The theoretical design of a class of time discretized operators emanating from the generalized family of time discretized operators given earlier in Algorithm 1 leading to explicit representations of the form 
(in a nonlinear sense) follow next.
Type 2-GInO-Explicit: Nth-order accurate representations
A family of Nth-order accurate explicit representations can be directly obtained from Algorithm 1 described in equations (3.110-3.114 ) by taking 
, 
, and 
( 
).
It is to be noted that this explicit time discretized operator is only conditionally stable when 
as in typical traditional explicit time integration operators (and considering Rayleigh damping with 
and 
diagonalized).
Type 2-GInO-Explicit: Nth-order accurate representations with artificial damping
On the other hand, setting 
, 
, 
, and 
in Algorithm 1 represented by equations (3.110-3.114), and 
is taken as
we then have the theoretical design leading to the representation described next. The resulting Nth-order accurate integration operators are explicit and L-stable for nonlinear structural dynamics [E-LNInO].
where we introduced 
which is termed as an artificial damping matrix and is taken as
and 
and 
may be determined by the following set of inequalities
and 
is termed an artificial damping ratio (alternative efforts are currently underway and we hope to disseminate these results in the near future).
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