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.

Structural Modal Parameter Recognition and Related Damage Identification Methods under Environmental Excitations: A Review

Modal parameters can accurately characterize the structural dynamic properties and assess the physical state of the structure.Therefore,it is particularly significant to identify the structural modal parameters according to the monitoring data information in the structural health monitoring(SHM)system,so as to provide a scientific basis for structural damage identification and dynamic model modification.In view of this,this paper reviews methods for identifying structural modal parameters under environmental excitation and briefly describes how to identify structural damages based on the derived modal parameters.The paper primarily introduces data-driven modal parameter recognition methods(e.g.,time-domain,frequency-domain,and time-frequency-domain methods,etc.),briefly describes damage identification methods based on the variations of modal parameters(e.g.,natural frequency,modal shapes,and curvature modal shapes,etc.)and modal validation methods(e.g.,Stability Diagram and Modal Assurance Criterion,etc.).The current status of the application of artificial intelligence(AI)methods in the direction of modal parameter recognition and damage identification is further discussed.Based on the pre-vious analysis,the main development trends of structural modal parameter recognition and damage identification methods are given to provide scientific references for the optimized design and functional upgrading of SHM systems.

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.

Some Trials of Typical Typhoon Tracks Using the Adjoint Method

In this paper, the adjoint method is applied to the statistical-dynamic model （SD-90） for the prediction of typhoon tracks along with the regularization thinking and optimal control techniques. The adjoint model and the gradient of objective function are deduced with the continual model respectively. For 4 typical typhoons, the forces and the initial velocity can be retrieved well, and the tracks of these typhoons are accurately fitted for an appropriate regularization parameter and optimal control parameter.

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.

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.

Numerical study on the spatially varying drag coefficient in simulation of storm surges employing the adjoint method

From the simulation of storm surges resulting from Typhoons 7203 and 8509 in the Bohai Sea, Yellow Sea and East China Sea, water level data at tide stations are assimilated into a two-dimensional storm surge model, to study the spatially varying drag coefficient (DC) by employing the adjoint method. In this study, the DC at some grid points is uniformly selected as the independent DC, while the DC at other grid points is obtained through linear interpolation of the independent DC. The DC at independent points is optimized by employing the adjoint assimilation method, and global optimization is achieved by optimizing the independent DC. To demonstrate the method's performance, three comparative experiments are carried out. In the first experiment, the DC is treated as a constant. In the second and third experiments, the DC is derived using an empirical formula. Comparing the experimental results, it is found that the simulation accuracy for both Typhoons 7203 and 8509 increases greatly when optimizing the independent DC. However, the number of independent points makes no great difference to the precision of simulation. Moreover, the DC inverted from Typhoons 7203 and 8509 differs in some sea areas because of the different typhoon tracks. However, the spatial distribution of the inverted DC, for both Typhoons 7203 and 8509, demonstrates a clear effect of the DC on the storm surge modeling near the coastal areas where the DC is highest or lowest.

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.

Design optimization of unsteady airfoils with continuous adjoint method

In this paper, a new unsteady aerodynamic design method is presented based on the Navier-Stokes equations and a continuous adjoint approach. A basic framework of time-accurate unsteady airfoil optimization which adopts time-averaged aerodynamic coefficients as objective functions is presented. The time-accurate continuous adjoint equation and its boundary conditions are derived. The flow field and the adjoint equation are simulated numerically by the finite volume method （FVM）. Feasibility and accuracy of the approach are perfectly validated by the design optimization results of the plunging NACA0012 airfoil.

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.

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.

Highly-Efficient Aerodynamic Optimal Design of Rotor Airfoil Using Viscous Adjoint Method

In order to overcome the efficiency problem of the conventional gradient-based optimal design method,a highly-efficient viscous adjoint-based RANS equations method is applied to the aerodynamic optimal design of hovering rotor airfoil.The C-shaped body-fitted mesh is firstly automatically generated around the airfoil by solving the Poisson equations,and the Navier-Stokes(N-S)equations combined with Spalart-Allmaras(S-A)one-equation turbulence model are used as the governing equations to acquire the reliable flowfield variables.Then,according to multi-constrained characteristics of the optimization of high lift/drag ratio for hovering rotor airfoil,its corresponding adjoint equations,boundary conditions and gradient expressions are newly derived.On these bases,two representative rotor airfoils,NACA0012 airfoil and SC1095 airfoil,are selected as numerical examples to optimize their synthesized aerodynamic characteristics about lift/drag ratio in hover,and better aerodynamic performance of optimal airfoils are obtained compared with the baseline.Furthermore,the new designed rotor with the optimized rotor airfoil has better hover aerodynamic characteristics compared with the baseline rotor.In contrast to the baseline airfoils optimized by the finite difference method,it is demonstrated that the adjoint optimal algorithm itself is practical and highly-efficient for the aerodynamic optimization of hover rotor airfoil.

