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Journal articles on the topic 'Adjoint-Based Optimization'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Qiu, Sheng. "A continuous adjoint-based aeroacoustic shape optimization for multi-mode duct acoustics." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 232, no. 21 (2017): 3897–914. http://dx.doi.org/10.1177/0954406217743273.

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A multi-mode adjoint-based optimization method is proposed for the noise reduction optimization in multi-mode duct acoustics problems. The objective is to minimize the amplitude of sound from an inlet duct on the wall and integral line while maintaining the aerodynamic performance. The complete detailed derivation of the adjoint equations and their corresponding adjoint boundary conditions are presented firstly based on the multi-mode linear Euler equations. With the solved adjoint variables, the final expression of the cost function gradient with respect to the design variables is formulated.

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Batay, Sagidolla, Bagdaulet Kamalov, Dinmukhamed Zhangaskanov, et al. "Adjoint-Based High-Fidelity Concurrent Aerodynamic Design Optimization of Wind Turbine." Fluids 8, no. 3 (2023): 85. http://dx.doi.org/10.3390/fluids8030085.

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To evaluate novel turbine designs, the wind energy sector extensively depends on computational fluid dynamics (CFD). To use CFD in the design optimization process, where lower-fidelity approaches such as blade element momentum (BEM) are more popular, new tools to increase the accuracy must be developed as the latest wind turbines are larger and the aerodynamics and structural dynamics become more complex. In the present study, a new concurrent aerodynamic shape optimization approach towards multidisciplinary design optimization (MDO) that uses a Reynolds-averaged Navier–Stokes solver in conjun

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Karpouzas, Georgios K., Evangelos M. Papoutsis-Kiachagias, Thomas Schumacher, Villers Eugene de, Kyriakos C. Giannakoglou, and Carsten Othmer. "Adjoint Optimization for Vehicle External Aerodynamics." International Journal of Automotive Engineering 7, no. 1 (2015): 1–7. https://doi.org/10.5281/zenodo.808434.

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Adjoint optimization is an exciting and fast growing field that has many applications in the automotive industry. We focus on the usage of adjoint for the optimization of aerodynami performance. While adjoint methods have come to the attention of mainstream CFD through inclusion in prominent commercial codes, most of the available tools are severely limited, precluding production use in this field. We detail a methodology that is based on the continuous adjoint method and is implemented in an Open Source framework. While more mathematically demanding in terms of its derivation, the continuous

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Nikolova, N. K., R. Safian, E. A. Soliman, M. H. Bakr, and J. W. Bandler. "Accelerated Gradient Based Optimization Using Adjoint Sensitivities." IEEE Transactions on Antennas and Propagation 52, no. 8 (2004): 2147–57. http://dx.doi.org/10.1109/tap.2004.832313.

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Bunting, Gregory, and Timothy F. Walsh. "Adjoint-based, large-scale optimization of metamaterials." Journal of the Acoustical Society of America 146, no. 4 (2019): 2830. http://dx.doi.org/10.1121/1.5136805.

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Wang, Jiahui, Yu Shi, Tyler Hughes, Zhexin Zhao, and Shanhui Fan. "Adjoint-based optimization of active nanophotonic devices." Optics Express 26, no. 3 (2018): 3236. http://dx.doi.org/10.1364/oe.26.003236.

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7

Nabi, S., P. Grover, and C. P. Caulfield. "Adjoint-based optimization of displacement ventilation flow." Building and Environment 124 (November 2017): 342–56. http://dx.doi.org/10.1016/j.buildenv.2017.07.030.

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Tsiakas, Konstantinos, Xenofon Trompoukis, Varvara Asouti, et al. "Discrete and Continuous Adjoint-Based Aerostructural Wing Shape Optimization of a Business Jet." Fluids 9, no. 4 (2024): 87. http://dx.doi.org/10.3390/fluids9040087.

