Variable-fidelity Optimization Research Articles

A parallel variable-fidelity algorithm for efficient constrained multi-objective aerodynamic design optimization

Modern aerodynamic design optimization aims to discover optimal configurations using computational fluid dynamics under complex flow conditions, which is a typical expensive multi-objective optimization problem. The multi-objective evolutionary algorithm based on decomposition (MOEA/D) combined with efficient global optimization is a promising method but requires enhanced efficiency and faces limitations in its application to multi-objective aerodynamic design optimization (MOADO). To address the issues, an efficient parallel MOEA/D assisted with variable-fidelity optimization (VFO) is proposed for solving MOADO, called the MOEA/D-VFO algorithm. Variable-fidelity surrogates are built for objectives and constraints, achieving higher accuracy using fewer high-fidelity samples and a great number of low-fidelity samples. By retaining more good candidates, the sub-optimization problems defined by decomposing original objectives are capable of discovering more favorable samples using MOEA/D, which prompts optimization convergence. A constraint-handling strategy is developed by incorporating the probability of feasibility functions in the sub-optimizations. The selection of new samples for parallel evaluation is improved by filtering out poor candidates and selecting effective promising samples, which improves the feasibility and diversity of solved Pareto solutions. A Pareto front (PF) can be efficiently found in a single optimization run. The proposed approach is demonstrated by four analytical test functions and verified by two aerodynamic design optimizations of airfoils with and without constraints, respectively. The results indicate that the MOEA/D-VFO approach can greatly improve optimization efficiency and obtain the PF satisfying constraints within an affordable computational budget.

An enhanced variable-fidelity optimization approach for constrained optimization problems and its parallelization

In this paper, a variable-fidelity constrained lower confidence bound (VF-CLCB) criterion is presented for computationally expensive constrained optimization problems (COPs) with two levels of fidelity. In VF-CLCB, the hierarchical Kriging model is adopted to model the objective and inequality constraints. Two infill sampling functions are developed based on the objective and the constraints, respectively, and an adaptive selection strategy is set to select the elite sample points. Moreover, based on the VF-CLCB criterion, a parallel optimization method noted as PVF-CLCB is subsequently developed to accelerate the optimization process. In PVF-CLCB, a VF influence function is defined to approximately evaluate the estimation error of the hierarchical Kriging models, based on which multiple promising points can be determined at each iteration. In addition, an allocation strategy is proposed to distribute the computation resources between the objective- and constraint-oriented functions properly. Lastly, the proposed VF-CLCB and PVF-CLCB approaches are compared with the alternative methods on 12 benchmark numerical cases, and their significant superiority in solving computationally expensive COPs is verified. Furthermore, the proposed methods are employed to optimize the global stability of the stiffened cylindrical shell, and the optimum structure is yielded.

A modified trust-region assisted variable-fidelity optimization framework for computationally expensive problems

PurposeVariable-fidelity optimization (VFO) frameworks generally aim at taking full advantage of high-fidelity (HF) and low-fidelity (LF) models to solve computationally expensive problems. The purpose of this paper is to develop a novel modified trust-region assisted variable-fidelity optimization (MTR-VFO) framework that can improve the optimization efficiency for computationally expensive engineering design problems.Design/methodology/approachThough the LF model is rough and inaccurate, it probably contains the gradient information and trend of the computationally expensive HF model. In the proposed framework, the extreme locations of the LF kriging model are firstly utilized to enhance the HF kriging model, and then a modified trust-region (MTR) method is presented for efficient local search. The proposed MTR-VFO framework is verified through comparison with three typical methods on some benchmark problems, and it is also applied to optimize the configuration of underwater tandem wings.FindingsThe results indicate that the proposed MTR-VFO framework is more effective than some existing typical methods and it has the potential of solving computationally expensive problems more efficiently.Originality/valueThe extreme locations of LF models are utilized to improve the accuracy of HF models and a MTR method is first proposed for local search without utilizing HF gradient. Besides, a novel MTR-VFO framework is presented which is verified to be more effective than some existing typical methods and shows great potential of solving computationally expensive problems effectively.

