An Efficient Aerodynamic Design Method Using Asymptotic Solution of Euler Equations

An efficient aerodynamic topology optimization method was constructed based on the parameterized level set method (PLSM) and large neighborhood search algorithm. The traditional aerodynamic optimization design process is to optimize the shape of a basic shape, and the aerodynamic configuration is determined. The determination of the basic shape relies on the experience of human designers, making it difficult to obtain innovative aerodynamic configurations. To this end, the PLSM is used to obtain aerodynamic configurations of supersonic Busemann biplane with different topological structures. In response to the characteristics of large sample size and low computational efficiency in aerodynamic topology optimization design, a large neighborhood search algorithm is adopted as the optimization algorithm. Aerodynamic topology optimization design was carried out to achieve good drag reduction effect for the Busemann biplane. Compared with shape optimization design, aerodynamic topology design can generate airfoils with different topological structures, greatly improving the design freedom and making it possible to find different aerodynamic configurations in a wide design space.

This study investigates the factors influencing the aerodynamic performance enhancement of the rear wing of the joined‐wing configuration UAVs by analyzing the flow field structure and flow mechanisms before and after airfoil optimization. Correspondingly, an optimization design concept tailored to joined‐wing aircraft is proposed. Focusing on the rear wing, a three‐dimensional design framework integrating geometric parameterization, aerodynamic simulation, and radar performance analysis was developed. The accuracy of the surrogate model was enhanced through multiround updating and optimization, and the optimal solution was derived using a genetic algorithm. The results demonstrate that the lift–drag ratio of the rear wing has been improved by approximately 9.7%. Specifically, the optimization outcomes confirm that adjusting the airfoil geometry and introducing a spanwise variation in the installation angle of control surfaces mitigates the adverse effects of front‐wing downwash on the rear wing, thereby improving the overall aerodynamic characteristics. By fully accounting for the interference of Frt‐wing downwash, this framework addresses the core challenges in rear‐wing aerodynamic optimization and provides an effective solution for enhancing the high‐altitude flight efficiency of UAVs.

We describe an asymptotic approach to gated ionic models of single-cell cardiac excitability. It has a form essentially different from the Tikhonov fast-slow form assumed in standard asymptotic reductions of excitable systems. This is of interest since the standard approaches have been previously found inadequate to describe phenomena such as the dissipation of cardiac wave fronts and the shape of action potential at repolarization. The proposed asymptotic description overcomes these deficiencies by allowing, among other non-Tikhonov features, that a dynamical variable may change its character from fast to slow within a single solution. The general asymptotic approach is best demonstrated on an example which should be both simple and generic. The classical model of Purkinje fibers (Noble in J. Physiol. 160:317-352, 1962) has the simplest functional form of all cardiac models but according to the current understanding it assigns a physiologically incorrect role to the Na current. This leads us to suggest an "Archetypal Model" with the simplicity of the Noble model but with a structure more typical to contemporary cardiac models. We demonstrate that the Archetypal Model admits a complete asymptotic solution in quadratures. To validate our asymptotic approach, we proceed to consider an exactly solvable "caricature" of the Archetypal Model and demonstrate that the asymptotic of its exact solution coincides with the solutions obtained by substituting the "caricature" right-hand sides into the asymptotic solution of the generic Archetypal Model. This is necessary, because, unlike in standard asymptotic descriptions, no general results exist which can guarantee the proximity of the non-Tikhonov asymptotic solutions to the solutions of the corresponding detailed ionic model.

Optimization and design of an aircraft’s morphing wing-tip demonstrator for drag reduction at low speed, Part I – Aerodynamic optimization using genetic, bee colony and gradient descent algorithms

In order to reduce the dependence on fossil fuels, the concept of distributed electric propulsion (DEP) has been introduced. At present, there are a few studies on the small electric propulsion engines of DEP vertical take-off and landing (VTOL) UAVs. According to the design requirements of DEP VTOL UAV ducted fans, the aerodynamic design of a ducted fan is carried out in this paper. Firstly, the initial model is established by referring to theoretical derivations. Then, Bezier curves are used to control the mid-arc and thickness distributions of the blade profiles at different blade heights in order to realize the parameterization as design variables. Finally, the objective function that can represent the performance of the rotor is selected, and a multi-objective genetic algorithm is used to optimize the performance. The optimal results are obtained, and the aerodynamic characteristics are analyzed. Compared with the initial model, the total pressure ratio and isentropic efficiency of the optimized model are increased by 5.3% and 7.8%, respectively, reaching 1.20 and 90.37%. Meanwhile, the aerodynamic performance of the optimized model is improved, indicating that the performance of the rotor is significantly enhanced and that the design task is completed.

