Sequential sampling designs based on space reduction

In the field of engineering design and optimization, metamodels are widely used to replace expensive simulation models in order to reduce computing costs. To improve the accuracy of metamodels effectively and efficiently, sequential sampling designs have been developed. In this article, a sequential sampling design using the Monte Carlo method and space reduction strategy (MCSR) is implemented and discussed in detail. The space reduction strategy not only maintains good sampling properties but also improves the efficiency of the sampling process. Furthermore, a local boundary search (LBS) algorithm is proposed to efficiently improve the performance of MCSR, which is called LBS-MCSR. Comparative results with several sequential sampling approaches from low to high dimensions indicate that the space reduction strategy generates samples with better sampling properties (and thus better metamodel accuracy) in less computing time.     

Methodology for metamodelling of microstructure evolution: precipitation kinetic case study

Numerical modeling is an important tool assisting in the designing and optimization of the production technology. The highest predictive capabilities are offered by multiscale modeling. The most important limitation of its wide application is computational cost. One of possible solutions is application of metamodels for fine scale modeling. In this paper, a systematic approach to development of metamodels is presented. All necessary steps, analyzing the model, selecting the metamodel inputs and outputs, gathering the training and testing datasets, choosing a metamodelling technique, training and testing the metamodel are described with a scientific background and practical examples. Development of the exemplary metamodel, replacing thermodynamic modeling of precipitation kinetic is presented.     

Metamodel: A key to intelligent CAD systems

We introduce the metamodel as a new modeling framework for design objects based on General Design Theory, a mathematical model of design. Using General Design Theory, the metamodel concept can serve three functions: (1) as a central modeling mechanism to integrate models, (2) as a mechanism for modeling physical phenomena, and (3) as a tool for describing evolving design objects. Modeling with multiple points of view is realized by representing physical phenomena that occur in the deisng object and by constructing models with knowledge of physics and design from the metamodel. We illustrate the first and second functions of metamodels with an example based on naive physics, and we illustrate the third function of the metamodel through design experiments. Finally, we present two systems to illustrate how the metamodel mechanism can be implemented.     

A new hybrid reliability-based design optimization method under random and interval uncertainties

This article proposes a new method for hybrid reliability-based design optimization under random and interval uncertainties (HRBDO-RI). In this method, Monte Carlo simulation (MCS) is employed to estimate the upper bound of failure probability, and stochastic sensitivity analysis (SSA) is extended to calculate the sensitivity information of failure probability in HRBDO-RI. Due to a large number of samples involved in MCS and SSA, Kriging metamodels are constructed to substitute true constraints. To avoid unnecessary computational cost on Kriging metamodel construction, a new screening criterion based on the coefficient of variation of failure probability is developed to judge active constraints in HRBDO-RI. Then a projection-outline-based active learning Kriging is achieved by sequentially select update points around the projection outlines on the limit-state surfaces of active constraints. Furthermore, the prediction uncertainty of Kriging metamodel is quantified and considered in the termination of Kriging update. Several examples, including a piezoelectric energy harvester design, are presented to test the accuracy and efficiency of the proposed method for HRBDO-RI.     

Hybrid metamodel-based design space management method for expensive problems

Metamodel-based global optimization methods have been extensively studied for their great potential in solving expensive problems. In this work, a design space management strategy is proposed to improve the accuracy and efficiency of metamodel-based optimization methods. In this strategy, the whole design space is divided into two parts: the important region constructed using several expensive points and the other region. Combined with a previously developed hybrid metamodel strategy, a hybrid metamodel-based design space management method (HMDSM) is developed. In this method, three representative metamodels are used simultaneously in the search of the global optimum in both the important region and the other region. In the search process, the important region is iteratively reduced and the global optimum is soon captured. Tests using a series of benchmark mathematical functions and a practical expensive problem demonstrate the excellent performance of the proposed method.     

Simulation metamodelling with neural networks: An experimental investigation

Artificial neural networks are often proposed as an alternative approach for formalizing various quantitative and qualitative aspects of complex systems. This paper examines the robustness of using neural networks as a simulation metamodel to estimate manufacturing system performances. Simulation models of a job shop system are developed for various configurations to train neural network metamodels. Extensive computational tests are carried out with the proposed models at various factor levels (study horizon, system load, initial system status, stochasticity, system size and error assessment methods) to see the metamodel accuracy. The results indicate that simulation metamodels with neural networks can be effectively used to estimate the system performances.     

