The tighten couplings of game strategies with adjoint methods for multi-criterion aerodynamic design optimization are ad-dressed. Its numerical implementation is also described in details. In cooperative game,adjoint ...The tighten couplings of game strategies with adjoint methods for multi-criterion aerodynamic design optimization are ad-dressed. Its numerical implementation is also described in details. In cooperative game,adjoint methods are coupled in parallel to compute Pareto front collaboratively. Conversely in a Nash game,adjoint methods are coupled in each player s decision making to achieve Nash equilibrium competitively. In Stackelberg game,adjoint methods used by players are nested hierarchically through incomp...展开更多
Propeller design is a highly intricate and interdisciplinary task that necessitates careful trade-offs between radiated noise levels and aerodynamic efficiency.To achieve efficient trade-off designs,an enhanced on-the...Propeller design is a highly intricate and interdisciplinary task that necessitates careful trade-offs between radiated noise levels and aerodynamic efficiency.To achieve efficient trade-off designs,an enhanced on-the-fly unsteady adjoint-based aerodynamic and aeroacoustic optimization methodology is developed,which maintains the fidelity of the Navier-Stokes solution for unsteady flow and of the moving-medium Ffowcs Williams-Hawkings(FW-H)formulation for capturing tonal noise.Furthermore,this on-the-fly approach enables a unified architecture for discreteadjoint sensitivity analysis encompassing both aerodynamics and aeroacoustics,facilitating effective multi-objective weighted optimizations.Subsequently,this proposed methodology is applied to perform trade-off optimizations between aerodynamics and aeroacoustics for a propeller by employing varying weighting factors to comprehend their influence on optimal configurations.The results demonstrate a positive correlation between efficiency and noise sensitivities,and thus indicate an inherent synchronicity where pursing noise reduction through purely aeroacoustic optimization inevitably entails sacrificing aerodynamic efficiency.However,by effectively incorporating appropriate weighting factors(recommended to range from 0.25 to 0.5)into the multi-objective function combined with both aerodynamics and aeroacoustics,it becomes feasible to achieve efficiency enhancement and noise reduction simultaneously.Key findings show that reducing blade planform size and equipping“rotated-S”shaped airfoil profiles in the tip region can effectively restrain noise levels while maintaining aerodynamic performance.展开更多
Manufactured blades are inevitably different from their design intent,which leads to a deviation of the performance from the intended value.To quantify the associated performance uncertainty,many approaches have been ...Manufactured blades are inevitably different from their design intent,which leads to a deviation of the performance from the intended value.To quantify the associated performance uncertainty,many approaches have been developed.The traditional Monte Carlo method based on a Computational Fluid Dynamics solver(MC-CFD)for a three-dimensional compressor is prohibitively expensive.Existing alternatives to the MC-CFD,such as surrogate models and secondorder derivatives based on the adjoint method,can greatly reduce the computational cost.Nevertheless,they will encounter’the curse of dimensionality’except for the linear model based on the adjoint gradient(called MC-adj-linear).However,the MC-adj-linear model neglects the nonlinearity of the performance function.In this work,an improved method is proposed to circumvent the lowaccuracy problem of the MC-adj-linear without incurring the high cost of other alternative models.The method is applied to the study of the aerodynamic performance of an annular transonic compressor cascade,subject to prescribed geometric variability with industrial relevance.It is found that the proposed method achieves a significant accuracy improvement over the MC-adj-linear with low computational cost,showing the great potential for fast uncertainty quantification.展开更多
Abstract A first study on the continuous adjoint formulation for aerodynamic optimization design of high pressure turbines based on S2 surface governed by the Euler equations with source terms is presented. The object...Abstract A first study on the continuous adjoint formulation for aerodynamic optimization design of high pressure turbines based on S2 surface governed by the Euler equations with source terms is presented. The objective function is defined as an integral function along the boundaries, and the adjoint equations and the boundary conditions are derived by introducing the adjoint variable vec- tors. The gradient expression of the objective function then includes only the terms related to phys- ical shape variations. The numerical solution of the adjoint equation is conducted by a finite- difference method with the Jameson spatial scheme employing the first and the third order dissipa- tive fluxes. A gradient-based aerodynamic optimization system is established by integrating the blade stagger angles, the stacking lines and the passage perturbation parameterization with the quasi-Newton method of Broyden Fletcher Goldfarb-Shanno （BFGS）. The application of the continuous adjoint method is validated through a single stage high pressure turbine optimization case. The adiabatic efficiency increases from 0.8875 to 0.8931, whilst the mass flow rate and the pressure ratio remain almost unchanged. The optimization design is shown to reduce the passage vortex loss as well as the mixing loss due to the cooling air injection.展开更多
