  We develop an efficient numerical method for the probabilistic quantification
of the response statistics of nonlinear multi-degree-of-freedom structural
systems under extreme forcing events, emphasizing accurate heavy-tail
statistics. The response is decomposed to a statistically stationary part and
an intermittent component. The stationary part is quantified using a
statistical linearization method while the intermittent part, associated with
extreme transient responses, is quantified through i) either a few carefully
selected simulations or ii) through the use of effective measures (effective
stiffness and damping). The developed approach is able to accurately capture
the extreme response statistics orders of magnitude faster compared with direct
methods. The scheme is applied to the design and optimization of small
attachments that can mitigate and suppress extreme forcing events delivered to
a primary structural system. Specifically, we consider the problem of
suppression of extreme responses in two prototype ocean engineering systems.
First, we consider linear and cubic springs and perform parametric optimization
by minimizing the forth-order moments of the response. We then consider a more
generic, possibly asymmetric, piecewise linear spring and optimize its
nonlinear characteristics. The resulting asymmetric spring design far
outperforms the optimal cubic energy sink and the linear tuned mass dampers.
