Multi-objective genetic programming approach for robust modeling of complex manufacturing processes having probabilistic uncertainty in experimental data

In this paper, a multi-objective uniform-diversity genetic programming (MUGP) algorithm deployed for robust Pareto modeling and prediction of complex nonlinear processes using some input-output data table. The uncertainties included in measured data are considered to obtain more robust models. The considered benchmarks are an explosive cutting and forming processes, in which the nonlinear behavior between the input and output of processes are detected using MUGP. For both case studies, a multi-objective modeling and prediction procedure firstly performed using deterministic data. Secondly, the same identification procedure carried out using probabilistic uncertainty in the experimental input-output data. The objective functions considered are namely, training error, prediction error and number of tree nodes (complexity of models) in the deterministic approach. Accordingly, the mean and standard deviation of training error and prediction error are considered in robust Pareto modeling and prediction of such processes. In this way, Pareto front of such modeling and prediction is first obtained for both explosive cutting and forming processes with deterministic data. Such Pareto front is then obtained using experimental input-output-data having probabilistic uncertainty in input parameters through a Monte Carlo simulation (MCS) approach. In addition, it has been shown that for both cases, the trade-off models obtained from deterministic data have significant biases when tested on data with probabilistic uncertainty. Finally, the obtained results of such multi-objective robust model identification show promising results in terms of compensating uncertainty in the experimental input-output-data.     

Multi‐objective design optimization of antennas for reflection,size, and gain variability using kriging surrogates and generalized domain segmentation

Cost‐efficient multi‐objective design optimization of antennas is presented. The framework exploits auxiliary data‐driven surrogates, a multi‐objective evolutionary algorithm for initial Pareto front identification, response correction techniques for design refinement, as well as generalized domain segmentation. The purpose of this last mechanism is to reduce the volume of the design space region that needs to be sampled in order to construct the surrogate model, and, consequently, limit the number of training data points required. The recently introduced segmentation concept is generalized here to allow for handling an arbitrary number of design objectives. Its operation is illustrated using an ultra‐wideband monopole optimized for best in‐band reflection, minimum gain variability, and minimum size. When compared with conventional surrogate‐based approach, segmentation leads to reduction of the initial Pareto identification cost by over 20%. Numerical results are supported by experimental validation of the selected Pareto‐optimal antenna designs.     

Multiobjective optimization for crash safety design of vehicles using stepwise regression model

In automotive industry, structural optimization for crashworthiness criteria is of special importance. Due to the high nonlinearities, however, there exists substantial difficulty to obtain accurate continuum or discrete sensitivities. For this reason, metamodel or surrogate model methods have been extensively employed in vehicle design with industry interest. This paper presents a multiobjective optimization procedure for the vehicle design, where the weight, acceleration characteristics and toe-board intrusion are considered as the design objectives. The response surface method with linear and quadratic basis functions is employed to formulate these objectives, in which optimal Latin hypercube sampling and stepwise regression techniques are implemented. In this study, a nondominated sorting genetic algorithm is employed to search for Pareto solution to a full-scale vehicle design problem that undergoes both the full frontal and 40% offset-frontal crashes. The results demonstrate the capability and potential of this procedure in solving the crashworthiness design of vehicles.     

A robust optimization approach based on multi-fidelity metamodel

Multi-fidelity (MF) metamodeling approaches have recently attracted a significant amount of attention in simulation-based design optimization due to their ability to conduct trade-offs between high accuracy and low computational expenses by integrating the information from high-fidelity (HF) and low-fidelity (LF) models. While existing MF metamodel assisted design optimization approaches may yield an inferior or even infeasible solution since they generally treat the MF metamodel as the real HF model and ignore the interpolation uncertainties from the MF metamodel. This situation will be more serious in non-deterministic optimization. Hence, in this work, a MF metamodel assisted robust optimization approach is developed, in which the interpolation uncertainty of the MF metamodel and design variable uncertainty are quantified and taken into consideration. To demonstrate the effectiveness and merits of the proposed approach, two numerical examples and a long cylinder pressure vessel design optimization problem are tested. Results show that for the test cases the proposed approach can obtain a solution that is both optimal and within the feasible region even with perturbation of the uncertain variables.     

