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Asia-Pacific International Symposium on Aerospace Technology (APISAT)

作者： Li, Bin Chen, Jiang Liu, Yi Xiang, Hang Beihang Univ Sch Energy & Power Engn Beijing 100191 Peoples R China

ISBN: (纸本)9789819740123;9789819740109;9789819740093

The full 3-dimensional (3D) aerodynamic optimization of axial flow compressors is a typical high-dimensional and expensive optimization problem, which has multiple variables and long evaluation time. To address this challenge, this study first establishes a novel aerodynamic optimization platform for multistage axial compressors, which reduces the time of single sample evaluation by a dimensionality reduction method which adoptsquasi-3-dimensional aerodynamic calculation instead of full 3D numerical simulation. The optimization platform also adopts radial basis function (RBF) parameterization technique. It can control the deformation of 3D geometry by fewer control points, and thus the control variables of single blade and the dimension of the optimization problem are reduced. Then the aerodynamic performance of a 3.5-stage highly loaded compressor is optimized based on the optimization platform. The results show that after optimization, within the basically unchanged total pressure ratio and surge margin, the efficiency of design point is increased by a maximum of 2.2% and the optimized characteristics curves are in good agreementwith the 3Dcalculation results. Compared to the full 3D aerodynamic optimization, the optimization platform in this article can save more than 10 times the time. Thus, the effectiveness of the optimization platform in compressor aerodynamic optimization is verified.

关键词： S2 throughflow analysis method Aerodynamic performance parameterization method Optimization algorithm

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Inverse Design method of Transonic Airfoil Based on Deep Neural Network

Inverse Design Method of Transonic Airfoil Based on Deep Neu...

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Asia-Pacific International Symposium on Aerospace Technology (APISAT)

作者： Bai, Zhiliang Zhang, Wei Wei, Ruyue Northwestern Polytech Univ Sch Aeronaut Xian 710072 Peoples R China

ISBN: (纸本)9789811926891;9789811926884

This research introduces a deep neural network-based transonic airfoil inverse design approach. A deep neural network is established using the predicted pressure coefficient distribution, angle of attack, and Mach number around the airfoil as inputs and parameterized airfoil parameters as outputs. The network solves the transonic flow field of an airfoil using the Euler equation. The neural network is trained to grasp the impacts of geometric changes on the pressure coefficient and shock wave at various places on the airfoil surface. Using the finite volume approach and multigrid acceleration method, the Euler equation is employed to batch compute the airfoil database and build the training set. As the training set's output vector, the CST parameterization technique is utilized to parameterize each synchronously generated airfoil into 10 parameters. The results demonstrate that the deep learning approach developed in this research is capable of achieving the reverse design of a transonic airfoil, and that the trained neural network can generate the airfoil with the requisite aerodynamic properties rapidly and directly. The generalization error of the anticipated airfoil geometry information is less than 1.5 percent when compared to the accurate results. The results reveal a high level of precision. The impact of the deep neural network's shape and the activation function of neurons on the subjects under investigation is also examined. The results reveal that the activation function used is critical to the outcome, and the design of the neural network also has an impact on the outcome.

关键词： Inverse design Transonic airfoil parameterization method Deep neural network

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parameterization of Unstable Manifolds for DDEs: Formal Series Solutions and Validated Error Bounds

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JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS 2022年第2期34卷 1285-1324页

作者： Henot, Olivier Lessard, Jean-Philippe James, J. D. Mireles McGill Univ Dept Math & Stat 805 Sherbrooke St West Montreal PQ H3A 0B9 Canada Florida Atlantic Univ Dept Math Sci Sci BldgRoom 234777 Glades Rd Boca Raton FL 33431 USA

This paper studies the local unstable manifold attached to an equilibrium solution of a system of delay differential equations (DDEs). Two main results are developed. The first is a general method for computing the formal Taylor series coefficients of a function parameterizing the unstable manifold. We derive linear systems of equations whose solutions are the Taylor coefficients, describe explicit formulas for assembling the linear equations for DDEs with polynomial nonlinearities. We also discuss a scheme for transforming non-polynomial DDEs into polynomial ones by appending auxiliary equations. The second main result is an a-posteriori theorem which-when combined with deliberate control of rounding errors-leads to mathematically rigorous computer assisted convergence results and error bounds for the truncated series. Our approach is based on the parameterization method for invariant manifolds and requires some mild non-resonance conditions between the unstable eigenvalues.

