等几何径向积分边界元法及其在功能梯度材料热应力分析中的应用

Due to the special advantages of boundary element method (BEM), BEM has become one of the important numerical methods for the thermal stress analysis of functionally graded materials. However, the analysis model used in the traditional BEM is the mesh model based on the Lagrange elements, which will introduce the geometric errors in the numerical analysis. In this project, the radial integration BEM (RIBEM) will be combined with isogeometric analysis to establish isogeometirc RIBEM for thermal stress analysis of functionally graded materials, in order to achieve the combination of boundary element analysis model and CAD model. The basic research idea is that the appropriate fundamental solution and normalized physical quantity are chosen to derive the boundary integral equation using the weighted residual method. And then, the domain integrals appearing in the integral equations are transformed into boundary integrals using radial integration method, in order to form the radial integration transformation technique based on isogeometric analysis. Finally, the non-uniform rational B-spline is adopted for the isogeometric discretization of boundary geometry and physical quantities to form the equations of system. In addition, the computational problems of singular and nearly singular integral as well as boundary thermal stress are also studied under the isogeometric discretization. Because of the numerical evaluation based on the precise geometry modeling data in CAD, this project will improve the computational accuracy and efficiency of BEM, and provide a robust computational tool for complex thermal stress problem of functionally graded materials.

边界元法以其具有的独特优势已成为功能梯度材料热应力分析的重要数值方法之一。然而，传统边界元法所使用的分析模型是基于拉格朗日单元的网格模型，会产生几何误差。本项目将径向积分边界元法与等几何分析相结合，建立适用于功能梯度材料热应力分析的等几何径向积分边界元法，实现边界元分析模型和CAD模型的结合。基本研究思路是：首先，选择合适的基本解和规格化物理量，通过加权余量法推导边界积分方程；其次，对积分方程中的域积分，采用径向积分法将其转换成边界积分，形成等几何分析下的径向积分转换技术；最后，采用非均匀有理B样条对边界和物理量进行等几何离散，形成系统方程组。此外，还将研究所涉及到的奇异和几乎奇异边界积分以及边界热应力在等几何离散下的计算。本项目将数值计算构架于CAD中的精确几何造型数据，提高了边界元法的计算精度和效率，为解决复杂的功能梯度材料热应力问题提供有力的分析手段。

径向积分边界元法在求解非线性和非均匀问题时，具有仅需基于模型边界数据进行边界离散的优点，因而数据准备简单、便于复杂边界建模，已广泛应用于求解各类物理问题。本项目发挥径向积分边界元法的优势，与等几何分析方法相结合，构建了等几何径向积分边界元法，对功能梯度材料进行力学分析。在研究过程中，选择对应均匀问题的开尔文基本解和规格化位移，推导了功能梯度材料弹性问题的边界域积分方程；采用增强四阶样条径向基函数近似域积分中的规格化位移，从而径向积分法将积分方程中的域积分转换成等效边界积分；采用非均匀有理B样条进行几何建模和物理量插值，并由Greville坐标定义确定边界配点、幂级数展开法计算强奇异边界积分和Telles转换法计算弱奇边界异积分，从而形成等几何径向积分边界元分析算法。本项目还将径向积分边界元法拓展应用于稳态/非稳态对流导热问题，考虑了空间变化热导率和速度、以及时空变化热源。采用拉普拉斯方程基本解格林函数和加权余量技术推导边界积分方程，并利用径向积分法将积分方程中的域积分转换成等效边界积分，从而形成了纯边界积分组成的仅边界离散的边界元算法。此外，通过界面相容条件，构建了径向积分边界元法和单元微分法耦合算法，用以分析多尺度结构力热问题，其中边界元法计算尺度较小结构，而其余部分使用单元微分法计算，从而提高计算效率。