固体和结构力学中的误差控制自适应边界元法

Abstract: Firstly, it is demonstrated that the residual of the hyper-singular integralequation can be used for the error estimation of an approximate solution of a known traction or displacement problem. It is also shown that this result can be further extended for mixed boundary problems. Based on these theoretical results, a new adaptive BEM algorithm for 2D and 3D elasticity problems is proposed, which is based on a measure of the residual of the hyper-singular integral equation. The innovative point is that the proposed algorithm is based on a good mathematical base and it is effective and can be performed in a simple way. In detail, the proposed a-posteriori error estimation is given by a measure of the residual of the hyper-singular integral equation, and this residual is provided by the difference of two boundary stress solutions. One of them is the normal boundary stress solution and the.other is calculated through the direct dicretization of the regularized hyper-singular boundary integral equation of displacement derivative. A number of 2D and 3D numerical examples show that the validity of the proposed algorithm. At last, a theoretical analysis of mechanical dissipation of an electroded quartz resonator is given and the ANSYS is used to analyze the vibration characters of a quartz crystal microbalance.

本项目将重点开展二维和三维固体力学边界元法的误差估计及基于误差控制的自适应网格划分的研究，探讨所提误差估计方法的数学基础及随网格细分的收敛速度问题。这些成果将被进一步扩展应用于板壳和智能材料的边界元分析。本项目的完成，对提高边界元法的分析能力、增进其稳定性和适用性、并向工程应用推进，将起到推动作用。