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
The primary concern in stealth aircraft design is the very large electrical size objects.However,the computational and storage requirements of these objects present significant obstacles for current highfidelity desig...The primary concern in stealth aircraft design is the very large electrical size objects.However,the computational and storage requirements of these objects present significant obstacles for current highfidelity design methods,particularly when addressing high-dimensional complex engineering design problems.To address these challenges,we developed a surface sensitivity technique based on the multilevel fast multipole algorithm(MLFMA).An access and storage of sparse partial derivative tensor was improved to significantly enhanced the computation performance.The far-field interactions of the surface sensitivity equation were realized by differential the multipole expansion.In addition,we proposed a fast far-field multiplication method to accelerate the multiplication process.The surface mesh derivative with respect to the design variables was calculated by analytical and complex variable methods,substantially improving computational efficiency.These advancements enabled the MLFMAbased surface sensitivity method to millions meshes and large-scale gradients,extending gradientbased optimization for very large electrical size problems.Test cases have verified the effectiveness of this method in optimizing very large electrical objects in terms of both accuracy and efficiency.展开更多
In recent years,incidents of simultaneous exceedance of PM_(2.5)and O_(3) concentrations,termed PM_(2.5)and O_(3) co-pollution events,have frequently occurred in China.This study conducted atmospheric circulation anal...In recent years,incidents of simultaneous exceedance of PM_(2.5)and O_(3) concentrations,termed PM_(2.5)and O_(3) co-pollution events,have frequently occurred in China.This study conducted atmospheric circulation analysis on two typical co-pollution events in Beijing,occurring from July 22 to July 28,2019,and from April 25 to May 2,2020.These events were categorized into pre-trough southerly airflow type(Type 1)and post-trough northwest flow type(Type 2).Subsequently,sensitivity analyses using the GRAPES-CUACE adjoint model were performed to quantify the contributions of precursor emissions from Beijing and surrounding areas to PM_(2.5)and O_(3) concentrations in Beijing for two types of co-pollution.The results indicated that the spatiotemporal distribution of sensitive source region varied among different circulation types.Primary PM_(2.5)(PPM_(2.5))emissions from Hebei contributed the most to the 24-hour average PM_(2.5)(24-h PM_(2.5))peak concentration(41.6%-45.4%),followed by Beijing emissions(31%-35.7%).The maximum daily 8-hour average ozone peak concentration was primarily influenced by the emissions from Hebei and Beijing,with contribution ratios respectively of 32.8%-44.8% and 29%-42.1%.Additionally,NO_(x)emissions were the main contributors in Type 1,while both NO_(x)and VOCs emissions contributed similarly in Type 2.The iterative emission reduction experiments for two types of co-pollution indicated that Type 1 required emission reductions in NO_(x)(52.4%-71.8%)and VOCs(14.1%-33.8%)only.In contrast,Type 2 required combined emission reductions in NO_(x)(37.0%-65.1%),VOCs(30.7%-56.2%),and PPM_(2.5)(31%-46.9%).This study provided a reference for controlling co-pollution events and improving air quality in Beijing.展开更多
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 ...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.展开更多
In this article,we explore the famous Selkov–Schnakenberg(SS)system of coupled nonlinear partial differential equations(PDEs)for Lie symmetry analysis,self-adjointness,and conservation laws.Moreover,miscellaneous sol...In this article,we explore the famous Selkov–Schnakenberg(SS)system of coupled nonlinear partial differential equations(PDEs)for Lie symmetry analysis,self-adjointness,and conservation laws.Moreover,miscellaneous soliton solutions like dark,bright,periodic,rational,Jacobian elliptic function,Weierstrass elliptic function,and hyperbolic solutions of the SS system will be achieved by a well-known technique called sub-ordinary differential equations.All these results are displayed graphically by 3D,2D,and contour plots.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
This work presents the “n<sup>th</sup>-Order Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (abbreviated as “n<sup>th</sup>-FASAM-N”), which will be shown to be the...This work presents the “n<sup>th</sup>-Order Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (abbreviated as “n<sup>th</sup>-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 n<sup>th</sup>-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 n<sup>th</sup>-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 n<sup>th</sup>-FASAM-N methodology becomes identical to the extant n<sup>th</sup> CASAM-N (“n<sup>th</sup>-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems”) methodology. Both the n<sup>th</sup>-FASAM-N and the n<sup>th</sup>-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 n<sup>th</sup>-FASAM-N and the n<sup>th</sup>-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.展开更多
This work highlights the unparalleled efficiency of the “n<sup>th</sup>-Order Function/ Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (n<sup>th</sup>-FASAM-N) by con...This work highlights the unparalleled efficiency of the “n<sup>th</sup>-Order Function/ Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (n<sup>th</sup>-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 n<sup>th</sup>-FASAM-N methodology is compared to the extant “n<sup>th</sup>-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (n<sup>th</sup>-CASAM-N) showing that: (i) the 1<sup>st</sup>-FASAM-N and the 1<sup>st</sup>-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” (1<sup>st</sup>-LASS);(ii) the 2<sup>nd</sup>-FASAM-N methodology is considerably more efficient than the 2<sup>nd</sup>-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 2<sup>nd</sup>-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 2<sup>nd</sup>-CASAM-N methodology requires 7 large-scale computations for obtaining these 28 second-order sensitivities;(iii) the 3<sup>rd</sup>-FASAM-N methodology is even more efficient than the 3<sup>rd</sup>-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 3<sup>rd</sup>-FASAM-N methodology, while the application of the 3<sup>rd</sup>-CASAM-N methodology requires at least 22 large-scale computations for computing the same 84 distinct third-order sensitivities. Together, the n<sup>th</sup>-FASAM-N and the n<sup>th</sup>-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.展开更多
