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| FE analysis based on Lagrange multipliers method and group theory for structures of cyclic symmetry under arbitrary displacement boundary conditions |
| Revised:November 21, 2002 |
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| DOI：10.7511/jslx20044075 |
| KeyWord:cyclic symmetry,FEM,group,lagrange multiplier |
| Liu Ling~1 Yang Haitian~ |
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| Abstract: |
| 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 problems subjected to symmetric essential boundary conditions. The present paper uses Lagrange multiplier method to develop FE equation. Stiffness matrix for cyclic symmetric structure is block-circulate so long as 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 used in this paper 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 substructuring technique. For the block-diagonal property of the generalized stiffness matrix, most computation can be carried out in a partitioning way. As a result, great efficiency can be gained, compared to basic FEM. The proposed algorithm can be easily applied into other analysis process for rotationally periodic structures, e.g. heat transfer problems, viscoelastic problems, etc. The contributions of the present 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. |
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