Aerodynamic/stealth design of S-duct inlet based on discrete adjoint method

It is a major challenge for the airframe-inlet design of modern combat aircrafts,as the flow and electromagnetic wave propagation in the inlet of stealth aircraft are very complex.In this study,an aerodynamic/stealth optimization design method for an S-duct inlet is proposed.The upwind scheme is introduced to the aerodynamic adjoint equation to resolve the shock wave and flow separation.The multilevel fast multipole algorithm(MLFMA)is utilized for the stealth adjoint equation.A dorsal S-duct inlet of flying wing layout is optimized to improve the aerodynamic and stealth characteristics.Both the aerodynamic and stealth characteristics of the inlet are effectively improved.Finally,the optimization results are analyzed,and it shows that the main contradiction between aerodynamic characteristics and stealth characteristics is the centerline and crosssectional area.The S-duct is smoothed,and the cross-sectional area is increased to improve the aerodynamic characteristics,while it is completely opposite for the stealth design.The radar cross section(RCS)is reduced by phase cancelation for low frequency conditions.The method is suitable for the aerodynamic/stealth design of the aircraft airframe-inlet system.

Introducing the nth-Order Features Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-FASAM-N): I. Mathematical Framework

This work presents the “nth-Order Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (abbreviated as “nth-FASAM-N”), which will be shown to be the most efficient methodology for computing exact expressions of sensitivities, of any order, of model responses with respect to features of model parameters and, subsequently, with respect to the model’s uncertain parameters, boundaries, and internal interfaces. The unparalleled efficiency and accuracy of the nth-FASAM-N methodology stems from the maximal reduction of the number of adjoint computations (which are considered to be “large-scale” computations) for computing high-order sensitivities. When applying the nth-FASAM-N methodology to compute the second- and higher-order sensitivities, the number of large-scale computations is proportional to the number of “model features” as opposed to being proportional to the number of model parameters (which are considerably more than the number of features).When a model has no “feature” functions of parameters, but only comprises primary parameters, the nth-FASAM-N methodology becomes identical to the extant nth CASAM-N (“nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems”) methodology. Both the nth-FASAM-N and the nth-CASAM-N methodologies are formulated in linearly increasing higher-dimensional Hilbert spaces as opposed to exponentially increasing parameter-dimensional spaces thus overcoming the curse of dimensionality in sensitivity analysis of nonlinear systems. Both the nth-FASAM-N and the nth-CASAM-N are incomparably more efficient and more accurate than any other methods (statistical, finite differences, etc.) for computing exact expressions of response sensitivities of any order with respect to the model’s features and/or primary uncertain parameters, boundaries, and internal interfaces.

Introducing the nth-Order Features Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-FASAM-N): II. Illustrative Example

This work highlights the unparalleled efficiency of the “nth-Order Function/ Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (nth-FASAM-N) by considering the well-known Nordheim-Fuchs reactor dynamics/safety model. This model describes a short-time self-limiting power excursion in a nuclear reactor system having a negative temperature coefficient in which a large amount of reactivity is suddenly inserted, either intentionally or by accident. This nonlinear paradigm model is sufficiently complex to model realistically self-limiting power excursions for short times yet admits closed-form exact expressions for the time-dependent neutron flux, temperature distribution and energy released during the transient power burst. The nth-FASAM-N methodology is compared to the extant “nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (nth-CASAM-N) showing that: (i) the 1st-FASAM-N and the 1st-CASAM-N methodologies are equally efficient for computing the first-order sensitivities;each methodology requires a single large-scale computation for solving the “First-Level Adjoint Sensitivity System” (1st-LASS);(ii) the 2nd-FASAM-N methodology is considerably more efficient than the 2nd-CASAM-N methodology for computing the second-order sensitivities since the number of feature-functions is much smaller than the number of primary parameters;specifically for the Nordheim-Fuchs model, the 2nd-FASAM-N methodology requires 2 large-scale computations to obtain all of the exact expressions of the 28 distinct second-order response sensitivities with respect to the model parameters while the 2nd-CASAM-N methodology requires 7 large-scale computations for obtaining these 28 second-order sensitivities;(iii) the 3rd-FASAM-N methodology is even more efficient than the 3rd-CASAM-N methodology: only 2 large-scale computations are needed to obtain the exact expressions of the 84 distinct third-order response sensitivities with respect to the Nordheim-Fuchs model’s parameters when applying the 3rd-FASAM-N methodology, while the application of the 3rd-CASAM-N methodology requires at least 22 large-scale computations for computing the same 84 distinct third-order sensitivities. Together, the nth-FASAM-N and the nth-CASAM-N methodologies are the most practical methodologies for computing response sensitivities of any order comprehensively and accurately, overcoming the curse of dimensionality in sensitivity analysis.

