The Resound technique can be viewed as a form of finite element substructuring which uses a fuzzy analytical description of the local modes in various regions of a structure that have a high modal density.  The following provides a descriptive overview of the Resound approach from this viewpoint.

Suppose we try to build a component mode synthesis (CMS) model of a structural-acoustic system, with each component being associated with a particular subsystem.   Suppose also that we have access to an infinitely fast computer so that computational expense is not an issue.  We could then create an extremely detailed FE model and extract (a very large number of) component modes associated with each subsystem. What we would find is that the first few component modes would tend to have long wavelength deformation that would be very dependent on the geometry of the subsystem.  However, as we go higher up the local modal series we would find that the component modes have more and more spatial variation and begin to look more and more like a simple standing wave system.

It is then natural to question whether we in fact need to use the exact component modes higher up in the modal series when synthesizing the response or whether we can approximate these modes with a simple standing wave system (i.e., analytically generated component modes which are based on the wavelength within a subsystem).

Suppose then that we represent the response of a structure in terms of an ‘exact’ set of component modes calculated numerically and an approximate set calculated analytically. If we want the model to be accurate at higher frequencies there may be a considerable number of analytical local modes (many more than the number of exact local modes). Suppose we then generate mass and stiffness matrices for the system – since there are a large number of analytical local modes we may still need our infinitely fast computer to accurately calculate and store all the entries in the mass and stiffness matrices. Suppose we also partition the mass and stiffness matrices into a local partition (associated with the component modes that have been calculated analytically) and a global partition (associated with the exact component modes and the constraint modes of the system). Due to the way that the modes have been calculated we would find that the local partition is associated with modes with short wavelength spatial deformation, whereas the global partition is associated with modes with relatively long wavelength spatial deformation. We can refer to each set as the set of local basis functions and the set of global basis functions respectively.

Suppose that we mass normalize the component modes. Since the component modes are mass and stiffness orthogonal with respect to each other we would find that the local partitions of the stiffness and mass matrices are diagonal matrices of the local natural frequencies squared and unity respectively. It turns out that the local basis functions are also stiffness orthogonal to the global basis functions (if we use fixed-interface CMS). The local and global basis functions are therefore only coupled by the off-diagonal terms in the mass matrix (the local-global partition). These terms depend on the shapes of the local basis functions, the shapes of the global basis functions and on the mass density within a subsystem. In effect they govern how the local basis functions and global basis functions interact and transfer energy between each other. In Resound these terms are referred to as the fuzzy coupling between the local and global basis functions.

Suppose we now generate a dynamic stiffness matrix for the system and reduce out the local degrees of freedom. The resulting global mass and stiffness matrices are relatively small (since we only considered a relatively small number of long-wavelength global basis functions).  The original global partition of the dynamic stiffness matrix is modified by the partitions of the dynamic stiffness matrix that involve the local basis functions.  Since the modification adds mass and damping to the global basis functions and since the approximations used in its derivation are similar to those adopted in fuzzy structure theory, it is referred to as fuzzy mass and damping.  It turns out that in order to calculate this modification we only need to know the local natural frequencies and the spatial correlation between the local mode shapes.  We can obtain analytical estimates of the local natural frequencies from the free wavenumber within a subsystem.  The spatial correlation between the local modes can also be estimated analytically by making certain assumptions about the local mode shapes.

If we apply excitations to the local and global degrees of freedom we can also calculate perturbations to the forcing terms that account for the fuzzy coupling in a similar manner. We can then solve for the global response by inverting the global dynamic stiffness matrix at each frequency of interest.

We are now at a point where we can ask whether it is completely necessary to use our infinitely fast computer.   Since we can efficiently calculate an analytical approximation to the fuzzy coupling between the local and global modes, we do not have to explicitly evaluate any of the local partitions of the dynamic stiffness matrix.  Also, since the global basis functions can be resolved using a relatively coarse finite element model the overall computational cost of the technique is relatively small and there is (fortunately) no need for an infinitely fast computer.

The total response in a given subsystem is obtained by summing the local and global responses. The local response is therefore determined from an SEA model with the subsystem input powers appropriately modified by the presence of the global modes.