关键词： Delay differential equations Invariant manifolds parameterization method Formal power series Validated numerics Computer assisted proof

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A novel algorithm based on parameterization method for calculation of curvature of the free surface flows

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APPLIED MATHEMATICAL MODELLING 2013年第1-2期37卷 570-585页

作者： Saghi, H. Ketabdari, M. J. Zamirian, M. Ferdowsi Univ Mashhad Dept Civil Engn Mashhad Iran Amirkabir Univ Technol Fac Marine Technol Tehran Iran Azad Univ Bojnourd Dept Math Bojnourd Iran

In this paper, a new approach based on parameterization method is presented for calculation of curvature on the free surface flows. In some phenomena such as droplet and bubble, surface tension is prominent. Therefore in these cases, accurate estimation of the curvature is vital. Volume of fluid (VOF) is a surface capturing method for free surface modeling. In this method, free surface curvature is calculated based on gradient of scalar transport parameter which is regarded as original method in this paper. However, calculation of curvature for a circle and other known geometries based on this method is not accurate. For instance, in practice curvature of a circle in interface cells is constant, while this method predicts different curvatures for it. In this research a novel algorithm based on parameterization method for improvement of the curvature calculation is presented. To show the application of parameterization method, two methods are employed. In the first approach denoted by, three line method, a curve is fitted to the free surface so that the distance between curve and linear interface approximation is minimized. In the second approach namely four point method, a curve is fitted to intersect points with grid lines for central and two neighboring cells. These approaches are treated as calculus of variation problems. Then, using the parameterization method, these cases are converted into the sequences of time-varying nonlinear programming problems. With some treatments a conventional equivalent model is obtained. It is finally proved that the solution of these sequences in the models tends to the solution of the calculus of variation problems. For verification of the presented methods, curvature of some geometrical shapes such as circle, elliptic and sinusoidal profile is calculated and compared with original method used in VOF process and analytical solutions. Finally, as a more practical problem, spurious currents are studied. The results showed that more

关键词： parameterization method Curvature Free surface flows Surface tension Volume of fluid

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Flow map parameterization methods for invariant tori in Hamiltonian systems

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2021年 101卷 105859-105859页

作者： Haro, Alex Mondelo, J. M. Univ Barcelona Dept Matemat & Informat Gran Via 585 Barcelona 08007 Spain Univ Barcelona BGSMath Gran Via 585 Barcelona 08007 Spain Univ Autonoma Barcelona Dept Matemat Av Eix CentEdifici C Bellaterra 08193 Barcelona Spain Univ Autonoma Barcelona CERES IEEC Av Eix CentEdifici C Bellaterra 08193 Barcelona Spain Univ Autonoma Barcelona BGSMath Av Eix CentEdifici C Bellaterra 08193 Barcelona Spain

The goal of this paper is to present a methodology for the computation of invariant tori in Hamiltonian systems combining flow map methods, parameterization methods, and symplectic geometry. While flow map methods reduce the dimension of the tori to be computed by one (avoiding Poincare maps), parameterization methods reduce the cost of a single step of the derived Newton-like method to be proportional to the cost of a FFT. Symplectic properties lead to some magic cancellations that make the methods work. The multiple shooting version of the methods are applied to the computation of invariant tori and their invariant bundles around librational equilibrium points of the Restricted Three Body Problem. The invariant bundles are the first order approximations of the corresponding invariant manifolds, commonly known as the whiskers, which are very important in the dynamical organization and have important applications in space mission design. (c) 2021 Elsevier B.V. All rights reserved.

