A Critical Review of Surrogate Assisted Robust Design Optimization

A number of multi-objective evolutionary algorithms have been proposed in recent years and many of them have been used to solve engineering design optimization problems. However, designs need to be robust for real-life implementation, i.e. performance should not degrade substantially under expected variations in the variable values or operating conditions. Solutions of constrained robust design optimization problems should not be too close to the constraint boundaries so that they remain feasible under expected variations. A robust design optimization problem is far more computationally expensive than a design optimization problem as neighbourhood assessments of every solution are required to compute the performance variance and to ensure neighbourhood feasibility. A framework for robust design optimization using a surrogate model for neighbourhood assessments is introduced in this article. The robust design optimization problem is modelled as a multi-objective optimization problem with the aim of simultaneously maximizing performance and minimizing performance variance. A modified constraint-handling scheme is implemented to deal with neighbourhood feasibility. A radial basis function (RBF) network is used as a surrogate model and the accuracy of this model is maintained via periodic retraining. In addition to using surrogates to reduce computational time, the algorithm has been implemented on multiple processors using a master–slave topology. The preliminary results of two constrained robust design optimization problems indicate that substantial savings in the actual number of function evaluations are possible while maintaining an acceptable level of solution quality.

Robust design optimization of the vibrating rotor-shaft system subjected to selected dynamic constraints

In the present paper, we explore issues in handling equality constraints in multiobjective robust design optimization (RDO) problems. Satisfying an equality constraint exactly under uncertainty can be a challenging task. The challenge of handling equality constraints is compounded in multiobjective RDO problems. Modeling the tradeoffs between the mean of the performance and the variation of the performance for each design objective in a multiobjective RDO problem is a complex task by itself. Equality constraints add to this complexity because of the additional tradeoffs that are introduced between constraint satisfaction under uncertainty and multiobjective performance. Satisfying equality constraints at their mean values, an approach typically followed, could lead to undesirable results. In this paper, we study the implications of equality constraint satisfaction in a multiobjective RDO problem, and provide a new problem formulation approach to resolve the above discussed tradeoffs. We illustrate that the proposed formulation can be used as an effective multiobjective design space exploration tool, with emphasis on equality constraint satisfaction under uncertainty. We present numerical examples to illustrate our theoretical developments.

:A robust multi-objective wing design optimization procedure using CFD is presented in this work. Instead of directly applying CFD to wing design optimization, a Kriging aerodynamic model that approximates CFD result is proposed to save computational cost. By introducing 6σ robust approach as a sub-objective function, multi-objective wing design problems can be solved with aerodynamic robustness. Non-Uniform Rational B-Spline curves (NURBS) are introduced to depict the wing geometry in design optimization process. The most important parameters of the wing geometry can then be screened by design of experiment and treated as design variables due to the local variation property of NURBS. Under this formulation, a multi-objective genetic algorithm based on non-dominated sorting is employed to handle multiple flight conditions in wing design. The optimum wing geometry is selected according to trade-off among design objects on the non-dominated front. To validate the proposed approach, two robust design optimization cases are studied for ONERA-M6-Wing. It turns out that the drag coefficient is insensitive to Mach number between Ma0.8–Ma0.9 after robust optimization. The result also verifies the effectiveness of our method in boosting performance and robustness of multiple flight conditions: the lift curve slope (Ma0.3) of optimum wing and the 6σ objective function of transonic drag (Ma0.8–Ma0.9) increase by 11.9% and 25.4%, respectively. Observations in the optimization cases are concluded as follows: 1) For wing aerodynamic design problems, the multi-objective robust optimization can both improve the performance at different flight conditions and provide robustness. 2) The Kriging aerodynamic model derived from CFD result can satisfy the precision requirement of wing design. 3) Even though the optimization result is subject to the weights of the 6σ function in sub-object, the influence is not comparable to the trade-off among design objects. 4) The optimum solution obtained by proposed approach is superior to gradient based optimization method, while the computational cost is acceptable.

The uncertainties of design variables, noise parameters, and metamodel are important factors in simulation-based robust design optimization. Most conventional metamodel construction methods only consider one or two uncertainties. In this paper, a new surrogate modeling method simultaneously measuring all the uncertainties is proposed for simulation-based robust design optimization of complex product. The effect of metamodel uncertainty on product performance uncertainty is quantified through uncertainty propagation analysis among design variables uncertainty, noise parameters uncertainty, metamodel uncertainty, and performance uncertainty. Then, the sampling points are selected and the metamodel is constructed based on the predictive interval of product performance and mean square error of the Kriging metamodel. The constructed metamodel is applied to robust design optimization considering multiple uncertainties. Results of two mathematical examples show that the proposed metamodel considering multiple uncertainties increases the result accuracy of robust design optimization. Finally, the proposed algorithm is applied to robust design optimization of a heat exchanger, and the total heat transfer rate is enhanced under uncertainties of fin structural parameters, operation conditions parameters and simulation metamodel.

