**1. Introduction**

From the time of the famous Wright brothers' first flight, global air traffic has grown continuously regardless of economic or political turmoil. In only the last two decades, it has more than doubled. Two major commercial aircraft suppliers, Boeing and Airbus, forecast further air traffic growth in the coming decades at a rate of nearly 4% annually, which will create a demand for global air fleet development of approximately 3% per year [1,2]. Such an expansion rate will result in doubling the number of passenger and freight airplanes by the end of the fourth decade of the 21st century.

The foreseen dynamic development coincides with a need to introduce new disruptive technologies supporting climate-neutral aviation. This target is imposed by the European Green Deal initiative [3], recently established by the European Commission. The ambitious goal of achieving net-zero emission air transport by 2050 requires the deployment of radical innovations within a short period. Such rapid anticipated progress forces aviation R&D entities to seek more streamlined design processes, which should be supported with advanced design tools and efficient optimization workflows. More efficient work schemes should be instituted simultaneously on an integrated aircraft level and for each subsystem and component.

This paper focuses on developing an optimization framework suited for airplane engine air-intake ducts. There is a long history of aerodynamic optimization studies on ducts

**Citation:** Dr˛ezek, P.S.; Kubacki, S.; ˙ Zółtak, J. Kriging-Based Framework ˙ Applied to a Multi-Point, Multi-Objective Engine Air-Intake Duct Aerodynamic Optimization Problem. *Aerospace* **2023**, *10*, 266. https://doi.org/10.3390/ aerospace10030266

Academic Editors: Spiros Pantelakis, Andreas Strohmayer and Jordi Pons-Prats

Received: 28 January 2023 Revised: 5 March 2023 Accepted: 6 March 2023 Published: 9 March 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

of various shapes in the literature. Although surrogate-assisted optimization is a visible trend nowadays, conventional direct strategies are still present in recent investigations. For instance, Furlan et al. [4] applied a Genetic Algorithm (GA) [5] to optimize an S-shaped channel parameterized using Bézier curves. D'Ambros et al. [6] employed the Tabu Search algorithm supported by the Free-Form Deformation (FFD) technique to a multi-objective optimization problem of a generic S-duct. Zeng et al. [7] used the GA method to enhance the aerodynamic performance of an S-duct scoop inlet. Sharma and Baloni [8] solved a multi-objective optimization problem in a turbofan engine compressor transition S-duct using the Particle Swarm Optimization (PSO) technique [9].

All the abovementioned investigations were successful in improving the corresponding objectives; however, direct strategies come at a significant computational cost resulting from multiple objective function evaluations. This drawback occurs notably in aerodynamic problems for which objective values are determined using expensive Computational Fluid Dynamics (CFD) codes.

Using surrogate-based frameworks allows for a significant reduction in costly evaluations by approximating the objective functions with simple analytical representations. These substitutes are commonly referred to as metamodels [10,11] or surrogates [12–14]. In such a procedure, conventional optimization techniques (e.g., steepest descent [15], as well as various quasi-Newton and population-based methods) are used to search for a superior solution in an artificial response landscape. CFD solver calls are required predominantly to build a database necessary for the surrogate construction and, to some extent, for its subsequent improvement.

Surrogates are usually categorized by the class of mathematical functions used for their creation. Popular types supporting aerodynamic optimization problems are low-order polynomial Response Surface Models (RSM) [16], regression splines [17], and various Radial Basis Functions (RBF) [18]. Polynomial-based surrogates have the advantages of simplicity and ease of use, although they have limited capabilities to approximate complex objective functions. RBF-based metamodels, instead, can model functions of high curvature with reasonably higher fitting effort. More complex models, such as Artificial Neural Networks (ANN) [19], use a nonlinear regression process to fit the surrogate. Although they are very efficient in applications with numerous variables, the ANN training process might be computationally expensive as it requires a solution to a high-dimensional optimization problem.

A characteristic class of metamodels has the ability to consider a stochastic component in the function approximation to quantify the confidence of the surrogate predictions. Moreover, this property is extensively used to boost the global search by identifying areas in the objective space with high improvement potential. These models are predominantly based on the Gaussian Process (GP); among them, the Kriging surrogate [20–23] is the most widely exploited.

The abovementioned metamodels have received the most prominent attention in the field of aerodynamic shape optimization of various ducts and channels. Lu et al. [24] employed a third-order polynomial RSM to optimize an S-shaped compressor transition duct. The authors were successful in reducing the pressure losses along the channel, together with an improvement in outlet pressure and velocity distribution. In the problem of annular S-duct shape optimization, Immonen [25] used fourth- and fifth-order RSM to minimize energy loss and improve flow uniformity. The multi-objective study resulted in considerable refinement in both examined parameters.

An RBF-based metamodel served to approximate the objective function in the optimization study on a stealth aircraft diffusing S-duct performed by Gan et al. [26]. The problem of simultaneous maximization of the duct pressure recovery and minimization of total pressure distortion was solved by a GA performing a search in a surrogate-based space. A significant distortion coefficient improvement and slight pressure recovery factor improvement characterized the solution located by the optimizer. A similar GA-based approach was used by Donghai et al. [27]; however, the surrogate was constructed using

ANN. With such a method, the authors suppressed flow separation in a strutted annular S-duct, which resulted in a considerable reduction in total pressure loss.