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This article presents single- and multi-disciplinary shape optimizations of a generic business jet wing at two transonic cruise flow conditions. The studies performed are based on two high-fidelity gradient-based optimization tools, assisted by the adjoint method (following both discrete and continuous approaches). Single discipline and coupled multi-disciplinary sensitivity derivatives computed from the two tools are compared and verified against finite differences. The importance of not making the frozen turbulence assumption in adjoint-based optimization is demonstrated. Then, a number of o

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Hekmat, Mohamad Hamed, and Masoud Mirzaei. "Development of Discrete Adjoint Approach Based on the Lattice Boltzmann Method." Advances in Mechanical Engineering 6 (January 1, 2014): 230854. http://dx.doi.org/10.1155/2014/230854.

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The purpose of this research is to present a general procedure with low implementation cost to develop the discrete adjoint approach for solving optimization problems based on the LB method. Initially, the macroscopic and microscopic discrete adjoint equations and the cost function gradient vector are derived mathematically, in detail, using the discrete LB equation. Meanwhile, for an elementary case, the analytical evaluation of the macroscopic and microscopic adjoint variables and the cost function gradients are presented. The investigation of the derivation procedure shows that the simplici

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10

Yang, L., and J. Yang. "Gradient-based aerodynamic shape optimization using a discrete adjoint approach on a graphics processing unit." Journal of Physics: Conference Series 2784, no. 1 (2024): 012008. http://dx.doi.org/10.1088/1742-6596/2784/1/012008.

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Abstract This paper presents a gradient-based aerodynamic shape optimization framework that utilizes a Graphics Processing Unit (GPU) for solving both flow and adjoint equations. It is built based on a GPU-accelerated flow solver that has been developed previously. Hence, the focus of this work is on how to solve the adjoint equations on the GPU and subsequently compute the gradients. The adjoint equations are right-preconditioned by a block Incomplete Lower Upper (ILU) preconditioner and solved by a restarted Generalized Minimum Residual (GMRES) method. The exact residual Jacobian matrix in t

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van Stratan, G. S. M., Sukanta Roy, and Yam Ke San. "Aerodynamic Shape Optimization of a NACA0018 Airfoil Using Adjoint Method and Gradient-Based Optimizer." MATEC Web of Conferences 377 (2023): 01016. http://dx.doi.org/10.1051/matecconf/202337701016.

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The purpose of this study is to demonstrate the suitability and efficacy of the adjoint method on the aerodynamic shape optimization on a simple symmetrical airfoil NACA0018 at low Reynolds number for wind turbine application. The adjoint method has been used in many pressure-based numerical simulations with various degrees of success leading to optimized geometries in their respective uses. ANSYS Fluent code was used in this simulation. Lift to drag ratio was defined as the observables for which adjoint sensitivities were formulated. The objective function of the optimization was set to maxim

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Chen, Lei, and Jiang Chen. "Continuous adjoint aerodynamic optimization design for multi-stage gas turbine with cooling air." Aircraft Engineering and Aerospace Technology 89, no. 3 (2017): 444–56. http://dx.doi.org/10.1108/aeat-05-2014-0058.

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Purpose This paper aims to conduct the optimization of the multi-stage gas turbine with the effect of the cooling air injection based on the adjoint method. Design/methodology/approach Continuous adjoint method is combined with the S2 surface code. Findings The optimization of the stagger angles, stacking lines and the passage can improve the attack angles and restrain the development of the boundary, reducing the secondary flow loss caused by the cooling air injection. Practical implications The aerodynamic performance of the gas turbine can be improved via the optimization of blade and passa

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13

Fan, Yan Hong. "Improving the Robustness of Adjoint-Based Airfoil Drag Reduction Design." Applied Mechanics and Materials 444-445 (October 2013): 259–63. http://dx.doi.org/10.4028/www.scientific.net/amm.444-445.259.

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The effect of steps in the line search on the consistence of adjoint-based drag reduction of airfoil is investigated in this paper. Constant step adopted in drag reduction design usually gives different optimization results and choice of constant step often depends on designers experience and optimization problem. Bracket method is applied to automatically give the optimal step in performing drag reduction design of airfoils RAE2822 and S73613 and the consistent optimal results are obtained. The results illustrate that the linear search method can automatically find the optimal step, and overc

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King, Ryan N., Katherine Dykes, Peter Graf, and Peter E. Hamlington. "Optimization of wind plant layouts using an adjoint approach." Wind Energy Science 2, no. 1 (2017): 115–31. http://dx.doi.org/10.5194/wes-2-115-2017.

