的计算精度和收敛率低，并且缺乏理论分析。本项目从影响等几何配点法的多重因素着手研究，完成了以下研究成果。第一、在计算域参数化构造和优化方面，提出了Gregory三变量参数体的构造方法，所构造的参数体可以有任意张边界面；另外，还提出了基于Polysquare简化的复杂计算域分割方法，用于分割复杂计算域，提高参数化质量。第二、在等几何配点法的配点格式和配点选取方法方面，设计了最小二乘等几何配点法，提出了通过拟合负载函数求解等几何配点问题的新方法，提高了计算精度和计算速度。第三、在等几何配点法的理论研究方面，我们推导了等几何配点法的收敛率公式，利用这个公式，阐明了影响等几何配点法的关键因素，并证明了等几何配点法相容性的充要条件。第四、在机翼翼型曲线优化系统实现方面，实现了基于势流模型与等几何配点法的机翼翼型曲线优化系统，在光滑几何外形的同时，进一步提升了升阻比这一性能指标。最后，我们撰写并发表综述文章，对等几何配点法的上述理论和应用成果进行了总结。在本项目执行过程中，在《Computer Methods in Applied Mechanics and Engineering》、《Computer-Aided Design》、《Computer Aided Geometric Design》、《Computers & Graphics》等国际知名期刊发表学术论文29篇，其中SCI收录21篇，EI收录6篇，授权软件著作权2项。培养博士生毕业4名，硕士生毕业3名。上述成果为几何设计和CAE提供了新理论、新方法、新工具，促进了CAD 和CAE的融合发展。