We consider modeling the presence of a substructure in a dynamical simulation in terms of the force the substructure exerts on the main structure of interest. The reaction force is given in terms of the displacement of the attachment points. We call the map between the displacement of the attachment points and the force at the attachment a Dirichlet to Neumann, or DtN, map. We use a regular perturbation expansion to derive an approximate DtN for the case when the substructure exhibits weakly nonlinear behavior. The representation is accurate to O(ϵ2t2) + O(t2/N2), where ϵ is a measure of the strength of the nonlinearity, N is the number of modes of the subsystem and t is the simulation time. We note that the representation is particularly suited to the situation when ϵ = O(N−1).

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