Unified uncertainty representation and quantification based on insufficient input data

Uncertainty quantification accuracy of system performance has an important influence on the results of reliability-based design optimization (RBDO). A new uncertain identification and quantification methodology is proposed considering the strong statistical variables, sparse variables, and interval variables simultaneously. Maximum likelihood function and Akaike information criterion (AIC) methods are used to identify the best-fitted distribution types and distribution parameters of sparse variables. The interval variables are represented with evidence theory. Finally, a unified uncertainty quantification framework considering the three types of uncertain design variables is put forward, and then the failure probability of system performance is quantified with belief and plausibility measures. The Kriging metamodel and random sampling method are used to reduce the computational complexity. Three examples are illustrated to verify the effectiveness of the proposed methodology.     

An approximate optimization strategy using refined hybrid metamodel

An approximate model called metamodel or surrogate model is a mathematical model that numerically approximates response of a system during an engineering simulation process or test. The introduction of a metamodel makes it possible to express response defined in the design problem as a simple mathematical function of design variables. A metamodel can be built with response surface method (RSM), kriging, neural network, radial basis function, and so on. Each method has its advantages and disadvantages. A combined metamodel called hybrid model, ensemble model, or multiple surrogates has been developed to maximize each metamodel's strength. The hybrid model of this research includes RSM and kriging. Besides, a strategy to refine the hybrid metamodel is implemented by reducing design space. In this process, information related to Hessian is utilized for an unconstrained optimization problem, on the contrary feasibility for a constrained optimization problem. This research presents a new hybrid metamodel-based optimization strategy called refined hybrid metamodel. Five mathematical test problems, two-bar design, spring design, and propeller shaft design problems are solved with the suggested method, verifying its usefulness. Most of the optimal results with the proposed method are closer to exact solutions with smaller function evaluations than existing methods.     

Uncertainty quantification in reliability estimation with limit state surrogates

Model-based reliability analysis is affected by different types of epistemic uncertainty, due to inadequate data and modeling errors. When the physics-based simulation model is computationally expensive, a surrogate has often been used in reliability analysis, introducing additional uncertainty due to the surrogate. This paper proposes a framework to include statistical uncertainty and model uncertainty in surrogate-based reliability analysis. Two types of surrogates have been considered: (1) general-purpose surrogate models that compute the system model output over the desired ranges of the random variables; and (2) limit-state surrogates. A unified approach to connect the model calibration analysis using the Kennedy and O’Hagan (KOH) framework to the construction of limit state surrogate and to estimating the uncertainty in reliability analysis is developed. The Gaussian Process (GP) general-purpose surrogate of the physics-based simulation model obtained from the KOH calibration analysis is further refined at the limit state (local refinement) to construct the limit state surrogate, which is used for reliability analysis. An efficient single-loop sampling approach using the probability integral transform is used for sampling the input variables with statistical uncertainty. The variability in the GP prediction (surrogate uncertainty) is included in reliability analysis through correlated sampling of the model predictions at different inputs. The Monte Carlo sampling (MCS) error, which represents the error due to limited Monte Carlo samples, is quantified by constructing a probability density function. All the different sources of epistemic uncertainty are quantified and aggregated to estimate the uncertainty in the reliability analysis. Two examples are used to demonstrate the proposed techniques.     

