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.     

Adaptive accuracy trust region: using cross-validation in the optimization process

This article presents an adaptive accuracy trust region (AATR) optimization strategy where cross-validation is used by the trust region to reduce the number of sample points needed to construct metamodels for each step of the optimization process. Lower accuracy metamodels are initially used for the larger trust regions, and higher accuracy metamodels are used for the smaller trust regions towards the end of optimization. Various metamodelling strategies are used in the AATR algorithm: optimal and inherited Latin hypercube sampling to generate experimental designs; quasi-Newton, kriging and polynomial regression metamodels to approximate the objective function; and the leave-k-out method for validation. The algorithm is tested with two-dimensional single-discipline problems. Results show that the AATR algorithm is a promising method when compared to a traditional trust region method. Polynomial regression in conjunction with a new hybrid inherited-optimal Latin hypercube sampling performed the best.     

Efficient infill sampling for unconstrained robust optimization problems

A novel infill sampling criterion is proposed for efficient estimation of the global robust optimum of expensive computer simulation based problems. The algorithm is especially geared towards addressing problems that are affected by uncertainties in design variables and problem parameters. The method is based on constructing metamodels using Kriging and adaptively sampling the response surface via a principle of expected improvement adapted for robust optimization. Several numerical examples and an engineering case study are used to demonstrate the ability of the algorithm to estimate the global robust optimum using a limited number of expensive function evaluations.     

Fast numerical methods for robust optimal design

A fast numerical approach for robust design optimization is presented. The core of the method is based on the state-of-the-art fast numerical methods for stochastic computations with parametric uncertainty. These methods employ generalized polynomial chaos (gPC) as a high-order representation for random quantities and a stochastic Galerkin (SG) or stochastic collocation (SC) approach to transform the original stochastic governing equations to a set of deterministic equations. The gPC-based SG and SC algorithms are able to produce highly accurate stochastic solutions with (much) reduced computational cost. It is demonstrated that they can serve as efficient forward problem solvers in robust design problems. Possible alternative definitions for robustness are also discussed. Traditional robust optimization seeks to minimize the variance (or standard deviation) of the response function while optimizing its mean. It can be shown that although variance can be used as a measure of uncertainty, it is a weak measure and may not fully reflect the output variability. Subsequently a strong measure in terms of the sensitivity derivatives of the response function is proposed as an alternative robust optimization definition. Numerical examples are provided to demonstrate the efficiency of the gPC-based algorithms, in both the traditional weak measure and the newly proposed strong measure.     

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.     

Metamodels for mixed variables based on moving least squares

Surrogate-based optimization has become a major field in engineering design, due to its capacity to handle complex systems involving expensive simulations. However, the majority of general-purpose surrogates (also called metamodels) are restricted to continuous variables, although versatile problems involve additional types of variables (discrete, integer, and even categorical to model technological options). Therefore, the main contribution of this paper consists in the development of metamodels specifically dedicated to handle mixed variables, in particular continuous and unordered categorical variables, and their comparison with state-of-the-art approaches. This task is performed in three steps: (i) considering an appropriate parametrization (integer mapping, regular simplex, dummy, effect codings) for the mixed variable design vector; (ii) defining metrics to compare pairs of design vectors; (iii) carrying out an ordinary or moving least square regression scheme based on the parametrization and metric previously defined. The proposed metamodels have been tested on six analytical benchmark test cases, and applied to the structural finite element analysis model of a rigid frame characterized by continuous and categorical variables. In particular, it is demonstrated that using a standard regular simplex representation for the nominal categorical variables usually outperforms a direct conversion of the nominal parameters to integer values, while offering an efficient and systematic way to encompass all types of variables in a common framework. It is also shown that the choice of a given variable representation has a higher impact on the results than the selected scheme (ordinary or moving least squares), or than the metric used for calculating distances between samples.     

Reliability-based design optimization of a car body using dimension- reduced Chebyshev polynomial

A reliability-based design optimization (RBDO) method of a car body is presented on basis of dimension-reduced Chebyshev polynomial method (DCM). To improve calculation efficiency and save computational time, complex models are often approximated by metamodels in reliability analysis. Traditional metamodels require a large number of sample points, which is time-consuming. To improve the efficiency, DCM is proposed to approximate the performance function of the car body. First, the performance function is decomposed by the dimension-reduction method into a sum of univariate functions, which are then fitted through Chebyshev polynomials. The reliability of the car body is predicted by the Taylor expansion method and the fourth-moment method. Finally, the result of RBDO is obtained using an improved adaptive genetic algorithm. The proposed method saves on the calculation time with high precision. Besides, the improved adaptive genetic algorithm reduces the number of iterations in the car body optimization and improves the efficiency.     

