INTRODUCTION
The computational speed of computers has increased exponentially during
the last 50 years. This has led to the development of largescale simulation
tools like the finite element methods, computational fluid dynamics codes,
etc., for analysis of complex engineering systems. The availability of
complex simulation models that provide a better representation of the
actual physical system has provided engineers with an opportunity to obtain
improved designs. The process of obtaining optimal designs is known as
design optimization. In a deterministic design optimization, the designs
are often driven to the limit of the design constraints, leaving little
or no latitude for uncertainties. The resulting deterministic optimal
solution is usually associated with a high chance of failure, of the artifact
being designed, due to the influence of uncertainties inherently present
during the modeling and manufacturing phases of the artifact and due to
uncertainties in the external operating conditions of the artifact. The
uncertainties include variations in certain parameters, which are either
controllable (e.g., dimensions) or uncontrollable (e.g., material properties)
and model uncertainties and errors associated with the simulation tools
used for simulation based design (Moses et al., 2001). Uncertainties
in simulation based design are inherently present and need to be accounted
for in the design optimization process. Uncertainties may lead to large
variations in the performance characteristics of the system and a high
chance of failure of the artifact. Optimized deterministic designs determined
without considering uncertainties can be unreliable and might lead to
catastrophic failure of the artifact being designed. Robust design optimization
and reliability based design optimization are methodologies that address
these problems. Robust designs are designs at which the variation in the
performance functions is minimal. Reliable designs are designs at which
the chance of failure of the system is low. It is extremely desirable
that the engineers design for robustness and reliability as it helps in
obtaining large market shares for products under competitive economic
conditions. The design of engineering structures on the basis of probability
concepts will leads to more consistent safety levels, the ReliabilityBased
Design Optimization (RBDO) aims to find the best comprise between cost
reduction and safety assurance. This approach has emerged in the last
years, but the culture of deterministic design still dominates in engineering
manufacture, because of significant computational effort is required to
perform RBDO procedure, which it is devoted to the probabilistic constraints
evaluation. The probabilistic constraints are the key in reliabilitybased
structural design (Ang and Tang, 2006), the evaluation of these constraints
create several numerical challenges associated with numerical efficiency,
stability and robustness. However, several approaches has been proposed
for probabilistic constraints evaluation. The probabilistic constraints
are controlled by the reliability analysis, both simulations techniques
and moment methods are available to carry out the reliability analysis.
However, simulation techniques such as the Monte Carlo simulation is prohibitively
expensive in RBDO procedure. Nevertheless, the moment methods and what
is known as the firstorder reliability method (FORM) (Ditlevsen and Madsen,
2005) is widely used in RBDO due to its simplicity and speed (Frangopol
and Moses, 1994). Two approach are developed to ensure the reliability
analysis in RBDO, it can classified into the Reliability Index Approach
(RIA) and the performance measure approach (PMA) (Youn and Choi, 2004).
In the RIA the probabilistic constraints are defined as the reliability
indexes, evaluated by FORM. However, RIA converges slowly or even fails
to converge for a number of problems. To alleviate this difficulty PMA
is introduced by solving an inverse problem for the FORM, which the probability
measure is converted to the performance measure. The conventional RBDO
procedure is to employ nested optimization and reliability analysis loops,
which the reliability analysis is estimated inside the optimization loop
and herself is realized by a particular optimization procedure. This makes
the computational cost of RBDO very expensive and is not suitable, especially
for a complex engineering applications. In this study, an efficient method
based on the transformation and the finite element method is proposed.
The goal of our method is to evaluate the probability of failure in closedform,
i.e., to avoid the reliability analysis loop in the classical RBDO.
RELIABILITY ANALYSIS
The design of structures and the prediction of their good functioning
lead to the verification of a certain number of rules resulting from the
knowledge of physical and mechanical experience of designers and constructors.
These rules traduce the necessity to limit the loading effects such as
stresses and displacements. Each rule represents an elementary event and
the occurrence of several events leads to a failure scenario. In addition
to the vector of deterministic variables {x} to be used in the system
control and optimization, the uncertainties are modeled by a vector of
stochastic physical variables affecting the failure scenario. The knowledge
of these variables is not, at best, more than statistical information
and we admit a representation in the form of random variables. For a given
design rule, the basic random variables are defined by their probability
distribution associated with some expected parameters; the vector of random
variables is noted herein {Y} whose realizations are written y. The safety
is the state where the structure is able to fulfill all the functioning
requirements: Mechanical and serviceability, for which it is designed.
To evaluate the failure probability with respect to a chosen failure scenario,
a performance function G(x, y) is defined by the condition of good functioning
of the structure. The limit between the state of failure G(x, y)≤0
and the state of safety G(x, y)>0 is known as the limit state surface
G(x, y) = 0. The failure probability is then calculated by:
where, P_{f} is the failure probability, f_{(y)}Y is
the joint density function of the random variables Y and Pr[.] is the
probability operator. The evaluation of integral (1) is not easy, because
it represents a very small quantity and all the necessary information
for the joint density function are not available. For these reasons, the
First and the second order reliability methods FORM/SORM (Madsen et
al., 2006) have been developed. They are based on the reliability
index concept, followed by an estimation of the failure probability. The
invariant reliability index β was introduced by Hasofer and Lind
(1974), who proposed to work in the space of standard independent Gaussian
variables instead of the space of physical variables.
RELIABILITY BASED DESIGN OPTIMIZATION
For deterministic optimization, many efficient numerical methods have been
developed and applied to different kinds of structures. But for RBDO problems,
the coupling between the mechanical modeling, the reliability analysis and the
optimization methods represents a very complex task and leads to very high computational
time. The major difficulty lies in the evaluation of system reliability, which
is carried out by a particular optimization procedure (Gheng et al.,
1998). Efforts were directed towards developing efficient techniques (Royset
et al., 2001; Tu et al., 1999) and general proposed programs
to integrate the reliability analysis for given uncertain information. These
programs and procedures compute the reliability index of a structure for the
defined failure modes, but do not provide an optimum set of the design parameters,
in order to improve the reliability of a structure. An enormous amount of computer
time is also involved in the whole design process. The sequential (or the classical)
RBDO procedure and the proposed concurrent approach presented, which is based
on the simultaneous solution of the reliability and optimization problems.
RBDO: Sequential approach: The sequential RBDO algorithm which
is shown in Fig. 1, is calculated by nesting the two
following subproblems (Kharmanda et al., 2002):
• 
Optimization problem under deterministic and reliability constraints: 
where, f(x) is the objective function, g_{k}(x) = 0 are the associated
deterministic constraints, β(x, u) is the reliability index of the
structure and β_{t} is the target reliability.
• 
Calculation of the reliability index β(x, u): 
where, d(u) is the distance in the normalized random space (Fig.
2) and H(x, u) is the limit state function in the normalized space.