Enhanced similarity-based metamodel updating strategy for reliability-based design optimization

Metamodels are becoming increasingly popular for handling large-scale optimization problems in product development. Metamodel-based reliability-based design optimization (RBDO) helps to improve the computational efficiency and reliability of optimal design. However, a metamodel in engineering applications is an approximation of a high-fidelity computer-aided engineering model and it frequently suffers from a significant loss of predictive accuracy. This issue must be appropriately addressed before the metamodels are ready to be applied in RBDO. In this article, an enhanced strategy with metamodel selection and bias correction is proposed to improve the predictive capability of metamodels. A similarity-based assessment for metamodel selection (SAMS) is derived from the cross-validation and similarity theories. The selected metamodel is then improved by Bayesian inference-based bias correction. The proposed strategy is illustrated through an analytical example and further demonstrated with a lightweight vehicle design problem. The results show its potential in handling real-world engineering problems.     

A pseudo-random number assignment strategy for the estimation of quadratic metamodels

Whereas an optimal Pseudo-Random Number (PRN) assignment strategy for simulation experiments involving the estimation of linear metamodels currently exists, no such optimal assignment strategy for quadratic metamodels has been proposed. This situation is now rectified by the introduction of a PRN assignment strategy for a quadratic metamodel for 3^k factorial designs. In addition to extending the theory from linear to quadratic metamodels, the proposed PRN strategy is shown to be superior to a number of existing and competing strategies in terms of various variance measures.     

基于Stochastic Kriging模型的不确定性序贯试验设计方法

不确定性研究中需要计算大量重复样本,这无疑对计算量较大的数值模拟提出了巨大的挑战.通过试验设计方法可以有效地减少不确定性研究中的计算量,然而,目前考虑不确定性的试验设计方法研究大多仍专注于传统试验设计方法.针对这一问题,为了通过更为合理的计算资源分配得到更精准的不确定性评估,基于有限样本的Stochastic Kriging模型提出了针对不确定性问题的三阶段序贯试验设计方法.首先,通过特定位置的采样对IMSE进行简化,构建了预选步进信息选取策略,通过预选增量样本总个数以及各取样位置处的分布信息,达到随机代理模型目标精度要求;同时,基于IMSE构建了基于步进信息的单轮选点试验设计准则,以同时考虑设计变量的取样位置及其分布信息.由算例与传统方法的对比分析可知,所建立方法通过等量的采样得到了精度更高的随机代理模型,验证了其在不确定性问题中的可行性和优势.     

Generation of a cokriging metamodel using a multiparametric strategy

In the course of designing structural assemblies, performing a full optimization is very expensive in terms of computation time. In order or reduce this cost, we propose a multilevel model optimization approach. This paper lays the foundations of this strategy by presenting a method for constructing an approximation of an objective function. This approach consists in coupling a multiparametric mechanical strategy based on the LATIN method with a gradient-based metamodel called a cokriging metamodel. The main difficulty is to build an accurate approximation while keeping the computation cost low. Following an introduction to multiparametric and cokriging strategies, the performance of kriging and cokriging models is studied using one- and two-dimensional analytical functions; then, the performance of metamodels built from mechanical responses provided by the multiparametric strategy is analyzed based on two mechanical test examples.     

An adaptive sampling method for variable-fidelity surrogate models using improved hierarchical kriging

Variable-fidelity (VF) modelling methods have been widely used in complex engineering system design to mitigate the computational burden. Building a VF model generally includes two parts: design of experiments and metamodel construction. In this article, an adaptive sampling method based on improved hierarchical kriging (ASM-IHK) is proposed to refine the improved VF model. First, an improved hierarchical kriging model is developed as the metamodel, in which the low-fidelity model is varied through a polynomial response surface function to capture the characteristics of a high-fidelity model. Secondly, to reduce local approximation errors, an active learning strategy based on a sequential sampling method is introduced to make full use of the already required information on the current sampling points and to guide the sampling process of the high-fidelity model. Finally, two numerical examples and the modelling of the aerodynamic coefficient for an aircraft are provided to demonstrate the approximation capability of the proposed approach, as well as three other metamodelling methods and two sequential sampling methods. The results show that ASM-IHK provides a more accurate metamodel at the same simulation cost, which is very important in metamodel-based engineering design problems.     

