Noise and vibration levels in structures like automobiles and aircraft have been reduced through the application of viscoelastic materials (VEMs) as damping treatments for many years [18, 34, 37]. Adding a VEM to a structure makes accurate prediction of a structure's response to harmonic excitations challenging. This is because the VEM's properties, including the Young's modulus, damping coefficient, and shear modulus, vary significantly as functions of both frequency of excitation and temperature [34]. The solution algorithm presented in this research takes advantage of the fact that the VEM properties typically vary smoothly with frequency by interpolating VEM property variations between known values at perhaps a half dozen frequencies.

The typical finite element (FE) discretization targeted by this work has millions of FE degrees of freedom in order to obtain acceptable accuracy over the frequency range of interest and is typically solved at hundreds of frequencies for tens to hundreds of load cases. Accurate approximate solutions to this large frequency response problem (FRP) can be computed efficiently on an approximating subspace. To decrease the cost of factoring the resulting reduced FRP at every frequency, the dimension of the approximating subspace is minimized by replacing hundreds to thousands of eigenvectors with a significantly smaller number of enrichment vectors called residual flexibility vectors (RFVs), damping deformation vectors (DDVs), and dynamic response vectors (DRVs). The RFVs and DDVs represent quasistatic response to loads and to dashpot forces, respectively, and including RFVs and DDVs in the approximating subspace is a common industrial practice.

The use of DRVs, which are corrections to approximate solutions of the FRP at select frequencies, is new. Because computing DRVs is very expensive on the FE subspace, we accurately approximate DRVs in a reduced subspace associated with the automated multilevel substructuring (AMLS) method. Also, we attempt to minimize the number of DRVs by computing those that will improve the accuracy of frequency response solutions the most. Overall, the cost of solving FRPs with VEMs can be reduced dramatically by including DRVs in the approximating subspace, because the accuracy obtained using DRVs could only be achieved otherwise by including a much larger number of global eigenvectors in the approximating subspace.