This study implements robust design optimization (RDO) of a self-centering viscous damper in the presence of seismic uncertainty. The RDO minimizes the effect of random input parameters on the output response, thus providing a robust solution. The optimal phenomenological parameters of the self-centering viscous damper are evaluated to ensure its robustness. The objective function of the RDO is cast as a sum of the mean and standard deviation of maximum floor acceleration and a constraint function of the inter-storey drift ratio of the floors. The mean component of the objective function minimizes the structural responses due to external loads, whereas the other component reduces the propagation of uncertainty from the input parameters. To reduce the computational cost of the RDO, a wavelet-based neural network is used as a surrogate model. The proposed optimization method is implemented in a 16-storey steel moment resisting frame with consideration of two ensembles of ground motions, i.e. far-field and near-field records, which are taken from FEMA P695 (ATC-63). To highlight the effectiveness of the proposed self-centering damper in mitigating maximum and residual drift ratios, two additional optimizations are undertaken using viscous and self-centering dampers. The numerical study shows that the neural network with a Gaussian wavelet function produces better accuracy compared to other wavelet functions. Also, the numerical analysis clearly elucidates the advantages of the proposed damper in reducing the residual drift of the floors over a viscous damper and a self-centering damper, which may lead to minimizing the post-earthquake economic losses.

Aiming at the structural optimization problem of a gun turret, the robust optimization model of the turret structure is established. This model takes the thickness of structural rib plate as the design variable, the minimization of the structural mass of the turret as the objective, the maximum displacement and the lowest natural frequency as the constraints. It also considers the influence of the structural instabilities and employs the 6σ robust optimization design method. The samples are obtained by the experimental design method. The radial basis function neural network is used to construct the surrogate model of the turret structure. The precision of the surrogate model is measured by the determination coefficient R2 and the Mean of Normalized Absolute Error MNAE. Finally, the robust optimization results are obtained by using the combinatorial optimization algorithm composed of Particle Swarm Optimization (PSO) algorithm and Sequential Quadratic Programming-NLPQL. The calculation results show that the 6σ robust optimization design method can effectively ensure the robustness of the optimization results. The accuracy of the radial basis function neural network surrogate model can meet the optimization requirements. The mass of structure is reduced by 9.85% after optimization.

Multiple parameter optimization methodology by integrating a game theory principle into priority-based decision making

This paper describes the methodologies that have been developed by ESTECO during the first phase of UMRIDA European Project, in the field of Uncertainty Management and Robust Design Optimization, and that have been implemented in the software platform modeFRONTIER. In particular, in the first part there are proposed two methodologies, one based on SS-ANOVA regression applied directly to the uncertainties variables and one based on a stepwise regression methodology applied to the Polynomial Chaos terms used for the uncertainty quantification. Aeronautical test cases proposed by UMRIDA consortium are used to verify the validity of the methodologies. In the second part, the state of art methodologies for Robust Design Optimization are compared with a new proposed approach, based on a min-max definition of the objectives, and the application of Polynomial Chaos coefficients for an accurate definition of percentiles (reliability-based robust design optimization). Also in this case an Aeronautical CFD test case is proposed to validate the methodologies.

Swarm intelligence hybridized with genetic search in multi-objective design optimization under constrained-Pareto dominance

One critical issue for manufacturing 3D MCM packages is the warpage induced during the process. The effects of several important parameters, including processing conditions, package geometry and materials, are specifically studied on the occurrence of warpage and optimized to reduce the warpage in many previous literatures. However, the design parameter values are not always precisely known but can contain a level of uncertainty to some extent due to, for example manufacturing tolerance. In some case, the design parameters variations can not be neglected. In this paper, the effect of structure geometry tolerance and material property tolerance on package warpage robustness was took into account and analyzed. The robust optimization mathematic model of the 3D packages, considering the uncertainty of the design parameters, was built. And the robust optimization method based on the Monte-carlo simulations and Genetic Algorithm was present to solve the optimization problem above. Firstly, the parametric design finite element model of the specific 3D MCM package was built. Secondly, the statistic distribution of the warpage induced by the design parameters variations was obtained based on the Monte-carlo simulations. Finally, the GA was implemented to optimize the mean value and standard deviation of the warpage. The resulting robust optimization design shows that the robust optimization design is insensitive to variations on design variables within a feasible region.

