Original Design Space Research Articles

Robust design space of compression parameters for initial and reprocessed granules

The design of experiments has been widely used for product and process development based on Quality by design (QbD) approach to achieve quality product. Sources of variability are identified and controlled during manufacturing. However, uncertainty due to uncontrollable variables in the process leads to quality issues and the need for reprocessing in some cases. This study proposes a systematic approach to determine the robust design space for process parameters embraced with the variation of initial and reprocessed materials to ensure that the tablet will meet the specification. The Taguchi L36 experiment design is applied to determine suitable tableting parameters for different granules to ensure physical specifications. Four types of calcium carbonate granules were produced with different moisture contents (ZM) and manufacturing conditions (ZT). Additionally, three critical parameters of tablet compression process—filling volume (XV), compression force (XF), and turret speed (XS)—each encompassing three levels, were selected as input variables. The responses were derived from the product specification, namely weight (YW), tensile strength (YTS), and disintegration time (YDT). The experiment results were embedded into two models: the single independence response model (SRM) and the multiple dependent response model (MRM). After a confirmation experiment using predicted parameters, a comparison of their %MAPE (mean average percentage error) revealed that MRM has lower accuracy in terms of YTS but dominates SRM in terms of the region of design space. This finding relates to the effect of data transformation capturing non-linear relationships between these factors and responses. The design space derived from the models allows manufacturers to adjust critical process parameters for getting desired specification tablets and also provides a framework for continuous improvement by enhancing their original design space and providing a rigorous reprocessing plan.

A dynamic aggregation strategy enhanced efficient global optimization algorithm for solving high-dimensional turbomachinery design problems

ABSTRACT To address challenges effectively in turbomachinery design optimization involving high-dimensional ( d ≥ 30 ) expensive black-box problems, a dedicated Efficient Global Optimization (EGO) algorithm is proposed with dynamic aggregation. This specialized approach efficiently navigates optimization tasks with limited sample evaluations. Specifically, the Dynamic Aggregate Efficient Global Optimization (DA-EGO) algorithm decomposes the original high-dimensional design space into low-dimensional subspaces for efficient surrogate-based optimization search, and the optimal solutions of subspaces are combined as an elite-point for the global search. Most importantly, the subspace variables are updated in each iteration, according to the variable interaction analyses in the sub- and full-spaces. The perturbation method and the analysis of variance are used to detect variable interactions. After being validated on 21 benchmark functions ranging from 30 to 90 dimensions, the DA-EGO is used for the optimization of a transonic compressor rotor with 28 variables and a multi-stage compressor optimization with 60 variables. With the above, the effectiveness of the proposed algorithm is well demonstrated.

Geometric space construction method combined of a spline-skinning based geometric variation method and PCA dimensionality reduction for ship hull form optimization

In the construction stage of ship hull geometric space, the representation and variation of hull forms pose challenges due to the complex nature of the geometry and the necessity for enhanced flexibility in space. In this study, we utilize T-spline technology for the representation of ship hulls, enabling a unified parameter domain with a reduced number of control points to effectively depict the entire hull surface. Furthermore, we introduce a method for geometric variation based on the T-spline parameter domain, which permits both local and global alterations, thus offering greater flexibility in modifying the geometry. By leveraging the T-spline representation and the geometric variation method, we developed a hybrid geometric approach for constructing the geometric space, which significantly increased the geometric flexibility in ship hull optimization. A geometric space characterized by a high degree of geometric freedom typically requires the use of numerous optimization variables and incurs considerable computational resource consumption. In this research, we apply a linear dimensionality reduction method, principal component analysis, to decrease the dimensionality of the geometric space. This method effectively preserves the original geometric design space, thereby eliminating the need for performance analyses. The numerical optimization results clearly indicate that the proposed method significantly enhances optimization efficiency.

