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INTRODUCTION
of physical systems characterized by uncertainty
models of small and medium sizes (less than 30
random variables).
In this work attention is directed to discrete
reliability-based optimization of structural sys-
tems under stochastic excitation. In particular,
excitation models defined in terms of hundreds
or thousands random variables (high dimensional
models) are considered here. One of the difficulties
in this type of problems is the high computational
cost involved in the reliability analyses required
during the optimization process. This is due to the
fact that the reliability estimation of stochastic
dynamical systems involves the estimation of
failure probabilities in high-dimensional uncertain
parameter spaces (Jensen et al., 2009; Valdebenito
and Schuëller, 2011). The objective of this work
is to present a general framework for solving this
class of reliability-based optimization problems.
The approach is based on the use of approximation
concepts, the application of an effective stochastic
sensitivity analysis, and the implementation of a
globally convergent optimization scheme. Special
attention is focused on the analysis and design of
structures protected by means of passive energy
dissipation systems. The basic function of these
devices when incorporated into the structure is
to absorb a portion of the input energy, thereby
reducing energy dissipation demand on structural
members and minimizing possible structural dam-
age. In this regard, the consideration of discrete
optimal design processes for protected structural
systems is one of the novel aspects of this work.
This type of problem is relevant from a practical
point of view since the potential advantages of
modern structural protective systems have lead
to the design and construction of an increasing
number of protected structures for the purpose
of mitigating seismic impact.
The reliability-based optimization problem is
formulated as the minimization of an objective
function subject to multiple reliability constraints.
All uncertainties involved in the problem (system
parameters and loading) are considered explicitly
In a large number of practical design situations
the design variables must be selected from a list
of discrete values. Standard methods address
discrete variable design optimization problems
by employing discrete or integer variable algo-
rithms to treat the problem directly in the primal
variable space (branch and bound techniques,
combinatorial methods, evolution-based optimiza-
tion techniques, etc.) (Kovács, 1980; Goldberg,
1989; Scharage, 1989).These methods are quite
general but are associated with a large number
of function calls (evaluation of objective and
constraint functions). On the other hand, in many
practical applications the applied loads and system
parameters may be subjected to uncertainty or
variability. Under uncertain conditions the field
of reliability-based optimization (RBO) provides
a realistic and rational framework for structural
design and optimization (Enevoldsen and Sø-
rensen, 1994). Schemes for discrete structural
optimization considering uncertainties have not
been addressed as frequently as their deterministic
counterpart. In most studies, ad hoc optimization
algorithms have been integrated directly with
a reliability method. For example, in (Hassan
and Crossley, 2008), the problem of optimiza-
tion under uncertainty has been approached by
means of genetic algorithms and direct Monte
Carlo simulation. As both the optimization and
the reliability algorithms require a large number
of function calls, numerical costs can be very
high. In order to reduce the computational efforts
related to reliability analysis, the application of
optimization methods in combination with the
first-order reliability methods was investigated
in, for example, (Tolson et al, 2004, Gunawan
and Papalambros, 2007). Another approach for
reducing computational efforts is the application
of meta-models, such as artificial neural networks
(see, e.g. (Papadrakakis and Lagaros, 2002;
Lagaros et al., 2008)). These approaches have
been applied to solve RBO problems of a series
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