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23 pages, 17910 KiB

Open AccessArticle

Aerodynamic Robust Design Research Using Adjoint-Based Optimization under Operating Uncertainties

by Yuhang Ma, Jiecheng Du, Tihao Yang, Yayun Shi, Libo Wang and Wei Wang

Aerospace 2023, 10(10), 831; https://doi.org/10.3390/aerospace10100831 - 24 Sep 2023

Cited by 1 | Viewed by 1489

Abstract

Robust optimization design (ROD) is playing an increasingly significant role in aerodynamic shape optimization and aircraft design. However, an efficient ROD framework that couples uncertainty quantification (UQ) and a powerful optimization algorithm for three-dimensional configurations is lacking. In addition, it is very important [...] Read more.

Robust optimization design (ROD) is playing an increasingly significant role in aerodynamic shape optimization and aircraft design. However, an efficient ROD framework that couples uncertainty quantification (UQ) and a powerful optimization algorithm for three-dimensional configurations is lacking. In addition, it is very important to reveal the maintenance mechanism of aerodynamic robustness from the design viewpoint. This paper first combines gradient-based optimization using the discrete adjoint-based approach with the polynomial chaos expansion (PCE) method to establish the ROD framework. A flying-wing configuration is optimized using deterministic optimization and ROD methods, respectively. The uncertainty parameters are Mach and the angle of attack. The ROD framework with the mean as an objective achieves better robustness with a lower mean (6.7% reduction) and standard derivation (Std, 18.92% reduction) compared to deterministic results. Moreover, we only sacrifice a minor amount of the aerodynamic performance (an increment of 0.64 counts in the drag coefficient). In comparison, the ROD with Std as an objective obtains a very different result, achieving the lowest Std and largest mean The far-field drag decomposition method is applied to compute the statistical moment variation of drag components and reveal how the ROD framework adjusts the drag component to realize better aerodynamic robustness. The ROD with the mean as the objective decreases the statistical moment of each drag component to improve aerodynamic robustness. In contrast, the ROD with Std as an objective reduces Std significantly by maintaining the inverse correlation relationship between the induced drag and viscous drag with an uncertainty parameter, respectively. The established ROD framework can be applied to future engineering applications that consider uncertainties. The unveiled mechanism for maintaining aerodynamic robustness will help designers understand ROD results more deeply, enabling them to reasonably construct ROD optimization problems. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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21 pages, 1862 KiB

Open AccessArticle

Adjoint and Direct Characteristic Equations for Two-Dimensional Compressible Euler Flows

by Kevin Ancourt, Jacques Peter and Olivier Atinault

Aerospace 2023, 10(9), 797; https://doi.org/10.3390/aerospace10090797 - 12 Sep 2023

Cited by 1 | Viewed by 1322

Abstract

The method of characteristics is a classical method for gaining understanding in the solution of a partial differential equation. It has recently been applied to the adjoint equations of the 2D steady-state Euler equations and the first goal of this paper is to [...] Read more.

The method of characteristics is a classical method for gaining understanding in the solution of a partial differential equation. It has recently been applied to the adjoint equations of the 2D steady-state Euler equations and the first goal of this paper is to present a linear algebra analysis that greatly simplifies the discussion of the number of independent characteristic equations satisfied along a family of characteristic curves. This method may be applied for both the direct and the adjoint problem. Our second goal is to directly derive in conservative variables the characteristic equations of 2D compressible inviscid flows. Finally, the theoretical results are assessed for a nozzle flow with a classical scheme and its dual consistent discrete adjoint. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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39 pages, 2760 KiB

Open AccessArticle

Parameter-Free Shape Optimization: Various Shape Updates for Engineering Applications

by Lars Radtke, Georgios Bletsos, Niklas Kühl, Tim Suchan, Thomas Rung, Alexander Düster and Kathrin Welker

Aerospace 2023, 10(9), 751; https://doi.org/10.3390/aerospace10090751 - 25 Aug 2023

Cited by 6 | Viewed by 2502

Abstract

In the last decade, parameter-free approaches to shape optimization problems have matured to a state where they provide a versatile tool for complex engineering applications. However, sensitivity distributions obtained from shape derivatives in this context cannot be directly used as a shape update [...] Read more.

