弹性力学问题自适应有限元及其局部多重网格法  

ADAPTIVE FINITE ELEMENT METHOD AND LOCAL MULTIGRID METHOD FOR ELASTICITY PROBLEMS

作　　者：刘春梅[1] 肖映雄[2] 舒适[1] 钟柳强[3] 

机构地区：[1]湘潭大学数学与计算科学学院,湖南湘潭411105 [2]湘潭大学土木工程与力学学院,湖南湘潭411105 [3]华南师范大学数学科学学院,广东广州510631

出　　处：《工程力学》2012年第9期60-67,91,共9页Engineering Mechanics

基　　金：国家自然科学基金重大研究计划重点支持项目(91130002);国家自然科学基金项目(10972191);长江学者和创新团队发展计划项目(IRT1179);湖南省研究生创新基金项目(CX2010B245;CX2010B246)

摘　　要：针对平面弹性力学问题,利用最新顶点二分法,设计了一种不需要标记振荡项和加密单元不需要满足"内节点"性质的自适应有限元法;利用自适应加密过程中每层网格上只有局部单元需要加密这一特性,设计了一种基于局部松弛的多重网格法。数值实验结果表明:该文设计的自适应有限元法具有一致收敛性和拟最优计算复杂度,基于局部松弛的多重网格法对求解弹性力学问题自适应网格下的有限元方程具有很好的计算效率和鲁棒性。In this paper, an adaptive finite element （AFEM） method is designed by using the newest vertex bisection for linear elasticity problems in two dimensions. This method marks exclusively according to the error estimator without special treatment of oscillation and performs a minimal element refinement without the interior node property. Furthermore, a type of multigrid method based on the local relaxation is applied to the AFEM discrete systems by using the special properties during refinement. The results of various numerical experiments are shown that the proposed AFEM method is uniformly convergent and has quasi-optimal numerical complexity. The resulting multigrid method is much more robust and efficient in CPU times than the usual multigrid methods.

关 键 词：弹性力学问题 自适应有限元 拟最优复杂性 局部松弛 多重网格法 

分 类 号：O343.3[理学—固体力学] TB115[理学—力学]