关键词： Invariant tori parameterization method KAM theory RTBP Lissajous orbits

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Rapid and accurate methods for computing whiskered tori and their manifolds in periodically perturbed planar circular restricted 3-body problems

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CELESTIAL MECHANICS & DYNAMICAL ASTRONOMY 2022年第1期134卷 1-38页

作者： Kumar, Bhanu Anderson, Rodney L. de la Llave, Rafael Georgia Inst Technol Sch Math 686 Cherry St NW Atlanta GA 30332 USA CALTECH Jet Prop Lab 4800 Oak Grove Dr Pasadena CA 91109 USA

When the planar circular restricted 3-body problem (RTBP) is periodically perturbed, families of unstable periodic orbits break up into whiskered tori, with most tori persisting into the perturbed system. In this study, we (1) develop a quasi-Newton method which simultaneously solves for the tori and their center, stable, and unstable directions;(2) implement continuation by both perturbation and rotation numbers;(3) compute Fourier-Taylor parameterizations of the stable and unstable manifolds;(4) regularize the equations of motion;and (5) globalize these manifolds. Our methodology improves on efficiency and accuracy compared to prior studies and applies to a variety of periodic perturbations. We demonstrate the tools near resonances in the planar elliptic RTBP.

关键词： Whiskered tori Invariant manifolds parameterization method Three-body problem Resonance

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Parametric generative schemes with geometric constraints for encoding and synthesizing airfoils

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ENGINEERING APPLICATIONS OF ARTIFICIAL INTELLIGENCE 2024年 128卷

作者： Xie, Hairun Wang, Jing Zhang, Miao Shanghai Aircraft Design & Res Inst Shanghai 200436 Peoples R China Shanghai Jiao Tong Univ Sch Aeronaut & Astronaut Shanghai 200240 Peoples R China

The modern aerodynamic optimization has a strong demand for parametric methods with high levels of intuitiveness, flexibility, and representative accuracy, which cannot be fully achieved through traditional airfoil parametric techniques. In this paper, two deep learning-based generative schemes are proposed to effectively capture the complexity of the design space while satisfying specific constraints. 1. Soft-constrained scheme: a Conditional Variational Autoencoder-based model that directly incorporates geometric constraints as part of the network. 2. Hard-constrained scheme: a Variational Autoencoder-based model for generating diverse airfoils, with projection onto given constraints using Free Form Deformation-based techniques. According to the statistical results, the reconstructed airfoils are both accurate and smooth, without any need for additional filters. The soft-constrained scheme generates airfoils that exhibit slight deviations from the expected geometric constraints, yet still converge to the reference airfoil in both geometry space and objective space with some degree of distribution bias. In contrast, the hard-constrained scheme produces airfoils with a wider range of geometric diversity while strictly adhering to the geometric constraints. The corresponding distribution in the objective space is also more diverse, with isotropic uniformity around the reference point and no significant bias. These proposed airfoil parametric methods can break through the boundaries of training data in the objective space, providing higher quality samples for random sampling and improving the efficiency of optimization design.

关键词： Deep learning parameterization method parametric generative schemes parametric airfoil geometric constraints

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Numerical Computation of Periodic Orbits and Isochrons for State-Dependent Delay Perturbation of an ODE in the Plane

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SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 2021年第3期20卷 1514-1543页

作者： Gimeno, Joan Yang, Jiaqi de la Llave, Rafael Univ Roma Tor Vergata Dept Math Via Ric Sci 1 I-00133 Rome Italy Georgia Inst Technol Sch Math Atlanta GA 30332 USA

