求解弱不连续问题的p型自适应有限元方法  

The p-Version Finite Element Method for Modeling Weak Discontinuity Problems

作　　者：王彪[1] 肖映雄[1] 李真有 

机构地区：[1]湘潭大学土木工程与力学学院,湘潭411105

出　　处：《固体力学学报》2016年第1期59-73,共15页Chinese Journal of Solid Mechanics

基　　金：国家自然科学基金项目(10972191);湖南省自然科学基金项目(14JJ2063);湖南省教育厅重点项目(15A183)资助

摘　　要：在实际工程计算中,存在大量的弱不连续问题,如含夹杂问题.利用通常的有限元方法,为确保界面上各点满足给定高精度,往往需要采用全域网格加密或全域提高单元阶次的方法,这将会导致计算机的物理内存和CPU时间的剧烈增长.p型自适应有限元方法是一种能通过自适应分析逐步增加单元阶次以改善计算精度的数值方法.论文针对弱不连续问题设计了相应的p型自适应有限元方法,重点讨论了容许误差控制标准对界面上各点计算结果的影响,并对几类典型的弱不连续问题进行了数值计算与模拟.数值结果表明,论文设计的p型自适应有限元方法对求解弱不连续问题是非常有效的,用较少的单元得到精度可靠的数值结果,可大大提高其有限元分析效率.There exist many weak discontinuity problems in practical engineering computations, such as the inclusion problems. In general finite element methods （FEM）, to achieve the desired high accuracy for the computational results of each point on the interface, either the meshes were refined, or the order of each element was increased throughout the domain. However, such methods caused a sharp rise in physical memory usage and CPU time. In this study, a p-version adaptive FEM for modeling the weak discontinuity problems was presented by combining higher-order elements with the adaptive method. The resulting method greatly improved the accuracy of calculation by adaptively increasing the order of each element in- volved in the FEM analysis. Discussions focused on the influences of different convergence criterions on the computational results of each point on the interface were made. To obtain better results near the interface, locally refined meshes and convergence criterion defined by the local stress error at the node were needed for this-version method. Moreover, numerical computations and simulations were carried out for several typical weak discontinuity problems, e. g. , elastic bodies with randomly distributed circular inclusions and the plane strain problem of multi-layered material. The numerical results showed that the proposed p-ver- sion adaptive FEM was very efficient in solving weak discontinuity problems. The efficiency was greatly improved and the reliable numerical results were obtained with fewer elements, even those of poor quality. The values of displacement and stress near the interface were of higher accuracy. Thus, the proposed method can be used in the self-consistent FEM analysis to calculate the equivalent elastic constants of composite materials.

关 键 词：弱不连续问题 有限元方法 P型 圆形夹杂 随机分布 

分 类 号：O241.82[理学—计算数学]