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 gam...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.展开更多
It is an inherent uncertainty problem that the application of laminar flow technology to the wing of large passenger aircraft is affected by flight conditions.In order to seek a more robust natural laminar flow contro...It is an inherent uncertainty problem that the application of laminar flow technology to the wing of large passenger aircraft is affected by flight conditions.In order to seek a more robust natural laminar flow control effect,it is necessary to develop an effective optimization design method.Meanwhile,attention must be given to the impact of crossflow(CF)instability brought on by the sweep angle.This paper constructs a robust optimization design framework based on discrete adjoint methods and non-intrusive polynomial chaos.Transition prediction is implemented by coupled Reynolds-Averaged Navier-Stokes(RANS)and simplified e^(N)method,which can consider both Tollmien-Schlichting(TS)wave and crossflow vortex instability.We have performed gradient enhancement processing on the general Polynomial Chaos Expansion(PCE),which is advantageous to reduce the computational cost of single uncertainty propagation.This processing takes advantage of the gradient information obtained by solving the coupled adjoint equations considering transition.The statistical moment gradient solution used for the robust optimization design also uses the derivatives of coupled adjoint equations.The framework is applied to the robust design of a 25°swept wing with infinite span in transonic flow.The uncertainty quantification and sensitivity analysis on the baseline wing shows that the uncertainty quantification method in this paper has high accuracy,and qualitatively reveals the factors that dominate in different flow field regions.By the robust optimization design,the mean and standard deviation of the drag coefficient can be reduced by 29%and 45%,respectively,and compared with the deterministic optimization design results,there is less possibility of forming shock waves under flight condition uncertainties.Robust optimization results illustrate the trade-off between the transition delay and the wave drag reduction.展开更多
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 ...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.展开更多
Adjoint method is widely used in aerodynamic design because only once solution of flow field is required for it to obtain the gradients of all design variables. However, the computational cost of adjoint vector is app...Adjoint method is widely used in aerodynamic design because only once solution of flow field is required for it to obtain the gradients of all design variables. However, the computational cost of adjoint vector is approximately equal to that of flow computation. In order to accelerate the solution of adjoint vector and improve the efficiency of adjoint-based optimization, machine learning for adjoint vector modeling is presented. Deep neural network (DNN) is employed to construct the mapping between the adjoint vector and the local flow variables. DNN can efficiently predict adjoint vector and its generalization is examined by a transonic drag reduction of NACA0012 airfoil. The results indicate that with negligible computational cost of the adjoint vector, the proposed DNN-based adjoint method can achieve the same optimization results as the traditional adjoint method.展开更多
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...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.展开更多
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 ...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.展开更多
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 adopt...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.展开更多
The goal of the arterial graft design problem is to find an optimal graft built on an occluded artery, which can be mathematically modeled by a fluid based shape optimization problem. The smoothness of the graft is on...The goal of the arterial graft design problem is to find an optimal graft built on an occluded artery, which can be mathematically modeled by a fluid based shape optimization problem. The smoothness of the graft is one of the important aspects in the arterial graft design problem since it affects the flow of the blood significantly. As an attractive design tool for this problem, level set methods are quite efficient for obtaining better shape of the graft. In this paper, a cubic spline level set method and a radial basis function level set method are designed to solve the arterial graft design problem. In both approaches, the shape of the arterial graft is implicitly tracked by the zero-level contour of a level set function and a high level of smoothness of the graft is achieved. Numerical results show the efficiency of the algorithms in the arterial graft design.展开更多
