  Structural damping is known to be approximately rate-independent in many
cases. Popular models for rate-independent dissipation are hysteresis models;
and a highly popular hysteresis model is the Bouc-Wen model. If such hysteretic
dissipation is incorporated in a refined finite element model, then the
mathematical model includes the usual structural dynamics equations along with
nonlinear nonsmooth ordinary differential equations for a large number of
internal hysteretic states at Gauss points, to be used within the virtual work
calculation for dissipation. For such systems, numerical integration becomes
difficult due to both the distributed non-analytic nonlinearity of hysteresis
as well as the very high natural frequencies in the finite element model. Here
we offer two contributions. First, we present a simple semi-implicit
integration approach where the structural part is handled implicitly based on
the work of Pich\'e, and where the hysteretic part is handled explicitly. A
cantilever beam example is solved in detail using high mesh refinement.
Convergence is good for lower damping and a smoother hysteresis loop. For a
less smooth hysteresis loop and/or higher damping, convergence is observed to
be roughly linear on average. Encouragingly, the time step needed for stability
is much larger than the time period of the highest natural frequency of the
structural model. Subsequently, data from several simulations conducted using
the above semi-implicit method are used to construct reduced order models of
the system, where the structural dynamics is projected onto a small number of
modes and the number of hysteretic states is reduced significantly as well.
Convergence studies of error against the number of retained hysteretic states
show very good results.
