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
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
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
The Regional Eta-coordinate Model（REM） has performed well in forecasting heavy rainfalls in China in recent years.A four-dimensional variational assimilation system（4DVar） is developed to improve the forecast skil...The Regional Eta-coordinate Model（REM） has performed well in forecasting heavy rainfalls in China in recent years.A four-dimensional variational assimilation system（4DVar） is developed to improve the forecast skill of the REM.The tangent linear model and adjoint model codes are written according to the＂code to code＂rule,and the establishment of the REM adjoint modeling system is introduced in detail in this paper.The tangent linear and adjoint models of the REM are validated against the observational data,and so is the gradient of the given cost function.It is shown that for the tangent linear model and cost function,when the magnitude of perturbations is reduced,the verification results approach 1.0;when the rounding error of computer is increased,the verification results depart off 1.0.In the validation of the adjoint model,the values on the left- and right-hand sides of the algebraic formula are equal with 13-digit accuracy.These results indicate that the tangent linear model and the adjoint model system of the REM are successfully coded,and the gradient of the cost function is correctly calculated.By using the REM adjoint modeling system,two 4DVar experiments and extended forecasts are performed using observational data for two real cases in June 1998 and August 2000.The results show that forecasts of temperature,wind speed, and specify humidity using the 4DVar-assimilated initial data are all improved at the end of the forecast period.However,the performance of the 4DVar in forcasting rainfall is different in these two cases.The prediction of location and amount of the accumulated rainfall is well improved in the first case,while in the second case the prediction has no significant improvement.The problem may result from the fact that the observational data used in the 4DVar for the second case are inadequate.This case will be studied further in future work.展开更多
This paper aims at a review of the work carried out to date on the adjoint assimilation of data in marine ecosys-tem models since 1995. The structure and feature of the adjoint assimilation in marine ecosystem models ...This paper aims at a review of the work carried out to date on the adjoint assimilation of data in marine ecosys-tem models since 1995. The structure and feature of the adjoint assimilation in marine ecosystem models are also introduced. To illustrate the application of the adjoint technique and its merits, a 4-variable ecosystem model coupled with a 3-D physical model is established for the Bohai Sea and the Yellow Sea. The chlorophyll concentration data derived from the SeaWiFS o-cean colour data are assimilated in the model with the technique. Some results are briefly presented.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
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
Normal form theory is a very effective method when we study degenerate bifurcations of nonlinear dynamical systems. In this paper by using adjoint operator method, normal forms of order 3 and 4 for nonlinear dynamical...Normal form theory is a very effective method when we study degenerate bifurcations of nonlinear dynamical systems. In this paper by using adjoint operator method, normal forms of order 3 and 4 for nonlinear dynamical system with nilpotent linear part and Z(2)-asymmetry are computed. According to normal forms obtained, universal unfoldings for some degenerate bifurcation cases of codimension 3 and simple global characterizations, are studied.展开更多
This paper mainly discusses the(2+1)-dimensional modified dispersive water-wave(MDWW) system which will be proved nonlinear self-adjointness. This property is applied to construct conservation laws corresponding to th...This paper mainly discusses the(2+1)-dimensional modified dispersive water-wave(MDWW) system which will be proved nonlinear self-adjointness. This property is applied to construct conservation laws corresponding to the symmetries of the system. Moreover, via the truncated Painlev′e analysis and consistent tanh-function expansion(CTE)method, the soliton-cnoidal periodic wave interaction solutions and corresponding images will be eventually achieved.展开更多
Control equation and adjoint equation are established by using block pulse functions, which transforms the linear time varying systems with time delays into a system of algebraic equations and the optimal control prob...Control equation and adjoint equation are established by using block pulse functions, which transforms the linear time varying systems with time delays into a system of algebraic equations and the optimal control problems are transformed into an optimization problem of multivariate functions thereby achieving the optimal control of linear systems with time delays.展开更多
Relative dispersion ratio(RDR)can be used to quantify the deviation behavior of a water parcel’s trajectory caused by a disturbance in a hydrodynamic system.It can be calculated by using a standard method for determi...Relative dispersion ratio(RDR)can be used to quantify the deviation behavior of a water parcel’s trajectory caused by a disturbance in a hydrodynamic system.It can be calculated by using a standard method for determining relative dispersion(RD),which accounts for the growth of the deviation of a cluster of particles from a specific initial time.However,the standard method for computing RD is time consuming.It involves numerous computations on tracing many water parcels.In this study,a new method based on the adjoint method is proposed to acquire a series of RDR fields in one round of tracing.Through this method,the continuous variation in the RDR corresponding to a time series of the disturbance time t can be obtained.The consistency and efficiency of the new method are compared with those of the standard method by applying it to a double-gyre flow and an unsteady Arnold-Beltrami-Childress flow field.Results show that the two methods have good consistency in a finite time span.The new method has a notable speedup for evaluating the RDR at multiple t.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
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
We propose a novel framework for learning a low-dimensional representation of data based on nonlinear dynamical systems,which we call the dynamical dimension reduction(DDR).In the DDR model,each point is evolved via a...We propose a novel framework for learning a low-dimensional representation of data based on nonlinear dynamical systems,which we call the dynamical dimension reduction(DDR).In the DDR model,each point is evolved via a nonlinear flow towards a lower-dimensional subspace;the projection onto the subspace gives the low-dimensional embedding.Training the model involves identifying the nonlinear flow and the subspace.Following the equation discovery method,we represent the vector field that defines the flow using a linear combination of dictionary elements,where each element is a pre-specified linear/nonlinear candidate function.A regularization term for the average total kinetic energy is also introduced and motivated by the optimal transport theory.We prove that the resulting optimization problem is well-posed and establish several properties of the DDR method.We also show how the DDR method can be trained using a gradient-based optimization method,where the gradients are computed using the adjoint method from the optimal control theory.The DDR method is implemented and compared on synthetic and example data sets to other dimension reduction methods,including the PCA,t-SNE,and Umap.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
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.
