How to Implement Bell-Shaped Damping Model?

The bell-shaped damping model has recently been proposed to overcome the limitations of the Rayleigh damping model in incorporating un-modelled energy dissipation when simulating the seismic response of large-scale structures. Using a sparse matrix implementation of the damping coefficient matrix, this model has the same order of computational efficiency as the Rayleigh model. It, unfortunately, requires intermediate equations and degrees of freedom, causing the number of equations to be much larger than the existing number of physical degrees of freedom. This expanded system of equations might be unsuitable to existing structural analysis software with a rigid architecture requiring the system of equations to be confined to only the physical degrees of freedom. To overcome this limitation, this study proposes a new implementation procedure that does not require expanding the system of equations and could fit into the existing structural analysis software. It is based on an iterative method. With examples, this paper will discuss this new procedure and its performance in terms of the rate of convergence.