Optimization design of antenna cover of deeply submerged body based onparallel variable-fidelity LCB algorithm

Objectives In order to reduce the weight while maintaining the performance of the antenna cover of a deeply submerged body, a parallel variable-fidelity lower confidence bound (PVF-LCB) approach is proposed to optimize the structure of the antenna cover. Methods The proposed algorithm adaptively allocates computational resources of different fidelities through a variable-fidelity LCB (VF-LCB) function, allowing it to select several candidate samples based on influence functions (IFs) constructed by the variable-fidelity Kriging model. Moreover, the proposed method is assisted with widely used constraint-handling methods to solve the structural optimization problem. Results The proposed approach obtains an optimized antenna cover structure which satisfies all constraints. Compared with the well-known variable-fidelity optimization method, the optimized structure is about 50% lower in weight. Additionally, the proposed approach reduces the weight of the antenna cover by about 30% compared with the results of single-fidelity parallel optimization methods. Conclusions The proposed method can not only reduce the design cycle of engineering optimization, but improve the quality of the optimal solution, giving it certain development prospects and guiding significance for engineering applications.

Aerodynamic/aeroacoustic variable-fidelity optimization of helicopter rotor based on hierarchical Kriging model

Rotor noise is one of the most important reasons for restricting helicopter development; hence, the optimization design of rotor blade considering aeroacoustic and aerodynamic performance at the same time has always been the focus of research attention. For complex rotor design problems with a large number of design variables, the efficiency of the traditional Kriging model needs to be improved. Thus, Hierarchical Kriging (HK) model is employed in this study for rotor optimization design. By using the validated RANS solver and acoustic method based on the FW–Hpds equation, an efficient aerodynamic/aeroacoustic optimization method for high-dimensional problem of rotors in hover based on HK model is developed. By using present HK model and new infill-sampling criteria, the number of design variables is increased from less than 20–53. Results of two analytical function test cases show that the HK model is efficient and accurate in calculation. Subsequently, the helicopter rotor blade is optimally designed for aerodynamic/aeroacoustic performance in hover based on the HK model with high dimensional design variables. The objective function is adopted to improve the rotational noise characteristics by reducing the absolute peak of the acoustic pressure. In addition, the constraints of thrust, hover efficiency, solidity, and airfoils thickness are strictly satisfied. Optimization results show that the Kriging model finds the objective of reducing the noise by 2.87 dB after 248 iterations while the HK model does it only after 164 iterations. The optimization efficiency of the HK model is significantly higher than that of the traditional Kriging model. In the case analyzed, the HK model saves 35% of the time used by the Kriging model.

Efficient aerodynamic shape optimization using variable-fidelity surrogate models and multilevel computational grids

A variable-fidelity method can remarkably improve the efficiency of a design optimization based on a high-fidelity and expensive numerical simulation, with assistance of lower-fidelity and cheaper simulation(s). However, most existing works only incorporate “two” levels of fidelity, and thus efficiency improvement is very limited. In order to reduce the number of high-fidelity simulations as many as possible, there is a strong need to extend it to three or more fidelities. This article proposes a novel variable-fidelity optimization approach with application to aerodynamic design. Its key ingredient is the theory and algorithm of a Multi-level Hierarchical Kriging (MHK), which is referred to as a surrogate model that can incorporate simulation data with arbitrary levels of fidelity. The high-fidelity model is defined as a CFD simulation using a fine grid and the lower-fidelity models are defined as the same CFD model but with coarser grids, which are determined through a grid convergence study. First, sampling shapes are selected for each level of fidelity via technique of Design of Experiments (DoE). Then, CFD simulations are conducted and the output data of varying fidelity is used to build initial MHK models for objective (e.g. CD) and constraint (e.g. CL, Cm) functions. Next, new samples are selected through infill-sampling criteria and the surrogate models are repetitively updated until a global optimum is found. The proposed method is validated by analytical test cases and applied to aerodynamic shape optimization of a NACA0012 airfoil and an ONERA M6 wing in transonic flows. The results confirm that the proposed method can significantly improve the optimization efficiency and apparently outperforms the existing single-fidelity or two-level-fidelity method.

An online variable-fidelity optimization approach for multi-objective design optimization

Multi-objective genetic algorithms (MOGAs) are effective ways for obtaining Pareto solutions of multi-objective design optimization problems. However, the high computational cost of MOGAs limits their applications to practical engineering optimization problems involving computational expensive simulations. To address this issue, a novel variable-fidelity (VF) optimization approach for multi-objective design optimization is proposed, in which a VF metamodel is embedded in the computation process of MOGA to replace the expensive simulation model. The VF metamodel is updated in the optimization process of MOGA, considering the cost of simulation models with different fidelity and the influence of the VF metamodel uncertainty. A normalized distance constraint is introduced to avoid selecting clustered sample points. Four numerical examples and two engineering cases are used to demonstrate the applicability and efficiency of the proposed approach. The results show that the proposed approach can obtain Pareto solutions with good quality and outperforms the other four approaches considered here as references in terms of computational efficiency.