约束阻尼结构的双向渐进拓扑优化

针对约束层阻尼板的拓扑优化问题,以模态损耗因子最大化为目标函数,约束阻尼材料体积分数为约束条件,建立了约束阻尼板的拓扑优化模型。基于模态应变能方法,推导了目标函数对设计变量的灵敏度。采用双向渐进优化算法(BESO)对约束阻尼材料的布局进行了拓扑优化,获得了约束阻尼材料的最优拓扑构型,并与渐进优化算法(ESO)进行了比较。研究结果表明：双向渐进优化算法相比单向渐进优化算法,获得的模态损耗因子更高,阻尼减振效果更好。     

The comparison of multi-objective particle swarm optimization and NSGA II algorithm: applications in centrifugal pumps

In the present study, multi-objective optimization of centrifugal pumps is performed in three steps. In the first step, efficiency (η) and the required net positive suction head (NPSHr) in a set of centrifugal pumps are numerically investigated using commercial software. Two meta-models based on the evolved group method of data handling (GMDH) type neural networks are obtained in the second step for modeling of η and NPSHr with respect to geometrical design variables. Finally, using the obtained polynomial neural networks, a multi-objective particle swarm optimization method (MOPSO) is used for Pareto-based optimization of centrifugal pumps considering two conflicting objectives, η and NPSHr. The Pareto results of the MOPSO method are also compared with those of a multi-objective genetic algorithm (NSGA II). It is shown that some interesting and important relationships as useful optimal design principles involved in the performance of centrifugal pumps can be discovered by Pareto-based multi-objective optimization of the obtained polynomial metamodels representing η and NPSHr characteristics.     

Distributed evolutionary algorithms with hierarchical evaluation

A distributed evolutionary algorithm is presented that is based on a hierarchy of (fitness or cost function) evaluation passes within each deme and is efficient in solving engineering optimization problems. Starting with non-problem-specific evaluations (using surrogate models or metamodels, trained on previously evaluated individuals) and ending up with high-fidelity problem-specific evaluations, intermediate passes rely on other available lower-fidelity problem-specific evaluations with lower CPU cost per evaluation. The sequential use of evaluation models or metamodels, of different computational cost and modelling accuracy, by screening the generation members to get rid of non-promising individuals, leads to reduced overall computational cost. The distributed scheme is based on loosely coupled demes that exchange regularly their best-so-far individuals. Emphasis is put on the optimal way of coupling distributed and hierarchical search methods. The proposed method is tested on mathematical and compressor cascade airfoil design problems.     

An efficient decision-making framework for hybrid metamodelling

In engineering design optimization, the usage of hybrid metamodels (HMs) can take full advantage of the individual metamodels, and improve robustness of the predictions by reducing the impact of a poor metamodel. When there are plenty of candidates, it is difficult to make decisions on which metamodels to choose before building an HM. The decisions should simultaneously take into account of the number, accuracy and diversity of the selected metamodels. To address this problem, this research developed an efficient decision-making framework based on partial least squares for metamodel screening. A new significance index is firstly derived from the view of fitting error in a regression model. Then, a desirable metamodel combination which consist of only the significant ones is subsequently configured for further constructing the final HM. The effectiveness of the proposed framework is demonstrated through several benchmark problems.     

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.     

Model-based process optimization in the presence of parameter uncertainty

In model-based process optimization one uses a mathematical model to optimize a certain criterion, for example the product yield of a chemical process. Models often contain parameters that have to be estimated from data. Typically, a point estimate (e.g. the least squares estimate) is used to fix the model for the optimization stage. However, parameter estimates are uncertain due to incomplete and noisy data. In this article, it is shown how parameter uncertainty can be taken into account in process optimization. To quantify the uncertainty, Markov Chain Monte Carlo (MCMC) sampling, an emerging standard approach in Bayesian estimation, is used. In the Bayesian approach, the solution to the parameter estimation problem is given as a distribution, and the optimization criteria are functions of that distribution. The formulation and implementation of the optimization is studied, and numerical examples are used to show that parameter uncertainty can have a large effect in optimization results.     

