Reliability Based Optimization

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General Remarks

The basic goal in any engineering discipline is to design and construct systems or components that satisfy certain performance objectives during their lifetime. Such objectives cover a wide range of possibilities, e.g. control of vibrations induced by wind or traffic loading on bridges, collapse prevention of buildings due to major earthquakes, minimization of the effects of multi-site damage in aerospace structures, etc. In almost any practical design situation it is impossible to comply with the performance objectives deterministically because of the inherent random nature of loading conditions, structural parameters and/or conditions of operation of the structures. Hence, the fulfillment of the performance objectives can be accomplished only by probabilistic means, i.e. with an associated reliability. In fact, high levels of reliability are usually associated with large economical costs, e.g. a structure with an enhanced reliability may require the use of an increased amount of construction material, more sophisticated construction procedures, thorough maintenance, etc. Considering that the available resources are always scarce, an adequate design procedure should offer an appropriate trade-off between an acceptable reliability level and economical design of the structure. Reliability-based optimization (RBO) provides the means for achieving such trade-off by offering an optimal design solution taking into account the effects of uncertainties [1][2][3][4][5][6].

Formulation of a Reliability-based Optimization problem

The RBO approach is an attractive and most useful design tool: it allows to determine the best design according to some predefined criterion. The formulation of an RBO problem requires the identification and definition of a number of items, namely the input variables of the system (i.e. design variables and uncertain parameters), the failure events of the system (i.e. violation of target performance), the constraints of the design problem and the objective function that allows to identify the most convenient design. Each of these items is briefly described below.

  • Definition of the design variables (\mathbf{x}). The design variables are those parameters that can be selected by the designer and that affect the performance of a system. Typical examples of design variables are the cross sections of structural members, interval of inspection and repair, topology parameters, etc. The design variables can be characterized as deterministic or uncertain; in the latter case, the mean value is usually set as the design variable.
  • Identification of the uncertain parameters. In any practical situation there are a number of parameters which are not known at the design stage and that affect the performance of a system. Such parameters may refer to loadings (e.g. wind loading, seismic events, water wave loading, etc.), structural parameters (e.g. geometry of a system, yield strength, etc.), and operation conditions (e.g. temperature, environmental conditions, etc.) among others. These parameters are characterized as uncertain variables (\mathbf{\theta}). The rational quantification of the effects of these parameters in the system performance requires an appropriate model to measure the plausibility of a given realization of \mathbf{\theta}, e.g. by means of the so-called non probabilistic approach (see, e.g. [7]) or by prescribing a joint probability density function f(\mathbf{\theta}).
  • Formulation of the failure (or critical) events associated with the performance of the system. As mentioned above, the system should fulfill certain performance requirements. The violation of any of these requirements causes a failure of the system. It should be noted that, in this context, failure does not necessarily imply collapse but rather an undesirable performance. A failure event is defined by means of the so-called performance function g, which depends on the design variables and the uncertain parameters, i.e. g=g(\mathbf{x},\mathbf{\theta}). The performance function is defined such that g(\mathbf{x},\mathbf{\theta}^*) is smaller or equal to zero when a specific realization of \mathbf{\theta}^*, in combination with a specific set of design variables \mathbf{x}, causes failure (i.e. an unacceptable performance of the system); in case of an acceptable performance, g(\mathbf{x},\mathbf{\theta}^*) is larger than zero. It is important to note that a realistic reliability model of a system may involve the definition of several failure events.
  • Definition of the constraints of the design problem. The objective of the constraints is to restrict or confine the design variable space in order to attain certain specific design requirements. The constraints can be of deterministic nature when they refer exclusively to the design variables. Deterministic constraints are formulated as a mathematical function (h(\mathbf{x})); usually, a deterministic constraint is defined such that h(\mathbf{x})\leq0 implies the satisfaction of a constraint. The constraints may also include both design variables and uncertain parameters. In such cases, the constraint is probabilistic and its fulfillment will be associated with one (or more) of the failure events defined above. Thus, a probabilistic constraint is satisfied when the probability of occurrence of the failure event (p_F(\mathbf{x})) is equal or smaller than a prescribed probability level (p_F^*), i.e. the constraint is satisfied if p_F(\mathbf{x})-p_F^*\leq0. The probability of occurrence of the failure event can be defined in terms of the classical multidimensional integral:

p_F(\mathbf{x})=\int_{g(\mathbf{x},\mathbf{\theta})\leq0}{f(\mathbf{\theta})\,d{\mathbf{\theta}}}


