适用于复杂外形产品的等几何造型理论及新型分析方法研究

Isogeometric analysis provides a new way to realize seamless integration of CAD/CAE. Till this moment, however, IGA is only utilized for the simulation of CAD products with simple geometry. To overcome this limitation, this project will further investigate new theory and efficient methodologies on isogeometric analysis for CAD products having complex engineering shapes. The proposed project has four major tasks: (1) to investigate methods for volumetric parametrization of CAD models with complex topology, to solve the construction problem of three-dimensional computational domain and enhance the application of isogeometric analysis in practice; (2) to propose the concept of analysis-reuse and further develop methods for topology-consistent volume partition for a set of CAD models, to achieve the analysis-reuse based on the integral computation and the structure of stiffness matrix and to improve the efficiency of isogeometric analysis solver; (3) to propose the generalized isogeometric analysis method in which the spline space of computational domain and the spline space of physical field can be chosen independently to improve the flexibility,local refinement property and solving efficiency of classical isogeometric analysis; (4)to investigate the shape optimization method based on the generalized isogeometric analysis approach in which the initial shape can be optimized directly to avoid the parametrization changing problem during the refinement process and improve the optimization efficiency. With the proposed research in this project, we expect to solve the bottleneck problems in isogeometric analysis, and to provide a solid theoretical and technical foundation for comprehensive applications of isogeometric analysis in practice.

等几何分析为实现CAD/CAE 的真正无缝融合提供了新途径，但目前仅适用于形状简单的CAD 模型.为克服这一局限性，本项目拟研究面向复杂外形CAD 产品的等几何造型理论及灵活高效的新型等几何分析方法.具体包括四项任务：1）研究适用于复杂拓扑CAD模型的体参数化方法,解决复杂三维计算域的构造难题，拓展等几何分析的应用广度；2)提出分析重用的概念，研究多个CAD 模型的拓扑一致体划分方法, 并实现基于积分计算与刚度矩阵结构的分析重用，提高求解效率；3) 提出计算域样条空间与物理场样条空间相异的广义等几何分析方法，可提升等几何分析的灵活性、局部加细性能与求解效率；4)研究基于广义等几何分析的外形优化方法，可实现对初始外形直接进行优化,避开了在加细过程中几何参数化发生改变的难题，提高优化效率.通过本项目的研究，可解决当前等几何分析面临的瓶颈问题，为将等几何分析进一步推向实用提供理论基础和技术支持.

围绕项目的预期研究内容与研究目标，本项目所取得的主要成果如下：1、针对复杂平面区域的参数化问题, 提出了基于区域分割与全局/局部优化相结合的高质量参数化方法，从而解决了复杂拓扑平面区域的参数化难题, 为等几何分析中任意平面计算域的参数化问题提供了一个基本框架; 2. 对于体参数化问题，提出了一种快速生成高质量CAD模型体参数化的离散模板方法，可通过求解一个稀疏的线性系统来得到内部控制顶点的最优位置；并研究了两种面向三维等几何分析的 Bézier 四面体与 Bézier 六面体之间的转化方法, 为不同体几何表示间的相互转化问题提供了有效的解决方案 ；3. 对于具有一致拓扑的CAD模型，提出了基于径向基函数的拓扑一致体参数化方法，并提出了基于预计算的等几何分析重用方法，可大大提高体参数化与等几何分析求解的效率；4. 提出了若干新型的等几何分析求解框架，例如计算域与物理场样条空间相异的新型等几何分析方法，基于广义细分方法的等几何分析框架及其收敛性分析等。目前已在国内外重要学术期刊上正式发表标注研究论文27篇,其中 SCI收录论文17篇，其中4篇SCI论文发表在浙江大学所遴选的TOP期刊上，EI收录论文6篇; 授权发明专利1项，获得学术科研奖励3项. 已圆满完成预定目标。

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