复杂产品模型等几何有限块理论方法及技术研究

Isogeometric analysis directly implements structural performance analysis based on exact CAD models, which is able to avoid the trivial operations caused by the difference between design and analysis models in the product design. However, the spline-parameterized models of complex product models are unavailable that makes isogeometric analysis difficult to solve the engineering models with complex geometries. This project will combine isogeometric analysis with finite cell method and immerse the complex model into a regular finite cell domain to solve the problems. The elements in the immersed domain are divided into solid elements, fictitious elements and trimmed elements by the geometric intersection approach, and a numerical integration method for irregular domains is proposed to compute the stiffness matrices of the trimmed elements, and then the isogeometric analysis for complex product models is achieved. In addition, a reusing strategy of element stiffness matrix is presented according to the element characteristics of the isogeometric finite cell method, which further improves the computational efficiency of the isogeometric finite cell method. The basic theory, core algorithm and key technology of this research retain the advantages of isogeometric analysis, e.g., high accuracy and high efficiency, and break the bottleneck that isogeometric analysis is unachievable in solving complex models, and solve the problem that the geometric boundary cannot be maintained in the conventional finite cell method. The research of this project not only provides technical support for the development of integrated 3D CAD/CAE software, but also promotes the innovative design ability in our manufacturing products.

等几何分析直接基于CAD精确模型进行结构性能分析，可避免产品设计中设计、分析模型差异引起的繁琐交互。然而，复杂产品模型的样条参数化模型获取极其困难，造成等几何分析难以求解形状复杂的工程实际模型。本项目将等几何分析与有限块法相结合，把复杂模型嵌入规整有限块区域进行求解，通过几何求交技术将嵌入域中的单元划分为实单元、虚单元和裁剪单元，并提出一种适用于不规则区域的数值积分方法以完成裁剪单元刚度矩阵计算，实现复杂产品模型的等几何分析。此外，根据等几何有限块法的单元特性，研究并提出一种单元刚度矩阵重用技术，进一步提高等几何有限块法的计算效率。本项目所研究的基础理论、核心算法及关键技术，既可保留等几何分析高精度、高效率等优点，又可克服等几何分析求解复杂模型的瓶颈，还能解决传统有限块法模型几何边界丢失的问题，为集成化的三维CAD/CAE设计分析软件研制提供技术支撑，有利于提升我国制造业产品创新设计能力。

等几何分析直接基于CAD精确模型进行结构性能分析，可避免产品设计中设计、分析模型差异引起的繁琐交互。然而，复杂产品模型的样条参数化模型获取极其困难，造成等几何分析难以求解形状复杂的工程实际模型。本项目将等几何分析与有限块法相结合，把复杂模型嵌入规整背景嵌入域进行求解。首先，通过计算CAD面片与嵌入域网格的最小有向距离，获取原几何模型的水平集隐式表达，从而根据该水平集值判断复杂模型在背景嵌入域中所占据的位置，实现复杂模型的有限块参数化表达。其次，基于嵌入域几何模型水平集函数值，将嵌入域单元划分为实单元、虚单元和裁剪单元，建立了不规则裁剪单元的自适应数值积分方法。然后，根据嵌入域形状规则的特点，提出了等几何有限块单元刚度矩阵重用方法，进一步提高了等几何有限块法的计算效率。最后，开发了等几何有限块分析原型系统，完成了复杂几何模型的实验验证，并将研究成果在结构拓扑优化中进行应用。本项目的基础理论、核心算法及关键技术，保留了等几何分析高精度、高效率的优点，突破了等几何分析求解复杂模型的瓶颈，为CAD/CAE设计分析一体化软件研制提供了技术支撑，有利于提升制造业设计能力。