With the stiffener present, there is an added stiffness S, and the structural equations take on the following form: There are, say, N degrees of freedom, and m of these are on Z, so S is an m X m matrix, and B an m X n rectangular matrix (B' is its transposed). If m is small with respect to N (this is what ‘slight' alteration of symmetry means), the term B‘ S B in (15) appears as a perturbation of A. If it was not there, the problem could be solved economically by equivariance. So the obvious thing to do is to transfer this disturbing term to the right-hand side. Let us introduce an auxiliary variable w (a vector of m components) and rewrite (15) as which is a ‘small' problem (of order m). One gets u by solving (16). One has to get the matrix entries in (18) first. But these are obtained by solving m + 1 linear systems whose matrix is A, for which the gain by a factor na~l promised by the theory can be obtained. So the procedure results in a net benefit if this gain exceeds m + 1. There is a simple mechanical interpretation of this method: w can be considered as the vector of forces exerted by the stiffener on the rest of the structure. One can get it as a function of the displacement of Z, taken as parameters, then treat it as just another load imposed on the membrane alone (thence Equation (16)). Thus the fact that no symmetry of the load is supposed is put to advantage. The previous idea is a well known one (‘Guyan's condensation' [7]). It is nothing else than the classical ‘Lagrange multiplier' approach of the calculus of variations. Conclusion Now, it remains to combine all these ideas into a strategy to tackle the problem of the space station. The key-remark is that in (15), A is ‘easier' to deal with than the whole stiffness matrix, be it due to symmetry or to any other cause. The antenna can be treated as a perturbation, just like the stiffener of our model problem. What plays the role of A is the stiffness matrix of the solar-cells array, which is not that of a symmetrical structure, but is ‘easy' nevertheless, because one may apply the equivariant imbedding idea to deal with it. Last, the remark on indices of section 2.3 is put to use, as suggested by Fig. 2(d). (The numbers shown there are not the indices, which would be large integers, but the orders of the small groups of the elements which are in totality or in part in the symmetry cell). The problem, which seemed to request a big computer, is thus replaced by a series of very small independent structural problems. A pocket calculator (and some patience) will do! Bibliographical Comments Space stations are extensively treated elsewhere from various viewpoints. Two keyreferences are [5] and the popular book [8]. Symmetry, in general and with respect to group theory, is a vast subject. The first thing to read is [11]. The elements of representation theory to which we had to refer
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