材料非线性问题的广义逆力法有限元格式表达

FINITE ELEMENT FORMULATION OF GENERALIZED INVERSE MATRIX FORCE METHOD FOR MATERIAL NONLINEARITY PROBLEMS

广义逆(generalized inverse matrix,GIM)力法是一种从经典力法的求解思路中引发而出的基于力法和广义逆矩阵理论的一种新的迭代解法,对于求解材料非线性问题具有其独特的特点和优势。由于该法在解决材料非线性问题时无需像传统的基于位移法的逐步增量法那样逐步求解,故又称特大增量步算法(large increment method,LIM)。算法的整个求解过程可以分为整体阶段和局部阶段。材料非线性问题的广义逆力法具有完整的用有限元格式表达的通用求解过程和公式,尤其是算法局部阶段的一致弹塑性柔度和刚度矩阵的表达式。算例为平面应力问题算例。此外,广义逆力法在结构并行计算方面具有不同于传统的子结构并行计算的新特点。该算法的意义主要集中在两点:一是结构计算不再只是采用位移元和杂交元,而是采用了更适应计算机计算的力法方法;二是结构并行计算不再是传统的子结构并行计算,而是新形式的并行计算。

The finite element formulation of generalized inverse matrix force method, or so-called large increment method (LIM) for material nonlinearity problems is proposed. LIM is a new iteration method, which is based on theories of the force method and generalized inverse matrix (GIM), and is of unique characteristics and advantages especially for material nonlinearity problems. Unlike the conventional incremental method based on the displacement method, which can not avoid time consumption and error accumulation, the proposed new method can work with very large increment up to several loading cycles as based on the force method. Unlike the classical force method, LIM does not need to find some basic structure any more. Consequently, this method sheds a new light to the force method in the computational mechanics. In addition, LIM is of intrinsic parallel-calculating characteristics, which are different from the traditional sub-structural algorithm based on displacement method. The algorithm can be divided into the global stage and the local stage. The finite element formulation with consideration of material nonlinearity is given. It includes the expression of consistent elastoplastic flexibility and stiffness matrix. An example of plane stress problem is also given to show the generality of this new method.