求解弱不连续问题的p型自适应有限元方法

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。本文，我们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，本文设计的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。

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

There exist many weak discontinuity problems such as inclusion problems in the practical engineering computations. For the commonly used finite element (FEM) methods, mesh refinement or the increase of element order throughout the domain is usually used in order to ensure that each point on the interface can satisfy the given high degree of accuracy. But this will lead to rapid growth of the computer’s physical memory and the CPU time. The p-version adaptive FEM method is an efficient numerical method which can greatly improve the accuracy of calculation through adaptively increasing the order of elements used in the FEM analysis. In this paper, we have designed the corresponding p-version adaptive FEM method for modeling the weak discontinuity problems and emphatically discussed the influence of different error control standards on the computational results of each point on the interface. Moreover, we have made the numerical computation and simulation for some typical weak discontinuity problems. The numerical results are shown that the p-version adaptive FEM method is very efficient for the solution of the weak discontinuity problems, and the efficiency can be greatly improved by obtaining the reliable numerical results with fewer elements.