A dynamic-mode-decomposition-based acceleration method for unsteady adjoint equations at low Reynolds numbers

The computational cost of unsteady adjoint equations remains high in adjoint-based unsteady aerodynamic op-timization.In this letter,the solution of unsteady adjoint equations is accelerated by dynamic mode decomposi-tion(DMD).The pseudo-time marching of every real-time step is approximated as an infinite-dimensional linear dynamical system.Thereafter,DMD is utilized to analyze the adjoint vectors sampled from these pseudo-time marching.First-order zero frequency mode is selected to accelerate the pseudo-time marching of unsteady adjoint equations in every real-time step.Through flow past a stationary circular cylinder and an unsteady aerodynamic shape optimization example,the efficiency of solving unsteady adjoint equations is significantly improved.Re-sults show that one hundred adjoint vectors contains enough information about the pseudo-time dynamics,and the adjoint dominant mode can be precisely predicted only by five snapshots produced from the adjoint vectors,which indicates DMD analysis for pseudo-time marching of unsteady adjoint equations is efficient.

Second-Order Adjoint Sensitivity Analysis Methodology for Computing Exactly Response Sensitivities to Uncertain Parameters and Boundaries of Linear Systems: Mathematical Framework

This work presents the “Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology (2nd-CASAM)” for the efficient and exact computation of 1st- and 2nd-order response sensitivities to uncertain parameters and domain boundaries of linear systems. The model’s response (i.e., model result of interest) is a generic nonlinear function of the model’s forward and adjoint state functions, and also depends on the imprecisely known boundaries and model parameters. In the practically important particular case when the response is a scalar-valued functional of the forward and adjoint state functions characterizing a model comprising N parameters, the 2nd-CASAM requires a single large-scale computation using the First-Level Adjoint Sensitivity System (1st-LASS) for obtaining all of the first-order response sensitivities, and at most N large-scale computations using the Second-Level Adjoint Sensitivity System (2nd-LASS) for obtaining exactly all of the second-order response sensitivities. In contradistinction, forward other methods would require (N2/2 + 3 N/2) large-scale computations for obtaining all of the first- and second-order sensitivities. This work also shows that constructing and solving the 2nd-LASS requires very little additional effort beyond the construction of the 1st-LASS needed for computing the first-order sensitivities. Solving the equations underlying the 1st-LASS and 2nd-LASS requires the same computational solvers as needed for solving (i.e., “inverting”) either the forward or the adjoint linear operators underlying the initial model. Therefore, the same computer software and “solvers” used for solving the original system of equations can also be used for solving the 1st-LASS and the 2nd-LASS. Since neither the 1st-LASS nor the 2nd-LASS involves any differentials of the operators underlying the original system, the 1st-LASS is designated as a “first-level” (as opposed to a “first-order”) adjoint sensitivity system, while the 2nd-LASS is designated as a “second-level” (rather than a “second-order”) adjoint sensitivity system. Mixed second-order response sensitivities involving boundary parameters may arise from all source terms of the 2nd-LASS that involve the imprecisely known boundary parameters. Notably, the 2nd-LASS encompasses an automatic, inherent, and independent “solution verification” mechanism of the correctness and accuracy of the 2nd-level adjoint functions needed for the efficient and exact computation of the second-order sensitivities.

提高基于Adjoint方法翼型优化设计鲁棒性的研究

通过引入线搜索方法,提高了基于Adjoint方法翼型优化设计的鲁棒性。针对给定的目标函数,推导了贴体坐标系下相应的Adjoint方程与边界条件的具体表达形式,以及梯度表达式。通过数值求解流动控制方程和Adjoint方程,得到目标函数对设计变量的梯度,并采用线搜索方法获得最优步长,由此提高了优化算法的鲁棒性。算例表明,线搜索方法可以自动寻找最优的步长,有效解决了传统的取常数步长优化步长选取受到限制,优化结果受步长影响的问题,使得优化方法对步长的依赖性变小,提高了基于Adjoint方法翼型优化设计的鲁棒性。

关于极面的ADJOINT收缩(英文)

高维代数簇的半线收缩已有很多研究 .将它们推广到极面收缩对高维簇的双有理分类理论是很有意义的 .设 X是非奇异的 n维射影簇 ,L是 X上的 ample除子 ,f:X→Y是以 KX(n- 3 ) L为支撑除子的极面收缩映射 .当 f 不是双有理映射时 ,Beltrametti等人系统的研究了 f 的结构 .本文主要研究 f 是双有理映射时的情形 .一个完整的结构定理被给出 .

