symmetry.) These operations constitute a group G. To each of them corresponds a matrix (here, 1 and S). The set of these matrics denoted by {Ug:geG} is therefore a group. The symmetry of the structure is expressed by (n is the order of G) where the pv (g) are complex coefficients. Functions pr (on G) are the irreducible representations (‘irreps') of G. The group with two elements of our previous example has two irreps, whose coefficients are those, mentioned above, which are displayed in Fig. 4. The projectors (one for each irrep index v) span mutually orthogonal subspaces of V, all invariant by A, hence a block-diagonalization. Therefore, the problem splits into as many independent subproblems as the number of irreps. Just as above, each of these is not to be solved over the whole structure, but only concerns the nodes of the symmetry cell (that is, a substructure which generates the whole structure by applying all symmetries). This is so because to each irrep pu corresponds a kind of displacements (called ‘D-symmetric') analogous to the even or odd displacements of the previous example: their defining relations are such that one can compute all their components from the components of cell nodes. So, if the group has n elements and n irreps, and if the given structure had N degrees of freedom, one has a trade-off: n problems of size N/n against one of size N. As the computing cost of such a problem is roughly Na, where a (which depends on how (5) is solved) is typically 1, 5 or 2, one may expect a gain by a factor na~l. Remark. Some assumptions in what precedes depend on G being commutative. When this is not so, the theory is more involved [3], but the main result (a gain in n“_|) is still valid. Remark. All of this is still valid if (5) is replaced by the eigenproblem i.e. the study of vibration modes. There is also a straightforward generalization to evolution problems. So there is a systematic way to take advantage of symmetry. This method is similar to Fourier analysis. Group representation theory, which can be done without in simple cases, is essential to this method and almost indispensable if the symmetry group is not commutative [3]. 2.3 The ‘assembly' of elements This section, which is more technical than the rest of the paper (but is independent and can be skipped), shows how essential some notions of abstract group theory may become. ‘Assembly' is the process by which the stiffness matrix of a structure is obtained from stiffness matrices (of small size) of its constitutive parts, or elements, like beams,
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