The Moore-Penrose metric generalized inverse T+ of linear operator T in Banach space is systematically investigated in this paper. Unlike the case in Hilbert space, even T is a linear operator in Banach Space, the Moo...The Moore-Penrose metric generalized inverse T+ of linear operator T in Banach space is systematically investigated in this paper. Unlike the case in Hilbert space, even T is a linear operator in Banach Space, the Moore-Penrose metric generalized inverse T+ is usually homogeneous and nonlinear in general. By means of the methods of geometry of Banach Space, the necessary and sufficient conditions for existence, continuitv, linearity and minimum property of the Moore-Penrose metric generalized inverse T+ will be given, and some properties of T+ will be investigated in this paper.展开更多
The proper orthogonal decomposition(POD)and the singular value decomposition(SVD) are used to study the finite difference scheme(FDS)for the nonstationary Navier-Stokes equations. Ensembles of data are compiled from t...The proper orthogonal decomposition(POD)and the singular value decomposition(SVD) are used to study the finite difference scheme(FDS)for the nonstationary Navier-Stokes equations. Ensembles of data are compiled from the transient solutions computed from the discrete equation system derived from the FDS for the nonstationary Navier-Stokes equations.The optimal orthogonal bases are reconstructed by the elements of the ensemble with POD and SVD.Combining the above procedures with a Galerkin projection approach yields a new optimizing FDS model with lower dimensions and a high accuracy for the nonstationary Navier-Stokes equations.The errors between POD approximate solutions and FDS solutions are analyzed.It is shown by considering the results obtained for numerical simulations of cavity flows that the error between POD approximate solution and FDS solution is consistent with theoretical results.Moreover,it is also shown that this validates the feasibility and efficiency of POD method.展开更多
A proper orthogonal decomposition (POD) method is applied to a usual finite element (FE) formulation for parabolic equations so that it is reduced into a POD FE formulation with lower dimensions and enough high accura...A proper orthogonal decomposition (POD) method is applied to a usual finite element (FE) formulation for parabolic equations so that it is reduced into a POD FE formulation with lower dimensions and enough high accuracy. The errors between the reduced POD FE solution and the usual FE solution are analyzed. It is shown by numerical examples that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is also shown that this validates the feasibility and efficiency of POD method.展开更多
Large eddy simulation(LES)is used to calculate the in-cylinder turbulent flow field in a direct injection spark ignition(DISI)engine.The computations are carried out for three different maximum valve lifts(MVL)and thr...Large eddy simulation(LES)is used to calculate the in-cylinder turbulent flow field in a direct injection spark ignition(DISI)engine.The computations are carried out for three different maximum valve lifts(MVL)and throughout 100 consecutive engine cycles.The simulated results as well as corresponding particle image velocimetry(PIV)measurement database are analyzed by the proper orthogonal decomposition(POD)method.Through a new developed POD quadruple decomposition the instantaneous in-cylinder flow fields are decomposed into four parts,named mean field,coherent field,transition field and turbulent field,respectively.Then the in-cylinder turbulent flow characteristics and cycle-to-cycle variations(CCV)are studied separately upon the four part flow fields.Results indicate that each part exhibits its specific characteristics and has close connection with others.The mean part contains more than 50%of the total kinetic energy and the energy cascade phenomenon occurs among the four part fields;the coherent field part possesses the highest CCV level which dominates CCV of the bulk flow.In addition,it is observed that a change in MVL affects significantly the in-cylinder flow behavior including CCV,especially for the coherent part.Furthermore,the POD analysis demonstrates that at least 25 sample cycles for the mean velocity and 50 sample cycles for the RMS velocity are necessary for obtaining converged and correct results in CCV.展开更多
We present dynamic mode decomposition (DMD) for studying the hairpin vortices generated by hemisphere protuberance measured by two-dimensional (2D) time-resolved (TR) particle image velocimetry (PIV) in a water channe...We present dynamic mode decomposition (DMD) for studying the hairpin vortices generated by hemisphere protuberance measured by two-dimensional (2D) time-resolved (TR) particle image velocimetry (PIV) in a water channel. The hairpins dynamic information is extracted by identifying their dominant frequencies and associated spatial structures. For this quasi-periodic data system, the resulting main Dynamic modes illustrate the different spatial structures associated with the wake vortex region and the near-wall region. By comparisons with proper orthogonal decomposition (POD), it can be concluded that the dynamic mode concentrates on a certain frequency component more effectively than the mode determined by POD. During the analysis, DMD has proven itself a robust and reliable algorithm to extract spatial-temporal coherent structures.展开更多
基金the National Natural Science Foundation of China(No.19971023)the Heilongjiang Provincial Natural Science Foundation of China.