Aerodynamic shape optimization based on proper orthogonal decomposition reparameterization under small training sets

Complex shape aerodynamic optimization frequently encounters the problem of dimensionality curse. Extracting geometric features from the original design space proves effective in this issue by reconstructing shapes with lower-dimension. While deep learning methods possess strong capabilities in dimensionality reduction, they suffer from longer training times and higher computational resource requirements. However, Proper Orthogonal Decomposition (POD) presents a more practical algorithm that can be effectively utilized for complex high-dimensional shapes. In this study, we propose a combination of the POD-based method and Gaussian Process Regression with an automatic kernel construction algorithm (AKC-GPR) to optimize complex shapes in a lower-dimensional reparameterization space using small training sets. This approach aims to explore the potential for reducing optimization cost. By reducing the design parameters from 56 to 18 with an error rate of 1.28 %, we establish surrogate models for optimization using AKC-GPR with progressively decreasing training sets. For samples below 100, the Mean Absolute Percentage Errors (MAPEs) remain below 2 %, while the Mean Relative Percentage Errors (MRPEs) approach 10 %. It emphasizes the proficiency of POD in geometric dimensionality reduction. The optimized shapes exhibit smoother contour, resulting in a 10 % improvement in lift and a relatively modest reduction in drag of less than 3 %. The lift-to-drag ratio experiences an increase of over 14 %. The shapes optimized by various training sets possess minimal differences in geometric shapes and aerodynamic characteristics. This confirms the AKC-GPR algorithm's high-precision fitting capability in effectively handling complex aerodynamic coefficients of multi-parameter shapes during small-sample shape optimization.

Acoustical and behavioral heuristics for fast interactive sound design.

During their creative process, designers routinely seek the feedback of end users. Yet, the collection of perceptual judgments is costly and time-consuming, since it involves repeated exposure to the designed object under elementary variations. Thus, considering the practical limits of working with human subjects, randomized protocols in interactive sound design face the risk of inefficiency, in the sense of collecting mostly uninformative judgments. This risk is all the more severe that the initial search space of design variations is vast. In this paper, we propose heuristics for reducing the design space considered during an interactive optimization process. These heuristics operate by using an approximation model, called surrogate model, of the perceptual quantity of interest. As an application, we investigate the design of pleasant and detectable electric vehicle sounds using an interactive genetic algorithm. We compare two types of surrogate models for this task, one based on acoustical descriptors gathered from the literature and the other based on behavioral data. We find that reducing by a factor of up to 64 an original design space of 4096 possible settings with the proposed heuristics reduces the number of iterations of the design process by up to 2 to reach the same performance. The behavioral approach leads to the best improvement of the explored designs overall, while the acoustical approach requires an appropriate choice of acoustical descriptor to be effective. Our approach accelerates the convergence of interactive design. As such, it is particularly suitable to tasks in which exhaustive search is prohibitively slow or expensive.

Improved stochastic subset optimization method for structural design optimization

The Stochastic Subset Optimization (SSO) algorithm was proposed for optimal reliability problems that minimizes the probability of system failure over the admissible space for the design parameters. It is based on the simulation of samples of the design parameters from an auxiliary Probability Density Function (PDF) and exploiting the information contained in these samples to identify subregions for the optimal design parameters within the original design space. This paper presents an improved version of SSO, named iSSO to overcome the shortcomings in the SSO. In the improved version, the Voronoi tessellation is implemented to partition the design space into non-overlapping subregions using the pool of samples distributed according to the auxiliary PDF. A double-sort approach is then used to identify the subregions for the optimal design. The iSSO is presented as a generalized design optimization approach primarily tailored for the stochastic structural systems but also adaptable to deterministic systems. Several optimization problems are considered to illustrate the effectiveness and efficiency of the proposed iSSO.