基于广义改进分解策略的多目标代理优化方法

为解决多目标代理优化方法中代理模型选择单一问题,提出基于广义改进函数分解策略的多目标代理优化方法.该方法充分利用模型预测信息构建广义改进多目标分解准则和广义改进R2指标准则,有效拓展多目标代理优化中代理模型的选择空间.所提两种准则通过随机均匀权重实现全局探索和局部搜索能力的自适应平衡.研究结果表明,所提方法在有限仿真条件下拥有良好的寻优性能,获得Pareto前沿在收敛性、多样性及空间分布性方面均具有一定优势.相比同类方法,该方法具有优势:1)不需要模型预测不确定性信息,适用于基于不同种类代理模型的代理优化方法; 2)实现简单且计算复杂度低,能够有效提升昂贵黑箱问题优化效率.     

Concurrent treatment of parametric uncertainty and metamodeling uncertainty in robust design

Robust design is an effective approach to design under uncertainty. Many works exist on mitigating the influence of parametric uncertainty associated with design or noise variables. However, simulation models are often computationally expensive and need to be replaced by metamodels created using limited samples. This introduces the so-called metamodeling uncertainty. Previous metamodel-based robust designs often treat a metamodel as the real model and ignore the influence of metamodeling uncertainty. In this study, we introduce a new uncertainty quantification method to evaluate the compound effect of both parametric uncertainty and metamodeling uncertainty. Then the new uncertainty quantification method is used for robust design. Simplified expressions of the response mean and variance is derived for a Kriging metamodel. Furthermore, the concept of robust design is extended for metamodel-based robust design accounting for both sources of uncertainty. To validate the benefits of our method, two mathematical examples without constraints are first illustrated. Results show that a robust design solution can be misleading without considering the metamodeling uncertainty. The proposed uncertainty quantification method for robust design is shown to be effective in mitigating the effect of metamodeling uncertainty, and the obtained solution is found to be more “robust” compared to the conventional approach. An automotive crashworthiness example, a highly expensive and non-linear problem, is used to illustrate the benefits of considering both sources of uncertainty in robust design with constraints. Results indicate that the proposed method can reduce the risk of constraint violation due to metamodel uncertainty and results in a “safer” robust solution.     

Effect of dimensional and material property uncertainties on thermal flexure microactuator response using probabilistic methods

Micromachining of microelectromechanical systems such as other fabrication processes has inherent variation that leads to uncertain dimensional and material properties. In this paper, the effect of material and feature dimension uncertainties due to fabrication process on electrothermal microactuator tip deflection is investigated. A simple and efficient uncertainty analysis method is used based on direct linearization method (DLM); uncertainty analysis is performed by creating second-order metamodel through Box-Behnken design and Monte Carlo simulation. The standard deviations of tip deflection obtained by these two probabilistic methods are very close. Simulation results have been validated by a comparison with experimental results in literature. Experimental results fall within 95% confidence boundary obtained by DLM method. Also, sensitivity analysis of microactuator has been explored; the results show that microactuator performance has been affected more by thermal expansion coefficient and microactuator gap uncertainties.     

Engineering computation under uncertainty - Capabilities of non-traditional models

This paper provides a review of various non-traditional uncertainty models for engineering computation and responds to the criticism of those models. This criticism imputes inappropriateness in representing uncertain quantities and an absence of numerically efficient algorithms to solve industry-sized problems. Non-traditional uncertainty models, however, run counter to this criticism by enabling the solution of problems that defy an appropriate treatment with traditional probabilistic computations due to non-frequentative characteristics, a lack of available information, or subjective influences. The usefulness of such models becomes evident in many cases within engineering practice. Examples include: numerical investigations in the early design stage, the consideration of exceptional environmental conditions and socio-economic changes, and the prediction of the behavior of novel materials based on limited test data. Non-traditional uncertainty models thus represent a beneficial supplement to the traditional probabilistic model and a sound basis for decision-making. In this paper non-probabilistic uncertainty modeling is discussed by means of interval modeling and fuzzy methods. Mixed, probabilistic/non-probabilistic uncertainty modeling is dealt with in the framework of imprecise probabilities possessing the selected components of evidence theory, interval probabilities, and fuzzy randomness. The capabilities of the approaches selected are addressed in view of realistic modeling and processing of uncertain quantities in engineering. Associated numerical methods for the processing of uncertainty through structural computations are elucidated and considered from a numerical efficiency perspective. The benefit of these particular developments is emphasized in conjunction with the meaning of the uncertain results and in view of engineering applications.     