文摘The Moore-Penrose metric generalized inverse T+ of linear operator T in Banach space is systematically investigated in this paper. Unlike the case in Hilbert space, even T is a linear operator in Banach Space, the Moore-Penrose metric generalized inverse T+ is usually homogeneous and nonlinear in general. By means of the methods of geometry of Banach Space, the necessary and sufficient conditions for existence, continuitv, linearity and minimum property of the Moore-Penrose metric generalized inverse T+ will be given, and some properties of T+ will be investigated in this paper.
基金the National Natural Science Foundation of China(Grant Nos.10471100,40437017,and 60573158)Beijing Jiaotong University Science and Technology Foundation
文摘The proper orthogonal decomposition(POD)and the singular value decomposition(SVD) are used to study the finite difference scheme(FDS)for the nonstationary Navier-Stokes equations. Ensembles of data are compiled from the transient solutions computed from the discrete equation system derived from the FDS for the nonstationary Navier-Stokes equations.The optimal orthogonal bases are reconstructed by the elements of the ensemble with POD and SVD.Combining the above procedures with a Galerkin projection approach yields a new optimizing FDS model with lower dimensions and a high accuracy for the nonstationary Navier-Stokes equations.The errors between POD approximate solutions and FDS solutions are analyzed.It is shown by considering the results obtained for numerical simulations of cavity flows that the error between POD approximate solution and FDS solution is consistent with theoretical results.Moreover,it is also shown that this validates the feasibility and efficiency of POD method.
基金supported by National Natural Science Foundation of China (Grant Nos. 10871022, 10771065,and 60573158)Natural Science Foundation of Hebei Province (Grant No. A2007001027)
文摘A proper orthogonal decomposition (POD) method is applied to a usual finite element (FE) formulation for parabolic equations so that it is reduced into a POD FE formulation with lower dimensions and enough high accuracy. The errors between the reduced POD FE solution and the usual FE solution are analyzed. It is shown by numerical examples that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is also shown that this validates the feasibility and efficiency of POD method.
基金supported by the National Natural Science Foundation of China(Grant Nos.51176020 and 51376029)
文摘Large eddy simulation(LES)is used to calculate the in-cylinder turbulent flow field in a direct injection spark ignition(DISI)engine.The computations are carried out for three different maximum valve lifts(MVL)and throughout 100 consecutive engine cycles.The simulated results as well as corresponding particle image velocimetry(PIV)measurement database are analyzed by the proper orthogonal decomposition(POD)method.Through a new developed POD quadruple decomposition the instantaneous in-cylinder flow fields are decomposed into four parts,named mean field,coherent field,transition field and turbulent field,respectively.Then the in-cylinder turbulent flow characteristics and cycle-to-cycle variations(CCV)are studied separately upon the four part flow fields.Results indicate that each part exhibits its specific characteristics and has close connection with others.The mean part contains more than 50%of the total kinetic energy and the energy cascade phenomenon occurs among the four part fields;the coherent field part possesses the highest CCV level which dominates CCV of the bulk flow.In addition,it is observed that a change in MVL affects significantly the in-cylinder flow behavior including CCV,especially for the coherent part.Furthermore,the POD analysis demonstrates that at least 25 sample cycles for the mean velocity and 50 sample cycles for the RMS velocity are necessary for obtaining converged and correct results in CCV.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10832001 and 10872145)the State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences
文摘We present dynamic mode decomposition (DMD) for studying the hairpin vortices generated by hemisphere protuberance measured by two-dimensional (2D) time-resolved (TR) particle image velocimetry (PIV) in a water channel. The hairpins dynamic information is extracted by identifying their dominant frequencies and associated spatial structures. For this quasi-periodic data system, the resulting main Dynamic modes illustrate the different spatial structures associated with the wake vortex region and the near-wall region. By comparisons with proper orthogonal decomposition (POD), it can be concluded that the dynamic mode concentrates on a certain frequency component more effectively than the mode determined by POD. During the analysis, DMD has proven itself a robust and reliable algorithm to extract spatial-temporal coherent structures.