Nowadays,the adjoint method has become a popular approach in the optimization of turbomachinery to further improve its aerodynamic performance.However,design variables in these adjoint optimization applications are ge...Nowadays,the adjoint method has become a popular approach in the optimization of turbomachinery to further improve its aerodynamic performance.However,design variables in these adjoint optimization applications are generally not direct design parameters of blade(such as wedge angles or maximum thickness),making the geometric variation by adjoint optimization can hardly be re-extracted as the change of each design parameter.By giving considerations to the G1 continuity constraint of adjoint method on its parameterization method,this manuscript shows how to apply a parameterization method in 3D blade design process into adjoint optimization.Nearly all design parameters can therefore be treated as design variables in the adjoint method and then participate in the sensitivity-based optimization.Finally,a fitted Rotor 67 blade is optimized and the adiabatic efficiency is significantly improved by nearly 0.91%.展开更多
This paper presents a thorough study of the effect of the Constant Eddy Viscosity(CEV)assumption on the optimization of a discrete adjoint-based design optimization system.First,the algorithms of the adjoint methods w...This paper presents a thorough study of the effect of the Constant Eddy Viscosity(CEV)assumption on the optimization of a discrete adjoint-based design optimization system.First,the algorithms of the adjoint methods with and without the CEV assumption are presented,followed by a discussion of the two methods’solution stability.Second,the sensitivity accuracy,adjoint solution stability,and Root Mean Square(RMS)residual convergence rates at both design and offdesign operating points are compared between the CEV and full viscosity adjoint methods in detail.Finally,a multi-point steady aerodynamic and a multi-objective unsteady aerodynamic and aeroelastic coupled design optimizations are performed to study the impact of the CEV assumption on optimization.Two gradient-based optimizers,the Sequential Least-Square Quadratic Programming(SLSQP)method and Steepest Descent Method(SDM)are respectively used to draw a firm conclusion.The results from the transonic NASA Rotor 67 show that the CEV assumption can deteriorate RMS residual convergence rates and even lead to solution instability,especially at a near stall point.Compared with the steady cases,the effect of the CEV assumption on unsteady sensitivity accuracy is much stronger.Nevertheless,the CEV adjoint solver is still capable of achieving optimization goals to some extent,particularly if the flow under consideration is benign.展开更多
The performance of turbomachinery is largely affected by the nonuniform boundary conditions caused by the coupling between neighboring parts,such as the inlet distortion and hot streak.Existing works study this proble...The performance of turbomachinery is largely affected by the nonuniform boundary conditions caused by the coupling between neighboring parts,such as the inlet distortion and hot streak.Existing works study this problem by comparing the flow fields with uniform and nonuniform boundary conditions,which is cost extensive.In this work a new and efficient method is developed by computing the sensitivities of arbitrary performance metric relative to the boundary conditions and this method quantitatively describes how the disturbance on the boundary surface affects the performance.The adjoint method popular in the design optimization is extended to current field for the efficient computation of the sensitivity.The extra cost is independent of the number of parameters describing the nonuniform distribution.Accuracy of the adjoint boundary sensitivity is validated against analytical results for a converging–diverging nozzle case.Its applications in a representative case are detailed and discussed in this paper.Current method provides a new tool,also direct guides to minimize the negative effect and improve the performance utilizing the adjoint boundary sensitivity.展开更多
The stability of the flow in the vaneless diffuser of a centrifugal compressor is studied with the linear theory. The characteristics of direct and adjoint perturbation modes are investigated,and the receptivity of th...The stability of the flow in the vaneless diffuser of a centrifugal compressor is studied with the linear theory. The characteristics of direct and adjoint perturbation modes are investigated,and the receptivity of the instability mode to momentum forcing or mass injection is identified based on the adjoint modes. Analysis shows that the perturbation with the largest amplitude is located at the outlet of the vaneless diffuser, while the highest-receptivity region is located in the middle of the vaneless diffuser along the radial direction. The large difference between the direct and adjoint modes indicates that the instability mechanism cannot be identified from a study of either eigenmode separately. Therefore, the structural sensitivity analysis is adopted to study the feedback of the instability mode. The structural sensitivity of the eigenvalue which is proportional to the perturbation pressure and velocity is used to explain the mechanism of flow control for the vaneless diffuser.展开更多
