Canonical representations of complex vibratory subsystems: time domain Dirichlet to Neumann maps

Abstract

Large scale dynamic simulations can often be simplified by appropriately replacing large portions of the domain by a Dirichlet to Neumann, or DtN map ( Givoli 1992. Numerical Methods for Problems on Infinite Domains, 1st ed. Elsevier, Amsterdam). Here we consider the problem of representing a linear dynamical subsystem by such a map. The exact DtN map is computed as a modal summation and its properties are studied. Bounds on the symbol of the DtN map in the Laplace domain are obtained. The exact map is then approximated, in particular in the high modal density regime. In the high modal density limit, we obtain the result that a subsystem can be accurately represented with just three parameters. Within such an approximation we obtain representations based on a maximum entropy representation, self-similar or fractal representation, and a rational function representation. The rational function representation leads to the interesting result that any complicated dynamical subsystem with a large number of degrees of freedom is asymptotically equivalent (in the limit of infinite modal density) to a single mass-dashpot-spring system. We end with numerical examples showing the efficiency of the rational function approximation.