STUDY ON THE ADJOINT METHOD IN DATA ASSIMILATION AND THE RELATED PROBLEMS

It is not reasonable that one can only use the adjoint of model in data assimilation. The simulated numerical experiment shows that for the tidal model, the result of the adjoint of equation is almost the same as that of the adjoint of model: the averaged absolute difference of the amplitude between observations and simulation is less than 5.0 cm and that of the phase-lag is less than 5.0°. The results are both in good agreement with the observed M2 tide in the Bohai Sea and the Yellow Sea. For comparison, the traditional methods also have been used to simulate M2 tide in the Bohai Sea and the Yellow Sea. The initial guess values of the boundary conditions are given first, and then are adjusted to acquire the simulated results that are as close as possible to the observations. As the boundary conditions contain 72 values, which should be adjusted and how to adjust them can only be partially solved by adjusting them many times. The satisfied results are hard to acquire even gigantic efforts are done. Here, the automation of the treatment of the open boundary conditions is realized. The method is unique and superior to the traditional methods. It is emphasized that if the adjoint of equation is used, tedious and complicated mathematical deduction can be avoided. Therefore the adjoint of equation should attract much attention.

Assimilated Tidal Results of Tide Gauge and TOPEX/POSEIDON Data over the China Seas Using a Variational Adjoint Approach with a Nonlinear Numerical Model

In order to obtain an accurate tide description in the China Seas, the 2-dimensional nonlinear numerical Princeton Ocean Model （POM） is employed to incorporate in situ tidal measurements both from tide gauges and TOPEX/POSEIDON （T/P） derived datasets by means of the variational adjoint approach in such a way that unknown internal model parameters, bottom topography, friction coefficients and open boundary conditions, for example, are adjusted during the process. The numerical model is used as a forward model. After the along-track T/P data are processed, two classical methods, i.e. harmonic and response analysis, are implemented to estimate the tide from such datasets with a domain covering the model area extending from 0° to 41°N in latitude and from 99°E to 142°E in longitude. And the results of these two methods are compared and interpreted. The numerical simulation is performed for 16 major constituents. In the data assimilation experiments, three types of unknown parameters （water depth, bottom friction and tidal open boundary conditions in the model equations） are chosen as control variables. Among the various types of data assimilation experiments, the calibration of water depth brings the most promising results. By comparing the results with selected tide gauge data, the average absolute errors are decreased from 7.9 cm to 6.8 cm for amplitude and from 13.0° to 9.0° for phase with respect to the semidiurnal tide M2 constituent, which is the largest tidal constituent in the model area. After the data assimilation experiment is performed, the comparison between model results and tide gauge observation for water levels shows that the RMS errors decrease by 9 cm for a total of 14 stations, mostly selected along the coast of China's Mainland, when a one-month period is considered, and the correlation coefficients improve for most tidal stations among these stations.

The Factorization of Adjoint Polynomials of E^G（i）-class Graphs and Chromatically Equivalence Analysis

Let Sn be the star with n vertices, and let G be any connected graph with p vertices. We denote by Eτp＋（r-1）^G（i） the graph obtained from Sr and rG by coinciding the i-th vertex of G with the vertex of degree r - 1 of S,, while the i-th vertex of each component of （r - 1）G be adjacented to r - 1 vertices of degree 1 of St, respectively. By applying the properties of adjoint polynomials, We prove that factorization theorem of adjoint polynomials of kinds of graphs Eτp＋（r-1）^G（i）∪（r - 1）K1 （1 ≤i≤p）. Furthermore, we obtain structure characteristics of chromatically equivalent graphs of their complements.

Estimation of eddy viscosity on the South China Sea shelf with adjoint assimilation method

The eddy viscosity of the ocean is an important parameter indicating the small-scale mixing process in the oceanic interior water column. Ekman wind-driven current model and adjoint assimilation technique are used to calculate the vertical profiles of eddy viscosity by fitting model results to the observation data. The data used in the paper include observed wind data and ADCP data obtained at Wenchang Oil Rig on the SCS （the South China Sea） shelf in August 2002. Different simulations under different wind conditions are analyzed to explore how the eddy viscosity develops with varying wind field. The results show that the eddy viscosity endured gradual variations in the range of 10^-3 -10^-2 m^2 /s during the periods of wind changes. The mean eddy viscosity undergoing strong wind could rise by about 25% as compared to the value under weak wind.