Shape-Informed Dimensional Reduction in Airfoil/Hydrofoil Modeling

Parametric models have been widely used in pertinent literature for reconstructing, modifying and representing a wide range of airfoil and/or hydrofoil profile geometries. Design spaces corresponding to these models can be exploited for modeling and profile-shape optimization under various performance criteria. Accuracy requirements, along with the need for modeling local features, often lead to high-dimensional design spaces that hinder the process of shape optimization and design through analysis. In this work, we propose a shape-informed dimensional reduction approach that attempts to tackle this deficiency by producing low-dimensional latent design spaces that can be efficiently used in shape representation and optimization. Furthermore, geometric moments are introduced in an attempt to cost-effectively capture analysis-relevant information that is generally expensive to produce. Specifically, geometric integral properties, although intrinsic features of the shape, are quite commonly related to performance indicators employed in performance optimization and therefore provide a cost-effective physics-informed component in the description of the design in the latent space. To this end, we employ the generalized Karhunen-Loève expansion to produce a shape- and physics-informed subspace retaining the highest possible geometric variance and robustness, that is, a lack of invalid designs. At the same time, a series of shape discretizations, encoding the foil’s shape profile, are examined with regard to their effect on the resulting latent space’s quality and efficiency. Our results demonstrate a significant reduction in the dimensionality of the original design space while maintaining a high representational capacity and a large percentage of valid geometries that facilitate fast convergence to optimal solutions in performance-based optimization.

A boundary identification approach for the feasible space of structural optimization using a virtual sampling technique-based support vector machine

To improve the computational efficiency of structural optimization, this study treats the feasibility evaluation of the solutions as a two-class classification problem and proposes a boundary identification approach (BIA) to identify the feasible region boundary of search space. The BIA includes a virtual sampling technique (VST), an improved Latin hypercube sampling (ILHS) method, and a support vector machine (SVM) classifier. The VST can generate cheap samples based on a mapping strategy from actual samples without time-consuming structural analysis. To enhance the global performance of the SVM, the ILHS yields the sample set on the normalized hypersphere of the original design space. An optimization framework based on the BIA hybridized with the harmony search algorithm is presented, and two numerical and three truss examples are utilized to examine the performances of the proposed optimization framework. The prediction accuracy of the BIA reaches about 99% in two numerical examples, and 97%, 90%, and 80% in the 10-bar, 72-bar, and 600-bar truss optimization examples, respectively. The results also show that the number of structural analyses required by the proposed optimization approach is reduced by more than 80% compared to the conventional metaheuristics optimization algorithms while obtaining a similar quality of optimal designs.

A new bandwidth-extended harmonic-tuned Class-E/F3 power amplifier based on continuous mode design philosophy

For Class-E/F3 power amplifiers (PAs) working in the radio frequency (RF) band, the freedom of broadband design is limited because the original design space is a single point. In this paper, a new bandwidth-extended Class-E/F3 PA is proposed, which is based on finite harmonic tuning and continuous mode design philosophy. In the theoretical derivation, the high-efficiency frequency domain equations are solved in conjunction with the inherent conditions of the transistor, resulting in a waveform cluster that can expand the design space while keeping the efficiency constant. This approach eliminates the need for tedious circuit derivation and is able to allow greater design freedom. With the extended Class-E/F3 design space, a broadband PA using GaN HEMT CGH40010F operating in the bandwidth of 1.9–3.3 GHz is designed and tested. The experimental results show that the designed PA demonstrates 65% to 74.8% drain efficiency (DE) and 40.1 to 41.7dBm output power over the target bandwidth.

Convolutional Neural Network Classification of Exhaled Aerosol Images for Diagnosis of Obstructive Respiratory Diseases

Aerosols exhaled from the lungs have distinctive patterns that can be linked to the abnormalities of the lungs. Yet, due to their intricate nature, it is highly challenging to analyze and distinguish these aerosol patterns. Small airway diseases pose an even greater challenge, as the disturbance signals tend to be weak. The objective of this study was to evaluate the performance of four convolutional neural network (CNN) models (AlexNet, ResNet-50, MobileNet, and EfficientNet) in detecting and staging airway abnormalities in small airways using exhaled aerosol images. Specifically, the model’s capacity to classify images inside and outside the original design space was assessed. In doing so, multi-level testing on images with decreasing similarities was conducted for each model. A total of 2745 images were generated using physiology-based simulations from normal and obstructed lungs of varying stages. Multiple-round training on datasets with increasing images (and new features) was also conducted to evaluate the benefits of continuous learning. Results show reasonably high classification accuracy on inbox images for models but significantly lower accuracy on outbox images (i.e., outside design space). ResNet-50 was the most robust among the four models for both diagnostic (2-class: normal vs. disease) and staging (3-class) purposes, as well as on both inbox and outbox test datasets. Variation in flow rate was observed to play a more important role in classification decisions than particle size and throat variation. Continuous learning/training with appropriate images could substantially enhance classification accuracy, even with a small number (~100) of new images. This study shows that CNN transfer-learning models could detect small airway remodeling (<1 mm) amidst a variety of variants and that ResNet-50 can be a promising model for the future development of obstructive lung diagnostic systems.