Pumping Optimization of Coastal Aquifers Using Seawater Intrusion Models of Variable-Fidelity and Evolutionary Algorithms

Variable-fidelity modelling has been utilized in several engineering optimization studies to construct surrogate models. However, similar approaches have received much less attention in coastal aquifer management problems. A variable-fidelity optimization framework was developed utilizing a lower-fidelity and computationally cheap model of seawater intrusion, based on the sharp interface assumption, and a simple correction process. The variable-fidelity method was compared to the direct optimization with the high-fidelity variable density and salt transport model and to conventional surrogate-based optimization. The surrogate-based approaches were embedded into the operations of an evolutionary algorithm to implement an efficient online update of the surrogate models and control the feasibility of the optimal solutions. Multiple independent optimization runs were performed to provide more insightful comparison outcomes. Although the variable-fidelity method found a better optimum than the conventional approach, the overall sample statistics showed that the surrogate-based optimization frameworks performed equally well and provided good approximations to the high-fidelity solution. Despite the potential for an improved exploration of the optimal search space by using the variable-fidelity method, the conventional approach had a 30% faster average convergence time.

Variable-fidelity shape optimization of dual-rotor wind turbines

PurposeDual-rotor wind turbines (DRWTs) are a novel type of wind turbines that can capture more power than their single-rotor counterparts. Because their surrounding flow fields are complex, evaluating a DRWT design requires accurate predictive simulations, which incur high computational costs. Currently, there does not exist a design optimization framework for DRWTs. Since the design optimization of DRWTs requires numerous model evaluations, the purpose of this paper is to identify computationally efficient design approaches.Design/methodology/approachSeveral algorithms are compared for the design optimization of DRWTs. The algorithms vary widely in approaches and include a direct derivative-free method, as well as three surrogate-based optimization methods, two approximation-based approaches and one variable-fidelity approach with coarse discretization low-fidelity models.FindingsThe proposed variable-fidelity method required significantly lower computational cost than the derivative-free and approximation-based methods. Large computational savings come from using the time-consuming high-fidelity simulations sparingly and performing the majority of the design space search using the fast variable-fidelity models.Originality/valueDue the complex simulations and the large number of designable parameters, the design of DRWTs require the use of numerical optimization algorithms. This work presents a novel and efficient design optimization framework for DRWTs using computationally intensive simulations and variable-fidelity optimization techniques.

Variable-Fidelity Optimization of Film-Cooling Hole Arrangements Considering Conjugate Heat Transfer

In spite of its crucial importance, conjugate heat transfer has been ignored in most film-hole-array optimization studies due to the burdensome associated with added complexity and computational requirements. To cope with this issue and to consider the conduction cooling effects, the hierarchical kriging model was applied to optimize the film-cooling hole array on a high-pressure turbine nozzle. The hierarchical kriging model used in this study consists of three levels and was constructed effectively using both adiabatic-condition-based analyses and conjugate-heat-transfer analyses. The expected-improvement-based efficient global optimization algorithm was coupled with the hierarchical kriging model to improve the efficiency of the whole optimization process. As a result, the required computation time to obtain the optimum solution was markedly reduced to a quarter compared to the time without the hierarchical kriging model, and the viability of the method for highly nonlinear and time-consuming problems was proved. Additionally, the conjugate-heat-transfer-based optimum film-hole array showed that it has improved the overall cooling effectiveness and could have a different shape compared to the array obtained under the adiabatic wall condition. The detailed causes of the resultant hole arrangement were thoroughly examined with respect to both the external and internal cooling aspects.

Variable-fidelity expected improvement method for efficient global optimization of expensive functions

The efficient global optimization method (EGO) based on kriging surrogate model and expected improvement (EI) has received much attention for optimization of high-fidelity, expensive functions. However, when the standard EI method is directly applied to a variable-fidelity optimization (VFO) introducing assistance from cheap, low-fidelity functions via hierarchical kriging (HK) or cokriging, only high-fidelity samples can be chosen to update the variable-fidelity surrogate model. The theory of infilling low-fidelity samples towards the improvement of high-fidelity function is still a blank area. This article proposes a variable-fidelity EI (VF-EI) method that can adaptively select new samples of both low and high fidelity. Based on the theory of HK model, the EI of the high-fidelity function associated with adding low- and high-fidelity sample points are analytically derived, and the resulting VF-EI is a function of both the design variables x and the fidelity level l. Through maximizing the VF-EI, both the sample location and fidelity level of next numerical evaluation are determined, which in turn drives the optimization converging to the global optimum of high-fidelity function. The proposed VF-EI is verified by six analytical test cases and demonstrated by two engineering problems, including aerodynamic shape optimizations of RAE 2822 airfoil and ONERA M6 wing. The results show that it can remarkably improve the optimization efficiency and compares favorably to the existing methods.

