正交各向异性弹性问题的规则化边界元法

本文致力于平面正交各向异性弹性问题的规则化边界元法研究，提出了新的规则化边界元法的理论和方法。对问题的基本解的特性进行了研究，确立基本解的积分恒等式，提出一种基本解的分解技术，在此基础上，结合转化域积分方程为边界积分方程的极限定理，建立了新颖的规则化边界积分方程。和现有方法比，本文不必将问题变换为各向同性的去处理，从而不含反演运算，也有别于Galerkin方法，无需计算重积分，因此所提方法不仅效率高，而且程序设计简单。特别是，所建方程可计算任何边界位移梯度，进而可计算任意边界应力，而不仅限于面力。数值实施时，采用二次单元和椭圆弧精确单元来描述边界几何，使用不连续插值逼近边界函数。数值算例表明，本文算法稳定、效率高，所取得的边界量数值结果与精确解相当接近。

A REGULARIZED BOUNDARY ELEMENT METHOD FOR ORTHOTROPIC ELASTIC PROBLEMS

This presentation is mainly devoted to the research on the regularization of indirect boundary integral equations (IBIEs) for orthotropic elastic problems and establishes the new theory and method of the regularized BEM. Some integral identities depicting the characteristics of the fundamental solution of the considered problems and a novel decomposition technique to the fundamental solution are proposed. Based on this, together with a limit theorem for the transformation from domain integral equations into boundary integral equations (BIEs), the regularized BIEs with indirect unknowns, which don’t involve the direct calculation of CPV and HFP integrals, are presented for orthotropic elastic problems. The presented method can solve the considered problems directly instead of transforming them into isotropic ones, and for this reason, no inverse transform is required. In addition, this method doesn’t require to calculate multiple integral as the Galerkin method. Furthermore, the proposed stress BIEs are suited for the computation of displacement gradients on the boundary, and not only limited to tractions. Also, they are independent of the displacement gradient BIEs and, as such, can be collocated at the same locations as the displacement gradient BIEs. This provides additional and concurrently useable equations for various purposes. A systematic approach for implementing numerical solutions is produced by adopting the discontinuous quadratic elements to approximate the boundary quantities and the quadratic elements to depict the boundary geometry. Especially, for the boundary value problems with elliptic boundary, an exact element is developed to model its boundary with almost no error. The convergence and accuracy of the proposed algorithm are investigated and compared for several numerical examples, demonstrating that a better precision and high computational efficiency can be achieved.