KT-EGO: a knowledge transfer assisted efficient global optimization algorithm for solving high-dimensional expensive black-box problems

Many engineering problems involve optimizing a high-dimensional expensive black-box (HEB) design space. To solve such problems efficiently, a knowledge transfer (KT) assisted efficient global optimization (EGO) algorithm is proposed, called the KT-EGO, which extends the EGO algorithm for solving problems over higher dimensions (i.e.d>20). Specifically, the original design space is divided into several low-dimensional subset design spaces. More importantly, in order to extract information from the subset design spaces to accelerate the progress of full optimization, a surrogate-based data fusion strategy is proposed in the KT-EGO. And further, a searching strategy with an adaptive variable range is devised to enhance the exploitation of promising areas. To show the effectiveness of the proposed algorithm, it is compared against state-of-the-art algorithms over 12 benchmark functions and a 28-dimensional engineering optimization for the design of a compressor blade, which fully validates the effectiveness of the KT-EGO for solving HEB problems.

Shape-supervised Dimension Reduction: Extracting Geometry and Physics Associated Features with Geometric Moments

In shape optimisation problems, subspaces generated with conventional dimension reduction approaches often fail to extract the intrinsic geometric features of the shape that would allow the exploration of diverse but valid candidate solutions. More importantly, they also lack incorporation of any notion of physics against which shape is optimised. This work proposes a shape-supervised dimension reduction approach. To simultaneously tackle these deficiencies, it uses higher-level information about the shape in terms of its geometric integral properties, such as geometric moments and their invariants. Their usage is based on the fact that moments of a shape are intrinsic features of its geometry, and they provide a unifying medium between geometry and physics. To enrich the subspace with latent features associated with shape’s geometrical features and physics, we also evaluate a set of composite geometric moments, using the divergence theorem, for appropriate shape decomposition. These moments are combined with the shape modification function to form a Shape Signature Vector (SSV) uniquely representing a shape. Afterwards, the generalised Karhunen–Loève expansion is applied to SSV, embedded in a generalised (disjoint) Hilbert space, which results in a basis of the shape-supervised subspace retaining the highest geometric and physical variance. Validation experiments are performed for a three-dimensional wing and a ship hull model. Our results demonstrate a significant reduction of the original design space’s dimensionality for both test cases while maintaining a high representation capacity and a large percentage of valid geometries that facilitate fast convergence to the optimal solution. The code developed to implement this approach is available at https://github.com/shahrozkhan66/SSDR.git . • Shape signature vector (SSV) comprises a shape modification vector and geometric moments (GM). • Karhunen–Loève expansion of SSV results in a shape-supervised space (SSS). • GM can induce the notion of physics resulting in a physics-informed SSS. • SSS retains higher geometric variance with fewer latent variables. • SSS is robust and compact, providing accelerated optimisation convergence.

An Inverse Design Framework for Isotropic Metasurfaces Based on Representation Learning

A hybrid framework for solving the non-uniqueness problem in the inverse design of isomorphic metasurfaces is proposed. The framework consists of a representation learning (RL) module and a variational autoencoder-particle swarm optimization (VAE-PSO) algorithm module. The RL module is used to reduce the complex high-dimensional space into a low-dimensional space with obvious features, with the purpose of eliminating the many-to-one relationship between the original design space and response space. The VAE-PSO algorithm first encodes all meta-atoms into a continuous latent space through VAE and then applies PSO to search for an optimized latent vector whose corresponding metasurface fulfills the target response. This framework gives the solution paradigm of the ideal non-uniqueness situation, simplifies the complexity of the network, improves the running speed of the PSO algorithm, and obtains the global optimal solution with 94% accuracy on the test set.