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 hover...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.展开更多
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 usin...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.展开更多
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...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.展开更多
High-performance compressor design is best achieved with a good trade-off between aerodynamic and structural considerations,which requires efficient and accurate multidisciplinary design and optimization tools.As adva...High-performance compressor design is best achieved with a good trade-off between aerodynamic and structural considerations,which requires efficient and accurate multidisciplinary design and optimization tools.As advanced compressors are defined with a large design space,their optimization is most efficiently achieved using a gradient-based approach,where the gradient can be computed using an adjoint method,at a cost nearly independent of the dimension of the design space.While the adjoint method has been widely used for aerodynamic shape optimization,its use for structural shape optimizations of compressor blades has not been as well studied.This paper discussed a discrete adjoint solver for structural sensitivity analysis developed within the opensource Computational Structural Mechanics(CSM)software CalculiX,and proposed an efficient stress sensitivity analysis method based on the Finite Element Method(FEM)using adjoint.The proposed method is applied to compute the stress sensitivity of a wide-chord fan blade in a highbypass-ratio engine.The accuracy of the adjoint-based stress sensitivity is verified against central finite differences.In terms of computational efficiency,the adjoint approach is about 4.5 times more efficient than the conventional approach using finite differences.This works marks an important step towards fluid-structural coupled adjoint optimization of wide-chord fan blades.展开更多
基金National Natural Science Foundation of China (10872093)
文摘The tighten couplings of game strategies with adjoint methods for multi-criterion aerodynamic design optimization are ad-dressed. Its numerical implementation is also described in details. In cooperative game,adjoint methods are coupled in parallel to compute Pareto front collaboratively. Conversely in a Nash game,adjoint methods are coupled in each player s decision making to achieve Nash equilibrium competitively. In Stackelberg game,adjoint methods used by players are nested hierarchically through incomp...
基金supported by the National Science and Technology Major Project,China(No.Y2019-I-0018-0017)the National Natural Science Foundation of China(No.11602200)+1 种基金Hunan Innovative Province Construction Special Fund,China(No.2021GK1020)the Priority Academic Program Development of Jiangsu Higher Education Institutions,China。
文摘Propeller design is a highly intricate and interdisciplinary task that necessitates careful trade-offs between radiated noise levels and aerodynamic efficiency.To achieve efficient trade-off designs,an enhanced on-the-fly unsteady adjoint-based aerodynamic and aeroacoustic optimization methodology is developed,which maintains the fidelity of the Navier-Stokes solution for unsteady flow and of the moving-medium Ffowcs Williams-Hawkings(FW-H)formulation for capturing tonal noise.Furthermore,this on-the-fly approach enables a unified architecture for discreteadjoint sensitivity analysis encompassing both aerodynamics and aeroacoustics,facilitating effective multi-objective weighted optimizations.Subsequently,this proposed methodology is applied to perform trade-off optimizations between aerodynamics and aeroacoustics for a propeller by employing varying weighting factors to comprehend their influence on optimal configurations.The results demonstrate a positive correlation between efficiency and noise sensitivities,and thus indicate an inherent synchronicity where pursing noise reduction through purely aeroacoustic optimization inevitably entails sacrificing aerodynamic efficiency.However,by effectively incorporating appropriate weighting factors(recommended to range from 0.25 to 0.5)into the multi-objective function combined with both aerodynamics and aeroacoustics,it becomes feasible to achieve efficiency enhancement and noise reduction simultaneously.Key findings show that reducing blade planform size and equipping“rotated-S”shaped airfoil profiles in the tip region can effectively restrain noise levels while maintaining aerodynamic performance.
