非线性有限元分析的非协调模式及存在的问题

利用非协调模式提高非线性有限元分析广泛采用的低阶单元的精度和性能,是国际计算力学界研究的热点和难点.阐述了国际上在非线性有限元分析中已广泛采用的增广假设应变法方法(the enhanced assumed strain, EAS)的基本原理,详细讨论了非协调模式用于非线性有限元分析保证收敛、稳定的条件及增广假设应变场插值函数的构造方法.介绍了国内学者关于几何非线性非协调模式的研究方法和研究成果: (1)从Hellinger-Reissner广义变分原理出发,提出了几何非线性非协调模式的收敛条件,并采用非线性计算的若干简化措施建立几何非线性非协调元的简化模型;(2)一类放松单元间协调要求的非线性广义变分原理,对几何非线性问题可以选择事先无协调约束的非协调函数建立非协调元,收敛性可以保证,并根据此非线性广义变分原理可建立C$^1$或C$^0$类几何非线性广义杂交元,C$^1$或C$^0$类精化杂交元和精化直接刚度法.指出了EAS方法用于非线性有限元分析存在的问题,即本构关系和求解方法的限制,并对非协调元应用于非线性有限元分析提出了展望. 

THE STATE-OF-THE-ART OF INCOMPATIBLE MODE OF NON-LINEAR FEM AND THE ISSUES

It is difficult and has been research focus to improve the accuracy and performance of the low order elements widely used in non-linear FEM analysis using incompatible mode. In this paper, the basic principles of EAS (The Enhanced Assumed Strain) method are systematically summarized, and the conditions to guarantee calculation convergence, the solution stability and the construction methods of the enhanced assumed strain interpolating function are elaborated in detail. The research methods and achievements about the incompatible mode geometrically non-linear FEM in China are reviewed, which include: 1) a convergence criteria of incompatible mode geometrically non-linear FEM was proposed based on Hellinger-Reissner generalized variational principle, and the simplified model of incompatible geometrically non-linear FEM was constructed by using some simplified methods; 2) a class of non-linear generalized variational principles was proposed, in which the compatible conditions between elements are relaxed. The incompatible models can be constructed by selecting incompatible functions without satisfying compatible conditions prior and the convergence can be guaranteed. The C1 or C0 geometrically non-linear generalized hybrid elements, C1 or C0 refined hybrid elements and directly refined stiffness methods can be constructed by applying the non-linear generalized variational principles. The problems with respect to the application of EAS method in the non-linear FEM analysis are pointed out, that is, the restriction of constitutive laws and the solving methods. Furthermore, the application prospect of the incompatible mode non-linear FEM is discussed.