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Abstract. Using adjoint optimization and three-dimensional steady-state Reynolds-averaged Navier–Stokes (RANS) simulations, we present a new gradient-based approach for optimally siting wind turbines within utility-scale wind plants. By solving the adjoint equations of the flow model, the gradients needed for optimization are found at a cost that is independent of the number of control variables, thereby permitting optimization of large wind plants with many turbine locations. Moreover, compared to the common approach of superimposing prescribed wake deficits onto linearized flow models, the c

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Abdulla, Mohammed, Seraj Alzhrani, Khalid A. Juhany, and Ibraheem AlQadi. "Application of Adjoint Aerodynamics Optimization for a High-Speed Vehicle Moving in a Tube." Journal of Physics: Conference Series 2772, no. 1 (2024): 012012. http://dx.doi.org/10.1088/1742-6596/2772/1/012012.

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Abstract This work explores the application of adjoint-based optimization for a self-propelled vehicle moving in a tube at a low-pressure of 10 KPa with a small suspension gap (d) between the vehicle and the tube. The objective is to minimize the drag force for air-ingested vehicles. The thrust was a design constraint to maintain the cruise conditions. The nose and tail parts of the model were parametrized using elliptical approximation. Gradient-based optimization was utilized in the adjoint-based optimization to deform the vehicle’s external shape. The optimization remarkably enhanced the ae

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Liu, Xiaodong, Peiliang Zhang, Guanghong He, Yongen Wang, and Xudong Yang. "Multi-objective aerodynamic optimization of flying-wing configuration based on adjoint method." Xibei Gongye Daxue Xuebao/Journal of Northwestern Polytechnical University 39, no. 4 (2021): 753–60. http://dx.doi.org/10.1051/jnwpu/20213940753.

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In order to solve the multi-objective multi-constraint design in aerodynamic design of flying wing, the aerodynamic optimization design based on the adjoint method is studied. In terms of the principle of the adjoint equation, the boundary conditions and the gradient equations are derived. The Navier-Stokes equations and adjoint aerodynamic optimization design method are adopted, the optimization design of the transonic drag reduction for the two different aspect ratio of the flying wing configurations is carried out. The results of the optimization design are as follows: Under the condition o

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17

Trompoukis, Xenofon, Konstantinos Tsiakas, Varvara Asouti, Marina Kontou, and Kyriakos Giannakoglou. "Continuous Adjoint-Based Optimization of an Internally Cooled Turbine Blade—Mathematical Development and Application." International Journal of Turbomachinery, Propulsion and Power 6, no. 2 (2021): 20. http://dx.doi.org/10.3390/ijtpp6020020.

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This paper presents an adjoint-based shape optimization framework and its demonstration in a conjugate heat transfer problem in a turbine blading. The gradient of the objective function is computed based on the continuous adjoint method, which also includes the adjoint to the turbulence model. Differences in the gradient resulting from making the frozen turbulence assumption are discussed. The developed software was used to optimize both the blade shape of the internally cooled linear C3X turbine blade and the position of cooling channels aiming at (a) minimum total pressure drop of the hot ga

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18

Şenol, Niyazi, Hasan U. Akay, and Şahin Yiğit. "A Gradient Enhanced Efficient Global Optimization-Driven Aerodynamic Shape Optimization Framework." Aerospace 12, no. 7 (2025): 644. https://doi.org/10.3390/aerospace12070644.

