Sequential RBF Surrogate-based Efficient Optimization Method for Engineering Design Problems with Expensive Black-Box Functions

As a promising technique, surrogate-based design and optimization（SBDO） has been widely used in modern engineering design optimizations. Currently, static surrogate-based optimization methods have been successfully applied to expensive optimization problems. However, due to the low efficiency and poor flexibility, static surrogate-based optimization methods are difficult to efficiently solve practical engineering cases. At the aim of enhancing efficiency, a novel surrogate-based efficient optimization method is developed by using sequential radial basis function（SEO-SRBF）. Moreover, augmented Lagrangian multiplier method is adopted to solve the problems involving expensive constraints. In order to study the performance of SEO-SRBF, several numerical benchmark functions and engineering problems are solved by SEO-SRBF and other well-known surrogate-based optimization methods including EGO, MPS, and IARSM. The optimal solutions, number of function evaluations, and algorithm execution time are recorded for comparison. The comparison results demonstrate that SEO-SRBF shows satisfactory performance in both optimization efficiency and global convergence capability. The CPU time required for running SEO-SRBF is dramatically less than that of other algorithms. In the torque arm optimization case using FEA simulation, SEO-SRBF further reduces 21% of thematerial volume compared with the solution from static-RBF subject to the stress constraint. This study provides the efficient strategy to solve expensive constrained optimization problems.

Surrogate-based modeling and dimension reduction techniques for multi-scale mechanics problems

Successful modeling and/or design of engineering systems often requires one to address the impact of multiple ＂design variables＂ on the prescribed outcome.There are often multiple,competing objectives based on which we assess the outcome of optimization.Since accurate,high fidelity models are typically time consuming and computationally expensive,comprehensive evaluations can be conducted only if an efficient framework is available.Furthermore,informed decisions of the model/hardware＇s overall performance rely on an adequate understanding of the global,not local,sensitivity of the individual design variables on the objectives.The surrogate-based approach,which involves approximating the objectives as continuous functions of design variables from limited data,offers a rational framework to reduce the number of important input variables,i.e.,the dimension of a design or modeling space.In this paper,we review the fundamental issues that arise in surrogate-based analysis and optimization,highlighting concepts,methods,techniques,as well as modeling implications for mechanics problems.To aid the discussions of the issues involved,we summarize recent efforts in investigating cryogenic cavitating flows,active flow control based on dielectric barrier discharge concepts,and lithium（Li）-ion batteries.It is also stressed that many multi-scale mechanics problems can naturally benefit from the surrogate approach for ＂scale bridging.＂

A surrogate-based optimization algorithm for network design problems

Network design problems （NDPs） have long been regarded as one of the most challenging problems in the field of transportation planning due to the intrinsic non-convexity of their bi-level programming form. Furthermore, a mixture of continuous/discrete decision variables makes the mixed network design problem （MNDP） more complicated and difficult to solve. We adopt a surrogate-based optimization （SBO） framework to solve three featured categories of NDPs （continuous, discrete, and mixed-integer）. We prove that the method is asymptotically completely convergent when solving continuous NDPs, guaranteeing a global optimum with probability one through an indefinitely long run. To demonstrate the practical performance of the proposed framework, numerical examples are provided to compare SBO with some existing solving algorithms and other heuristics in the literature for NDP. The results show that SBO is one of the best algorithms in terms of both accuracy and efficiency, and it is efficient for solving large-scale problems with more than 20 decision variables. The SBO approach presented in this paper is a general algorithm of solving other optimization problems in the transportation field.

Optimal agricultural spreading scheduling through surrogate-based optimization and MINLP models

The most commonly used definition of climate smart agriculture(CSA)is provided by the Food and Agricultural Organisation of the United Nations,which defines CSA as“agriculture that sustainably increases productivity,enhances resilience,reduces/removes greenhouse gas where possible,and enhances achievement of national food security and development goals”.In this definition,the principal goal of CSA is identified as food security and development,while productivity,adaptation,and mitigation are identified as the three interlinked pillars necessary for achieving this goal.In the context provided by the CSA,soils are seen as a lever to improve the carbon footprint of agriculture,namely through their role as carbon sinks.Improving soils and in particular agricultural soils’content in soil organic carbon(SOC)in one of the measures enabling to improve the environmental impact of agricultural practices.In this context,composts can be seen as an important feedstock for sustainable farming.To support the development of organic amendment strategies enabling to increase soils’SOC content,this work proposes a novel methodology to optimize the monthly scheduling of composts and mineral fertilizers amendments.The schedule proposed maximizes soil health-via improved SOC content-while ensuring optimal gross operating surplus from agriculture.This problem is subjected to certain operational,regulatory and soil-dynamics constraints which leads to a complex optimization problem and has to be solved in a relatively short time period for decision-making purposes.This is a nonlinear optimization problem(NLP)which is based on a soil-simulation model from which the analytic functions are not explicitly available for the optimization model.Operational and regulatory constraints are explicitly defined and integer and continuous variables are needed in the modeling.In order to effectively solve it in a deterministic way,a novel surrogate-modeling approach for the objective functions and constraints is proposed.Another novelty comes from the implementation of a continuous-variable-reduction procedure in order to build effective surrogates.The optimisation model results obtained from a real case study show that local optimal solutions can be identified with short computation times.The evaluated scenario shows that the optimized strategy could increase by 8%the carbon in soil after 13 years while increasing by 7%the estimated agricultural profit when compared to expert based application schedules.

