Reliability-based structural optimization under hazard loads

The present paper studies the reliability-based structural optimization of the civil engineering in the seismic zone. The objective is to minimize the sum of construction material cost and the expected failure loss under severe earthquake, which is obtained by the sum of the products of the failure probability and its failure losses for the important failure modes. The set of constraints includes the deterministic constraints, and the constraints based on structural reliability—the reliability index constraints of structural element failure for the serviceability state under minor earthquake and the failure probability of the structural system for the ultimate limit state under severe earthquake. By introducing the load roughness index, the structural system reliability computation under hazard load can be greatly simplified, which is approximately determined by its weakest failure mode. Finally, the numerical example of high rising shear RC frame is calculated.     

Reliability-based design optimization of aeroelastic structures

Aeroelastic phenomena are most often either ignored or roughly approximated when uncertainties are considered in the design optimization process of structures subject to aerodynamic loading, affecting the quality of the optimization results. Therefore, a design methodology is proposed that combines reliability-based design optimization and high-fidelity aeroelastic simulations for the analysis and design of aeroelastic structures. To account for uncertainties in design and operating conditions, a first-order reliability method (FORM) is employed to approximate the system reliability. To limit model uncertainties while accounting for the effects of given uncertainties, a high-fidelity nonlinear aeroelastic simulation method is used. The structure is modelled by a finite element method, and the aerodynamic loads are predicted by a finite volume discretization of a nonlinear Euler flow. The usefulness of the employed reliability analysis in both describing the effects of uncertainties on a particular design and as a design tool in the optimization process is illustrated. Though computationally more expensive than a deterministic optimum, due to the necessity of solving additional optimization problems for reliability analysis within each step of the broader design optimization procedure, a reliability-based optimum is shown to be an improved design. Conventional deterministic aeroelastic tailoring, which exploits the aeroelastic nature of the structure to enhance performance, is shown to often produce designs that are sensitive to variations in system or operational parameters.     

Reliability-based multi-scale design optimization of composite frames considering structural compliance and manufacturing constraints

Structural and Multidisciplinary Optimization - The paper proposes an efficient methodology for concurrent reliability-based multi-scale design optimization (RBMDO) of composite frames to minimize...     

Particle swarm approach for structural design optimization

This paper presents in detail the background and implementation of a particle swarm optimization algorithm suitable for constraint structural optimization tasks. Improvements, effect of the different setting parameters, and functionality of the algorithm are shown in the scope of classical structural optimization problems. The effectiveness of the approach is illustrated by three benchmark structural optimization tasks. Results show the ability of the proposed methodology to find better optimal solutions for structural optimization tasks than other optimization algorithms.     

Reliability-based design optimization for elastoplastic mechanical structures

In this paper, two special formulations to carry out a reliability-based design optimization of elastoplastic mechanical structures are introduced. The first approach is based on a well-known two-level method where the first level involves the optimization for the design parameters whereas the evaluation of the probabilistic constraints is carried out in a sub-optimization level. Because the evaluation of the probabilistic constraints in a sub-optimization level causes non-convergence behavior for some problems as indicated in the literature, an alternative formulation based on one-level is developed considering the optimality conditions of the β-computation by which the probabilistic constraint appears in the first level reliability-based design optimization formulation. In both approaches, an explicit parameter optimization problem is proposed for the computation of a design point for elastoplastic structures.Three examples in this paper demonstrate that the one-level reliability-based design optimization formulation is superior in terms of convergence to an optimal design than the two-level reliability-based design optimization formulation.     

Reliability-based design optimization using probabilistic sufficiency factor

A probabilistic sufficiency factor approach is proposed that combines safety factor and probability of failure. The probabilistic sufficiency factor approach represents a factor of safety relative to a target probability of failure. It provides a measure of safety that can be used more readily than the probability of failure or the safety index by designers to estimate the required weight increase to reach a target safety level. The probabilistic sufficiency factor can be calculated from the results of Monte Carlo simulation with little extra computation. The paper presents the use of probabilistic sufficiency factor with a design response surface approximation, which fits it as a function of design variables. It is shown that the design response surface approximation for the probabilistic sufficiency factor is more accurate than that for the probability of failure or for the safety index. Unlike the probability of failure or the safety index, the probabilistic sufficiency factor does not suffer from accuracy problems in regions of low probability of failure when calculated by Monte Carlo simulation. The use of the probabilistic sufficiency factor accelerates the convergence of reliability-based design optimization.     