Design of manufacturing systems by a hybrid approach with neural network metamodelling and stochastic local search

Metamodels are models of simulation models. Metamodels are able to estimate the simulation responses corresponding to a given combination of input variables. A simulation metamodel is easier to manage and provides more insights than simulation alone. Traditionally, the multiple regression analysis is utilized to develop the metamodel from a set of simulation experiments. Simulation can consequentially benefit from the metamodelling in post-simulation analysis. A backpropagation (BP) neural network is a proven tool in providing excellent response predictions in many application areas and it outperforms regression analysis for a wide array of applications. In this paper, a BP neural network is used to generate metamodels for simulated manufacturing systems. For the purpose of optimal manufacturing systems design, mathematical models can be formulated by using the mapping functions generated from the neural network metamodels. The optimization model is then solved by a stochastic local search approach, simulated annealing (SA), to obtain an optimal configuration with respect to the objective of the systems design. Instead of triggering the detailed simulation programs, the SA-based optimization procedure evaluates the simulation outputs by the neural network metamodels. By using the SA-based optimization algorithm, the solution space of the studied problem is extensively exploited to escape the entrapment of local optima while the number of time consuming simulation runs is reduced. The proposed methodology is illustrated to be both effective and efficient in solving a manufacturing systems design problem through an example.     

Analyzing transformation-based simulation metamodels

To analyze a simulation (response surface) metamodel that involves a variance-stabilizing transformation of the original simulation-generated response, we present two techniques. In the first technique we compute an approximate percentile-type confidence interval for the mean of the original response at a selected factor-level combination (design point) as follows: we compute the usual confidence interval for the mean of the transformed response at that design point; and then we untransform the corresponding endpoints to obtain the desired confidence interval for the untransformed metamodel. In the second technique we compute the Maximum Likelihood Estimator (MLE) for the mean of the untransformed response based on standard distributional properties of the transformed metamodel; then using the delta method to approximate the MLE's variance, we construct for the untransformed metamodel an asymptotically exact confidence interval centered on the MLE. We illustrate these techniques in a case study on manufacturing cell design, comparing them with a more conventional approach for analyzing transformed-based simulation metamodels. A Monte Carlo performance evaluation shows that significantly better confidence-interval coverage is maintained with the second proposed technique (called the "MLE-delta method") over a wide range of values for the residual variance of the transformed metamodel.     

A comparative study of the scalability of alternative metamodelling techniques

Metamodels, also known as surrogate models, can be used in place of computationally expensive simulation models to increase computational efficiency for the purposes of design optimization or design space exploration. The accuracy of these metamodels varies with the scale and complexity of the underlying model. In this article, three metamodelling methods are evaluated with respect to their capabilities for modelling high-dimensional, nonlinear, multimodal functions. Methods analyzed include kriging, radial basis functions, and support vector regression. Each metamodelling technique is used to model a set of single output functions with dimensionality ranging from fifteen to fifty independent variables and modality ranging from one to ten local maxima. The number of points used to train the models is increased until a predetermined error threshold is met. Results show that kriging metamodels perform most consistently across a variety of functions, although radial basis functions and support vector regression are very competitive for highly multimodal functions and functions with large local gradients, respectively. Support vector regression metamodels consistently offer the shortest build and prediction times when applied to large scale multimodal problems.     

Estimation of input parameters in complex simulation using a Gaussian process metamodel

The unknown input parameters of a simulation code are usually adjusted by the nonlinear least squares estimation (NLSE) method which minimizes the sum of differences between computer responses and real observations. However, when a simulation program is very complex and takes several hours for one execution, the NLSE method may not be computationally feasible. In this case, one may build a statistical metamodel which approximates the complex simulation code. Then this metamodel is used as if it is the true simulation code in the NLSE method, which makes the problem computationally feasible. This ‘approximated’ NLSE method is described in this article. A Gaussian process model is used as a metamodel of complex simulation code. The proposed method is validated through a toy-model study where the true parameters are known a priori. An application to nuclear fusion device is presented.     