Multi-infill strategy for kriging models used in variable fidelity optimization

In this paper, a computationally efficient optimization method for aerodynamic design has been developed. The low-fidelity model and the multi-infill strategy are utilized in this approach. Low-fidelity data is employed to provide a good global trend for model prediction, and multiple sample points chosen by different infill criteria in each updating cycle are used to enhance the exploitation and exploration ability of the optimization approach. Take the advantages of low-fidelity model and the multi-infill strategy, and no initial sample for the high-fidelity model is needed. This approach is applied to an airfoil design case and a high-dimensional wing design case. It saves a large number of high-fidelity function evaluations for initial model construction. What’s more, faster reduction of an aerodynamic function is achieved, when compared to ordinary kriging using the multi-infill strategy and variable-fidelity model using single infill criterion. The results indicate that the developed approach has a promising application to efficient aerodynamic design when high-fidelity analyses are involved.

Rapid design optimization of antennas using variable-fidelity EM models and adjoint sensitivities

Purpose Development of techniques for expedited design optimization of complex and numerically expensive electromagnetic (EM) simulation models of antenna structures validated both numerically and experimentally. The paper aims to discuss these issues. Design/methodology/approach The optimization task is performed using a technique that combines gradient search with adjoint sensitivities, trust region framework, as well as EM simulation models with various levels of fidelity (coarse, medium and fine). Adaptive procedure for switching between the models of increasing accuracy in the course of the optimization process is implemented. Numerical and experimental case studies are provided to validate correctness of the design approach. Findings Appropriate combination of suitable design optimization algorithm embedded in a trust region framework, as well as model selection techniques, allows for considerable reduction of the antenna optimization cost compared to conventional methods. Research limitations/implications The study demonstrates feasibility of EM-simulation-driven design optimization of antennas at low computational cost. The presented techniques reach beyond the common design approaches based on direct optimization of EM models using conventional gradient-based or derivative-free methods, particularly in terms of reliability and reduction of the computational costs of the design processes. Originality/value Simulation-driven design optimization of contemporary antenna structures is very challenging when high-fidelity EM simulations are utilized for performance utilization of structure at hand. The proposed variable-fidelity optimization technique with adjoint sensitivity and trust regions permits rapid optimization of numerically demanding antenna designs (here, dielectric resonator antenna and compact monopole), which cannot be achieved when conventional methods are of use. The design cost of proposed strategy is up to 60 percent lower than direct optimization exploiting adjoint sensitivities. Experimental validation of the results is also provided.

Objective Selection of Minimum Acceptable Mesh Refinement for EMC Simulations

Optimization of computational electromagnetics (CEM) simulation models can be costly in both time and computing resources. Mesh refinement is a key parameter in determining the number of unknowns to be processed. In turn, this controls the time and memory required for a simulation. Hence, it is important to use only a mesh that is good enough for the objectives of the simulation, whether for direct handling of high-fidelity EM models or, even more importantly, for setting up low-fidelity models in a variable-fidelity optimization. On the other hand, in the early stages of an optimization process, a relatively coarse mesh can show whether the governing parameters of the simulation are being appropriately modeled. As the simulation geometry approaches its target, the mesh definition becomes more refined. This letter presents initial results for an approach to identifying the minimum acceptable mesh coarseness based on the projected evolution of FSV's global difference measure when a model is refined from a very crude representation rather than the more usual high-fidelity model. Future work to verify the generality of this letter could provide substantial savings in time and effort for CEM analysis in EMC.

Improving surrogate-assisted variable fidelity multi-objective optimization using a clustering algorithm

Surrogate-assisted evolutionary optimization has proved to be effective in reducing optimization time, as surrogates, or meta-models can approximate expensive fitness functions in the optimization run. While this is a successful strategy to improve optimization efficiency, challenges arise when constructing surrogate models in higher dimensional function space, where the trade space between multiple conflicting objectives is increasingly complex. This complexity makes it difficult to ensure the accuracy of the surrogates. In this article, a new surrogate management strategy is presented to address this problem. A k-means clustering algorithm is employed to partition model data into local surrogate models. The variable fidelity optimization scheme proposed in the author's previous work is revised to incorporate this clustering algorithm for surrogate model construction. The applicability of the proposed algorithm is illustrated on six standard test problems. The presented algorithm is also examined in a three-objective stiffened panel optimization design problem to show its superiority in surrogate-assisted multi-objective optimization in higher dimensional objective function space. Performance metrics show that the proposed surrogate handling strategy clearly outperforms the single surrogate strategy as the surrogate size increases.

