Improved group-theoretical method for eigenvalue problems of special symmetric structures, using graph theory

Group-theoretical methods for decomposition of eigenvalue problems of skeletal structures with symmetry employ the symmetry group of the structures and block-diagonalize their matrices. In some special cases, such decompositions can further be continued. This particularly happens when submatrices resulted from the decomposition process, correspond to substructures with new symmetrical properties which are not among the properties of the original structure. Thus, a group-theoretical method is not able to recognize such additional symmetry from the original problem. In this paper, an algorithm is presented based upon a combination of group-theoretical ideas and graph-methods. This algorithm identifies the cases where the structure has the potential of being further decomposed, and also finds the symmetry group, and subsequently the transformation which can further decompose the system. It is also possible to find out when the block-diagonalization is complete and no further decomposition is possible. This is particularly useful for large eigenvalue problems such as calculation of the buckling load or natural frequencies of vibrating systems with special symmetries.