In the last decade, parameter-free approaches to shape optimization problems have matured to a state where they provide a versatile tool for complex engineering applications. However, sensitivity distributions obtained from shape derivatives in this context cannot be directly used as a shape update in gradient-based optimization strategies. Instead, an auxiliary problem has to be solved to obtain a gradient from the sensitivity. While several choices for these auxiliary problems were investigated mathematically, the complexity of the concepts behind their derivation has often prevented their application in engineering. This work aims to explain several approaches to compute shape updates from an engineering perspective. We introduce the corresponding auxiliary problems in a formal way and compare the choices by means of numerical examples. To this end, a test case and exemplary applications from computational fluid dynamics are considered. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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14 pages, 3178 KiB

Open AccessArticle

Aeroacoustic and Aerodynamic Adjoint-Based Shape Optimization of an Axisymmetric Aero-Engine Intake

by Morteza Monfaredi, Varvara Asouti, Xenofon Trompoukis, Konstantinos Tsiakas and Kyriakos Giannakoglou

Aerospace 2023, 10(9), 743; https://doi.org/10.3390/aerospace10090743 - 22 Aug 2023

Cited by 1 | Viewed by 1870

Abstract

A continuous adjoint-based aeroacoustic optimization, based on a hybrid model including the Ffowcs Williams–Hawkings (FW–H) acoustic analogy, to account for the multidisciplinary design of aero-engine intakes with an axisymmetric geometry, is presented. To optimize such an intake, the generatrix of its lips is [...] Read more.

A continuous adjoint-based aeroacoustic optimization, based on a hybrid model including the Ffowcs Williams–Hawkings (FW–H) acoustic analogy, to account for the multidisciplinary design of aero-engine intakes with an axisymmetric geometry, is presented. To optimize such an intake, the generatrix of its lips is parameterized using B-Splines, and the energy contained in the sound pressure spectrum, at the blade passing frequency at receivers located axisymmetrically around the axis of the engine, is minimized. The engine is not included in the optimization and manifests its presence through an independently computed time-series of static pressure over the annular boundary of the simulation domain that corresponds to the inlet to the fan. Taking advantage of the case axisymmetry, the steady 3D RANS equations are solved in the rotating frame of reference and post-processed to compute the flow quantities’ time-series required by the FW–H analogy. The numerical solution of the unsteady flow equations and the otherwise excessive overall cost of the optimization are, thus, avoided. The objective function gradient is computed using the continuous adjoint method, coupled with the analytical differentiation of the FW–H analogy. The adjoint equations are also solved in the rotating frame via steady solver. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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25 pages, 11448 KiB

Open AccessArticle

Effects of Static Stability Margin on Aerodynamic Design Optimization of Truss-Braced Wing Aircraft

by Li Li, Lei Qiao, Jiakuan Xu and Junqiang Bai

Aerospace 2023, 10(7), 603; https://doi.org/10.3390/aerospace10070603 - 30 Jun 2023

Cited by 1 | Viewed by 3160

Abstract

Currently, the aviation industry is facing an oil and energy crisis and is contributing much more greenhouse gas emissions to the environment. Aircraft design approaches, such as aerodynamic shape optimization, new configuration concepts, and active control technology, have been the primary and effective [...] Read more.

Currently, the aviation industry is facing an oil and energy crisis and is contributing much more greenhouse gas emissions to the environment. Aircraft design approaches, such as aerodynamic shape optimization, new configuration concepts, and active control technology, have been the primary and effective means of achieving goals concerning fuel burn, noise, and emissions. For now, the design problems of relaxed static stability (RSS, an active control technique) and truss-braced wing (TBW) configurations with high-fidelity aerodynamic shape optimization methods have been investigated widely to promote aerodynamic performance. Nevertheless, they are studied almost always separately, and the combination of exploration and refined design is rarely presented. Therefore, the purposes of this work are to evaluate the benefits of RSS on a full TBW wing–body–tail configuration under various flight conditions and the effects on multi-components and to further explore the potential and analyze the aerodynamic features with the combination of shape optimization and RSS. To address these issues, on the one hand, a range of seven static stability margins are adopted to evaluate its effects with a high-fidelity Reynolds-averaged Navier–Stokes solver. On the other hand, seven cases of drag minimization multipoint aerodynamic design optimization are performed, which are with 600 shape variables and 13 twist variables, subject to lift coefficient, trim, and thickness constraints. The results indicate that with RSS only, the initial configuration has a 2.39% drag reduction under cruise conditions and a 3.01% and a 5.24% drag reduction under two off-design conditions. Additionally, the effects on the multi-components are observed and analyzed. Moreover, all of the optimized configurations with RSS have 2.13%, 2.42%, and 2.12% drag reductions under cruise conditions, drag divergence conditions, and near-buffet-onset conditions, respectively. The most promising optimized configuration has a lift-to-drag ratio of 24.48 with an aerodynamic efficiency of 17.14. The evaluations with a series of off-design points also present high-level aerodynamic efficiency. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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17 pages, 5076 KiB

