Reliability based multidisciplinary design optimization of cooling turbine blade considering uncertainty data statistics

Considering the coupling among aerodynamic, heat transfer and strength, a reliability based multidisciplinary design optimization method for cooling turbine blade is introduced. Multidisciplinary analysis of cooling turbine blade is carried out by sequential conjugated heat transfer analysis and strength analysis with temperature and pressure interpolation. Uncertainty data including the blade wall, rib thickness, elasticity Modulus and rotation speed is collected. Data statistics display the probability models of uncertainty data follow three-parameter Weibull distribution. The thickness of blade wall, thickness and height of ribs are chosen as design variables. Kriging surrogate model is introduced to reduce time-consuming multidisciplinary reliability analysis in RBMDO loop. The reliability based multidisciplinary design optimization of a cooling turbine blade is carried out. Optimization results shows that the RBMDO method proposed in this work improves the performance of cooling turbine blade availably.

     

On wind turbine blade design optimization

A wind turbine with a horizontal rotation axis is considered in the stationary airflow. A problem of choosing the law of change of the setting angle of the cross section of the wind turbine blade and the value of the angular speed so that the wind energy utilization coefficient is maximal is discussed. Maximization of the respective functional is considered as a variational problem with a priori unknown parameter. Once it is solved numerically or analytically, the optimal value of this parameter is determined, with the sought variables given by rather simple formulas. The results are analyzed on a qualitative level. In particular, the setting angle is found to change monotonically along the blade, which is confirmed by practical wind turbine design. Examples of using the proposed approach to wind turbine blade design are considered.     

Structural design optimization of wind turbine towers

This paper describes several optimization models for the design of a typical wind turbine tower structure. The main tower body is considered to be built from uniform segments where the effective design variables are chosen to be the cross-sectional area, radius of gyration and height of each segment. The nacelle/rotor combination is regarded as a rigid non-rotating mass attached at the top of the tower. Five optimization strategies are developed and tested. The last one concerning reduction of vibration level by direct maximization of the system natural frequencies works very well and has shown excellent results for both tower alone and the combined tower/rotor model. Extensive computer experimentation has shown that global optimality is attainable from the proposed discretized model and a new mathematical concept is given for the exact placement of the system frequencies. The normal mode method is applied to obtain forced response for different types of excitations. The optimization problem is formulated as a nonlinear mathematical programming problem solved by the interior penalty function technique. Finally, the model is applied to the design of a 100-kW horizontal axis wind turbine (ERDA-NASA MOD-0). It has succeeded in arriving at the optimum solutions showing significant improvements in the overall system performance as compared with a reference or baseline design.     

Reliability-based design optimization of multidisciplinary system under aleatory and epistemic uncertainty

Structural and Multidisciplinary Optimization - This paper proposes formulations and algorithms for reliability-based design optimization (RBDO) of both single and multidisciplinary systems under...     

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 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...     

Reliability-based design optimization with confidence level under input model uncertainty due to limited test data

For obtaining a correct reliability-based optimum design, the input statistical model, which includes marginal and joint distributions of input random variables, needs to be accurately estimated. However, in most engineering applications, only limited data on input variables are available due to expensive testing costs. The input statistical model estimated from the insufficient data will be inaccurate, which leads to an unreliable optimum design. In this paper, reliability-based design optimization (RBDO) with the confidence level for input normal random variables is proposed to offset the inaccurate estimation of the input statistical model by using adjusted standard deviation and correlation coefficient that include the effect of inaccurate estimation of mean, standard deviation, and correlation coefficient.     

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 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.     