Robust design seeks an optimal solution where the design objective is insensitive to the variations of input variables while the design feasibility under the variations is maintained. Accurate robustness assessment for both design objective and feasibility usually requires an intensive computational effort. In this paper, an accurate robustness assessment method with a moderate computational effort is proposed. The numerical Gauss–Hermite integration technique is employed to calculate the mean and standard deviation of the objective and constraint functions. To effectively use the Gauss–Hermite integration technique, a transformation from a general random variable into a normal variable is performed. The Gauss–Hermite integration and the transformation result in concise formulas and produce an accurate approximation to the mean and standard deviation. This approach is then incorporated into the framework of robust design optimization. The design of a two-bar truss and an automobile torque arm is used to demonstrate the effectiveness of the proposed method. The results are compared with the commonly used Taylor expansion method and Monte Carlo simulation in terms of accuracy and efficiency. Copyright © 2005 John Wiley & Sons, Ltd.

This paper presents a new approach aims to solve robust multidisciplinary design optimization MDO problem called Improved Multi-objective Robust Collaborative Optimization . This method combines the Multi-objective Robust Collaborative Optimization method, the Worst Possible Point constraint cuts and the Genetic algorithm NSGA-II type as an optimizer to solve the robust optimization problem of complex structure named Y-stiffened panel under interval uncertainty . The proposed approach hierarchically decomposes the optimization problem into a structure level considered as an upper level in the Y-stiffened panel and a second level considered as a lower level of the studied panel. A robust multi-objective optimization problem intended to optimize the eigenfrequency, the global mass and the displacement at a fixed point of the Y-stiffened panel at the first level and each structure’s robust optimization problem allows optimizing its eigenfrequency and mass limited by their local constraint functions at the second one. Tor demonstrate our method, an engineering example of Y-stiffened panel is treated. A good performance of proposed method is proved by a comparison between obtained results and Non-Distributed Multi-objective Robust Optimization .

There are two general methods of robust design optimization (RDO) and reliability-based robust design optimization (RBRDO) for the optimal design of structures with non-deterministic variables. In the problem-solving process of both RDO and RBRDO, assessing the probabilistic constraints of the problem and the standard deviation of the structural responses is time-consuming and costly. In this study, by presenting a multi-step approach, for a probable optimal solution, the deterministic constraints are first investigated. Then the limit state functions related to probabilistic constraints are computed based on the mean value of random variables (in the RBRDO problem) and in the step of uncertainties effects simulation on the response of the structure, probabilistic constraints are evaluated (in the RBRDO problem). Designs that not meeting the deterministic constraints or in the unsafe area for the mean values of the random variables are not included in calculating the first and second terms of the objective function and reliability assessment. This leads to reduce the volume of calculations. The Monte Carlo simulation method is used to assess the probabilistic constraints, and the EVPS metaheuristic algorithm is used for the optimization process. Three benchmark trusses of 10, 25, and 72 bars were studied to assess the efficiency of the proposed approach. The first objective function in these problems was the mean weight of trusses and the second objective function was the standard deviation of the displacement of a specific node in a certain direction.

A new method is presented for an application of the Gaussian mixture model (GMM) to a multi-objective robust design optimization (RDO) of planar steel frame structures under aleatory (stochastic) uncertainty in material properties, external loads, and discrete design variables. Uncertainty in the discrete design variables is modeled in the wide range between the smallest and largest values in the catalog of the cross-sectional areas. A weighted sum of Gaussians is statistically trained based on the sampled training data to capture an underlying joint probability distribution function (PDF) of random input variables and the corresponding structural response. A simple regression function for predicting the structural response can be found by extracting the information from a conditional PDF, which is directly derived from the captured joint PDF. A multi-objective RDO problem is formulated with three objective functions, namely, the total mass of the structure, and the mean and variance values of the maximum inter-story drift under some constraints on design strength and serviceability requirements. The optimization problem is solved using a multi-objective genetic algorithm utilizing the trained GMM for calculating the statistical values of objective and constraint functions to obtain Pareto-optimal solutions. Since the three objective functions are highly conflicting, the best trade-off solution is desired and found from the obtained Pareto-optimal solutions by performing fuzzy-based compromise programming. The robustness and feasibility of the proposed method for finding the RDO of planar steel frame structures with discrete variables are demonstrated through two design examples.