Open AccessArticle

Shape Transformation Approaches for Fluid Dynamic Optimization

by Peter Marvin Müller, Georgios Bletsos and Thomas Rung

Aerospace 2023, 10(6), 519; https://doi.org/10.3390/aerospace10060519 - 31 May 2023

Cited by 1 | Viewed by 1523

Abstract

The contribution is devoted to combined shape- and mesh-update strategies for parameter-free (CAD-free) shape optimization methods. Three different strategies to translate the shape sensitivities computed by adjoint shape optimization procedures into simultaneous updates of both the shape and the discretized domain are employed [...] Read more.

The contribution is devoted to combined shape- and mesh-update strategies for parameter-free (CAD-free) shape optimization methods. Three different strategies to translate the shape sensitivities computed by adjoint shape optimization procedures into simultaneous updates of both the shape and the discretized domain are employed in combination with a mesh-morphing strategy. Considered methods involve a linear Steklov–Poincaré (Hilbert space) approach, a recently suggested highly non-linear p-Laplace (Banach space) method, and a hybrid variant which updates the shape in Hilbert space. The methods are scrutinized for optimizing the power loss of a two-dimensional bent duct flow using an unstructured, locally refined grid that initially displays favorable grid properties. Optimization results are compared with respect to the optimization convergence, the computational effort, and the preservation of the mesh quality during the optimization sequence. Results indicate that all methods reach, approximately, the same converged optimal solution, which reduces the objective function by about 18% for this classical benchmark example. However, as regards the preservation of the mesh quality, more advanced Banach space methods are advantageous in comparison to Hilbert space methods even when the shape update is performed in Hilbert space to save costs. In specific, while the computational cost of the Banach space method and the hybrid method is about 3.5 and 2.5 times the cost of the pure Hilbert space method, respectively, the grid quality metrics are 2 times and 1.7 times improved for the Banach space and hybrid method, respectively. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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35 pages, 25211 KiB

Open AccessArticle

Aerodynamic Optimization Design of Supersonic Wing Based on Discrete Adjoint

by Hanyue Rao, Yayun Shi, Junqiang Bai, Yifu Chen, Tihao Yang and Junfu Li

Aerospace 2023, 10(5), 420; https://doi.org/10.3390/aerospace10050420 - 29 Apr 2023

Cited by 1 | Viewed by 3139

Abstract

Reducing fuel consumption and improving the economy by effectively reducing cruising drag is the main objective of the aerodynamic design of supersonic civil aircraft. In this paper, the aerodynamic optimization design system based on the Reynolds-Averaged Navier–Stokes (RANS) equation and discrete adjoint theory [...] Read more.

Reducing fuel consumption and improving the economy by effectively reducing cruising drag is the main objective of the aerodynamic design of supersonic civil aircraft. In this paper, the aerodynamic optimization design system based on the Reynolds-Averaged Navier–Stokes (RANS) equation and discrete adjoint theory is applied to supersonic wing design. Based on this system, a single-point optimization design study of aerodynamic drag reduction in cruise conditions was carried out for two typical supersonic wing layouts, subsonic leading edge and supersonic leading edge, and the drag reduction reached 3.78% and 4.53%, respectively. The aerodynamic design characteristics of different types of supersonic wings were explored from the perspectives of wing load, twist angle distribution, pressure distribution, airfoil shape characteristics, and flow field characteristics. The optimization results show that the drag reduction of the subsonic leading edge configuration is dominated by the induced drag, while the optimizer mainly focuses on reducing the shock wave drag for the supersonic leading edge configuration. By comparing the sensitivity analysis of lift and drag coefficients to airfoil deformation with the optimization results, the optimized dominant directions of two types of supersonic wings are qualitatively analyzed. The derivatives obtained from discrete adjoint equations are useful to elaborate the design tendency and the reason for the trade-off generation of supersonic wings under specific layouts and engineering constraints, which provides a reference for the design of supersonic wings in the future. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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21 pages, 3929 KiB

Open AccessArticle

Explaining the Lack of Mesh Convergence of Inviscid Adjoint Solutions near Solid Walls for Subcritical Flows

by Carlos Lozano and Jorge Ponsin

Aerospace 2023, 10(5), 392; https://doi.org/10.3390/aerospace10050392 - 24 Apr 2023

Viewed by 1481

Abstract

Numerical solutions to the adjoint Euler equations have been found to diverge with mesh refinement near walls for a variety of flow conditions and geometry configurations. The issue is reviewed, and an explanation is provided by comparing a numerical incompressible adjoint solution with [...] Read more.