Adjoint-based Sensitivity Analysis of a Mesoscale Low on the Mei-yu Front and Its Implications for Adaptive Observation

An adjoint sensitivity analysis of one mesoscale low on the mei-yu Front is presented in this paper. The sensitivity gradient of simulation error dry energy with respect to initial analysis is calculated. And after verifying the ability of a tangent linear and adjoint model to describe small perturbations in the nonlinear model, the sensitivity gradient analysis is implemented in detail. The sensitivity gradient with respect to different physical fields are not uniform in intensity, simulation error is most sensitive to the vapor mixed ratio. The localization and consistency are obvious characters of horizontal distribution of the sensitivity gradient, which is useful for the practical implementation of adaptive observation. The sensitivity region tilts to the northwest with height increasing; the singular vector calculation proves that this tilting characterizes a quick-growing structure, which denotes that using the leading singular vectors to decide the adaptive observation region is proper. When connected with simulation of a mesoscale low on the mei-yu Front, the sensitivity gradient has the following physical characters： the obvious sensitive region is mesoscale, concentrated in the middle-upper troposphere, and locates around the key system; and the sensitivity gradient of different physical fields correlates dynamically.

Multi-objective optimization strategies using adjoint method and game theory in aerodynamics

There are currently three different game strategies originated in economics： （1） Cooperative games （Pareto front）, （2） Competitive games （Nash game） and （3） Hierarchical games （Stackelberg game）. Each game achieves different equilibria with different performance, and their players play different roles in the games. Here, we introduced game concept into aerodynamic design, and combined it with adjoint method to solve multi- criteria aerodynamic optimization problems. The performance distinction of the equilibria of these three game strategies was investigated by numerical experiments. We computed Pareto front, Nash and Stackelberg equilibria of the same optimization problem with two conflicting and hierarchical targets under different parameterizations by using the deterministic optimization method. The numerical results show clearly that all the equilibria solutions are inferior to the Pareto front. Non-dominated Pareto front solutions are obtained, however the CPU cost to capture a set of solutions makes the Pareto front an expensive tool to the designer.

Mesoscale Moist Adjoint Sensitivity Study of a Mei-yu Heavy Rainfall Event

The mesoscale moist adjoint sensitivities related to the initiation of mesoscale convective systems （MCSs） are evaluated for a mei-yu heavy rainfall event. The sensitivities were calculated on a realistic background gained from a four-dimensional variational data assimilation of precipitation experiment to make the sensitivity computation possible and reasonable within a strong moist convective event at the mesoscale. The results show that the computed sensitivities at the mesoscale were capable of capturing the factors affecting MCS initiation. The sensitivities to the initial temperature and moisture are enhanced greatly by diabatic processes, especially at lower levels, and these sensitivities are much larger than those stemming from the horizontal winds, which implies that initiation of MCSs is more sensitive to low-level temperature and moisture perturbations rather than the horizontal winds. Moreover, concentration of sensitivities at low levels reflects the characteristics of the mei-yu front. The results provide some hints about how to improve quantitative precipitation forecasts of mei-yu heavy rainfall, such as by conducting mesoscale targetted observations via the adjoint-based method to reduce the low-level errors in the initial temperature and moisture.

典型李超代数的极大Adjoint二次迷向空间的维数

对复数域上的典型李超代数定义了极大Adjoint二次迷向空间,这是典型李超代数的内蕴特征;对每一类典型李超代数的极大Adjoint二次迷向空间做了细致的刻画,并确定它们各自的维数.

Adjoint Sensitivity Experiments of a Meso-β-scale Vortex in the Middle Reaches of the Yangtze River

A relatively independent and small-scale heavy rainfall event occurred to the south of a slow eastwardmoving meso-α-scale vortex. The analysis shows that a meso-β-scale system is heavily responsible for the intense precipitation. An attempt to simulate it met with some failures. In view of its small scale, short lifetime and relatively sparse observations at the initial time, an adjoint model was used to examine the sensitivity of the meso-β-scale vortex simulation with respect to initial conditions. The adjoint sensitivity indicates how small perturbations of initial model variables anywhere in the model domain can influence the central vorticity of the vortex. The largest sensitivity for both the wind and temperature perturbation is located below 700 hPa, especially at the low level. The largest sensitivity for the water vapor perturbation is located below 500 hPa, especially at the middle and low levels. The horizontal adjoint sensitivity for all variables is mainly located toward the upper reaches of the Yangtze River with respect to the simulated meso-β-scale system in Hunan and Jiangxi provinces with strong locality. The sensitivity shows that warm cyclonic perturbations in the upper reaches can have a great effect on the development of the meso-β-scale vortex. Based on adjoint sensitivity, forward sensitivity experiments were conducted to identify factors influencing the development of the meso-β-scale vortex and to explore ways of improving the prediction. A realistic prediction was achieved by using adjoint sensitivity to modify the initial conditions and implanting a warm cyclone at the initial time in the upper reaches of the river with respect to the meso-β-scale vortex, as is commonly done in tropical cyclone prediction.