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The aerodynamic optimization of airfoil shapes remains a critical research area for enhancing aircraft performance under various flight conditions. In this study, the RAE 2822 airfoil was selected as a benchmark case to investigate and compare the effectiveness of surrogate-based methods under an Efficient Global Optimization (EGO) framework and an adjoint-based approach in both single-point and multi-point optimization settings. Prior to optimization, the computational fluid dynamics (CFD) model was validated against experimental data to ensure accuracy. For the surrogate-based methods, Krigi

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He, Yijun, Jifang Qiu, Bowen Zhang, et al. "Experimental demonstration of a performance-enhanced optical quantizer based on adjoint shape optimization." Chinese Optics Letters 23, no. 4 (2025): 041301. https://doi.org/10.3788/col202523.041301.

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Wei, Jiasen, Jan Oscar Pralits, and Alessandro Bottaro. "Optimization of a thermoacoustic system." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 265, no. 7 (2023): 552–63. http://dx.doi.org/10.3397/in_2022_0076.

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Clean combustion such as hydrogen combustion for the reduction of NOx emission is prone to thermoacoustic instabilities, which may cause structural vibrations and equipment failures. In the present work, an adjoint-based sensitivity analysis is applied to a low-order thermoacoustic model with a non-zero mean flow. The adjoint-based analysis can be a powerful tool to yield information for optimization techniques to eliminate or control such instabilities. With quick and low-cost calculations, structural sensitivity and base-state sensitivity are achieved and the conditions most sensitive to ins

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Hu, Rui, Antony Jameson, and Qiqi Wang. "Adjoint-Based Aerodynamic Optimization of Supersonic Biplane Airfoils." Journal of Aircraft 49, no. 3 (2012): 802–14. http://dx.doi.org/10.2514/1.c031417.

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22

Zandvliet, Maarten, Martijn Handels, Gijs van Essen, Roald Brouwer, and Jan-Dirk Jansen. "Adjoint-Based Well-Placement Optimization Under Production Constraints." SPE Journal 13, no. 04 (2008): 392–99. http://dx.doi.org/10.2118/105797-pa.

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Summary Determining the optimal location of wells with the aid of an automated search method can significantly increase a project's net present value (NPV) as modeled in a reservoir simulator. This paper has two main contributions: first, to determine the effect of production constraints on optimal well locations and, second, to determine optimal well locations using a gradient-based optimization method. Our approach is based on the concept of surrounding the wells whose locations have to be optimized by so-called pseudowells. These pseudowells produce or inject at a very low rate and, thus, h

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Yuan, Shijin, Shicheng Wen, Hongyu Li, Xinfeng Zhang, and Qin Liu. "An optimization framework for adjoint-based climate simulations." International Journal of High Performance Computing Applications 28, no. 2 (2013): 174–82. http://dx.doi.org/10.1177/1094342013495096.

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24

Treweek, Benjamin C., Timothy Walsh, Clay M. Sanders, Wilkins Aquino, Samuel D. Parker, and Michael R. Haberman. "Adjoint-based optimization to minimize scattered pressure fields." Journal of the Acoustical Society of America 148, no. 4 (2020): 2637. http://dx.doi.org/10.1121/1.5147331.

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Carpentieri, G., B. Koren, and M. J. L. van Tooren. "Adjoint-based aerodynamic shape optimization on unstructured meshes." Journal of Computational Physics 224, no. 1 (2007): 267–87. http://dx.doi.org/10.1016/j.jcp.2007.02.011.

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Protas, Bartosz, and Wenyuan Liao. "Adjoint-based optimization of PDEs in moving domains." Journal of Computational Physics 227, no. 4 (2008): 2707–23. http://dx.doi.org/10.1016/j.jcp.2007.11.014.

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Maruyama, Takayuki, Kota Watanabe, and Hajime Igarashi. "A robust optimization based on adjoint variable method." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 29, no. 6 (2010): 1524–32. http://dx.doi.org/10.1108/03321641011078599.

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Li, D., and R. Hartmann. "Adjoint-based airfoil optimization with discretization error control." International Journal for Numerical Methods in Fluids 77, no. 1 (2014): 1–17. http://dx.doi.org/10.1002/fld.3971.

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Youness, Rtimi, and Frederic Messine. "An implementation of adjoint-based topology optimization in magnetostatics." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 38, no. 3 (2019): 1023–35. http://dx.doi.org/10.1108/compel-09-2018-0379.