  • Statement of an objective function, which defines the goals pursued by the design procedure. The spectrum of possible goals is rather wide and it is problem dependent, e.g. the benefits of a system during its operation period are maximized, the downtime of a critical facility is minimized, etc. The objective function (denoted by F(\cdot)) depends on the design variables and, eventually, on the uncertain parameters, i.e. F(\mathbf{x}) and F(\mathbf{x},\mathbf{\theta}), respectively. In the latter case, the uncertain objective function can be replaced by a substitute deterministic problem (F^*(\cdot) - see, e.g. [8]), i.e.:

F^*(\mathbf{x})=\int_{}{F(\mathbf{x},\mathbf{\theta})f(\mathbf{\theta})\,d{\mathbf{\theta}}}

Some remarks about the solution of a Reliability-based Optimization problem

In problems of engineering interest, the number of uncertain parameters (\mathbf{\theta}) associated with RBO problems can be rather large, i.e. hundreds or even thousands. Moreover, the performance function(s), which is(are) associated with the definition the failure event(s), may be a nonlinear implicit function of \mathbf{\theta}; therefore, its evaluation can be numerically quite involved, i.e. a finite element (FE) model may be required. Hence, to compute the integrals associated with probability by means of analytical solutions or quadratures is, in almost all cases, unfeasible [9]. Thus, it is necessary to apply approximate reliability techniques, e.g. the First Order Reliability Approximation (FORM - see, e.g. [10]), or simulation techniques, e.g. Monte Carlo Simulation (MCS - see, e.g. [11]), in order to estimate such integrals. In both approaches, it is necessary to repeatedly evaluate the performance function, which may require considerable numerical efforts. Furthermore, the optimization step in the solution of an RBO problem requires the repeated evaluation of the objective function and constraints for different values of the design variables in order to identify the optimal design. Thus, the direct solution of the RBO problem may render infeasible even for academic problems due to the tremendous numerical costs involved, caused by the repeated assessment of the system response for different realizations of (\mathbf{x},\mathbf{\theta}). Then, it is necessary to resort to specific techniques in order to decrease the computational costs. Such techniques include the introduction of approximation concepts, procedures to reduce the dimension of the structure (e.g. meta-models), and the use of advanced simulation techniques, among others.

Notes

  1. I. Enevoldsen and J.D. Sørensen. Reliability-based optimization in structural engineering. Structural Safety, 15(3):169–196, 1994.
  2. M. Gasser and G.I. Schuëller. Reliability-based optimization of structural systems.Mathematical Methods of Operations Research, 46(3):287–307, 1997.[1]
  3. H.A. Jensen. Design and sensitivity analysis of dynamical systems subjected to stochastic loading. Computers and Structures, 83:1062–1075, 2005.[2]
  4. M. Papadrakakis, N.D. Lagaros, and V. Plevris. Design optimization of steel structures considering uncertainties. Engineering Structures, 27(9):1408–1418, 2005.[3]
  5. Z. Tao, R.B. Corotis, and J.H. Ellis. Reliability-based structural design with Markov decision processes. Journal of Structural Engineering, 121(6):971–980, 1995.[4]
  6. B.D. Youn, K.K. Choi, and L. Du. Adaptive probability analysis using an enhanced hybrid mean value method. Structural and Multidisciplinary Optimization, 29(2):134–148, 2005.[5][6]
  7. D. Moens and D. Vandepitte. A survey of non-probabilistic uncertainty treatment in finite element analysis. Computer Methods in Applied Mechanics and Engineering, 194(12-16):1527–1555, 2005.[7]
  8. K. Marti and G. Stoeckl. Stochastic linear programming methods in limit load analysis and optimal plastic design under stochastic uncertainty. ZAMM, 84(10-11):666–677, 2004.[8]
  9. G.I. Schuëller, H.J. Pradlwarter and P.S. Koutsourelakis. A critical appraisal of reliability estimation procedures for high dimensions. Probabilistic Engineering Mechanics, 19(4):463-474, 2004.[9][10]
  10. O. Ditlevsen and O.H. Madsen. Structural Reliability Methods. JohnWiley and Sons, 1996. ISBN: 978-0471960867 [11]
  11. N. Metropolis and S. Ulam. The Monte Carlo method. Journal of the American Statistical Association, 44(247):335-341, 1949.[12]


See Also