Implementation of Discrete Capability into the Enhanced Multipoint Approximation Method for Solving Mixed Integer-Continuous Optimization Problems

Multipoint approximation method (MAM) focuses on the development of metamodels for the objective and constraint functions in solving a mid-range optimization problem within a trust region. To develop an optimization technique applicable to mixed integer-continuous design optimization problems in which the objective and constraint functions are computationally expensive and could be impossible to evaluate at some combinations of design variables, a simple and efficient algorithm, coordinate search, is implemented in the MAM. This discrete optimization capability is examined by the well established benchmark problem and its effectiveness is also evaluated as the discreteness interval for discrete design variables is increased from 0.2 to 1. Furthermore, an application to the optimization of a lattice composite fuselage structure where one of design variables (number of helical ribs) is integer is also presented to demonstrate the efficiency of this capability.     

Optimizing Latin hypercube design for sequential sampling of computer experiments

Space-filling and projective properties are desired features in the design of computer experiments to create global metamodels to replace expensive computer simulations in engineering design. The goal in this article is to develop an efficient and effective sequential Quasi-LHD (Latin Hypercube design) sampling method to maintain and balance the two aforementioned properties. The sequential sampling is formulated as an optimization problem, with the objective being the Maximin Distance, a space-filling criterion, and the constraints based on a set of pre-specified minimum one-dimensional distances to achieve the approximate one-dimensional projective property. Through comparative studies on sampling property and metamodel accuracy, the new approach is shown to outperform other sequential sampling methods for global metamodelling and is comparable to the one-stage sampling method while providing more flexibility in a sequential metamodelling procedure.     

Robust optimization based on analytical evaluation of uncertainty propagation

Optimization under uncertainty requires proper handling of those input parameters that contain scatter. Scatter in input parameters propagates through the process and causes scatter in the output. Stochastic methods (e.g. Monte Carlo) are very popular for assessing uncertainty propagation using black-box function metamodels. However, they are expensive. Therefore, in this article a direct method of calculating uncertainty propagation has been employed based on the analytical integration of a metamodel of a process. Analytical handling of noise variables not only improves the accuracy of the results but also provides the gradients of the output with respect to input variables. This is advantageous in the case of gradient-based optimization. Additionally, it is shown that the analytical approach can be applied during sequential improvement of the metamodel to obtain a more accurate representative model of the black-box function and to enhance the search for the robust optimum.     

Metamodel-based inverse method for parameter identification: elastic–plastic damage model

This article proposed a metamodel-based inverse method for material parameter identification and applies it to elastic–plastic damage model parameter identification. An elastic–plastic damage model is presented and implemented in numerical simulation. The metamodel-based inverse method is proposed in order to overcome the disadvantage in computational cost of the inverse method. In the metamodel-based inverse method, a Kriging metamodel is constructed based on the experimental design in order to model the relationship between material parameters and the objective function values in the inverse problem, and then the optimization procedure is executed by the use of a metamodel. The applications of the presented material model and proposed parameter identification method in the standard A 2017-T4 tensile test prove that the presented elastic–plastic damage model is adequate to describe the material's mechanical behaviour and that the proposed metamodel-based inverse method not only enhances the efficiency of parameter identification but also gives reliable results.     

Reliability-based design optimization using a moment method and a kriging metamodel

Reliability-based design optimization (RBDO) has been used for optimizing engineering systems with uncertainties in design variables and system parameters. RBDO involves reliability analysis, which requires a large amount of computational effort, so it is important to select an efficient method for reliability analysis. Of the many methods for reliability analysis, a moment method, which is called the fourth moment method, is known to be less expensive for moderate size problems and requires neither iteration nor the computation of derivatives. Despite these advantages, previous research on RBDO has been mainly based on the first-order reliability method and relatively little attention has been paid to moment-based RBDO. This article considers difficulties in implementing the moment method into RBDO; they are solved using a kriging metamodel with an active constraint strategy. Three numerical examples are tested and the results show that the proposed method is efficient and accurate.