A new resolution method for solving linear systems resulting from the discrete adjoint approach in aerodynamic shape optimization with a single objective and multiple constraint functions is presented. The steady flow is governed by the three-dimensional compressible Favre-averaged Navier–Stokes equations combined with a two-equation turbulence model. Block flexible variants of the generalized minimal residual method are Krylov methods designed for the solution of linear systems with multiple right-hand sides allowing variable preconditioning. Then, they can be applied to the discrete adjoint equations of Favre-averaged Navier–Stokes equations for all objective and constraint functions as a right-hand side. The paper focuses on the development of a new algorithm combining the three following features: flexibility, multiple right-hand-side concurrent processing, and spectral deflation. This latter property enables to recycle approximate spectral information from one restart to the next one in order to avoid generalized minimal residual stagnation when short restart recurrences are performed. This novel block method is named block flexible generalized minimal residual with deflated restarting. It is evaluated on two aerodynamic configurations showing its capabilities to reduce computational times over standard Krylov methods.

Aerodynamic optimization in vehicle body design has become a critical focus in the automotive industry, driven by the growing demands for energy efficiency, extended electric vehicle range, and enhanced driving stability. This review systematically summarizes the state-of-the-art research on aerodynamic considerations and optimization strategies for vehicle bodies, structured around three core themes. First, it elaborates on the fundamental aerodynamic principles underlying body design, including key forces (drag, lift, side force), flow field characteristics (boundary layer separation, vortex generation), and critical design parameters (overall body shape, detailed components, aerodynamic add-ons) that govern aerodynamic performance. Second, it discusses the construction of aerodynamic models and problem-solving approaches, covering numerical simulation (CFD) with common software and turbulence models, experimental testing (wind tunnel and road tests), and optimization frameworks integrating parametric modeling, sensitivity analysis, and advanced algorithms (e.g., genetic algorithms, machine learning). Third, a typical case study of aerodynamic optimization for a passenger vehicle is presented, followed by an analysis of future trends, such as AI-driven automated design, active aerodynamic systems, multi-physics coupling simulation, and the synergistic optimization of aerodynamics with lightweight design and electric vehicle-specific requirements. This review synthesizes the current knowledge of aerodynamic design and optimization for vehicle bodies, identifies key challenges in balancing performance, styling, and manufacturing feasibility, and provides insights for future research and engineering practice in the automotive sector.

Aerodynamic shape optimization of large-span coal sheds for wind-induced effect mitigation using surrogate models

