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 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.     

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 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 optimization under random vibration environment

A methodology of formulating the optimum design problem for structural systems with random parameters and subjected to random vibration as a mathematical programming problem is presented. The proposed method is applied to the optimum design of a cantilever beam with a tip mass and a truss structure supporting a water tank. The excitations are assumed to be Gaussian processes and the geometric and material properties are taken to be normally distributed random variables. The probabilistic constraints are specified for individual failure modes since it is easier to specify the reliability level for each failure mode keeping in view the consequences of failure in that particular mode. The time parameter appearing in the random vibration based constraints is eliminated by replacing the probabilities of failure by suitable upper bounds. The numerical results demonstrate the feasibility and effectiveness of applying the reliability-based design concepts to structures with random parameters and operating in random vibration environment.     

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 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.     

Reliability-based topology optimization of continuum structures subject to local stress constraints

Topology optimization of continuum structures is a challenging problem to solve, when stress constraints are considered for every finite element in the mesh. Difficulties are compounding in the reliability-based formulation, since a probabilistic problem needs to be solved for each stress constraint. This paper proposes a methodology to solve reliability-based topology optimization problems of continuum domains with stress constraints and uncertainties in magnitude of applied loads considering the whole set of local stress constrains, without using aggregation techniques. Probabilistic constraints are handled via a first-order approach, where the principle of superposition is used to alleviate the computational burden associated with inner optimization problems. Augmented Lagrangian method is used to solve the outer problem, where all stress constraints are included in the augmented Lagrangian function; hence sensitivity analysis may be performed only for the augmented Lagrangian function, instead of for each stress constraint. Two example problems are addressed, for which crisp black and white topologies are obtained. The proposed methodology is shown to be accurate by checking reliability indices of final topologies with Monte Carlo Simulation.     

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.     

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...     

Sequential approximate reliability-based design optimization for structures with multimodal random variables

Structural and Multidisciplinary Optimization - For practical engineering design problems, random variables tend to follow multimodal probability distributions when working at multiple operating...     

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 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 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...     

Optimal design of interference fit assemblies subjected to fatigue loads

This paper presents a methodology for an optimal design of interference fit subjected to fatigue loads. Optimization consists in finding a trade-off between mass and competing safety factors at hub and shaft contact zone as well as in shaft fillet. Developing an effective calculation method for fatigue strength of an interference fitted assembly using the finite element method is one of the main steps of the procedure. Meanwhile, coupling the finite elements model of interference fit with an optimization algorithm is not adequate considering the computing time and the significant number of calculations necessary to portrait the assembly behavior. Therefore, a sequential approximate multi-objective optimization algorithm (SAMOO) is presented. The method involves Design Of Experiments (DOE), interpolation with kriging functions, and multi-objective optimization. Preliminary study of parameter variance, and advanced post-processing of multi-objective optimization, provide engineers with valuable information for identifying an optimal design of interference fit assembly using fewer finite element calculations.     

A review of optimization of structures subjected to transient loads

Various aspects of structural optimization techniques under transient loads are extensively reviewed. The main themes of the paper are treatment of time-dependent constraints, calculation of design sensitivity, and approximation. Each subject is reviewed with corresponding papers that have been published since the 1970s. The treatment of time-dependent constraints in both the direct method and the transformation method is discussed. Two ways of calculating design sensitivity of a structure under transient loads are discussed—direct differentiation method and adjoint variable method. The approximation concept mainly focuses on the response surface method in crashworthiness and local approximation with the intermediate variables. Especially, a method using the equivalent static load is discussed as an approximation method. It takes advantage of the well-established static response optimization. The structural optimization in flexible multibody dynamic systems is reviewed in the viewpoint of the above three themes.     

Reliability-based shape optimization of structures undergoing fluid–structure interaction phenomena

Fluid–structure interaction phenomena are often roughly approximated when the stochastic nature of a system is considered in the design optimization process, leading to potentially significant epistemic uncertainty. In this paper, after reviewing the state-of-the-art methods in robust and reliability-based design optimization of problems undergoing fluid–structure interaction phenomena, a computational framework is presented that integrates a high-fidelity aeroelastic model into reliability-based design optimization. The design optimization problem is formulated pursuant to the reliability index and performance measure approaches. The system reliability is evaluated by a first-order reliability analysis method. The steady-state aeroelastic problem is described by a three-field formulation and solved by a staggered procedure, coupling a potentially detailed structural finite element model and a finite volume discretization of the Euler flow. The design and imperfection sensitivities are computed by evaluating the analytically derived direct and adjoint coupled aeroelastic sensitivity equations. The computational framework is verified by the optimization of three-dimensional wing structures. The lift-to-drag ratio is maximized, subject to stress constraints, by varying shape, thickness, and material properties. Uncertainties in structural parameters, including design parameters, operating conditions, and modeling uncertainties are considered. The results demonstrate the need for reliability-based optimization methods, for the design of structures undergoing fluid–structure interaction phenomena, and the applicability of the proposed framework to realistic design problems. Comparing the optimization results for different levels of uncertainty shows the importance of accounting for uncertainties in a quantitative manner.