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Purpose In magnetostatics, topology optimization (TO) addresses the problem of finding the distributions of both current densities and ferromagnetic materials to comply with fixed magnetic specifications. The purpose of this paper is to develop TO in order to design Hall-effect Thrusters (HETs). Design/methodology/approach In fact, TO problems are known to be large-scale optimization problems. The authors therefore adopt the adjoint method to reduce the computation time required to obtain the gradient information. In this paper, they illustrate the continuous variant of the adjoint method in t

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Qiu, S., WB Song, and H. Liu. "Shape optimization of a general bypass duct for tone noise reduction using continuous adjoint method." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 228, no. 1 (2013): 119–34. http://dx.doi.org/10.1177/0954406213481915.

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A novel continuous adjoint-based acoustic propagation method is proposed for low-noise turbofan duct design. A fan bypass duct tonal noise propagation model that is verified by comparison with an analytical solution of the modal radiation from a semi-infinite duct with the shear layer is enhanced with its continuous adjoint formulation, having been applied to design the bypass duct. First, this article presents the complete formulation of the time-dependent optimal design problem. Second, a continuous adjoint-based acoustic propagation method for two-dimensional bypass duct configurations is d

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Popovac, Mirza. "Continuous Adjoint Topology Optimization of Duct Flow Configurations with Explicit Volume Constraint for Design Variable Update." Energies 15, no. 19 (2022): 7349. http://dx.doi.org/10.3390/en15197349.

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The adjoint method is a very promising gradient-based optimisation framework for computational fluid dynamics (CFD), because of its independence of the computation cost from the number of design variables. With the aim of improving the robustness of the continuous adjoint topology optimization, this paper presents a direct method for the design variable update. Based on the intrinsic feature of the topology optimization, this straightforward update method explicitly controls the design variable target volume, precluding any case-dependent parameters. The details of its implementation are discu

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Xu, Zhaoke, and Jian Xia. "Aerodynamic Optimization Based on Continuous Adjoint Method for a Flexible Wing." International Journal of Aerospace Engineering 2016 (2016): 1–16. http://dx.doi.org/10.1155/2016/4706925.

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Aerodynamic optimization based on continuous adjoint method for a flexible wing is developed using FORTRAN 90 in the present work. Aerostructural analysis is performed on the basis of high-fidelity models with Euler equations on the aerodynamic side and a linear quadrilateral shell element model on the structure side. This shell element can deal with both thin and thick shell problems with intersections, so this shell element is suitable for the wing structural model which consists of two spars, 20 ribs, and skin. The continuous adjoint formulations based on Euler equations and unstructured me

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Elham, Ali, and Michel J. L. van Tooren. "Discrete adjoint aerodynamic shape optimization using symbolic analysis with OpenFEMflow." Structural and Multidisciplinary Optimization 63, no. 5 (2021): 2531–51. http://dx.doi.org/10.1007/s00158-020-02799-7.

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AbstractThe combination of gradient-based optimization with the adjoint method for sensitivity analysis is a very powerful and popular approach for aerodynamic shape optimization. However, differentiating CFD codes is a time consuming and sometimes a challenging task. Although there are a few open-source adjoint CFD codes available, due to the complexity of the code, they might not be very suitable to be used for educational purposes. An adjoint CFD code is developed to support students for learning adjoint aerodynamic shape optimization as well as developing differentiated CFD codes. To achie

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Li, Ming, Junqiang Bai, and Feng Qu. "Radar Cross Section Reduction and Shape Optimization using Adjoint Method and Automatic Differentiation." Applied Computational Electromagnetics Society 36, no. 3 (2021): 320–35. http://dx.doi.org/10.47037/2020.aces.j.360312.