A Bi-stage Multi-objective Reliability-based Design Optimization Using Surrogate Model for Reusable Thrust Chambers

To reduce the computational burden of Multi-Objective Reliability-Based Design Optimization (MORBDO) and promote its application in complex practical systems, a bi-stage MORBDO procedure using surrogate models is proposed in this paper. The novel process consists of two stages: in the first stage, a narrowed solution space named as MORBDO design space is obtained by a pre-multi-objective deterministic design optimization in the whole original design space; In the second stage, the MORBDO design space is extended to form an augmented space by considering uncertainties, and then the MORBDO is executed with the double-loop strategy over the augmented space. Besides, the multiple response surface (MRS) is adopted in the bi-stage MORBDO procedure to replace the time-consuming finite element analysis. The proposed method is validated by conducting MORBDO on a lab-scale reusable LOX/H2 thrust chamber aiming at maximum residual strength and cyclic life. Compared with the traditional MORBDO implemented in the whole original design space, the bi-stage MORBDO can significantly reduce the computational burden of the surrogate model by 49.33% and only one-quarter of the optimization population is required in NSGA II to search for the optimal solution. Finally, a sensitivity analysis is performed to quantify the importance ranking of design variables of thrust chambers. The results indicate that the geometry of cooling channels near the throat plays the most important role in the thrust chamber's residual strength and cyclic life. The proposed method is easy to be conducted and holds great potentials in the MORBDO of other complex practical systems.

A two-step parametric generative method for heat exchangers design in additive manufacturing

Additive Manufacturing(AM) liberates a lot for the hands of designers to create complex geometry and avoid many traditional manufacturing constrains. A set of advanced generative design and topological optimization tools are emerging to support the design for AM. However, many existing methods are based on traditional topological optimization methods, which are focusing more on lightweight design but have difficulty to optimize component’s functions. In heat exchanger design, the functional features, e.g. fluid tunnels, are usually fixed without changing during the lightweight optimization. This cannot guarantee an optimal functional performance for the design. To solve this problem, this paper proposes a two-step generative design method that can both help optimizing the functional features for performance improvement and lightweight design. This method has two generative optimization loops, including functional feature group optimization and lightweight optimization. A set of finite optimized candidate alternative solutions from the first optimization loop are inputs for the second lightweight optimization loop to form boundary conditions and freezing regions in lightweight design. With the topological optimization in the second loop, the pre-selected functional features can be combined with the optimal volume layouts in the original design space to form a final component with improved function performance and reduced mass. To demonstrate the proposed method, a design case of a heat exchanger with conformal cooling channels is presented. The method can be adopted for other similar design cases in additive manufacturing applications.

Linear reduced order method for design-space dimensionality reduction and flow-field learning in hull form optimization

In the earlier stage of hull form optimization design, a series of design variables is usually needed to control the hull shape, so as to find optimal hull forms with better performance. In the surrogate-based hydrodynamic performance optimization for ships, with the increase of the dimensionality of design space, the number of new sample hulls to construct surrogate model needs to be larger, which will bring a large amount of calculation. Through reduced order method, the dimensionality of the optimization design space can be reduced while keeping the deformation range of the original design space to a great extent, for instance, using the linear combination of a smaller number of bases to represent the deformation range. In addition, in the later stage of hull form optimization design, flow field results of the new sample hulls can be fully utilized to do the dimensionality reduction multi-physics field learning. In this paper, the principle of the Proper Orthogonal Decomposition method is used and briefly introduced, the steps of dimensionality reduction of the design space are shown then, and some important problems for the design-space dimensionality reduction in the specific field of hull form optimization, such as retainability of fixed control points, irrelevance of the relative order of data to dimensionality reduction results, and decision of the new design space range after dimensionality reduction, are deep analyzed. Furthermore, taking the resistance optimization of the modified Wigley ship as an example, the specific application and error analysis of the dimensionality reduction method for design-space dimensionality reduction in the earlier stage of hull form optimization and the multi-physics field learning in the later stage of hull form optimization are given, and the applicability and reliability of the method are demonstrated by analyzing the influence of mode order and sample number on reconstruction effect of the hull shape or flow field, and the prediction effect of flow field for not-in-the-database new hull form in detail. Results show that the linear dimensionality reduction method can reduce samples needed for optimization, thus reduce the amount of calculation for the surrogate-based hull form optimization, and be used for quick prediction of multi-physics fields of any new form in the design space. Furthermore, it can not only be applied to the sensitivity analysis or a Pareto frontier selection in comprehensive performance optimization of hull form based on CFD, but also be implemented in the real-time forecast of the flow field and influence analysis of the ship performance when adjusting the hull form (or hull appendages).