Multiobjective crashworthiness optimization design of functionally graded foam-filled tapered tube based on dynamic ensemble metamodel

Foam-filled thin-walled structures have recently gained attention with increasing interest due to their excellent energy absorption capacity. In this study, a new type of foam-filled thin-walled structure called as functionally graded foam-filled tapered tube (FGFTT) is proposed. FGFTT consists of graded density foam and thin-walled tapered tube. In order to investigate the energy absorption characteristics of FGFTTs, the numerical simulations for two kinds of FGFTTs subjected to axial dynamical loading are carried out by nonlinear finite element code LS-DYNA. In addition, a new kind of multiobjective crashworthiness optimization method employing the dynamic ensemble metamodeling method together with the multiobjective particle swarm optimization (MOPSO) algorithm is presented. This new kind of multiobjective crashworthiness optimization method is then used to implement the crashworthiness optimization design of FGFTTs. Meanwhile, the crashworthiness optimization designs of FGFTTs are implemented by using traditional multiobjective crashworthiness optimization method, which employs metamodels such as polynomial response surface (PRS), radial basis function (RBF), kriging (KRG), support vector regression (SVR) or the ensemble with the static design of experiment (DOE). Finally, by comparing the optimal designs of FGFTTs obtained by using the new multiobjective crashworthiness optimization method and the traditional one, the results show that the proposed new crashworthiness optimization method is more feasible.     

CO-EVOLUTIONARY HYBRID DIFFERENTIAL EVOLUTION FOR MIXED-INTEGER OPTIMIZATION PROBLEMS

Evolutionary algorithms are promising candidates for obtaining the global optimum. Hybrid differential evolution is one or the evolutionary algorithms, which has been successfully applied to many real-world nonlinear programming problems. This paper proposes a co-evolutionary hybrid differential evolution to solve mixed-integer nonlinear programming (MINLP) problems. The key ingredients of the algorithm consist of an integer-valued variable evolution and a real-valued variable co-evolution, so that the algorithm can be used to solve MINLP problems or pure integer programming problems. Furthermore, the algorithm combines a local search heuristic (called acceleration) and a widespread search heuristic (called migration) to promote the search for a global optimum. Some numerical examples are tested to illustrate the performance of the proposed algorithm. Numerical examples show that the proposed algorithm converges to better solutions than the conventional MINLP optimization methods     

Multi-objective aerodynamic shape optimization of small livestock trailers

This article presents a formal optimization study of the design of small livestock trailers, within which the majority of animals are transported to market in the UK. The benefits of employing a headboard fairing to reduce aerodynamic drag without compromising the ventilation of the animals’ microclimate are investigated using a multi-stage process involving computational fluid dynamics (CFD), optimal Latin hypercube (OLH) design of experiments (DoE) and moving least squares (MLS) metamodels. Fairings are parameterized in terms of three design variables and CFD solutions are obtained at 50 permutations of design variables. Both global and local search methods are employed to locate the global minimum from metamodels of the objective functions and a Pareto front is generated. The importance of carefully selecting an objective function is demonstrated and optimal fairing designs, offering drag reductions in excess of 5% without compromising animal ventilation, are presented.     

A comparative study of expected improvement-assisted global optimization with different surrogates

Efficient global optimization (EGO) uses the surrogate uncertainty estimator called expected improvement (EI) to guide the selection of the next sampling candidates. Theoretically, any modelling methods can be integrated with the EI criterion. To improve the convergence ratio, a multi-surrogate efficient global optimization (MSEGO) was suggested. In practice, the EI-based optimization methods with different surrogates show widely divergent characteristics. Therefore, it is important to choose the most suitable algorithm for a certain problem. For this purpose, four single-surrogate efficient global optimizations (SSEGOs) and an MSEGO involving four surrogates are investigated. According to numerical tests, both the SSEGOs and the MSEGO are feasible for weak nonlinear problems. However, they are not robust for strong nonlinear problems, especially for multimodal and high-dimensional problems. Moreover, to investigate the feasibility of EGO in practice, a material identification benchmark is designed to demonstrate the performance of EGO methods. According to the tests in this study, the kriging EGO is generally the most robust method.     

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.     

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.     

A hybrid cellular automaton–bi-directional evolutionary optimization algorithm for topological optimization of crashworthiness

This article proposes a new algorithm for topological optimization under dynamic loading which combines cellular automata with bi-directional evolutionary structural optimization (BESO). The local rules of cellular automata are used to update the design variables, which avoids the difficulty of obtaining gradient information under nonlinear collision conditions. The intermediate-density design problem of hybrid cellular automata is solved using the BESO concept of 0–1 binary discrete variables. Some improvement strategies are also proposed for the hybrid algorithm to solve certain problems in nonlinear topological optimization, e.g. numerical oscillation. Some typical examples of crashworthiness problems are provided to illustrate the efficiency of the proposed method and its ability to find the final optimal solution. Finally, numerical results obtained using the proposed algorithms are compared with reference examples taken from the literature. The results show that the hybrid method is computationally efficient and stable.