基金funded by the National Natural Science Foundation of China(No.52006177)National Science and Technology Major Project,China(No.2017-II-0009-0023)。
文摘Manufactured blades are inevitably different from their design intent,which leads to a deviation of the performance from the intended value.To quantify the associated performance uncertainty,many approaches have been developed.The traditional Monte Carlo method based on a Computational Fluid Dynamics solver(MC-CFD)for a three-dimensional compressor is prohibitively expensive.Existing alternatives to the MC-CFD,such as surrogate models and secondorder derivatives based on the adjoint method,can greatly reduce the computational cost.Nevertheless,they will encounter’the curse of dimensionality’except for the linear model based on the adjoint gradient(called MC-adj-linear).However,the MC-adj-linear model neglects the nonlinearity of the performance function.In this work,an improved method is proposed to circumvent the lowaccuracy problem of the MC-adj-linear without incurring the high cost of other alternative models.The method is applied to the study of the aerodynamic performance of an annular transonic compressor cascade,subject to prescribed geometric variability with industrial relevance.It is found that the proposed method achieves a significant accuracy improvement over the MC-adj-linear with low computational cost,showing the great potential for fast uncertainty quantification.
基金funded by the Aeronautical Science Foundation of China–China(No.2010ZB51023)
文摘Abstract A first study on the continuous adjoint formulation for aerodynamic optimization design of high pressure turbines based on S2 surface governed by the Euler equations with source terms is presented. The objective function is defined as an integral function along the boundaries, and the adjoint equations and the boundary conditions are derived by introducing the adjoint variable vec- tors. The gradient expression of the objective function then includes only the terms related to phys- ical shape variations. The numerical solution of the adjoint equation is conducted by a finite- difference method with the Jameson spatial scheme employing the first and the third order dissipa- tive fluxes. A gradient-based aerodynamic optimization system is established by integrating the blade stagger angles, the stacking lines and the passage perturbation parameterization with the quasi-Newton method of Broyden Fletcher Goldfarb-Shanno （BFGS）. The application of the continuous adjoint method is validated through a single stage high pressure turbine optimization case. The adiabatic efficiency increases from 0.8875 to 0.8931, whilst the mass flow rate and the pressure ratio remain almost unchanged. The optimization design is shown to reduce the passage vortex loss as well as the mixing loss due to the cooling air injection.
基金The project supported by the National Natural Science Foundation of China (10372040)Scientific Research Foundation (SRF) for Returned Oversea's Chinese Scholars (ROCS) (2003-091). The English text was polished by Yunming Chen
文摘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.
文摘It is an inherent uncertainty problem that the application of laminar flow technology to the wing of large passenger aircraft is affected by flight conditions.In order to seek a more robust natural laminar flow control effect,it is necessary to develop an effective optimization design method.Meanwhile,attention must be given to the impact of crossflow(CF)instability brought on by the sweep angle.This paper constructs a robust optimization design framework based on discrete adjoint methods and non-intrusive polynomial chaos.Transition prediction is implemented by coupled Reynolds-Averaged Navier-Stokes(RANS)and simplified e^(N)method,which can consider both Tollmien-Schlichting(TS)wave and crossflow vortex instability.We have performed gradient enhancement processing on the general Polynomial Chaos Expansion(PCE),which is advantageous to reduce the computational cost of single uncertainty propagation.This processing takes advantage of the gradient information obtained by solving the coupled adjoint equations considering transition.The statistical moment gradient solution used for the robust optimization design also uses the derivatives of coupled adjoint equations.The framework is applied to the robust design of a 25°swept wing with infinite span in transonic flow.The uncertainty quantification and sensitivity analysis on the baseline wing shows that the uncertainty quantification method in this paper has high accuracy,and qualitatively reveals the factors that dominate in different flow field regions.By the robust optimization design,the mean and standard deviation of the drag coefficient can be reduced by 29%and 45%,respectively,and compared with the deterministic optimization design results,there is less possibility of forming shock waves under flight condition uncertainties.Robust optimization results illustrate the trade-off between the transition delay and the wave drag reduction.