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.
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.
基金Supported by the National Natural Science Foundation of China(40631002 and 40875043)the National Basic Research Program of China(2006CB400500)
文摘The Regional Eta-coordinate Model（REM） has performed well in forecasting heavy rainfalls in China in recent years.A four-dimensional variational assimilation system（4DVar） is developed to improve the forecast skill of the REM.The tangent linear model and adjoint model codes are written according to the＂code to code＂rule,and the establishment of the REM adjoint modeling system is introduced in detail in this paper.The tangent linear and adjoint models of the REM are validated against the observational data,and so is the gradient of the given cost function.It is shown that for the tangent linear model and cost function,when the magnitude of perturbations is reduced,the verification results approach 1.0;when the rounding error of computer is increased,the verification results depart off 1.0.In the validation of the adjoint model,the values on the left- and right-hand sides of the algebraic formula are equal with 13-digit accuracy.These results indicate that the tangent linear model and the adjoint model system of the REM are successfully coded,and the gradient of the cost function is correctly calculated.By using the REM adjoint modeling system,two 4DVar experiments and extended forecasts are performed using observational data for two real cases in June 1998 and August 2000.The results show that forecasts of temperature,wind speed, and specify humidity using the 4DVar-assimilated initial data are all improved at the end of the forecast period.However,the performance of the 4DVar in forcasting rainfall is different in these two cases.The prediction of location and amount of the accumulated rainfall is well improved in the first case,while in the second case the prediction has no significant improvement.The problem may result from the fact that the observational data used in the 4DVar for the second case are inadequate.This case will be studied further in future work.
文摘This paper aims at a review of the work carried out to date on the adjoint assimilation of data in marine ecosys-tem models since 1995. The structure and feature of the adjoint assimilation in marine ecosystem models are also introduced. To illustrate the application of the adjoint technique and its merits, a 4-variable ecosystem model coupled with a 3-D physical model is established for the Bohai Sea and the Yellow Sea. The chlorophyll concentration data derived from the SeaWiFS o-cean colour data are assimilated in the model with the technique. Some results are briefly presented.
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.
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Normal form theory is a very effective method when we study degenerate bifurcations of nonlinear dynamical systems. In this paper by using adjoint operator method, normal forms of order 3 and 4 for nonlinear dynamical system with nilpotent linear part and Z(2)-asymmetry are computed. According to normal forms obtained, universal unfoldings for some degenerate bifurcation cases of codimension 3 and simple global characterizations, are studied.
基金Supported by National Natural Science Foundation of China under Grant Nos.11371293,11505090the Natural Science Foundation of Shaanxi Province under Grant No.2014JM2-1009+1 种基金Research Award Foundation for Outstanding Young Scientists of Shandong Province under Grant No.BS2015SF009the Science and Technology Innovation Foundation of Xi’an under Grant No.CYX1531WL41
文摘This paper mainly discusses the(2+1)-dimensional modified dispersive water-wave(MDWW) system which will be proved nonlinear self-adjointness. This property is applied to construct conservation laws corresponding to the symmetries of the system. Moreover, via the truncated Painlev′e analysis and consistent tanh-function expansion(CTE)method, the soliton-cnoidal periodic wave interaction solutions and corresponding images will be eventually achieved.
文摘Control equation and adjoint equation are established by using block pulse functions, which transforms the linear time varying systems with time delays into a system of algebraic equations and the optimal control problems are transformed into an optimization problem of multivariate functions thereby achieving the optimal control of linear systems with time delays.
文摘Relative dispersion ratio(RDR)can be used to quantify the deviation behavior of a water parcel’s trajectory caused by a disturbance in a hydrodynamic system.It can be calculated by using a standard method for determining relative dispersion(RD),which accounts for the growth of the deviation of a cluster of particles from a specific initial time.However,the standard method for computing RD is time consuming.It involves numerous computations on tracing many water parcels.In this study,a new method based on the adjoint method is proposed to acquire a series of RDR fields in one round of tracing.Through this method,the continuous variation in the RDR corresponding to a time series of the disturbance time t can be obtained.The consistency and efficiency of the new method are compared with those of the standard method by applying it to a double-gyre flow and an unsteady Arnold-Beltrami-Childress flow field.Results show that the two methods have good consistency in a finite time span.The new method has a notable speedup for evaluating the RDR at multiple t.
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.
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We propose a novel framework for learning a low-dimensional representation of data based on nonlinear dynamical systems,which we call the dynamical dimension reduction(DDR).In the DDR model,each point is evolved via a nonlinear flow towards a lower-dimensional subspace;the projection onto the subspace gives the low-dimensional embedding.Training the model involves identifying the nonlinear flow and the subspace.Following the equation discovery method,we represent the vector field that defines the flow using a linear combination of dictionary elements,where each element is a pre-specified linear/nonlinear candidate function.A regularization term for the average total kinetic energy is also introduced and motivated by the optimal transport theory.We prove that the resulting optimization problem is well-posed and establish several properties of the DDR method.We also show how the DDR method can be trained using a gradient-based optimization method,where the gradients are computed using the adjoint method from the optimal control theory.The DDR method is implemented and compared on synthetic and example data sets to other dimension reduction methods,including the PCA,t-SNE,and Umap.
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...
