Efficient Robust Optimization of Metal Forming Processes using a Sequential Metamodel Based Strategy

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.

: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.

<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>

Product improvement and cost reduction have always been important goals in the metal forming industry. The rise of finite element (FEM) simulations for processes has contributed to these goals in a major way. More recently, coupling FEM simulations to mathematical optimisation techniques has shown the potential to make a further giant contribution to product improvement and cost reduction. Much research on the optimisation of metal forming processes has been published during the last couple of years. Although the results are impressive, the optimisation techniques are generally only applicable to specific optimisation problems for specific products and specific metal forming processes. As a consequence, applying optimisation techniques to other metal forming problems requires a lot of optimisation expertise, which forms a barrier for more general industrial application of these techniques. In this paper, we overcome this barrier by proposing a generally applicable optimisation strategy that makes use of FEM simulations of metal forming processes. It consists of a structured methodology for modelling optimisation problems related to metal forming. Subsequently, screening is applied to reduce the size of the optimisation problem by selecting only the most important design variables. Finally, the reduced optimisation problem is solved by an efficient optimisation algorithm. The strategy is generally applicable in a sense that it is not constrained to a certain type of metal forming problem, product or process. Also, any FEM code may be included in the strategy. Furthermore, the structured approach for modelling and solving optimisation problems should enable non-optimisation specialists to apply optimisation techniques to improve their products and processes. The optimisation strategy has been successfully applied to a hydroforming process, which demonstrates the potential of the optimisation of metal forming processes in general and more specific the proposed optimisation strategy.

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.

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 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 .

The purpose of robust design optimization is to minimize variations in design performances and therefore to make the design insensitive to uncertainties. Current robust design methods fall into two types — probabilistic robust design and worst-case (interval) robust design. The former method is used when random variables are involved. In this method, robustness is measure by standard deviations of design performances. The later method is used when uncertainties are represented by intervals. The widths of design performances are then used to measure robustness. In many engineering application, both random variables and interval variables may exist simultaneously. In this paper, a general approach to robust design optimization is proposed. The generality comes from the ability to handle both random and interval variables. To alleviate the computational burden, we employ a previously developed general robustness assessment method — semi-second-order Taylor expansion method, to evaluate the maximum and minimum standard deviations of a design performance. An efficient integration strategy of the general robustness assessment and optimization is proposed. With the integration strategy, the number of function calls can be reduced while good accuracy is maintained. A robust shaft design problem is given for demonstration.

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 and reliability-based aeroelastic tailoring framework for composite aircraft wings

This study focuses on the robust design optimization of permanent magnet linear synchronous motors (PMLSM) considering the effect of uncertainties (noise factors) such as manufacturing errors, installation errors and working wears. A robust design optimization approach based on quantified constraint satisfaction problem (QCSP) is proposed. It establishes a robust PMLSM thrust ripple model considering noise factors by QCSP, and obtains the optimal design parameters that have lower thrust ripple and smaller thrust ripple fluctuation within their error range by branch-and-prune algorithm. First, an analytical model of thrust ripple for PMLSM is established. Second, the robust QCSP model of thrust ripple is established by quantifier concept based on analytical model. It defines the design parameters and noise variables as existential and universal quantifiers, respectively, defines the variables of QCSP model as quantifier parameters that are composed of existential and universal quantifiers, then QCSP model are established by quantifier parameters. Third, the algorithm of branch-and-prune based on interval arithmetic is proposed to optimize the QCSP model. This approach is proven to be robust and superior compared with traditional deterministic approach and another robust approach. Finally, Finite element analysis and motor prototype experiments confirm the feasibility and validity of the proposed method.

Robustness, optimization and Finite Element (FE) simulations are of major importance for achieving better products and cost reductions in the metal forming industry. In this paper, a metamodel‐based robust optimization strategy is proposed for metal forming processes. The applicability of the strategy is demonstrated by application to an analytical test function and an industrial V‐bending process. The results of both applications underline the importance of including uncertainty and robustness explicitly in the optimization procedure.

Application of deep learning based multi-fidelity surrogate model to robust aerodynamic design optimization

Development of a deviation package method for low-cost robust optimization in compressor blade design

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.