Numerical solutions to the adjoint Euler equations have been found to diverge with mesh refinement near walls for a variety of flow conditions and geometry configurations. The issue is reviewed, and an explanation is provided by comparing a numerical incompressible adjoint solution with an analytic adjoint solution, showing that the anomaly observed in numerical computations is caused by a divergence of the analytic solution at the wall. The singularity causing this divergence is of the same type as the well-known singularity along the incoming stagnation streamline, and both originate at the adjoint singularity at the trailing edge. The argument is extended to cover the fully compressible case, in subcritical flow conditions, by presenting an analytic solution that follows the same structure as the incompressible one. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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29 pages, 43083 KiB

Open AccessArticle

Adjoint-Based Aerodynamic Design Optimization and Drag Reduction Analysis of a Military Transport Aircraft Afterbody

by Hanyue Rao, Yifu Chen, Yayun Shi, Tihao Yang and Hongyang Liu

Aerospace 2023, 10(4), 331; https://doi.org/10.3390/aerospace10040331 - 27 Mar 2023

Cited by 2 | Viewed by 2352

Abstract

Based on the adjoint method, the afterbody of a military transport aircraft was optimized and designed to meet engineering constraints under real flight conditions. Guidance for the key design parameters of the afterbody of the military transport aircraft is given. The vortex dynamics [...] Read more.

Based on the adjoint method, the afterbody of a military transport aircraft was optimized and designed to meet engineering constraints under real flight conditions. Guidance for the key design parameters of the afterbody of the military transport aircraft is given. The vortex dynamics and boundary layer extraction methods were used to analyze the optimization results of military transport aircraft. It was found that, upstream of the vortex shedding point, the circumferential accumulation process of the vorticity is weakened. The position of the vortex shedding and the appearance of the saddle line are delayed by reducing the circumferential inverse pressure gradient and the intensity of the crossflow. The afterbody vortex system of the optimized configuration is further away from the surface. Meanwhile, the distance between the counter-rotated vortex decreases, and the upwashing speed of the vortex core is smaller. Therefore, vortex-induced drag is reduced. Finally, compared with the initial configuration, the optimized configuration has a relative drag reduction of 23.2%. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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18 pages, 1883 KiB

Open AccessArticle

Uncertainty Quantification of Compressor Map Using the Monte Carlo Approach Accelerated by an Adjoint-Based Nonlinear Method

by Shenren Xu, Qian Zhang, Dingxi Wang and Xiuquan Huang

Aerospace 2023, 10(3), 280; https://doi.org/10.3390/aerospace10030280 - 11 Mar 2023

Cited by 3 | Viewed by 1912

Abstract

Precise and inexpensive uncertainty quantification (UQ) is crucial for robust optimization of compressor blades and to control manufacturing tolerances. This study looks into the suitability of MC−adj−nonlinear, a nonlinear adjoint-based approach, to precisely and rapidly assess the performance discrepancies of a transonic compressor [...] Read more.

Precise and inexpensive uncertainty quantification (UQ) is crucial for robust optimization of compressor blades and to control manufacturing tolerances. This study looks into the suitability of MC−adj−nonlinear, a nonlinear adjoint-based approach, to precisely and rapidly assess the performance discrepancies of a transonic compressor blade section, arising from geometric alterations, and building upon previous research. In order to assess the practicality and illustrate the benefits of the adjoint-based nonlinear approach, its proficiency and precision are gauged against two other methodologies, the adjoint-based linear approach (MC−adj−linear) and the high-fidelity nonlinear Computational Fluid Dynamics (MC−CFD) method. The MC−adj−nonlinear methodology exhibits impressive generalization capabilities. The MC−adj−nonlinear method offers a great balance between precision and time efficiency, since it is more precise than the MC−adj−linear method in both design and near-stall conditions, yet requires approximately a thirtieth of the time of the MC−CFD method. Finally, the MC−adj−nonlinear method was utilized to conduct fast UQ analyses of the section at four distinct speeds to quantify the performance uncertainty for the compressor map. It is found that aerodynamic performance is more sensitive to geometric deviations at high speeds than at low speeds. The impact of the geometric deviations is generally detrimental to the mean efficiency. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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19 pages, 7322 KiB

Open AccessArticle

Shock Equations and Jump Conditions for the 2D Adjoint Euler Equations

by Carlos Lozano and Jorge Ponsin

Aerospace 2023, 10(3), 267; https://doi.org/10.3390/aerospace10030267 - 10 Mar 2023

Cited by 1 | Viewed by 1571

Abstract

This paper considers the formulation of the adjoint problem in two dimensions when there are shocks in the flow solution. For typical cost functions, the adjoint variables are continuous at shocks, wherein they have to obey an internal boundary condition, but their derivatives [...] Read more.

