Non-Gradient Based Parameter Sensitivity Estimation for Robust Design Optimization

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 .

A new robust design optimization method to automatically search multiple optimal solutions and to estimate robust objective variations is presented. The method does not require a presumed probability distribution for parameters, and is applicable even when the objective function is non-differen-tiable and/or discontinuous and when parameter variations are large. The method is developed based on the relationships between performance variation and their sensitivity regions using regions using a worst-case scenario analysis. Three loops are included in the method. Inner loop optimizes objective function subject to the robustness constraints at a robust index; Middle loop conducts the search for the robust objective variation using a iterative approach; Outer loop performs the search for the multiple optimal solutions by means of adjusting the thresholds of robustness indexes. The method provides the objective values and their robust objective variations of multiple optimal solutions. By using analytical interpretations and providing a illustrative example, the method shows advantages that it does not require designers to cautiously determine the thresholds of robustness indexes and yields more information that enable designers to carry out a trade-offs analysis between objective values and their robustness.

: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.

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.

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.

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.

A robust bi-level optimization modelling approach for municipal solid waste management; a real case study of Iran

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.

The primary contribution of this paper lies in the development of a bi-level optimization framework designed to model a retailer’s power trading strategies within the context of multi-microgrids. Another noteworthy aspect of this research is managing the risk associated with the retailer’s decision-making process, achieved by addressing the unpredictability of market prices through a robust bi-level structure. In this model, the upper-level problem focuses on the retailer’s decision-making, while the lower-level problem addresses microgrid considerations. Furthermore, a load-shifting strategy is integrated into the microgrid problem to enhance the profitability-to-cost ratio for both the retailer and the microgrids. The results demonstrate a 3% increase in the retailer’s profits and a significant 30% reduction in the microgrids’ costs using the load-shifting strategy. In the most unfavorable scenario, the retailer experiences a profit reduction of 21.67%, while the maximum increase in the microgrids’ costs reaches 238.3%.

Robust design optimization (RDO) has been eminent, ascertaining optimal configuration of engineering systems in presence of uncertainties. However, computational aspect of conventional RDO can often get computationally intensive as neighborhood assessments of every solution are required to compute the performance variance and ensure feasibility. Surrogate assisted optimization is one of the efficient approaches in order to mitigate this issue of computational expense. However, the performance of a surrogate model plays a key factor in determining the optima in multi-modal and highly non-linear landscapes, in presence of uncertainties. In other words, the approximation accuracy of the model is principal in yielding the actual optima and thus, avoiding any misguide to the decision maker on the basis of false or, local optimum points. Therefore, an extensive survey has been carried out by employing most of the well-known surrogate models in the framework of RDO. It is worth mentioning that the numerical study has revealed consistent performance of a model out of all the surrogates utilized. Finally, the best performing model has been utilized in solving a large-scale practical RDO problem. All the results have been compared with that of Monte Carlo simulation results.

Decentralized Decision System for Closed-Loop Supply Chain: A Bi-Level Multi-Objective Risk-Based Robust Optimization Approach

Comparative Study of Metamodeling and Sampling Design for Expensive and Semi-Expensive Simulation Models Under Uncertainty

<div class="section abstract"><div class="htmlview paragraph">Existing multi-axle steering system designs generally use the deterministic optimization method without considering the uncertainties during the design process; therefore an actual steering movement may deviate from the ideal movement calculated by some mathematical models. In order to make design results have less sensitive to the uncertainties in the design process, some uncertainties need be taken into account at the early design stage. This paper proposes a robust optimization design method for a double front axle steering system (DFASS) of heavy trucks based on Monte Carlo method. The DFASS consists of two trapezoidal steering mechanisms (TSM) and one rocker system, and the optimization objectives of DFASS include the minimum mean value and variance of the maximum turning angle error of the TSM and rocker system. In addition, the robust optimization model includes 13 design variables which are all geometry parameters of DFASS and represented by normal distribution. Through the orthogonal experiment, we obtain the important factors affecting optimization objectives and build the response surface models of optimization objective. Based on the response surfaces, robustness design of DFASS is performed. By simulating analysis, the improved robustness of DFASS based on robust design method is approved.</div></div>

The coupling of finite element simulations to mathematical optimization techniques has contributed significantly to product improvements and cost reductions in the metal forming industries. The next challenge is to bridge the gap between deterministic optimization techniques and the industrial need for robustness. This paper introduces a generally applicable strategy for modeling and efficiently solving robust optimization problems based on time consuming simulations. Noise variables and their effect on the responses are taken into account explicitly. The robust optimization strategy consists of four main stages: modeling, sensitivity analysis, robust optimization and sequential robust optimization. Use is made of a metamodel-based optimization approach to couple the computationally expensive finite element simulations with the robust optimization procedure. The initial metamodel approximation will only serve to find a first estimate of the robust optimum. Sequential optimization steps are subsequently applied to efficiently increase the accuracy of the response prediction at regions of interest containing the optimal robust design. The applicability of the proposed robust optimization strategy is demonstrated by the sequential robust optimization of an analytical test function and an industrial V-bending process. For the industrial application, several production trial runs have been performed to investigate and validate the robustness of the production process. For both applications, it is shown that the robust optimization strategy accounts for the effect of different sources of uncertainty onto the process responses in a very efficient manner. Moreover, application of the methodology to the industrial V-bending process results in valuable process insights and an improved robust process design.