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An efficient Radar Cross Section (RCS) gradient evaluation method based on the adjoint method is presented. The Method of Moments is employed to solve the Combined Field Integral Equation (CFIE) and the corresponding derivatives computing routines are generated by the program transformation Automatic Differentiation (AD) technique. The differential code is developed using three kinds of AD mode: tangent mode, multidirectional tangent mode, and adjoint mode. The differential code in adjoint mode is modified and optimized by changing the “two-sweeps” architecture into the “inner-loop two-sweeps”

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Qiu, S., H. Liu, and WP Li. "Turbofan duct geometry optimization for low noise using remote continuous adjoint method." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 229, no. 1 (2014): 69–90. http://dx.doi.org/10.1177/0954406214532631.

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In this paper, a remote continuous adjoint-based acoustic propagation (RABAP) method is proposed for low noise turbofan duct design. The goal is to develop a set of adjoint equations and their corresponding boundary conditions in order to quantify the influence of geometry modifications on the amplitude of sound pressure at a near-field location. The governing equations for the 2.5D acoustic perturbation equation solver (APE) formulation for duct acoustic propagation is first introduced. This is followed by the formulation and discretization of the remote continuous adjoint equations based on

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Ma, Chuang, Jiangtao Huang, Daochun Li, et al. "Discrete Adjoint Optimization Method for Low-Boom Aircraft Design Using Equivalent Area Distribution." Aerospace 11, no. 7 (2024): 545. http://dx.doi.org/10.3390/aerospace11070545.

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This paper introduces a low-boom aircraft optimization design method guided by equivalent area distribution, which effectively improves the intuitiveness and refinement of inverse design. A gradient optimization method based on discrete adjoint equations is proposed to achieve the fast solution of the gradient information of target equivalent area distribution relative to design variables and to drive the aerodynamic shape update to the optimal solution. An optimization experiment is carried out based on a self-developed supersonic civil aircraft configuration with engines. The results show th

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Jun, Deng, Gao Zhenghong, Huang Jiangtao, Zhao Ke, and Xia Lu. "Design method of aircraft boundary characteristics based on upwind adjoint equation." Journal of Applied Artificial Intelligence 1, no. 3 (2024): 192–206. http://dx.doi.org/10.59782/aai.v1i3.324.

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The boundary characteristics of an aircraft determine its safety and flight performance, and have always been the difficulty and focus of aircraft design. This paper aims to improve the accuracy, efficiency and robustness of the adjoint equation solution for complex flow problems by introducing the adjoint equation of the upwind format and different flux limiter processing forms, thereby expanding the application scope of the adjoint optimization method in the design of the boundary characteristics of aircraft. Firstly, the basic principle of discrete adjoint gradient solution is introduced. O

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Agarwal, Dheeraj, Christos Kapellos, Trevor T. Robinson, and Cecil G. Armstrong. "Using Parametric Effectiveness for Efficient CAD-Based Adjoint Optimization." Computer-Aided Design and Applications 16, no. 4 (2018): 703–19. https://doi.org/10.14733/cadaps.2019.703-719.

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Parametric effectiveness is a measure of the ability of the parameters defining a CAD model to be used for optimization. It compares the optimum change in performance that can be achieved using a CAD model’s parameterization, to the maximum performance improvement that could be obtained if the model is free to move. In this work, the approach is applied to the shape optimization of an S-Bend duct for minimizing the power-loss and an automotive car mirror for minimizing the noise perceived by the driver of the car.

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Ito, Shin-ichi, Takeru Matsuda, and Yuto Miyatake. "Adjoint-based exact Hessian computation." BIT Numerical Mathematics 61, no. 2 (2021): 503–22. http://dx.doi.org/10.1007/s10543-020-00833-0.

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AbstractWe consider a scalar function depending on a numerical solution of an initial value problem, and its second-derivative (Hessian) matrix for the initial value. The need to extract the information of the Hessian or to solve a linear system having the Hessian as a coefficient matrix arises in many research fields such as optimization, Bayesian estimation, and uncertainty quantification. From the perspective of memory efficiency, these tasks often employ a Krylov subspace method that does not need to hold the Hessian matrix explicitly and only requires computing the multiplication of the H

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Hekmat, Mohamad Hamed, and Masoud Mirzaei. "Continuous and Discrete Adjoint Approach Based on Lattice Boltzmann Method in Aerodynamic Optimization Part I: Mathematical Derivation of Adjoint Lattice Boltzmann Equations." Advances in Applied Mathematics and Mechanics 6, no. 5 (2014): 570–89. http://dx.doi.org/10.4208/aamm.2013.m226.