This work presents the “Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology (2<sup>nd</sup>-CASAM)” for the efficient and exact computation of 1<sup>st</sup>- and 2<sup>...This work presents the “Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology (2<sup>nd</sup>-CASAM)” for the efficient and exact computation of 1<sup>st</sup>- and 2<sup>nd</sup>-order response sensitivities to uncertain parameters and domain boundaries of linear systems. The model’s response (<em>i.e.</em>, 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 2<sup>nd</sup>-CASAM requires a single large-scale computation using the First-Level Adjoint Sensitivity System (1<sup>st</sup>-LASS) for obtaining all of the first-order response sensitivities, and at most N large-scale computations using the Second-Level Adjoint Sensitivity System (2<sup>nd</sup>-LASS) for obtaining exactly all of the second-order response sensitivities. In contradistinction, forward other methods would require (<em>N</em>2/2 + 3 <em>N</em>/2) large-scale computations for obtaining all of the first- and second-order sensitivities. This work also shows that constructing and solving the 2<sup>nd</sup>-LASS requires very little additional effort beyond the construction of the 1<sup>st</sup>-LASS needed for computing the first-order sensitivities. Solving the equations underlying the 1<sup>st</sup>-LASS and 2<sup>nd</sup>-LASS requires the same computational solvers as needed for solving (<em>i.e.</em>, “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 1<sup>st</sup>-LASS and the 2<sup>nd</sup>-LASS. Since neither the 1<sup>st</sup>-LASS nor the 2<sup>nd</sup>-LASS involves any differentials of the operators underlying the original system, the 1<sup>st</sup>-LASS is designated as a “<u>first-level</u>” (as opposed to a “first-order”) adjoint sensitivity system, while the 2<sup>nd</sup>-LASS is designated as a “<u>second-level</u>” (rather than a “second-order”) adjoint sensitivity system. Mixed second-order response sensitivities involving boundary parameters may arise from all source terms of the 2<sup>nd</sup>-LASS that involve the imprecisely known boundary parameters. Notably, the 2<sup>nd</sup>-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.展开更多
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
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
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
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 TO...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.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
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 ...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.展开更多
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 ...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.展开更多
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 ve...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.展开更多
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supported by the National Key Research and Development Program of China(Grant No.2023YFB3002800).
文摘The primary concern in stealth aircraft design is the very large electrical size objects.However,the computational and storage requirements of these objects present significant obstacles for current highfidelity design methods,particularly when addressing high-dimensional complex engineering design problems.To address these challenges,we developed a surface sensitivity technique based on the multilevel fast multipole algorithm(MLFMA).An access and storage of sparse partial derivative tensor was improved to significantly enhanced the computation performance.The far-field interactions of the surface sensitivity equation were realized by differential the multipole expansion.In addition,we proposed a fast far-field multiplication method to accelerate the multiplication process.The surface mesh derivative with respect to the design variables was calculated by analytical and complex variable methods,substantially improving computational efficiency.These advancements enabled the MLFMAbased surface sensitivity method to millions meshes and large-scale gradients,extending gradientbased optimization for very large electrical size problems.Test cases have verified the effectiveness of this method in optimizing very large electrical objects in terms of both accuracy and efficiency.
基金supported by the National Key Research and Development Program of China(No.2022YFC3701205)the National Natural Science Foundation of China(No.41975173)the Science and Technology Development Fund of the Chinese Academy of Meteorological Sciences(No.2021KJ011)。
文摘In recent years,incidents of simultaneous exceedance of PM_(2.5)and O_(3) concentrations,termed PM_(2.5)and O_(3) co-pollution events,have frequently occurred in China.This study conducted atmospheric circulation analysis on two typical co-pollution events in Beijing,occurring from July 22 to July 28,2019,and from April 25 to May 2,2020.These events were categorized into pre-trough southerly airflow type(Type 1)and post-trough northwest flow type(Type 2).Subsequently,sensitivity analyses using the GRAPES-CUACE adjoint model were performed to quantify the contributions of precursor emissions from Beijing and surrounding areas to PM_(2.5)and O_(3) concentrations in Beijing for two types of co-pollution.The results indicated that the spatiotemporal distribution of sensitive source region varied among different circulation types.Primary PM_(2.5)(PPM_(2.5))emissions from Hebei contributed the most to the 24-hour average PM_(2.5)(24-h PM_(2.5))peak concentration(41.6%-45.4%),followed by Beijing emissions(31%-35.7%).The maximum daily 8-hour average ozone peak concentration was primarily influenced by the emissions from Hebei and Beijing,with contribution ratios respectively of 32.8%-44.8% and 29%-42.1%.Additionally,NO_(x)emissions were the main contributors in Type 1,while both NO_(x)and VOCs emissions contributed similarly in Type 2.The iterative emission reduction experiments for two types of co-pollution indicated that Type 1 required emission reductions in NO_(x)(52.4%-71.8%)and VOCs(14.1%-33.8%)only.In contrast,Type 2 required combined emission reductions in NO_(x)(37.0%-65.1%),VOCs(30.7%-56.2%),and PPM_(2.5)(31%-46.9%).This study provided a reference for controlling co-pollution events and improving air quality in Beijing.