Recently, we have proposed a robust and computationally efficient numerical algorithm for the optimization of 2-D and 3 -D catalysts distributions in Proton Exchange Membrane Fuel Cells (PEMFCs) using a gradient-based technique [1-3]. The algorithm is based on the computation of the catalyst sensitivity functions of the cell current and cell voltage by using the adjoint method. This method was initially developed by the applied mathematics community to solve optimization problems in fluid dynamics, climate, and heat transfer problems and, later, by other research groups to solve optimization problems in magnetics, model parameter determination, and semiconductor devices [4-6]. It was only recently that COMSOL, a popular finite element solver, has introduced the adjoint method, which has started to be used by the electrochemical community for calculation of sensitivities of platinum distribution in PEMFCs [7,8]. In this presentation we analyze the cell current, cell voltage, and spectral electrochemical impendence properties of PEMFCs by using the catalyst sensitivity functions of cell voltage, γPt(r). These functions show how much the cell voltage is increasing by adding 1 unit amount of platinum at a given location inside the catalyst layer (and assuming the cell is operating at the same current density). For instance, the figure below shows the 2-D catalysis sensitivity functions of the cell voltage for two cells with membrane thicknesses of 60 mm and 25 mm, in which the catalyst is distributed uniformly inside the cell. It is apparent from these figures that in order to increase the cell voltage and consequently the power of the cell, one needs to increase the concentration of Pt particles in the regions under the oxygen openings. As it will be shown at the conference the optimum catalyst distribution also depends on the operating temperature, catalyst and gas diffusion layers porosities, Nafion content, and cell current. In addition to the above analysis, we will also introduce a technique for the analysis of variability of polarization curves induced by the randomness of pore sizes and pore shapes in the catalyst and gas diffusion layers in PEMFCs. The technique is based on the evaluation of sensitivity functions of the cell voltage and can also be extended to the analysis of variability induced by the random platinum distribution in the catalyst layers (which is assumed to have a uniform random distribution). More details about the method for the analysis of variability effects, as well as about the numerical implementation of the adjoint method in the case of finite element simulations and its computational efficiency will be presented at the conference. [1] J. Lamb, G. Mixon, and P. Andrei, “Optimization of polymer electrolyte membrane fuel cells with a large number of degrees of freedom”, ECS 231Meeting, New Orleans, LA, 2017. [2] J. Lamb, G. Mixon, and P. Andrei, “Mathematical optimization of the spatial distribution of platinum particles in the catalyst layer of PEMFCs”, PRIME 2016 Meeting, Honolulu, HI, 2016. [3] P. Andrei, G. Mixon, M. Mehta, and V. Bevara, “Design of the catalyst layers in PEMFCs using an adjoint sensitivity analysis approach”, ECS Transactions 66(8), 91-128, 2015. [4] D. Cacuci, M. Ionescu-Bujor and M. N. Ionel, Sensitivity and Uncertainty Analysis, Volume 2: Applications to Large-Scale Systems, Chapman & Hall/CRC (2005) [5] W. K. Anderson and V. Venkatakrishnan, “Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation”, Computers & Fluids, 28, 443 (1999). [6] P. Andrei, I. Mayergoyz, “Quantum mechanical effects on random oxide thickness and random doping induced fluctuations in ultrasmall semiconductor devices”, Journal of Applied Physics, 94, 7163-7172 (2003). [7] COMSOL Multiphysics Software, https://www.comsol.com. [8] F. C. Cetinbas, S. G. Advani and A. K. Prasad, “Three dimensional proton exchange membrane fuel cell cathode model using a modified agglomerate approach based on discrete catalyst particles”, Electrochimica Acta, 174, 787 (2015). Figure 1

The assumption of the existence of steady state, linearized, inhomogeneous solar wind flow imposes a relation between the perturbation variables such that the solutions are regular at the critical points. The critical point regularity conditions are derived for the case of hydromagnetic flow with no rotation, for which there are two critical points, and the case of hydrodynamic flow with rotation and one critical point. It is possible the organization imposed by the critical points is reflected in correlations observed between solar wind parameters near the orbit of Earth. Analytic asymptotic solutions are also derived to complete the description given previously of the intermediate range flow obtained by numerical integrations. The asymptotic solutions can be described in terms of pressure waves that in general propagate radially with respect to the zero-order flow. The pressure perturbation for expected solar wind conditions is found to increase with distance and the velocity perturbations to decrease.

With rapid development of urban rail transit, maglev trains, benefiting from its comfortable, energy-saving and environmentally friendly merits, have gradually entered people's horizons. In this paper, aiming at improving the aerodynamic performance of an urban maglev train, the aerodynamic optimization design has been performed. An improved two-point infill criterion has been adopted to construct the cross-validated Kriging model. Meanwhile, the multi-objective genetic algorithm and complex three-dimensional geometric parametrization method have been used, to optimize the streamlined head of the train. Several optimal shapes have been obtained. Results reveal that the optimization strategy used in this paper is sufficiently accurate and time-efficient for the optimization of the urban maglev train, and can be applied in practical engineering. Compared to the prototype of the train, optimal shape benefits from higher lift of the leading car and smaller drag of the whole train. Sensitivity analysis reveals that the length and height of the streamlined head have a great influence on the aerodynamic performance of the train, and strong nonlinear relationships exist between these design variables and aerodynamic performance. The conclusions drawn in this study offer the chance to derive critical reference values for the optimization of the aerodynamic characteristics of urban maglev trains.Graphic abstract