Fast Low-fidelity Wing Aerodynamics Model for Surrogate-based Shape Optimization

Variable-fidelity optimization (VFO) can be efficient in terms of the computational cost when compared with traditional approaches, such as gradient-based methods with adjoint sensitivity information. In variable-fidelity methods, the direct optimization of the expensive high-fidelity model is replaced by iterative re-optimization of a physics-based surrogate model, which is constructed from a corrected low-fidelity model. The success of VFO is dependent on the reliability and accuracy of the low-fidelity model. In this paper, we present a way to develop a fast and reliable low-fidelity model suitable for aerodynamic shape of transonic wings. The low-fidelity model is component based and accounts for the zero-lift drag, induced drag, and wave drag. The induced drag can be calculated by a proper method, such lifting line theory or a panel method. The zero-lift drag and the wave drag can be calculated by two-dimensional flow model and strip theory. Sweep effects are accounted for by simple sweep theory. The approach is illustrated by a numerical example where the induced drag is calculated by a vortex lattice method, and the zero-lift drag and wave drag are calculated by MSES (a viscous-inviscid method). The low-fidelity model is roughly 320 times faster than a high-fidelity computational fluid dynamics models which solves the Reynolds-averaged Navier-Stokes equations and the Spalart-Allmaras turbulence model. The responses of the high-and low-fidelity models compare favorably and, most importantly, show the same trends with respect to changes in the operational conditions (Mach number, angle of attack) and the geometry (the airfoil shapes).

Variable-fidelity optimization with design space reduction

Advanced engineering systems, like aircraft, are defined by tens or even hundreds of design variables. Building an accurate surrogate model for use in such high-dimensional optimization problems is a difficult task owing to the curse of dimensionality. This paper presents a new algorithm to reduce the size of a design space to a smaller region of interest allowing a more accurate surrogate model to be generated. The framework requires a set of models of different physical or numerical fidelities. The low-fidelity (LF) model provides physics-based approximation of the high-fidelity (HF) model at a fraction of the computational cost. It is also instrumental in identifying the small region of interest in the design space that encloses the high-fidelity optimum. A surrogate model is then constructed to match the low-fidelity model to the high-fidelity model in the identified region of interest. The optimization process is managed by an update strategy to prevent convergence to false optima. The algorithm is applied on mathematical problems and a two-dimensional aerodynamic shape optimization problem in a variable-fidelity context. Results obtained are in excellent agreement with high-fidelity results, even with lower-fidelity flow solvers, while showing up to 39% time savings.

An Improved Parallel Optimization Framework for Transonic Airfoil Design

Variable Fidelity Optimization (VFO) is used to obtain a minimum drag transonic airfoil, at constant lift, subject to thickness and pitching moment constraints using several low fidelity solvers. VFO has emerged as an attractive method of performing, both, high-speed and high-fidelity optimization. VFO uses computationally inexpensive low-fidelity models, complemented by a surrogate to account for the difference between the high- and low-fidelity models, to obtain the optimum of the function efficiently and accurately. The authors’ original Variable Fidelity (VF) framework is modified for increased efficiency and accuracy by incorporating parallel evaluation and more constraints to the optimization problem. The method is found to be efficient and capable of finding the optimum that closely agrees with the results of high-fidelity optimization alone.

A multi-objective variable-fidelity optimization method for genetic algorithms

A novel variable-fidelity optimization (VFO) scheme is presented for multi-objective genetic algorithms. The technique uses a low- and high-fidelity version of the objective function with a Kriging scaling model to interpolate between them. The Kriging model is constructed online through a fixed updating schedule. Results for three standard genetic algorithm test cases and a two-objective stiffened panel optimization problem are presented. For the stiffened panel problem, statistical analysis of four performance metrics are used to compare the Pareto fronts between the VFO method, full high-fidelity optimizer runs, and Pareto fronts developed by enumeration. The fixed updating approach is shown to reduce the number of high-fidelity calls significantly while approximating the Pareto front in an efficient manner.

WITHDRAWN: Variable fidelity optimization with design space reduction

The Publisher regrets that this article is an accidental duplication of an article that has already been published in CJA, 26/2 (2013) 435-441, http://dx.doi.org/10.1016/j.cja.2013.02.021. The duplicate article has therefore been withdrawn.