学位论文简介

相比于有限元法，边界元法具有应力精度高和降维的优势，本文将基于边界元法来研究奇异性问题。奇异性问题包括两个方面：一、是边界积分方程中本身存在的奇异性，即基本解的奇异性问题；二、是边界元法求解存在应力奇异性的具体问题，包括裂纹问题和V形切口问题。

本文的主要研究内容和创新点如下：

（1）提出了二维和三维类奇异域积分解决方案。本文首先分析了时域基本解的特性，即当时间步长很小时，时域基本解具有类似于奇异积分的性质。然后采用非线性变换来处理类奇异域积分。最后确定了二维和三维类奇异域积分解决方案。

（2）应用双层插值边界元法计算了二维裂纹的应力强度因子以及模拟裂纹扩展。本文首先采用局部坐标近似展开的方法来计算强奇异积分和超奇异积分。然后推导了基于双层插值边界元法的裂尖奇异单元形函数。最后使用张开位移计算应力强度因子，以最大周向应力准则确定裂纹扩展方向，采用帕里斯公式计算裂纹扩展速率，最终模拟了单边斜裂纹和双边裂纹的扩展路径。

（3）针对顶点奇异性，提出了奇异点单元。本文首先介绍了双边界元法的基本思路，推导了间断位移法的边界积分方程。然后分析了顶点奇异性的阶次，得出在绝大多数情况下，顶点奇异性的阶次接近-0.5的结论。基于这一结论，推导了奇异点单元的形函数。最后采用双边界元法与奇异点单元结合的方法计算了三维穿透型裂纹的应力强度因子。

（4）通过分析V形切口的应力奇异性阶次，提出了针对V形切口的新型奇异单元。V形切口问题不同于裂纹问题，它的应力具有多重奇异性，并且随切口角变化。通过分析第一阶和第二阶特征值随切口角的变化情况，发现应力奇异性主要由第一阶特征值决定，尤其是对于切口角较大的情况。基于这一特性，提出了针对V形切口的新型奇异单元。

主要学术成果

[1]Jianming Zhang, Yunqiao Dong, Chuanming Ju, Weicheng Lin. A new singular element for evaluating stress intensity factors of V-shaped notches under mixed-mode load. Engineering Analysis with Boundary Elements, 2018, 93: 161-166.

[2]Yunqiao Dong, Chenjun Lu, Yuan Li, Jianming Zhang, Guizhong Xie, Yudong Zhong. Accurate numerical evaluation of domain integrals in 3D boundary element method for transient heat conduction problem. Engineering Analysis with Boundary Elements, 2015, 60: 89-94.

[3]Yunqiao Dong, Jianming Zhang, Guizhong Xie, Chenjun Lu, Lei Han and Pan Wang. A general algorithm for the numerical evaluation of domain integrals in 3D boundary element method for transient heat conduction. Engineering Analysis with Boundary Elements, 2015, 51: 30-36.

[4]Yunqiao Dong, Jianming Zhang, Guizhong Xie, Chenjun Lu, Yuan Li, Xu Han and Guangyao Li. A general algorithm for evaluating domain integrals in 2D boundary element method for transient heat conduction. International Journal of Computational Methods, 2015, 12(2): 1550010 (13 pages).

[5]Jianming Zhang, Yudong Zhong, Yunqiao Dong, Weicheng Lin. Expanding element interpolation method for analysis of thin-walled structures. Engineering Analysis with Boundary Elements, 2018, 86: 82-88.

[6]Jianming Zhang, Weicheng Lin, Yunqiao Dong, Chuanming Ju. A double-layer interpolation method for implementation of BEM analysis of problems in potential theory. Applied Mathematical Modelling, 2017, 51: 250-269.

[7]Yudong Zhong, Jianming Zhang, Yunqiao Dong, Yuan Li, Weicheng Lin, Jinyuan Tang. A serendipity triangular patch for evaluating weakly singular boundary integrals. Engineering Analysis with Boundary Elements, 2016, 69: 86-92.

[8]Guizhong Xie, Jianming Zhang, Yunqiao Dong, Cheng Huang, Guangyao Li. An improved exponential transformation for nearly singular boundary element integrals in elasticity problems. International Journal of Solids and Structures, 2014, 51: 1322-1329.