This study aims to optimize the impeller geometry of a backward-curved blade centrifugal fan in a helicopter oil cooler to improve the aerodynamic performance of the impeller. The study consists of three main parts: aerodynamic design and parametrization of the impeller, numerical analysis of impeller performance, and multi-objective design optimization to maximize fan static pressure and total to static isentropic efficiency. In the first part, a one-dimensional impeller design was performed using quasi-experimental methods and a direct optimization method with the multi-objective genetic algorithm, and the impeller geometry was parameterized through a commercial tool. The three-dimensional flow through the impeller was solved in the second part with a commercial computational fluid dynamics tool. The Reynolds-averaged Navier-Stokes equations were solved on a multi-block grid, and a second-order accurate finite volume method is employed. The most appropriate turbulence model and grid size were selected considering time, cost, and fidelity. In the third part, a sensitivity analysis was performed with the design-of-experiment method, and the parameters that affect the objective function most significantly were determined. A multi-objective design optimization based on a non-dominated sorting genetic algorithm was performed with the Kriging response surface method, and Pareto-optimal solutions were obtained. Results show that, at the design point, there is a 9.6% and 0.96% increase in the fan static pressure and total to static isentropic efficiency, respectively. Trained kriging response surface model predicted the fan static pressure and total to static isentropic efficiency with an error of 0.12% and 0.33%, respectively.

*† This paper deals with the use of the adjoint equation for aerodynamic shape optimization of complex configurations with overset grids methods. The use of overset grid eases the grid generation process, but the nontrivial task of ensuring communication between the overlapping grids needs careful attention. This need is effectively addressed by using a practically useful technique known as Implicit Hole Cutting (IHC) method first introduced by Lee and Baeder 1 and further demonstrated by the authors 2 in a previous study. The method depends on the simplicity of the cell selection process based on the main criterion of cell size, and all grid points including hole interior points and hole fringe points are treated indiscriminately in the flow computation. The simplicity of the IHC method is exploited for the adjoint equation solver. Similar to the flow solver, the adjoint equations are solved on the conventional point-matched grids and overlapped grids within a multi-block framework. Parallel computing with MPI is also used to improve the overall efficiency of the design optimization process. The method is successfully applied to several two and threedimensional shape optimization cases for both external and internal flows problems.

Adaptive morphing trailing-edge wings have the potential to reduce the fuel burn of transport aircraft. However, to take full advantage of this technology and to quantify its benefits, design studies are required. To address this need, the aerodynamic performance benefits of a morphing trailing-edge wing are quantified using aerodynamic design optimization. The aerodynamic model solves the Reynolds-averaged Navier–Stokes equations with a Spalart–Allmaras turbulence model. A gradient-based optimization algorithm is used in conjunction with an adjoint method that computes the required derivatives. The baseline geometry is optimized using a multipoint formulation and 192 shape design variables. The average drag coefficient is minimized subject to lift, pitching moment, geometric constraints, and a maneuver bending moment constraint. The trailing edge of the wing is optimized based on the multipoint optimized wing. The trailing-edge morphing is parameterized using 90 design variables that are optimized independently for each flight condition. A total of 407 trailing-edge optimizations are performed at different flight conditions to span the entire cruise flight envelope. A 1% drag reduction at on-design conditions and a 5% drag reduction near off-design conditions are observed. The effectiveness of the trailing-edge morphing is demonstrated by comparing it with the optimized results of a hypothetical fully morphing wing. In addition, the fuel-burn reductions for a number of flights are computed using the optimization results. A 1% cruise fuel-burn reduction is achieved using an adaptive morphing trailing edge for a typical long-haul twin-aisle mission.

This paper presents the fundamentals of a continuous adjoint method and the applications of this method to the aerodynamic design optimization of both external and internal flows. General formulation of the continuous adjoint equations and the corresponding boundary conditions are derived. With the adjoint method, the complete gradient information needed in the design optimization can be obtained by solving the governing flow equations and the corresponding adjoint equations only once for each cost function, regardless of the number of design parameters. An inverse design of airfoil is firstly performed to study the accuracy of the adjoint gradient and the effectiveness of the adjoint method as an inverse design method. Then the method is used to perform a series of single and multiple point design optimization problems involving the drag reduction of airfoil, wing, and wing-body configuration, and the aerodynamic performance improvement of turbine and compressor blade rows. The results demonstrate that the continuous adjoint method can efficiently and significantly improve the aerodynamic performance of the design in a shape optimization problem.