A reduced order data-driven method for resistance prediction and shape optimization of hull vane

Hull Vane (HV) is an energy-saving appendage introduced by Hull Vane BV company to reduce total ship resistance. Shapewise, HV is a hydrofoil wing transversely fixed at the transom bottom of the hull.In this paper, a data-driven shape optimization method is proposed for HV. To avoid the time-consuming resistance evaluation of designs via a viscous flow solver, we develop a Machine-Learning (ML) based model that predicts the hull's total resistance in the presence of an HV. For this purpose, Principal-Component Analysis (PCA) is first implemented to reduce the dimensionality of the problem, and then the prediction model is trained with the most influential of the Principal Components (PCs). Given that these PCs capture the maximum geometric variance of the original design space, higher accuracy can be achieved at the expense of a few training samples. After the training phase, the model is integrated with an optimizer, which searches in a dimensionally-reduced design space for the optimal design of the HV. The obtained results achieved a 70% dimensionality reduction with the aid PCA and an approximately 98% accuracy for predicting total resistance. Compared with the reference HV, the optimized one reduced the total resistance by 1.2%.

An Approach to Bayesian Optimization for Design Feasibility Check on Discontinuous Black-Box Functions

Abstract The paper presents a novel approach to applying Bayesian Optimization (BO) in predicting an unknown constraint boundary, also representing the discontinuity of an unknown function, for a feasibility check on the design space, thereby representing a classification tool to discern between a feasible and infeasible region. Bayesian optimization is a low-cost black-box global optimization tool in the Sequential Design Methods where one learns and updates knowledge from prior evaluated designs, and proceeds to the selection of new designs for future evaluation. However, BO is best suited to problems with the assumption of a continuous objective function and does not guarantee true convergence when having a discontinuous design space. This is because of the insufficient knowledge of the BO about the nature of the discontinuity of the unknown true function. In this paper, we have proposed to predict the location of the discontinuity using a BO algorithm on an artificially projected continuous design space from the original discontinuous design space. The proposed approach has been implemented in a thin tube design with the risk of creep-fatigue failure under constant loading of temperature and pressure. The stated risk depends on the location of the designs in terms of safe and unsafe regions, where the discontinuities lie at the transition between those regions; therefore, the discontinuity has also been treated as an unknown creep-fatigue failure constraint. The proposed BO algorithm has been trained to maximize sampling toward the unknown transition region, to act as a high accuracy classifier between safe and unsafe designs with minimal training cost. The converged solution has been validated for different design parameters with classification error rate and function evaluations at an average of <1% and ∼150, respectively. Finally, the performance of our proposed approach in terms of training cost and classification accuracy of thin tube design is shown to be better than the existing machine learning (ML) algorithms such as Support Vector Machine (SVM), Random Forest (RF), and Boosting.

