基于拉格朗日乘子法和群论的具有任意位移边界条件的旋转周期对称结构有限元分析

FE analysis based on Lagrange multipliers method and group theory for structures of cyclic symmetry under arbitrary displacement boundary conditions

对任意位移边界条件下的旋转周期对称结构,由拉格朗日乘子法建立有限元方程。在对称适应的坐标系下,由结构刚度矩阵的块循环性质,利用群变换给出一种新的求解方法。数值验证给出令人满意的结果。

Cyclic symmetry can be found in many engineering structures. When analyze behaviors of these structures, computing efficiency can be greatly improved if structural symmetry is fully exploited. However, it seems that most of the existing algorithms utilizing symmetry only relate to the problems subjected to symmetric essential boundary conditions. This paper uses Lagrange multiplier method to develop FE equation. Stiffness matrix for cyclic symmetric structure is block-circulate unless a kind of symmetry-adapted reference coordinate system is adopted. By a group transformation, structure is then analyzed in a group space. Base vector of this space is orthogonal with respect to group representation matrix. As a consequence, generalized stiffness matrix is block-diagonal. A matrix transformation is then proposed to make the generalized stiffness matrix nonsingular. Solve the whole equation system by a method similar to substructure technique. For the block-diagonal property of the generalized stiffness matrix, the most computation can be carried out in a partitioning way. As a result, great efficiency can be gained, compared with basic FEM. The proposed algorithm can be easily applied to other analysis process for rotationally periodic structures, e.g. heat transfer problems, viscoelastic problems, etc. The contributions of this paper are twofold. Firstly, a matrix transformation combined with group theory and numerical methods is proposed to analyze structures of cyclic symmetry subjected to arbitrary boundary conditions. Secondly, the computational convenience and efficiency are fully discussed and demonstrated by means of three numerical examples.