文摘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.
文摘In this article,we explore the famous Selkov–Schnakenberg(SS)system of coupled nonlinear partial differential equations(PDEs)for Lie symmetry analysis,self-adjointness,and conservation laws.Moreover,miscellaneous soliton solutions like dark,bright,periodic,rational,Jacobian elliptic function,Weierstrass elliptic function,and hyperbolic solutions of the SS system will be achieved by a well-known technique called sub-ordinary differential equations.All these results are displayed graphically by 3D,2D,and contour plots.
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.
文摘This work presents the “n<sup>th</sup>-Order Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (abbreviated as “n<sup>th</sup>-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 n<sup>th</sup>-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 n<sup>th</sup>-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 n<sup>th</sup>-FASAM-N methodology becomes identical to the extant n<sup>th</sup> CASAM-N (“n<sup>th</sup>-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems”) methodology. Both the n<sup>th</sup>-FASAM-N and the n<sup>th</sup>-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 n<sup>th</sup>-FASAM-N and the n<sup>th</sup>-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.
文摘This work highlights the unparalleled efficiency of the “n<sup>th</sup>-Order Function/ Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (n<sup>th</sup>-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 n<sup>th</sup>-FASAM-N methodology is compared to the extant “n<sup>th</sup>-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (n<sup>th</sup>-CASAM-N) showing that: (i) the 1<sup>st</sup>-FASAM-N and the 1<sup>st</sup>-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” (1<sup>st</sup>-LASS);(ii) the 2<sup>nd</sup>-FASAM-N methodology is considerably more efficient than the 2<sup>nd</sup>-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 2<sup>nd</sup>-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 2<sup>nd</sup>-CASAM-N methodology requires 7 large-scale computations for obtaining these 28 second-order sensitivities;(iii) the 3<sup>rd</sup>-FASAM-N methodology is even more efficient than the 3<sup>rd</sup>-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 3<sup>rd</sup>-FASAM-N methodology, while the application of the 3<sup>rd</sup>-CASAM-N methodology requires at least 22 large-scale computations for computing the same 84 distinct third-order sensitivities. Together, the n<sup>th</sup>-FASAM-N and the n<sup>th</sup>-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.
文摘This work presents the “Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology (2<sup>nd</sup>-CASAM)” for the efficient and exact computation of 1<sup>st</sup>- and 2<sup>nd</sup>-order response sensitivities to uncertain parameters and domain boundaries of linear systems. The model’s response (<em>i.e.</em>, 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 2<sup>nd</sup>-CASAM requires a single large-scale computation using the First-Level Adjoint Sensitivity System (1<sup>st</sup>-LASS) for obtaining all of the first-order response sensitivities, and at most N large-scale computations using the Second-Level Adjoint Sensitivity System (2<sup>nd</sup>-LASS) for obtaining exactly all of the second-order response sensitivities. In contradistinction, forward other methods would require (<em>N</em>2/2 + 3 <em>N</em>/2) large-scale computations for obtaining all of the first- and second-order sensitivities. This work also shows that constructing and solving the 2<sup>nd</sup>-LASS requires very little additional effort beyond the construction of the 1<sup>st</sup>-LASS needed for computing the first-order sensitivities. Solving the equations underlying the 1<sup>st</sup>-LASS and 2<sup>nd</sup>-LASS requires the same computational solvers as needed for solving (<em>i.e.</em>, “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 1<sup>st</sup>-LASS and the 2<sup>nd</sup>-LASS. Since neither the 1<sup>st</sup>-LASS nor the 2<sup>nd</sup>-LASS involves any differentials of the operators underlying the original system, the 1<sup>st</sup>-LASS is designated as a “<u>first-level</u>” (as opposed to a “first-order”) adjoint sensitivity system, while the 2<sup>nd</sup>-LASS is designated as a “<u>second-level</u>” (rather than a “second-order”) adjoint sensitivity system. Mixed second-order response sensitivities involving boundary parameters may arise from all source terms of the 2<sup>nd</sup>-LASS that involve the imprecisely known boundary parameters. Notably, the 2<sup>nd</sup>-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.
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.
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.
文摘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.
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.
文摘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.
基金The National Key Basic Research Program of China under contract No. 2005CB422303the International Cooperation Program Project under contract No. 2004DFB02700the National Natural Science Foundation of China under contract No. 40552002
文摘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.
基金supported by the National Natural Science Foundation of China under Grant No.40405020.
文摘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.