Among contemporary literature sources on duct shape optimization, the use of Kriging has noticeable dominance, chiefly due to its ability to interpolate complex functions and inherent features supporting global optimization. Zerbinati et al. [28] used the so-called Multiple-Gradient Descent Algorithm to search for an optimum in an objective space approximated using the Kriging technique. The algorithm successfully reduced the pressure loss and velocity variance in an air-cooling S-duct. Verstraete et al. [29] compared the behavior of Kriging-based and ANN-based surrogates assisting a Differential Evolution (DE) algorithm [30] in the minimization of U-shaped cooling channel pressure loss. The authors reported a superior performance by the Kriging metamodel. The study was broadened by Verstraete and Li [31] to a multi-objective problem of pressure loss minimization and heat transfer maximization. Due to the previous research outcomes, Kriging solely assisted the DE algorithm. Despite having objectives of competing nature, the optimizer was successful in improving both measures. Koo et al. [32] executed a similar comparison of Kriging and ANN metamodels in a multi-objective optimization problem of a heat exchanger inlet duct. The study outcomes resulted in improved pressure loss and flow rate uniformity achieved by both metamodels, although preferences towards specific objectives were different for particular surrogates. Wang and Wang [33] employed a Kriging-assisted GA technique in a multi-objective optimization problem of a UAV S-shaped intake duct. The authors were successful in the combined improvement of the aerodynamic and electromagnetic performance of the S-duct diffuser.

The studies referenced in the previous paragraph used merely Kriging-based objective landscape approximation to search for the optimum. The full potential of Kriging, however, manifests itself in its ability to not only predict the objective value but also assess the uncertainty of this prediction. This property is extensively used to balance local and global searches by probing the objective space in regions with a high likelihood of improvement. The search strategy based on this unique Kriging feature constitutes the Efficient Global Optimization (EGO) proposed by Jones et al. [23]. This algorithm is proven to balance the exploration and exploitation properties efficiently and, as such, has been applied in numerous shape optimization studies.

Bea et al. [34] used the Kriging-based EGO strategy to successfully improve the pressure recovery factor in a diffusing S-duct. A similar objective measure was optimized by Dehghani et al. [35]. The authors employed EGO to enhance the performance of an axisymmetric diffuser channel. Marchlewski et al. [36] employed a similar search technique utilizing Kriging's uncertainty assessment in multi-objective optimization of U-shaped engine intake. Using Pareto front delimitation in an objective space, the authors reduced the duct pressure loss while maintaining the initial level of exiting flow uniformity. Dr˛ezek ˙ et al. [37] optimized analogous intake geometry using GA-assisted EGO. The two objectives were combined using the Achievement Scalarizing Function (ASF) [38]. The optimizer was successful in the simultaneous improvement of the duct's pressure loss and flow distortion.

Every aerodynamic optimization problem requires design variables representing geometry as the subject of improvement. Expressing generic models using independent parameters (variables) is usually referred to as parameterization, for which a variety of methods is possible. The choice of a specific technique may profoundly impact the optimization process's computational cost and the final result. The vast majority of literature sources referenced above use parameterization techniques that focus on modifying geometry definition, namely, direct engineering parameters [24,32], Bézier curves [4,8], B-splines [27–29,31], Non-Uniform Rational B-Splines (NURBS) [7,35], and Free-Form Deformation [6]. Such an approach requires a subsequent mesh regeneration at each reshaping step. The mesh morphing technique is an efficient alternative to reduce the overall computational cost of the process. This method requires only initial generation of a mesh, which is adjusted in the subsequent steps while preserving the original grid topology. Although some recent literature sources describe the combined use of surrogates and mesh morphing (applied

to, e.g., fuselage–wing junction [39,40], generic car model [40], effusion cooling plate [41], and cooling channel rib [42]), only a few studies report such an application to intake duct optimization. To the best of our knowledge, these are [36,37].

Most intake duct optimization studies available in the literature consider only one operating condition, usually the nominal on-design point. Such an approach is commonly called Single-Point Optimization (SPO). As the conditions may change substantially at various mission stages, such a strategy may lead to a suboptimal performance at offdesign points. Some studies (e.g., [33,43]) evaluate the off-design conditions in the postoptimization phase to secure the design against severe performance deterioration. A few literature sources report simultaneous optimization under multiple flight conditions—often referred to as Multi-Point Optimization (MPO). Brahmachary et al. [44] and Fujio and Ogawa [45] executed multi-point studies on axisymmetric scramjet intake. The authors employed a surrogate-assisted GA algorithm to improve multiple performance parameters via Pareto front evaluation. Chiang et al. [46] enhanced the shape of a Boundary-Layer Ingesting (BLI) engine S-duct intake. The performance objectives for cruise, descent, and climb conditions were combined using a simple weighted sum function. The foreseen importance of MPO in prospective design processes should lead to investigations into advanced scalarization methods allowing simultaneous involvement of multiple objectives and flight conditions while overcoming the well-known deficiencies of the weighted sum technique. Such studies are absent in the state-of-the-art literature.

This paper concentrates on designing an optimization scheme combining the advantages of the Kriging metamodel, mesh morphing technique, and advanced objectives scalarization method. The components are duly integrated to create a synergistic effect of improvements in the economic efficiency of the design process. To assess the usefulness of the proposed framework in practical engineering problems, the algorithm is applied to multi-point aerodynamic optimization of an I-31T turboprop aircraft's air intake. The procedure simultaneously reduces the air-duct pressure losses and mitigates the flow distortion at the engine compressor's front face while considering multiple flight conditions.

The novelty of this study is the use of the augmented Chebyshev-type ASF to combine multiple performance objectives under multiple flight conditions. Such a strategy integrated with the Kriging surrogate and the mesh morphing technique creates a synergistic effect of cost-efficient MPO.