Parametric Hull Form Optimization of Containerships for Minimum Resistance in Calm Water and in Waves

This paper described the process of generating the optimal parametric hull shape with a fully parametric modeling method for three containerships of diferent sizes.The newly created parametric ship hull was applied to another ship with a similar shape,which greatly saved time cost.A process of selecting design variables was developed,and during this process,the infuence of these variables on calm water resistance was analyzed.After we obtained the optimal hulls,the wave added resistance and motions of original hulls and optimal hulls in regular head waves were analyzed and compared with experi-mental results.Computations of the fow around the hulls were obtained from a validated nonlinear potential fow boundary element method.Using the multi-objective optimization algorithm,surrogate-based global optimization(SBGO)reduced the computational efort.Compared with the original hull,wave resistance of the optimal hulls was signifcantly reduced for the two larger ships at Froude numbers corresponding to their design speeds.Optimizing the hull of the containerships slightly reduced their wave added resistance and total resistance in regular head waves,while optimization of their hulls hardly afected wave-induced motions.

Data-driven surrogate model for aerodynamic design using separable shape tensor method

In the context of increasing dimensionality of design variables and the complexity of constraints, the efficacy of Surrogate-Based Optimization(SBO) is limited. The traditional linear and nonlinear dimensionality reduction algorithms are mainly to decompose the mathematical matrix composed of design variables or objective functions in various forms, the smoothness of the design space cannot be guaranteed in the process, and additional constraint functions need to be added in the optimization, which increases the calculation cost. This study presents a new parameterization method to improve both problems of SBO. The new parameterization is addressed by decoupling affine transformations(dilation, rotation, shearing, and translation) within the Grassmannian submanifold, which enables a separate representation of the physical information of the airfoil in a highdimensional space. Building upon this, Principal Geodesic Analysis(PGA) is employed to achieve geometric control, compress the design space, reduce the number of design variables, reduce the dimensions of design variables and enhance predictive performance during the surrogate optimization process. For comparison, a dimensionality reduction space is defined using 95% of the energy,and RAE 2822 for transonic conditions are used as demonstrations. This method significantly enhances the optimization efficiency of the surrogate model while effectively enabling geometric constraints. In three-dimensional problems, it enables simultaneous design of planar shapes for various components of the aircraft and high-order perturbation deformations. Optimization was applied to the ONERA M6 wing, achieving a lift-drag ratio of 18.09, representing a 27.25% improvement compared to the baseline configuration. In comparison to conventional surrogate model optimization methods, which only achieved a 17.97% improvement, this approach demonstrates its superiority.

Adaptive optimization methodology based on Kriging modeling and a trust region method

Surrogate-Based Optimization(SBO) is becoming increasingly popular since it can remarkably reduce the computational cost for design optimizations based on high-fidelity and expensive numerical analyses. However, for complicated optimization problems with a large design space, many design variables, and strong nonlinearity, SBO converges slowly and shows imperfection in local exploitation. This paper proposes a trust region method within the framework of an SBO process based on the Kriging model. In each refinement cycle, new samples are selected by a certain design of experiment method within a variable design space, which is sequentially updated by the trust region method. A multi-dimensional trust-region radius is proposed to improve the adaptability of the developed methodology. Further, the scale factor and the limit factor of the trust region are studied to evaluate their effects on the optimization process. Thereafter, different SBO methods using error-based exploration, prediction-based exploitation, refinement based on the expected improvement function, a hybrid refinement strategy, and the developed trust-regionbased refinement are utilized in four analytical tests. Further, the developed optimization methodology is employed in the drag minimization of an RAE2822 airfoil. Results indicate that it has better robustness and local exploitation capability in comparison with those of other SBO

Sequential ensemble optimization based on general surrogate model prediction variance and its application on engine acceleration schedule design

The Efficient Global Optimization(EGO)algorithm has been widely used in the numerical design optimization of engineering systems.However,the need for an uncertainty estimator limits the selection of a surrogate model.In this paper,a Sequential Ensemble Optimization(SEO)algorithm based on the ensemble model is proposed.In the proposed algorithm,there is no limitation on the selection of an individual surrogate model.Specifically,the SEO is built based on the EGO by extending the EGO algorithm so that it can be used in combination with the ensemble model.Also,a new uncertainty estimator for any surrogate model named the General Uncertainty Estimator(GUE)is proposed.The performance of the proposed SEO algorithm is verified by the simulations using ten well-known mathematical functions with varying dimensions.The results show that the proposed SEO algorithm performs better than the traditional EGO algorithm in terms of both the final optimization results and the convergence rate.Further,the proposed algorithm is applied to the global optimization control for turbo-fan engine acceleration schedule design.

Aerodynamic optimization design of general parameters for cycloidal propeller in hover based on surrogate model

A surrogate-model-based aerodynamic optimization design method for cycloidal propeller in hover was proposed,in order to improve its aerodynamic efficiency,and analyze the basic criteria for its aerodynamic optimization design.The reliability and applicability of overset mesh method were verified.An optimization method based on Kriging surrogate model was proposed to optimize the geometric parameters for cycloidal propeller in hover with the use of genetic algorithm.The optimization results showed that the thrust coefficient was increased by 3.56%,the torque coefficient reduced by 12.05%,and the figure of merit(FM)increased by 19.93%.The optimization results verified the feasibility of this design idea.Although the optimization was only carried out at a single rotation speed,the aerodynamic efficiency was also significantly improved over a wide range of rotation speeds.The optimal configuration characteristics for micro and small-sized cycloidal propeller were:solidity of 0.2-0.22,maximum pitch angle of 25°-35°,pitch axis locating at 35%-45% of the blade chord length.