A multi-surrogate approximation method for metamodeling

Metamodeling methods have been widely used in engineering applications to create surrogate models for complex systems. In the past, the input–output relationship of the complex system is usually approximated globally using only a single metamodel. In this research, a new metamodeling method, namely multi-surrogate approximation (MSA) metamodeling method, is developed using multiple metamodels when the sample data collected from different regions of the design space are of different characteristics. In this method, sample data are first classified into clusters based on their similarities in the design space, and a local metamodel is identified for each cluster of the sample data. A global metamodel is then built using these local metamodels considering the contributions of these local metamodels in different regions of the design space. Compared with the traditional approach of global metamodeling using only a single metamodel, this MSA metamodeling method can improve the modeling accuracy considerably. Applications of this metamodeling method have also been demonstrated in this research.     

一类不确定时滞系统的最优控制

对控制能量存在约束条件下一类不确定时滞系统的最优控制问题进行了研究．首先基于一类随机模型误差的描述．定义了一个平均意义上的包含跟踪误差和控制能量在内的性能指标;然后通过谱分解极小化该性能指标,为一类不确定时滞系统导出了一种最优的控制器设计方法,可以兼顾模型不确定性和控制能量约束。仿真研究进一步说明了所提出方法的有效性．     

元模型误差导致的仿真过程不可重现性研究

混沌动力学系统内在的不可预测性限制了元模型技术的应用。分析了元模型重现原始模型输入/输出映射关系的基本特性,以及混沌动力学系统采用元模型技术时内在的不可重现性;为使元模型能够重现混沌动力学系统的输入/输出映射关系,提出了控制原始模型运行特征和控制元模型输出参数误差2种控制策略;最后针对逻辑斯蒂映射混沌动力学系统,研究了2种元模型构建控制策略的具体应用。     

Randomization-based control design for Mini-UAVs

The area of research on probabilistic and randomized methods for analysis and design of uncertain systems is fairly recent and is focused both on algorithmic as well as theoretical developments. In this paper a framework for randomization-based control design is presented and applied to a Mini-UAV platform. The proposed approach makes use of random search and uncertainty randomization for controller synthesis and probabilistic robustness analysis. Several structured uncertain parameters, related to the plant and to the operating conditions, are taken into account to design a robust flight control system. A selection criterion, based on estimated probability and its degradation function, is proposed in order to match stability and performance metrics fulfillment. Computational issues associated to the specific application, integration of a priori domain knowledge and human designer interaction with automated design are also addressed.     

Metamodeling by symbolic regression and Pareto simulated annealing

The subject of this paper is a new approach to symbolic regression. Other publications on symbolic regression use genetic programming. This paper describes an alternative method based on Pareto simulated annealing. Our method is based on linear regression for the estimation of constants. Interval arithmetic is applied to ensure the consistency of a model. To prevent overfitting, we merit a model not only on predictions in the data points, but also on the complexity of a model. For the complexity, we introduce a new measure. We compare our new method with the Kriging metamodel and against a symbolic regression metamodel based on genetic programming. We conclude that Pareto-simulated-annealing-based symbolic regression is very competitive compared to the other metamodel approaches.     