Gaussian Process-Driven, Nested Experimental Co-Design: Theoretical Framework and Application to an Airborne Wind Energy System

Abstract This paper presents and experimentally evaluates a nested combined plant and controller optimization (co-design) strategy that is applicable to complex systems that require extensive simulations or experiments to evaluate performance. The proposed implementation leverages principles from Gaussian process (GP) modeling to simultaneously characterize performance and uncertainty over the design space within each loop of the co-design framework. Specifically, the outer loop uses a GP model and batch Bayesian optimization to generate a batch of candidate plant designs. The inner loop utilizes recursive GP modeling and a statistically driven adaptation procedure to optimize control parameters for each candidate plant design in real-time, during each experiment. The characterizations of uncertainty made available through the GP models are used to reduce both the plant and control parameter design space as the process proceeds, and the optimization process is terminated once sufficient design space reduction has been achieved. The process is validated in this work on a lab-scale experimental platform for characterizing the flight dynamics and control of an airborne wind energy (AWE) system. The proposed co-design process converges to a design space that is less than 8% of the original design space and results in more than a 50% increase in performance.

Application of Dynamic Space Reduction Method Based on Partial Correlation Analysis in Hull Optimization

Hull optimization design based on computational fluid dynamics (CFD) is a highly computationally intensive complex engineering problem. Because of reasons such as many variables, spatially complex design performance, and huge computational workload, hull optimization efficiency is low. To improve the efficiency of hull optimization, a dynamic space reduction method based on a partial correlation analysis is proposed in this study. The proposed method dynamically uses hull-form optimization data to analyze and reduce the range of values for relevant design variables and, thus, considerably improves the optimization efficiency. This method is used to optimize the wave-making resistance of an S60 hull, and its feasibility is verified through comparison. 1. Introduction In recent years, to promote the rapid development of green ships, hull optimization methods based on computational fluid dynamics (CFD) have been widely used by many researchers, such as Tahara et al. (2011), Peri and Diez (2013), Kim and Yang (2010), Yang and Huang (2016), Chang et al. (2012), and Feng et al. (2009). However, hull optimization design is a typically complex engineering problem. It requires many numerical simulation calculations, and the design performance space is complex, which has resulted in low optimization efficiency and difficulty in obtaining a global optimal solution. Commonly used solutions include 1) efficient optimization algorithms, 2) approximate model techniques, and 3) high-performance cluster computers. However, these methods still cannot satisfy the engineering application requirements in terms of efficiency and quality of the solution. To solve the problem of low optimization efficiency and difficulty in obtaining an optimal solution in engineering optimization problems, many scholars have conducted research on design space reduction technology. Reungsinkonkarn and Apirukvorapinit (2014) applied the search space reduction (SSR) algorithm to the particle swarm optimization (PSO) algorithm, eliminating areas in which optimal solutions may not be found through SSR to improve the optimization efficiency of the algorithm. Chen et al. (2015) and Diez et al. (2014, 2015) used the Karhunen–Loeve expansion to evaluate the hull, eliminating the less influential factors to achieve space reduction modeling with fewer design variables. Further extensions to nonlinear dimensionality reduction methods can be found in D'Agostino et al. (2017) and Serani et al. (2019). Jeong et al. (2005) applied space reduction techniques to the aerodynamic shape optimization of the vane wheel, using the rough set theory and decision trees to extract aerofoil design rules to improve each target. Gao et al. (2009) and Wang et al. (2014) solved the problem of low optimization efficiency in the aerodynamic shape optimization design of an aircraft, by using analysis results of partial correlation, which reduced the range of values of relevant design variables to reconstruct the optimized design space. Li et al. (2013) divided the design space into several smaller cluster spaces using the clustering method, which is a global optimization method based on an approximation model, thus achieving design space reduction. Chu (2010) combined the rough set theory and the clustering method for application to the concept design stage of bulk carriers, thus realizing the exploration and reduction of design space. Feng et al. (2015) applied the rough set theory and the sequential space reduction method to the resistance optimization of typical ship hulls to achieve the reduction of design space. Wu et al. (2016) used partial correlation analysis to reduce the design space of variables of a KCS container ship to improve optimization efficiency. Most of the above space reduction methods need to sample and calculate the original design space in the early stage of optimization and then obtain the reduced design space through data mining. This process increases the computational cost of sampling, making it difficult to control optimization efficiency.