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 with cooperation between support vector machine and particle swarm optimization

Reliability-based design optimization (RBDO) is concerned with designing an engineering system to minimize a cost function subject to the reliability requirement that failure probability should not exceed a threshold. Conventional RBDO methods are less than satisfactory in dealing with discrete design parameters and complex limit state functions (nonlinear and non-differentiable). Methods that are flexible enough to address the concerns above, however, come at a high computational cost. To enhance computational efficiency without sacrificing model flexibility, we propose a new RBDO framework: PS2, which combines Particle Swarm Optimization (PSO), Support Vector Machine (SVM), and Subset Simulation (SS). SS can efficiently estimate small failure probabilities, based on which SVM is adopted to evaluate the reliability of candidate solutions using binary classification. PSO is employed to solve the discrete optimization problem. Primary emphasis is placed upon the cooperation between SVM and PSO. The cooperation is mutually beneficial since the SVM classifier helps PSO evaluate the feasibility of solutions with high efficiency while the optimal solutions obtained by PSO assist in retraining the SVM classifier to attain better accuracy. The PS2 framework is implemented to find the optimal design of a ten-bar truss, whose component sizes are selected from a commercial standard. The reliability constraints are non-differentiable with two failure modes: yield stress and buckling stress. The interactive process between PSO and SVM contributes greatly to the success of the PS2 framework. It is shown that in various trials the PS2 framework consistently outperforms both the double-loop and single-loop approaches in terms of computational efficiency, solution quality, and model flexibility.     

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 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 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.

     

Reliability-based design optimization of crane bridges using Kriging-based surrogate models

Cranes as indispensable and important hoisting machines of modern manufacturing and logistics systems have been wildly used in factories, mines, and custom ports. For crane designs, the crane bridge is one of the most critical systems, as its mechanical skeleton bearing and transferring the operational load and the weight of the crane itself thus must be designed with sufficient reliability in order to ensure safe crane services. Due to extremely expensive computational costs, current crane bridge design has been primarily focused either on deterministic design based on conventional design formula with empirical parameters from designers’ experiences or on reliability-based design by employing finite-element analysis. To remove this barrier, the paper presents the study of using an advanced surrogate modeling technique for the reliability-based design of the crane bridge system to address the computational challenges and thus enhance design efficiency. The Kriging surrogate models are first developed for the performance functions for the crane system design and used for the reliability-based design optimization. Comparison studies with existing crane design methods indicated that employing the surrogate models could substantially improve the design efficiency while maintaining good accuracy.

     

Reliability-based design optimization of problems with correlated input variables using a Gaussian Copula

The reliability-based design optimization (RBDO) using performance measure approach for problems with correlated input variables requires a transformation from the correlated input random variables into independent standard normal variables. For the transformation with correlated input variables, the two most representative transformations, the Rosenblatt and Nataf transformations, are investigated. The Rosenblatt transformation requires a joint cumulative distribution function (CDF). Thus, the Rosenblatt transformation can be used only if the joint CDF is given or input variables are independent. In the Nataf transformation, the joint CDF is approximated using the Gaussian copula, marginal CDFs, and covariance of the input correlated variables. Using the generated CDF, the correlated input variables are transformed into correlated normal variables and then the correlated normal variables are transformed into independent standard normal variables through a linear transformation. Thus, the Nataf transformation can accurately estimates joint normal and some lognormal CDFs of the input variable that cover broad engineering applications. This paper develops a PMA-based RBDO method for problems with correlated random input variables using the Gaussian copula. Several numerical examples show that the correlated random input variables significantly affect RBDO results.     

Reliability-based design optimization using a family of methods of moving asymptotes

In this study, an effective method for reliability-based design optimization is proposed enhancing sequential optimization and reliability assessment (SORA) method by a family of methods of moving asymptotes (MMA) approximations. In SORA, reliability estimation and deterministic optimization are performed sequentially. And the sensitivity and function value of probabilistic constraint at the most probable point (MPP) are obtained in the process of finding reliability information. In this study, a family of MMA approximations are constructed by utilizing the sensitivity and function value of the probabilistic constraint at the MPP. So, no additional evaluation of the probabilistic constraint is required in constructing MMA approximations. Moreover, no additional evaluation of the probabilistic constraint is required in the deterministic optimization of SORA by using a family of MMA approximations. The efficiency and accuracy of the proposed method were verified through numerical examples.