Numerical study on spatially varying control parameters of a marine ecosystem dynamical model with adjoint method

Based on the simulation of a marine ecosystem dynamical model in the Bohai Sea, the Yellow Sea and the East China Sea, chlorophyll data are assimilated to study the spatially varying control parameters （CPs） by using the adjoint method. In this study, the CPs at some grid points are selected as the independent CPs, while the CPs at other grid points can be obtained through linear interpolation with the independent CPs. The independent CPs are uniformly selected from each 30′ × 30′area, and we confirm that the optimal influence radius is 1.2° by a twin experiment. In the following experiments, when only the maximum growth rate of phytoplankton （Vm） is estimated by two given types of spatially varying CPs, the mean relative errors of Vm are 1.22% and 0.94% while the decrease rates of the mean error of chlorophyll in the surface are 94.6% and 95.8%, respectively. When the other four CPs are estimated respectively, the results are also satisfactory, which indicates that the adjoint method has a strong ability of optimizing the prescribed CP with spatial variations. However, when all these five most important CPs are estimated simultaneously, the collocation of the changing trend of each parameter influences the estimation results remarkably. Only when the collocation of the changing trend of each parameter is consistent with the ecological mechanisms which influence the growth of the phytoplankton in marine ecosystem, could the five most important CPs be estimated more accurately.

Deep Time Reconstructions of Internal Earth Structure with the Adjoint Method

The adjoint method is a powerful tool to obtain gradient information in a computational model relative to unknown model parameters,allowing one to solve inverse problems where analytical solutions are not available or the cost to determine prohibitive.

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.

Theoretical framework for geoacoustic inversion by adjoint method

Traditional geoacoustic inversions are generally solved by matched-field processing in combination with metaheuristic global searching algorithms which usually need massive computations. This paper proposes a new physical framework for geoacoustic retrievals. A parabolic approximation of wave equation with non-local boundary condition is used as the forward propagation model. The expressions of the corresponding tangent linear model and the adjoint operator are derived, respectively, by variational method. The analytical expressions for the gradient of the cost function with respect to the control variables can be formulated by the adjoint operator, which in turn can be used for optimization by the gradient-based method.

Adjoint Method-Based Algorithm for Calculating the Relative Dispersion Ratio in a Hydrodynamic System

Relative dispersion ratio(RDR)can be used to quantify the deviation behavior of a water parcel’s trajectory caused by a disturbance in a hydrodynamic system.It can be calculated by using a standard method for determining relative dispersion(RD),which accounts for the growth of the deviation of a cluster of particles from a specific initial time.However,the standard method for computing RD is time consuming.It involves numerous computations on tracing many water parcels.In this study,a new method based on the adjoint method is proposed to acquire a series of RDR fields in one round of tracing.Through this method,the continuous variation in the RDR corresponding to a time series of the disturbance time t can be obtained.The consistency and efficiency of the new method are compared with those of the standard method by applying it to a double-gyre flow and an unsteady Arnold-Beltrami-Childress flow field.Results show that the two methods have good consistency in a finite time span.The new method has a notable speedup for evaluating the RDR at multiple t.

ADJOINT OPERATOR METHOD AND NORMAL FORMS OF HIGHER ORDER FOR NONLINEAR DYNAMICAL SYSTEM

Normal form theory is a very effective method when we study degenerate bifurcations of nonlinear dynamical systems. In this paper by using adjoint operator method, normal forms of order 3 and 4 for nonlinear dynamical system with nilpotent linear part and Z(2)-asymmetry are computed. According to normal forms obtained, universal unfoldings for some degenerate bifurcation cases of codimension 3 and simple global characterizations, are studied.