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AbstractThe significance of flow optimization utilizing the lattice Boltzmann (LB) method becomes obvious regarding its advantages as a novel flow field solution method compared to the other conventional computational fluid dynamics techniques. These unique characteristics of the LB method form the main idea of its application to optimization problems. In this research, for the first time, both continuous and discrete adjoint equations were extracted based on the LB method using a general procedure with low implementation cost. The proposed approach could be performed similarly for any optimiz

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Zhang, Zhengfang, Weifeng Chen, and Xiaoliang Cheng. "A Penalty Optimization Algorithm for Eigenmode Optimization Problem Using Sensitivity Analysis." Communications in Computational Physics 15, no. 3 (2014): 776–96. http://dx.doi.org/10.4208/cicp.190313.090913a.

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AbstractThis paper investigates the eigenmode optimization problem governed by the scalar Helmholtz equation in continuum system in which the computed eigenmode approaches the prescribed eigenmode in the whole domain. The first variation for the eigenmode optimization problem is evaluated by the quadratic penalty method, the adjoint variable method, and the formula based on sensitivity analysis. A penalty optimization algorithm is proposed, in which the density evolution is accomplished by introducing an artificial time term and solving an additional ordinary differential equation. The validit

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Walther, Benjamin, and Siva Nadarajah. "Adjoint-Based Constrained Aerodynamic Shape Optimization for Multistage Turbomachines." Journal of Propulsion and Power 31, no. 5 (2015): 1298–319. http://dx.doi.org/10.2514/1.b35433.

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Djeddi, Reza, and Kivanc Ekici. "Efficient One-Shot Technique for Adjoint-Based Unsteady Optimization." AIAA Journal 59, no. 9 (2021): 3448–64. http://dx.doi.org/10.2514/1.j060142.

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Jones, Martin, and Nail K. Yamaleev. "Adjoint-Based Optimization of Three-Dimensional Flapping-Wing Flows." AIAA Journal 53, no. 4 (2015): 934–47. http://dx.doi.org/10.2514/1.j053239.

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Agarwal, Dheeraj, Kapellos Christos, Trevor Robinson, and Cecil Armstrong. "Using Parametric Effectiveness for Efficient CAD-Based Adjoint Optimization." Computer-Aided Design and Applications 16, no. 4 (2018): 703–19. http://dx.doi.org/10.14733/cadaps.2019.703-719.

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Freund, Jonathan B. "Adjoint-based optimization for understanding and suppressing jet noise." Procedia Engineering 6 (2010): 54–63. http://dx.doi.org/10.1016/j.proeng.2010.09.007.

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Freund, Jonathan B. "Adjoint-based optimization for understanding and suppressing jet noise." Journal of Sound and Vibration 330, no. 17 (2011): 4114–22. http://dx.doi.org/10.1016/j.jsv.2011.02.009.

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Suwartadi, Eka, Stein Krogstad, and Bjarne Foss. "Adjoint-based surrogate optimization of oil reservoir water flooding." Optimization and Engineering 16, no. 2 (2014): 441–81. http://dx.doi.org/10.1007/s11081-014-9268-4.

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Protas, Bartosz. "Adjoint-based optimization of PDE systems with alternative gradients." Journal of Computational Physics 227, no. 13 (2008): 6490–510. http://dx.doi.org/10.1016/j.jcp.2008.03.013.

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Sayed, Ehab, Mohamed H. Bakr, Berker Bilgin, and Ali Emadi. "Adjoint-Based Design Optimization of Nonlinear Switched Reluctance Motors." Electric Power Components and Systems 47, no. 19-20 (2019): 1705–16. http://dx.doi.org/10.1080/15325008.2020.1731864.

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