This paper considers the formulation of the adjoint problem in two dimensions when there are shocks in the flow solution. For typical cost functions, the adjoint variables are continuous at shocks, wherein they have to obey an internal boundary condition, but their derivatives may be discontinuous. The derivation of the adjoint shock equations is reviewed and detailed predictions for the behavior of the gradients of the adjoint variables at shocks are obtained as jump conditions for the normal adjoint gradients in terms of the tangent gradients. Several numerical computations on a very fine mesh are used to illustrate the behavior of numerical adjoint solutions at shocks. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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26 pages, 2553 KiB

Open AccessArticle

Multi-Row Turbomachinery Aerodynamic Design Optimization by an Efficient and Accurate Discrete Adjoint Solver

by Hangkong Wu, Xuanlong Da, Dingxi Wang and Xiuquan Huang

Aerospace 2023, 10(2), 106; https://doi.org/10.3390/aerospace10020106 - 21 Jan 2023

Cited by 6 | Viewed by 1946

Abstract

This paper proposes an approach that combines manual differentiation (MD) and automatic differentiation (AD) to develop an efficient and accurate multi-row discrete adjoint solver. In this approach, the structures of adjoint codes generated using an AD tool are first analyzed. Then, the AD-generated [...] Read more.

This paper proposes an approach that combines manual differentiation (MD) and automatic differentiation (AD) to develop an efficient and accurate multi-row discrete adjoint solver. In this approach, the structures of adjoint codes generated using an AD tool are first analyzed. Then, the AD-generated codes are manually adjusted to reduce memory and CPU time consumption. This manual adjustment is performed by replacing the automatically generated low-efficient differentiated codes with manually developed ones. To demonstrate the effectiveness of the proposed approach, the single-stage transonic compressor–NASA Stage 35 and the 1.5-stage Aachen turbine–are used. The solution information exchange at a rotor-stator/stator-rotor interface is achieved by a conservative, non-reflective, and robust discrete adjoint mixing plane method. The results show that the discrete adjoint solver developed by hybrid automatic and manual differentiation is more economical in computational cost than that developed purely by an AD tool and has higher sensitivity accuracy than the adjoint solver with the constant eddy viscosity (CEV) assumption. Moreover, the multi-row turbomachinery design optimizations can be efficiently performed by the discrete adjoint solver developed by the hybrid automatic and manual differentiation. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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Review

Jump to: Research

38 pages, 1935 KiB

Open AccessReview

A Review of Solution Stabilization Techniques for RANS CFD Solvers

by Shenren Xu, Jiazi Zhao, Hangkong Wu, Sen Zhang, Jens-Dominik Müller, Huang Huang, Mohammad Rahmati and Dingxi Wang

Aerospace 2023, 10(3), 230; https://doi.org/10.3390/aerospace10030230 - 26 Feb 2023

Cited by 3 | Viewed by 3515

Abstract

Nonlinear, time-linearized and adjoint Reynolds-averaged Navier-Stokes (RANS) computational fluid dynamics (CFD) solvers are widely used to assess and improve the aerodynamic and aeroelastic performance of aircrafts and turbomachines. While RANS CFD solver technologies are relatively mature for applications at design conditions where the [...] Read more.

Nonlinear, time-linearized and adjoint Reynolds-averaged Navier-Stokes (RANS) computational fluid dynamics (CFD) solvers are widely used to assess and improve the aerodynamic and aeroelastic performance of aircrafts and turbomachines. While RANS CFD solver technologies are relatively mature for applications at design conditions where the flow is benign, their use in off-design conditions, featuring flow instabilities, such as separations and shock wave/boundary layer interactions, still faces many challenges, with tight residual convergence being a major difficulty. To cope with this, several solver stabilization techniques have been proposed. However, a systematic and comparative study of these techniques has not been reported, to some extent hindering the wide deployment of these methods for industrial applications. In this paper, we critically review the existing methods for solver convergence stabilization, with the main purpose of explaining the rationale behind the algorithms and providing a systematic view of the seemingly different methods. Specifically, mathematical formulations and implementation details of these methods, example applications, and the pros and cons of the methods are discussed in detail, along with suggestions for further improvements. This review is expected to give CFD method developers an overview of the various solution stabilization methods and application engineers an idea how to choose a suitable method for their respective applications. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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