基金This work was supported by the National Natural Science Foundation of China under Grant No.40075009 and 40505011.The authors would like to thank the computer center of the College of Science at Zhejiang University for computer support of the 0rigin2K.
文摘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.
基金This work was supported by the National Numerical Wind tunnel Project(Grant NNW2018-ZT1B01)the National Natural Science Foundation of China(Grant 91852115).
文摘Adjoint method is widely used in aerodynamic design because only once solution of flow field is required for it to obtain the gradients of all design variables. However, the computational cost of adjoint vector is approximately equal to that of flow computation. In order to accelerate the solution of adjoint vector and improve the efficiency of adjoint-based optimization, machine learning for adjoint vector modeling is presented. Deep neural network (DNN) is employed to construct the mapping between the adjoint vector and the local flow variables. DNN can efficiently predict adjoint vector and its generalization is examined by a transonic drag reduction of NACA0012 airfoil. The results indicate that with negligible computational cost of the adjoint vector, the proposed DNN-based adjoint method can achieve the same optimization results as the traditional adjoint method.
基金Supported by the State Ministry of Science and Technology of China (Nos. 2007AA09Z118, 2008AA09A402)the National Natural Science Foundation of China (No. 41076006)the Ministry of Education's 111 Project (No. B07036)
文摘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.
基金This work was supported jointly by the Typhoon Foundation of Shanghaiby LASC of the Institute of Atmospheric Physics of the Chinese Academy of Sciencesby the National Natural Science Foundation of China under Grant No. 40633030.
文摘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.
基金supported by the Shanghai Municipal Natural Science Foundation(No.13ZR1415700)
文摘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.
基金Supported by National Foundation of Natural Science(11471092)Natural Science Foundation of Zhejiang Province(LZ13A010003)Foundation of Zhejiang Educational Committee(Y201121891)
文摘The goal of the arterial graft design problem is to find an optimal graft built on an occluded artery, which can be mathematically modeled by a fluid based shape optimization problem. The smoothness of the graft is one of the important aspects in the arterial graft design problem since it affects the flow of the blood significantly. As an attractive design tool for this problem, level set methods are quite efficient for obtaining better shape of the graft. In this paper, a cubic spline level set method and a radial basis function level set method are designed to solve the arterial graft design problem. In both approaches, the shape of the arterial graft is implicitly tracked by the zero-level contour of a level set function and a high level of smoothness of the graft is achieved. Numerical results show the efficiency of the algorithms in the arterial graft design.
基金supported by the National Major Science and Technology Project of China(Nos.2017-II-0006-0020 and J2019-II-0003-0023)。
文摘Nowadays,the adjoint method has become a popular approach in the optimization of turbomachinery to further improve its aerodynamic performance.However,design variables in these adjoint optimization applications are generally not direct design parameters of blade(such as wedge angles or maximum thickness),making the geometric variation by adjoint optimization can hardly be re-extracted as the change of each design parameter.By giving considerations to the G1 continuity constraint of adjoint method on its parameterization method,this manuscript shows how to apply a parameterization method in 3D blade design process into adjoint optimization.Nearly all design parameters can therefore be treated as design variables in the adjoint method and then participate in the sensitivity-based optimization.Finally,a fitted Rotor 67 blade is optimized and the adiabatic efficiency is significantly improved by nearly 0.91%.
基金supported by the National Science and Technology Major Project,China(No.2017-II-0009-0023)China’s 111 project(No.B17037)sponsored by Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University,China.