频域结构不确定性模型有效性分析——正切结构Nevanlinna-Pick 插值法

利用正切结构Ｎｅｖａｎｌｉｎｎａ－Ｐｉｃｋ插值理论,研究了扰动集有结构的具有线性分式传递函数的模型有效性分析问题。模型有效性与参数辨识相结合晃本领域目前的研究热点问题,故这里讨论的模型集的不确定性不仅仍扰动集的不稳定,还有不易观测的部分名义模型的不确定。我们将这类模型集的有效性分析问题转化为双线性矩阵不等式（ＢＭＩ）求解问题,构造了双迭代算法进行求解,并给出了算法的理论分析,提到了在有限步迭代后可     

Robust optimization of constrained mechanical system with joint clearance and random parameters using multi-objective particle swarm optimization

Joint clearance and uncertainty are inevitable in mechanical systems due to design tolerance, abrasion, manufacture error, assembly error and imperfections. In this study, kinematic analysis and robust optimization of constrained mechanical systems with joint clearance and random parameters were performed. Joint clearance was modeled by Lankarani-Nikravesh contact force model, and probability space was applied for characterizing uncertain parameters. A kinematic analysis method based on Baumgarte approach and confidence region method was presented to predict kinematic error of the mechanical system. Slider-crank mechanism, an illustrative example was presented to show the influence of clearance and uncertainty on the kinematic accuracy. Then, a novel multi-objective robust optimization methodology was presented for kinematic accuracy robust optimization design of the constrained mechanical system. In this approach, a multi-objective robust optimization model derived from 95% confidence region is constructed to reduce the effects of clearance and parameter uncertainty on 95% confidence region of kinematic error. The robust optimization model is a double-loop process. A multi-objective robust optimization strategy, combing Kriging surrogate model, multi-objective particle swarm optimization, confidence region and Monte Carlo methods, was proposed to search the design variables for minimizing the optimization objectives derived from confidence region while balancing computational accuracy and efficiency of the optimization process. The optimal results of the slider-crank mechanism demonstrated the validity and feasibility of the proposed robust optimization method.     

A method for selecting surrogate models in crashworthiness optimization

Surrogate model or response surface based design optimization has been widely adopted as a common process in automotive industry, as large-scale, high fidelity models are often required. However, most surrogate models are built by using a limited number of design points without considering data uncertainty. In addition, the selection of surrogate model in the literature is often arbitrary. This paper presents a Bayesian metric to complement root mean square error for selecting the best surrogate model among several candidates in a library under data uncertainty. A strategy for automatically selecting the best surrogate model and determining a reasonable sample size was proposed for design optimization of large-scale complex problems. Lastly, a vehicle example with full-frontal and offset-frontal impacts was presented to demonstrate the proposed methodology.     

A two-stage adaptive multi-fidelity surrogate model-assisted multi-objective genetic algorithm for computationally expensive problems

Surrogate model-assisted multi-objective genetic algorithms (MOGA) show great potential in solving engineering design problems since they can save computational cost by reducing the calls of expensive simulations. In this paper, a two-stage adaptive multi-fidelity surrogate (MFS) model-assisted MOGA (AMFS-MOGA) is developed to further relieve their computational burden. In the warm-up stage, a preliminary Pareto frontier is obtained relying only on the data from the low-fidelity (LF) model. In the second stage, an initial MFS model is constructed based on the data from both LF and high-fidelity (HF) models at the samples, which are selected from the preliminary Pareto set according to the crowding distance in the objective space. Then the fitness values of individuals are evaluated using the MFS model, which is adaptively updated according to two developed strategies, an individual-based updating strategy and a generation-based updating strategy. The former considers the prediction uncertainty from the MFS model, while the latter takes the discrete degree of the population into consideration. The effectiveness and merits of the proposed AMFS-MOGA approach are illustrated using three benchmark tests and the design optimization of a stiffened cylindrical shell. The comparisons between the proposed AMFS-MOGA approach and some existing approaches considering the quality of the obtained Pareto frontiers and computational efficiency are made. The results show that the proposed AMFS-MOGA method can obtain Pareto frontiers comparable to that obtained by the MOGA with HF model, while significantly reducing the number of evaluations of the expensive HF model.