Efficient Computational Strategies for Reliability-Based Optimization
Sankaran Mahadevan
Professor, Civil and Environmental Engineering
Box 1831-B, Vanderbilt University, Nashville, TN 37235
Phone: 615-322-3040, Email: sankaran.mahadevan@vanderbilt.edu
During reliability-based optimization, reliability analysis is required during each
iteration of the optimizer in order to evaluate reliability constraints. (Additional analysis
may also be required to evaluate the derivatives of reliability constraints when gradientbased optimization algorithms are used.) Reliability analysis is also an iterative process;
so traditional reliability-based optimization is a nested loop analysis (i.e. a reliability
analysis loop nested within optimization). This methodology requires prohibitive
computational effort, especially when the optimum design has to satisfy a number of
reliability constraints and the function evaluations are time-consuming, as in the case of
large finite element analyses.
Two approaches have been developed in the literature to decouple the reliability
analysis and optimization loops to achieve computational efficiency (Roysett and Der
Kiureghian, 2001; Du and Chen, 2002). In this paper, we extend the sequential
optimization reliability analysis (SORA) approach of Du and Chen (2002) to solve
several problems: (1) problems where FORM-based reliability analysis is infeasible and
Monte Carlo simulation needs to be used; (2) problems using standard deviations of some
random variables as design parameters to achieve robustness; and (3) problems where the
various limit states are evaluated in different codes and there is feedback coupling
between the codes (typically encountered in multi-disciplinary optimization). In the
SORA method, optimization is performed using a ‘deterministic equivalent’ for the
reliability constraint. Reliability analysis follows optimization sequentially as opposed
to being nestted within the optimization. The reliability analysis provides an update to the
deterministic equivalent of the reliability constraint; then optimization using the new
constraint follows, and the process is repeated until convergence.
The reliability analysis used in the original SORA approach is limited to an
inverse FORM analysis, i.e., solving for the minimum distance point on a contour of the
performance function in the space of uncorrelated standard normal variables, given the
target value of the reliability index. In practical applications, one might encounter
situations where FORM is either inaccurate or infeasible, and might have to use other
methods such as Monte Carlo simulation. The proposed methodology in this paper
develops a modular approach, which allows different reliability methods to be used for
different limit states. The method also extracts sensitivity information from Monte Carlo
simulation for use in the optimization.
The paper extends the modular decoupled methodology to reliability-based robust
design optimization, of particular current interest in the automotive industry, where it is
not only important to optimize the mean value of an objective but also minimize its
variance. In this context, standard deviations of some variables are used as design
parameters (referred to as tolerance design). The twin objectives of reliability and
robustness lead to a multi-objective optimization problem. Several methods available for
such optimization are combined with the decoupled methodology for computational
efficiency.
In multidisciplinary systems, the evaluation of different limit states may require
different individual disciplinary analyses. For example, an aerospace system may have
limit states related to aerodynamics, structural and thermal disciplines, and each
disciplinary analysis is done separately. However, each analysis depends on the output of
other disciplinary analyses. Several deterministic optimization approaches have been
developed for such coupled systems. This paper develops a methodology for extending
the decoupled approach to the reliability-based optimization of coupled multidisciplinary
systems.
The proposed methodology is illustrated with several example applications to
structural, automotive and aerospace systems.
References
1. Royset, J.O., A. Der Kiureghian, and E. Polak, "Reliability-Based Structural Design
by the Decoupled Approach," Reliability Engineering and Structural Safety, Vol. 73,
pp. 213-221, 2001.
2. Du, X., and W. Chen, "Sequential Optimization and Reliability Assessment Method
for Efficient Probabilistic Design," Proceedings of the ASME Design Engineering
Technical Conferences, Montreal, Canada, Paper number DETC 2002/DAC-34127,
2002.
Acknowledgement
The study reported in this paper is being supported by funding from two sources, NASA
Langley Research Center and General Motors. The support is gratefully acknowledged.