文摘This paper presents a thorough study of the effect of the Constant Eddy Viscosity(CEV)assumption on the optimization of a discrete adjoint-based design optimization system.First,the algorithms of the adjoint methods with and without the CEV assumption are presented,followed by a discussion of the two methods’solution stability.Second,the sensitivity accuracy,adjoint solution stability,and Root Mean Square(RMS)residual convergence rates at both design and offdesign operating points are compared between the CEV and full viscosity adjoint methods in detail.Finally,a multi-point steady aerodynamic and a multi-objective unsteady aerodynamic and aeroelastic coupled design optimizations are performed to study the impact of the CEV assumption on optimization.Two gradient-based optimizers,the Sequential Least-Square Quadratic Programming(SLSQP)method and Steepest Descent Method(SDM)are respectively used to draw a firm conclusion.The results from the transonic NASA Rotor 67 show that the CEV assumption can deteriorate RMS residual convergence rates and even lead to solution instability,especially at a near stall point.Compared with the steady cases,the effect of the CEV assumption on unsteady sensitivity accuracy is much stronger.Nevertheless,the CEV adjoint solver is still capable of achieving optimization goals to some extent,particularly if the flow under consideration is benign.
基金supported by the National Natural Science Foundation of China(Nos.51876098 and 51911540475)。
文摘The performance of turbomachinery is largely affected by the nonuniform boundary conditions caused by the coupling between neighboring parts,such as the inlet distortion and hot streak.Existing works study this problem by comparing the flow fields with uniform and nonuniform boundary conditions,which is cost extensive.In this work a new and efficient method is developed by computing the sensitivities of arbitrary performance metric relative to the boundary conditions and this method quantitatively describes how the disturbance on the boundary surface affects the performance.The adjoint method popular in the design optimization is extended to current field for the efficient computation of the sensitivity.The extra cost is independent of the number of parameters describing the nonuniform distribution.Accuracy of the adjoint boundary sensitivity is validated against analytical results for a converging–diverging nozzle case.Its applications in a representative case are detailed and discussed in this paper.Current method provides a new tool,also direct guides to minimize the negative effect and improve the performance utilizing the adjoint boundary sensitivity.
文摘The stability of the flow in the vaneless diffuser of a centrifugal compressor is studied with the linear theory. The characteristics of direct and adjoint perturbation modes are investigated,and the receptivity of the instability mode to momentum forcing or mass injection is identified based on the adjoint modes. Analysis shows that the perturbation with the largest amplitude is located at the outlet of the vaneless diffuser, while the highest-receptivity region is located in the middle of the vaneless diffuser along the radial direction. The large difference between the direct and adjoint modes indicates that the instability mechanism cannot be identified from a study of either eigenmode separately. Therefore, the structural sensitivity analysis is adopted to study the feedback of the instability mode. The structural sensitivity of the eigenvalue which is proportional to the perturbation pressure and velocity is used to explain the mechanism of flow control for the vaneless diffuser.
文摘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.
基金The State Ministry of Science and Technology of China under contract No. 2007AA09Z118the National Natural Science Foundation of China under contract No. 41076006the Ministry of Education’s 111 Project under contract No. B07036
文摘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.
文摘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.
基金Supported by the Science Center for Gas Turbine Project,China(No.P2022-C-II-001-001).
文摘High-performance compressor design is best achieved with a good trade-off between aerodynamic and structural considerations,which requires efficient and accurate multidisciplinary design and optimization tools.As advanced compressors are defined with a large design space,their optimization is most efficiently achieved using a gradient-based approach,where the gradient can be computed using an adjoint method,at a cost nearly independent of the dimension of the design space.While the adjoint method has been widely used for aerodynamic shape optimization,its use for structural shape optimizations of compressor blades has not been as well studied.This paper discussed a discrete adjoint solver for structural sensitivity analysis developed within the opensource Computational Structural Mechanics(CSM)software CalculiX,and proposed an efficient stress sensitivity analysis method based on the Finite Element Method(FEM)using adjoint.The proposed method is applied to compute the stress sensitivity of a wide-chord fan blade in a highbypass-ratio engine.The accuracy of the adjoint-based stress sensitivity is verified against central finite differences.In terms of computational efficiency,the adjoint approach is about 4.5 times more efficient than the conventional approach using finite differences.This works marks an important step towards fluid-structural coupled adjoint optimization of wide-chord fan blades.