Reliability-based design optimization using SORM and SQP

In this work a second order approach for reliability-based design optimization (RBDO) with mixtures of uncorrelated non-Gaussian variables is derived by applying second order reliability methods (SORM) and sequential quadratic programming (SQP). The derivation is performed by introducing intermediate variables defined by the incremental iso-probabilistic transformation at the most probable point (MPP). By using these variables in the Taylor expansions of the constraints, a corresponding general first order reliability method (FORM) based quadratic programming (QP) problem is formulated and solved in the standard normal space. The MPP is found in the physical space in the metric of Hasofer-Lind by using a Newton algorithm, where the efficiency of the Newton method is obtained by introducing an inexact Jacobian and a line-search of Armijo type. The FORM-based SQP approach is then corrected by applying four SORM approaches: Breitung, Hohenbichler, Tvedt and a recent suggested formula. The proposed SORM-based SQP approach for RBDO is accurate, efficient and robust. This is demonstrated by solving several established benchmarks, with values on the target of reliability that are considerable higher than what is commonly used, for mixtures of five different distributions (normal, lognormal, Gumbel, gamma and Weibull). Established benchmarks are also generalized in order to study problems with large number of variables and several constraints. For instance, it is shown that the proposed approach efficiently solves a problem with 300 variables and 240 constraints within less than 20 CPU minutes on a laptop. Finally, a most well-know deterministic benchmark of a welded beam is treated as a RBDO problem using the proposed SORM-based SQP approach.     

A hyper-heuristic approach to aircraft structural design optimization

The conceptual design of an aircraft is a challenging problem in which optimization can be of great importance to the quality of design generated. Mass optimization of the structural design of an aircraft aims to produce an airframe of minimal mass whilst maintaining satisfactory strength under various loading conditions due to flight and ground manoeuvres. Hyper-heuristic optimization is an evolving field of research wherein the optimization process is continuously adapted in order to provide greater improvements in the quality of the solution generated. The relative infancy of hyper-heuristic optimization has resulted in limited application within the field of aerospace design. This paper describes a framework for the mass optimization of the structural layout of an aircraft at the conceptual level of design employing a novel hyper-heuristic approach. This hyper-heuristic approach encourages solution space exploration, thus reducing the likelihood of premature convergence, and improves the feasibility of and convergence upon the best solution found. A case study is presented to illustrate the effects of hyper-heuristics on the problem for a large commercial aircraft. Resulting solutions were generated of considerably lighter mass than the baseline aircraft. A further improvement in solution quality was found with the use of the hyper-heuristics compared to that obtained without, albeit with a penalty on computation time.     

Reliability-based design optimization by adaptive-sparse polynomial dimensional decomposition

This paper puts forward two new methods for reliability-based design optimization (RBDO) of complex engineering systems. The methods involve an adaptive-sparse polynomial dimensional decomposition (AS-PDD) of a high-dimensional stochastic response for reliability analysis, a novel integration of AS-PDD and score functions for calculating the sensitivities of the failure probability with respect to design variables, and standard gradient-based optimization algorithms, encompassing a multi-point, single-step design process. The two methods, depending on how the failure probability and its design sensitivities are evaluated, exploit two distinct combinations built on AS-PDD: the AS-PDD-SPA method, entailing the saddlepoint approximation (SPA) and score functions; and the AS-PDD-MCS method, utilizing the embedded Monte Carlo simulation (MCS) of the AS-PDD approximation and score functions. In both methods, the failure probability and its design sensitivities are determined concurrently from a single stochastic simulation or analysis. When applied in collaboration with the multi-point, single-step framework, the proposed methods afford the ability of solving industrial-scale design problems. Numerical results stemming from mathematical functions or elementary engineering problems indicate that the new methods provide more computationally efficient design solutions than existing methods. Furthermore, shape design of a 79-dimensional jet engine bracket was performed, demonstrating the power of the AS-PDD-MCS method developed to tackle practical RBDO problems.     

Robust formulation for Reliability-based design optimization of structures

The reliability-based design optimization (RBDO) has been widely recognized as a powerful optimization tool under probabilistic constraints, through appropriate modeling of uncertainties. However, the drawback of RBDO is that it does not reflect the ability of the structure to comply with large data variations, unforeseen actions or deterioration mechanisms. On the other hand, the robust design optimization (RDO) reduces the variability of the structural performance, in addition to its mean level. However, RDO does not take direct advantage of the interaction between controllable (product design values) and noise variables (environmental random values), and the obtained results do not accurately indicate what parameter has the highest effect on the performance characteristics. The purpose of this paper is to propose a robust formulation for reliability-based design optimization (RRBDO) that combines the advantages of both optimization procedures and overcomes their weaknesses. The optimization model proposed overcomes the limitations of the existing models without compromising the reliability level, by considering a robust convex objective function and a performance variation constraint. The proposed formulation can consider the total cost of structures and can control structural parameter variations. It takes into account uncertainty and variability in the same mathematical formulation. A numerical solution procedure is also developed, for which results are analyzed and compared with RBDO for several examples of concrete and steel structures.     