Adjoint Assimilation in Marine Ecosystem Models and an Example of Application

This paper aims at a review of the work carried out to date on the adjoint assimilation of data in marine ecosys-tem models since 1995. The structure and feature of the adjoint assimilation in marine ecosystem models are also introduced. To illustrate the application of the adjoint technique and its merits, a 4-variable ecosystem model coupled with a 3-D physical model is established for the Bohai Sea and the Yellow Sea. The chlorophyll concentration data derived from the SeaWiFS o-cean colour data are assimilated in the model with the technique. Some results are briefly presented.

The Study of Single Station Inverting the Sea Surface Current by HF Ground Wave Radar Based on Adjoint Assimilation Technology

This paper introduces the assimilation technology in an ocean dynamics model and discusses the feasibility of inverting the sea surface current in the detection zone by assimilating the sea current radial velocity detected by single station HF ground wave radar in ocean dynamics model. Based on the adjoint assimilation and POM model, the paper successfully inverts the sea surface current through single station HF ground wave radar in the Zhoushan sea area. The single station HF radar inversion results are also compared with the bistatic HF radar composite results and the fixed point measured results by Annderaa current meter. The error analysis shows that acquisition of flow velocity and flow direction data from the single station HF radar based on adjoint assimilation and POM model is viable and the data obtained have a high correlation and consistency with the flow field observed by HF radar.

Inversion Study on Pollutant Discharges in the Bohai Sea with the Adjoint Method

The adjoint method is presented which could be used to estimate the time-varying contamination concentration(CC) from pollution source(PS). Then the pollutant flux is calculated. In order to guarantee the continuity of pollutant distribution and make the calculated results more accurate, the independent point(IP) scheme is proposed. The contamination concentrations(CCs) at some time steps are selected as the independent points(IPs), and only CCs at these IPs are optimized while CCs at other points are calculated through linear interpolation of the independent CCs. In twin numerical experiments, all the given distributions are successfully inverted with the adjoint method. The cost functions and the mean absolute errors(MAEs) in concentrations and pollutant fluxes decrease greatly after assimilation, and the cost functions are reduced by about 5 orders of magnitude compared with their initial values. The results indicate that the adjoint method is computationally efficient to recover CCs from PS. It is easier to invert the given distribution which is less complex. The inversion efficiency with IP scheme is raised compared to that without this scheme. The IP scheme is significant for the inversion result, in which appropriate IP number could yield better inversion results. More work will be done to apply this method to real experiment.

An Application of the Adjoint Method to a Statistical-Dynamical Tropical-Cyclone Prediction Model （SD-90）Ⅱ：Real Tropical Cyclone Cases

In the first paper in this series, a variational data assimilation of ideal tropical cyclone （TC） tracks was performed for the statistical-dynamical prediction model SD-90 by the adjoint method, and a prediction of TC tracks was made with good accuracy for tracks containing no sharp turns. In the present paper, the cases of real TC tracks are studied. Due to the complexity of TC motion, attention is paid to the diagnostic research of TC motion. First, five TC tracks are studied. Using the data of each entire TC track, by the adjoint method, five TC tracks are fitted well, and the forces acting on the TCs are retrieved. For a given TC, the distribution of the resultant of the retrieved force and Coriolis force well matches the corresponding TC track, i.e., when a TC turns, the resultant of the retrieved force and Coriolis force acts as a centripetal force, which means that the TC indeed moves like a particle; in particular, for TC 9911, the clockwise looping motion is also fitted well. And the distribution of the resultant appears to be periodic in some cases. Then, the present method is carried out for a portion of the track data for TC 9804, which indicates that when the amount of data for a TC track is sufficient, the algorithm is stable. And finally, the same algorithm is implemented for TCs with a double-eyewall structure, namely Bilis （2000） and Winnie （1997）, and the results prove the applicability of the algorithm to TCs with complicated mesoscale structures if the TC track data are obtained every three hours.