接触问题的自适应有限元方法研究

项目名称： 接触问题的自适应有限元方法研究

项目编号： No.11261040

项目类型： 地区科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 袁达明

作者单位： 南昌航空大学

项目金额： 45万元

中文摘要： 自适应有限元方法是当前科学和工程计算的重要方法。本项目利用自适应有限元方法快速求解力学中的摩擦接触问题，包括连续接触状态和滑动接触状态下的两个物体间带单边条件的满足Columb摩擦定律的摩擦接触问题及带自由边界的Signorini问题。该方法的特点是，在经典迭代分析框架下，结合罚方法或增广Lagrangian技巧将反映接触条件的几何约束及服从的摩擦定律引入能量泛函，利用重构型误差估计方法中的Zienkiewicz-Zhu方法给出区域内部压力在能量范数下的后验误差估计，在接触区域上建立反映其上表面压力在L1范数下的误差估计。在上述两种误差策略下，给出摩擦接触问题的自适应算法。在求解自由边界的Signorini问题时，我们利用虚拟区域思想，将原问题转化为扩大光滑区域Ω上的辅助问题。该辅助问题的容许解集为L2(Ω)上的满足Neumann边值条件函数集，利于应用有限元方法求解。

中文关键词： 接触问题；变分不等式；迭代算法；保正性；三对角矩阵

英文摘要： Adaptive Finite Element Method is an important approach for scientific and engineering computing.In this project,we plan to apply this method to solve frictional contact problem with unilateral condition and Columb law between two bodies,where the continuous contact and sliding contact are considered.Besides, Signorini problem, which belongs to contact problem with free boundary, is also considered. Our method is based on the frame of the classical iterative analysis method. We introduce the energy functional which can reflect the geometry restriction and the friction law on the contact zone.The a posteriori error estimate in the inner region is derived by Zienkiewicz-Zhu technique,which is a recovery type method for error estimate. Simultaneously, the error associates with the pressure on the contact zone uner L1-norm is presented. The adaptive process is controlled by these two strategies. Besides,we associate the method of fictitious domains with adaptive finite element method for solving Signorini problems. We extent the original problem to an auxiliary problems with Neumann boundary conditions and the admissible solution set is L2(Ω) in the bigger smooth domain Ω. Obviously, it is easy to implement the numerical approximation.

英文关键词： contact problems；variational inequality；iteration alogrithm；positivity-preserving；diagonal matrix