Reliability-based design optimization using the directional bat algorithm

Neural Computing and Applications - Reliability-based design optimization (RBDO) problems are important in engineering applications, but it is challenging to solve such problems. In this study, a...     

Reliability-based robust design optimization: A general methodology using genetic algorithm

In this paper, we present an improved general methodology including four stages to design robust and reliable products under uncertainties. First, as the formulation stage, we consider reliability and robustness simultaneously to propose the new formulation of reliability-based robust design optimization (RBRDO) problems. In order to generate reliable and robust Pareto-optimal solutions, the combination of genetic algorithm with reliability assessment loop based on the performance measure approach is applied as the second stage. Next, we develop two criteria to select a solution from obtained Pareto-optimal set to achieve the best possible implementation. Finally, the result verification is performed with Monte Carlo Simulations and also the quality improvement during manufacturing process is considered by identifying and controlling the critical variables. The effectiveness and applicability of this new proposed methodology is demonstrated through a case study.     

A hybridized angle-encouragement-based decomposition approach for many-objective optimization problems

Due to the large objective space when handling many-objective optimization problems (MaOPs), it is a challenging work for multi-objective evolutionary algorithms (MOEAs) to balance convergence and diversity during the search process. Although a number of decomposition-based MOEAs have been designed for the above purpose, some difficulties are still encountered for tackling some difficult MaOPs. As inspired by the existing decomposition approaches, a new Hybridized Angle-Encouragement-based (HAE) decomposition approach is proposed in this paper, which is embedded into a general framework of decomposition-based MOEAs, called MOEA/D-HAE. Two classes of decomposition approaches, i.e., the angle-based decomposition and the proposed encouragement-based boundary intersection decomposition, are sequentially used in HAE. The first one selects appropriate solutions for association in the feasible region of each subproblem, which is expected to well maintain the diversity of associated solutions. The second one acts as a supplement for the angle-based one under the case that no solution is located in the feasible region of subproblem, which aims to ensure the convergence and explore the boundaries. By this way, HAE can effectively combine their advantages, which helps to appropriately balance convergence and diversity in evolutionary search. To study the effectiveness of HAE, two series of well-known test MaOPs (WFG and DTLZ) are used. The experimental results validate the advantages of HAE when compared to other classic decomposition approaches and also confirm the superiority of MOEA/D-HAE over seven recently proposed many-objective evolutionary algorithms.     

Reliability-based design optimization for problems with interval distribution parameters

Traditional reliability-based design optimization (RBDO) generally describes uncertain variables using random distributions, while some crucial distribution parameters in practical engineering problems can only be given intervals rather than precise values due to the limited information. Then, an important probability-interval hybrid reliability problem emerged. For uncertain problems in which interval variables are included in probability distribution functions of the random parameters, this paper establishes a hybrid reliability optimization design model and the corresponding efficient decoupling algorithm, which aims to provide an effective computational tool for reliability design of many complex structures. The reliability of an inner constraint is an interval since the interval distribution parameters are involved; this paper thus establishes the probability constraint using the lower bound of the reliability degree which ensures a safety design of the structure. An approximate reliability analysis method is given to avoid the time-consuming multivariable optimization of the inner hybrid reliability analysis. By using an incremental shifting vector (ISV) technique, the nested optimization problem involved in RBDO is converted into an efficient sequential iterative process of the deterministic design optimization and the hybrid reliability analysis. Three numerical examples are presented to verify the proposed method, which include one simple problem with explicit expression and two complex practical applications.     

Reliability-based structural optimization with probability and convex set hybrid models

For structural systems exhibiting both probabilistic and bounded uncertainties, it may be suitable to describe these uncertainties with probability and convex set models respectively in the design optimization problem. Based on the probabilistic and multi-ellipsoid convex set hybrid model, this paper presents a mathematical definition of reliability index for measuring the safety of structures in presence of parameter or load uncertainties. The optimization problem incorporating such reliability constraints is then mathematically formulated. By using the performance measure approach, the optimization problem is reformulated into a more tractable one. Moreover, the nested double-loop optimization problem is transformed into an approximate single-loop minimization problem by considering the optimality conditions and linearization of the limit-state function, which further facilitates efficient solution of the design problem. Numerical examples demonstrate the validity of the proposed formulation as well as the efficiency of the presented numerical techniques.     

