作　　者： ; ; ; ;

机构地区： 广东工业大学机电工程学院

出　　处： 《力学进展》 2010年第1期1-12,共12页

摘　　要： 基于有限覆盖技术的数值流形方法是一种新的广义的数值方法.该方法的场函数近似原理和有限元、无网格、单位分解等方法相似,但在网格划分、覆盖形式、近似函数等方面有其自身的特点和优势.对该方法近年来在理论研究和应用方面取得的重要进展进行了综述.在理论研究方面,日前已对不同形式物理覆盖流形单元的性能进行了研究,结果表明流形单元的精度较有限单元高,且提高覆盖函数的阶次能提高单元的精度;同时理论研究已由二维低阶流形方法推广到三维高阶流形方法,由线性流形方法推广到非线性流形方法,由基于能量原理的流形方法推广到基于加权余量的流形方法,非协调流形方法、无网格流形方法等也已开展了研究;此外,覆盖系统的自动生成、覆盖函数的形式以及边界条件的处理方法等流形方法相关理论的研究也取得了进展.在应用方面,开展了有关岩石破坏和裂纹扩展等非连续变形分析更深入的研究,并已逐步推广到金属塑性变形分析、多孔介质变形分析以及温度场的数值分析等多个领域.针对日前流形方法的研究和应用现状,该文展望了流形方法理论及实现方法的研究方向、及其在计算流体力学、金属成形等大变形问题、多物理场分析等领域的应用前景. Numerical manifold method（NMM） is a new and generalized numerical method based on finite cover technique. It is similar to finite element method, meshless method and partition of unit method on the approximation principle of field functions, but has its own characteristics and advantages on mesh generation, cover system and approximation function. The important advances on numerical manifold method and its applications in recent years were reviewed in this paper. Performance assessment for manifold elements with different physical cover showed that manifold elements have higher computational accuracy than finite element, and the computational accuracy can be improved by employing high order cover functions. Theoretical studies on manifold method have also been extend to three dimension high order manifold method from two dimension low order manifold method, to nonlinear manifold method from linear manifold method, to Galerkin weighted residual manifold method from based on energy principle. Incompatible manifold method and meshless manifold method have been investigated as well. Moreover, the researches on some relative theories of NMM, such as automatic generalization of cover system, the form of cover functions and treatments of boundary conditions, have been developed. In applications of manifold method, discontinuous deformation problems on rock damage and crack propagation have been analyzed more deeply than before, and it has been extend to apply in various fields of engineering gradually, such as metal forming, porous media mechanics, temperature field etc. Based on the recent work, some perspectives in research topics for NMM and its applications in computational fluid dynamics （CFD）, metal forming, multiphysical field and other domains are presented.

关 键 词： 数值方法 数值流形 有限覆盖 覆盖函数 有限元

领　　域： [理学] [理学] [建筑科学] [建筑科学]