Fig. 1: 
Sequential RBDO algorithm 

Fig. 2: 
Physical space v/s normalized space 
The constrained minimization of the objective function f(x) is carried
out in the physical space of design variables {x} but the reliability
index β is calculated in the normalized space of random variables
{u}, which are the image of {y} in the standard space. In the physical
space, the image of H(x, u) is the limit state function G(x, y).
TRANSFORMATION OF RANDOM VARIABLE
The theory of the Transformation Method is based on the following theorem
(Soong, 1973):
Theorem: Suppose that X is a random variable with PDF (probability
density function) f(x) and A⊂
is the onedimensional space where, f(x)>0. Consider the random variable
(function of x) Y = u(X), where, y = u(x) defines a onetoone transformation
that maps the set A onto a set B⊂ so that
the Eq. Y = u(x) can be uniquely solved for x in terms of y, say x = w(y).
Then, the PDF of Y is h(y):
where,
is the Jacobian of the transformation.
A new approach: Transformationfinite element method (TFEM):
The transformation method together with the finite element method is used
to obtain the probability density function (PDF), in closedform, using
the random variable transformation between the input random variables
and the output variable. Once the PDF is evaluated the calculation of
the probability of failure P_{f} is straight forward.

Fig. 3: 
Algorithm of TEM 
The accuracy
of the solution is increased when increasing the number of elements as
usual. The algorithm of this method is shown in Fig. 3.
Proposed Method: TRBDO: In this study, the inverse problem of
the probabilistic transformation method has been developed. It`s shown
that, if we proposed the distribution of the output variable we can calculate
the exact probability density function of the input which is very helpful
in the design process of a mechanical product. The accuracy of our method
has been verified by 10000 simulations of MonteCarlo.
In order to avoid the calculation of the reliability index and the separation
of the solution in two spaces which leads to very large computational
time, especially for largescale structures, the transformation approach
with the Finite Element Method consists in finding in one step the probability
of failure i.e., the calculation of reliability index for each iteration
of outer loop is not needed in other word the inner loop is not needed
(Fig. 4). The optimization problem under deterministic
and probability of failure constraints is:
where, f(x) is the objective function, g_{k}(x)≤0
are the associated deterministic constraints, P_{f}(x, u) is the
probability of failure of the structure and P_{f, t} is the target
probability of failure.

Fig. 4: 
Algorithm of TRBDO 
TRBDO algorithm
Advantage of TRBDO:
• 
Using our proposed technique TFEM, we can find the
probability of failure in closed form, which is not known in general

• 
The combination of the TFEM and the RBDO eliminate the inner loop
to find the reliability index 
• 
Work with P_{f} in deterministic form is more efficient
and accuracy to work with the reliability index which need a numerical
method to evaluate it 
Application: To illustrate the efficiency of our proposed approach
(TRBDO), the 6bar truss structure shown in Fig. 5 is
analyzed:
Following, step by step the application of TRBDO:
Step 1: Modeling the Truss using FEM.
Using finite element method to obtain the displacement of node 1 and
2:

Fig. 5: 
Trusses structural of aircraft hangar (left), 6bar
truss structure (right) 
Bar 31
Bar 42
Bar 43
Bar 21
Bar 32
Bar 41
The assembly leads, to the global equilibrium system:
Where the solution gives:

Fig. 6: 
Exact PDF(u_{1}) using TFEM and approximate
using 10000 MC simulations 
Step 2: Using transformation technique to calculate the probability
density function of the first node PDF(u_{1}) where the Young`s
modulus E is random (Fig. 6).
(a) Calculate the inverse function
(b) Calculate the Jacobean
let us suppose E~U(10^{8}, 3.10^{8})
Step 3: Let us consider the displacement of node 1 is considered
as limitstate function. Find the probability of failure P_{f}
where, for example, u_{1,limit} = 2.10^{7}
Step 4: TRBDO formulation: It is request to minimize the
cross section (A)of the bar, where F = 10 N and L = 10 mm.
The TRBDO approach is now formulated as:
The solution of the above problem is w = 2.4781, t = 3.8421. The comparison
with classical RBDO shows that the computation time is divided by more
than 100, when the PTM is applied.
CONCLUSION
The coupling of optimization and reliability problems allows us to obtain
the best compromise between cost and safety. But this coupling represents
a very complex task and leads to very high computational time. The major
difficulty lies in the evaluation of system reliability which is carried
out by a particular optimization procedure. The proposed method, which
we call it TRBDO, avoid this coupling and avoid the searching of reliability
index. The proposed method gives us the probability of failure in closed
form in one step. The efficiency of the new method is confirmed by several
applications on structures; a 6bar truss problem is shown in this study.