Reliability-based design optimization using kriging surrogates and subset simulation

The aim of the present paper is to develop a strategy for solving reliability-based design optimization (RBDO) problems that remains applicable when the performance models are expensive to evaluate. Starting with the premise that simulation-based approaches are not affordable for such problems, and that the most-probable-failure-point-based approaches do not permit to quantify the error on the estimation of the failure probability, an approach based on both metamodels and advanced simulation techniques is explored. The kriging metamodeling technique is chosen in order to surrogate the performance functions because it allows one to genuinely quantify the surrogate error. The surrogate error onto the limit-state surfaces is propagated to the failure probabilities estimates in order to provide an empirical error measure. This error is then sequentially reduced by means of a population-based adaptive refinement technique until the kriging surrogates are accurate enough for reliability analysis. This original refinement strategy makes it possible to add several observations in the design of experiments at the same time. Reliability and reliability sensitivity analyses are performed by means of the subset simulation technique for the sake of numerical efficiency. The adaptive surrogate-based strategy for reliability estimation is finally involved into a classical gradient-based optimization algorithm in order to solve the RBDO problem. The kriging surrogates are built in a so-called augmented reliability space thus making them reusable from one nested RBDO iteration to the other. The strategy is compared to other approaches available in the literature on three academic examples in the field of structural mechanics.     

Reliability-based optimization of design variance to identify critical tolerances

Reliability-based design optimization (RBDO) is a topic of interest for research in both academia and industry. RBDO typically involves adjusting the mean values of the design variables while fixing the spread parameters, often measured as variance, in order to accomplish a given objective within the stated constraints. This paper proposes an alternate way to meet given design criteria by fixing the mean values of the statistical inputs and allowing the spread parameters to become design variables. To do this, product cost models are proposed in terms of statistical variables. By performing this type of optimization, the design changes are kept to a minimum, and the focus is instead shifted to variance control. An initial study is performed on a three-bar truss subject to static loading with material variability. A more complex example is performed involving the cost minimization of an unmanned undersea vehicle subjected to hydrostatic buckling.     

Reliability-based design optimization for crashworthiness of vehicle side impact

With the advent of powerful computers, vehicle safety issues have recently been addressed using computational methods of vehicle crashworthiness, resulting in reductions in cost and time for new vehicle development. Vehicle design demands multidisciplinary optimization coupled with a computational crashworthiness analysis. However, simulation-based optimization generates deterministic optimum designs, which are frequently pushed to the limits of design constraint boundaries, leaving little or no room for tolerances (uncertainty) in modeling, simulation uncertainties, and/or manufacturing imperfections. Consequently, deterministic optimum designs that are obtained without consideration of uncertainty may result in unreliable designs, indicating the need for Reliability-Based Design Optimization (RBDO).Recent development in RBDO allows evaluations of probabilistic constraints in two alternative ways: using the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA). The PMA using the Hybrid Mean Value (HMV) method is shown to be robust and efficient in the RBDO process, whereas RIA yields instability for some problems. This paper presents an application of PMA and HMV for RBDO for the crashworthiness of a large-scale vehicle side impact. It is shown that the proposed RBDO approach is very effective in obtaining a reliability-based optimum design.     

Reliability-based design optimization of trusses under dynamic shakedown constraints

Structural and Multidisciplinary Optimization - A reliability-based design optimization problem under dynamic shakedown constraints for elastic perfectly plastic truss structures subjected to...     

Reliability-based design optimization of composite stiffened panels in post-buckling regime

This paper focuses on Deterministic and Reliability Based Design Optimization (DO and RBDO) of composite stiffened panels considering post-buckling regime and progressive failure analysis. The ultimate load that a post-buckled panel can hold is to be maximised by changing the stacking sequence of both skin and stringers composite layups. The RBDO problem looks for a design that collapses beyond the shortening of failure obtained in the DO phase with a target reliability while considering uncertainty in the elastic properties of the composite material. The RBDO algorithm proposed is decoupled and hence separates the Reliability Analysis (RA) from the deterministic optimization. The main code to drive both the DO and RBDO approaches is written in MATLAB and employs Genetic Algorithms (GA) to solve the DO loops because discrete design variables and highly nonlinear response functions are expected. The code is linked with Abaqus to perform parallel explicit nonlinear finite element analyses in order to obtain the structural responses at each generation. The RA is solved through an inverse Most Probable failure Point (MPP) search algorithm that benefits from a Polynomial Chaos Expansion with Latin Hypercube Sampling (PCE-LHS) metamodel when the structural responses are required. The results led to small reductions in the maximum load that the panels can bear but otherwise assure that they will collapse beyond the shortening of failure imposed with a high reliability.