We present algorithms and their implementation to compute limit cycles and their isochrons for state-dependent delay differential equations (SDDEs) which are perturbed from a planar ordinary differential equation (ODE) with a limit cycle. Note that the space of solutions of an SDDE is infinite dimensional. We compute a two parameter family of solutions of the SDDE which converges to the solutions of the ODE as the perturbation goes to zero in a neighborhood of the limit cycle. The method we use formulates functional equations among periodic functions (or functions converging exponentially to periodic). The functional equations express that the functions solve the SDDE. Therefore, rather than evolving initial data and finding solutions of a certain shape, we consider spaces of functions with the desired shape and require that they are solutions. The mathematical theory of these invariance equations is developed in a companion paper [J. Yang, J. Gimeno, and R. de la Llave, SIAM J. Math. Anal., 53 (2021), pp. 4031-4067], which provides proofs of a posteriori theorems. They show that if there is a sufficiently approximate solution (with respect to some explicit condition numbers), then there is a true solution close to the approximate one. Since the numerical methods produce an approximate solution, and provide estimates of the condition numbers, we can make sure that the numerical solutions we consider approximate true solutions. In this paper, we choose a systematic way to approximate functions by a finite set of numbers (Taylor-Fourier series) and develop a toolkit of algorithms that implement the operators-notably composition-that enter into the theory. We also present several results obtained by running the implementations in some representative cases.

关键词： state-dependent delay perturbation theory parameterization method

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Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

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JOURNAL OF NONLINEAR SCIENCE 2023年第3期33卷 46-46页

作者： Lessard, Jean-Philippe Matsue, Kaname Takayasu, Akitoshi McGill Univ Dept Math & Stat 805 Sherbrooke St West Montreal PQ H3A 0B9 Canada Kyushu Univ Inst Math Ind Fukuoka 8190395 Japan Kyushu Univ Int Inst Carbon Neutral Energy Res WPI I2CNER Fukuoka 8190395 Japan Univ Tsukuba Fac Engn Informat & Syst 1-1-1 Tennodai Tsukuba Ibaraki 3058573 Japan

In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial points, referred to as saddle-type blow-up solutions, are studied. Combining dynamical systems machinery (e.g., compactifications, timescale desingularizations of vector fields) with tools from computer-assisted proofs (e.g., rigorous integrators, the parameterization method for invariant manifolds), these blow-up solutions are obtained as trajectories on local stable manifolds of hyperbolic saddle equilibria at infinity. With the help of computer-assisted proofs, global trajectories on stable manifolds, inducing blow-up solutions, provide a global picture organized by global-in-time solutions and blow-up solutions simultaneously. Using the proposed methodology, intrinsic features of saddle-type blow-ups are observed: locally smooth dependence of blow-up times on initial points, level set distribution of blow-up times and decomposition of the phase space playing a role as separatrixes among solutions, where the magnitude of initial points near those blow-ups does not matter for asymptotic behavior. Finally, singular behavior of blow-up times on initial points belonging to different family of blow-up solutions is addressed.

关键词： Saddle-type blow-up solutions Rigorous numerics Compactifications Desingularization parameterization method Separatrix

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A Numerical method for Solving Boundary Value Problem for Essentially Loaded Differential Equations

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LOBACHEVSKII JOURNAL OF MATHEMATICS 2021年第3期42卷 551-559页

作者： Kadirbayeva, Zh M. Inst Math & Math Modeling Dept Differential Equat A26G7T4 Alma Ata 050010 Kazakhstan Int Informat Technol Univ Dept Math & Comp Modeling Alma Ata 050040 Kazakhstan

A linear boundary value problem for essentially loaded differential equations is investigated. Using the properties of essentially loaded differential equation and assuming the invertibility of the matrix compiled through the coefficients at the values of the derivative of the desired function at load points, we reduce the considering problem to a two-point boundary value problem for loaded differential equations. The method of parameterization is used for solving the problem. The linear boundary value problem for loaded differential equations by introducing additional parameters at the loading points is reduced to equivalent boundary value problem with parameters. The equivalent boundary value problem with parameters consists of the Cauchy problem for the system of ordinary differential equations with parameters, boundary condition and continuity conditions. The solution of the Cauchy problem for the system of ordinary differential equations with parameters is constructed using the fundamental matrix of differential equation. The system of linear algebraic equations with respect to the parameters are composed by substituting the values of the corresponding points in the built solutions to the boundary condition and the continuity condition. Numerical method for finding solution of the problem is suggested, which based on the solving the constructed system and the Bulirsch-Stoer method for solving Cauchy problem on the subintervals.

关键词： essentially loaded differential equation parameterization method algorithm

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