The localized differential quadrature (LDQ) method is a numerical technique with high accuracy for solving most kinds of nonlinear problems in engineering and can overcome the difficulties of other methods (such as di...The localized differential quadrature (LDQ) method is a numerical technique with high accuracy for solving most kinds of nonlinear problems in engineering and can overcome the difficulties of other methods (such as difference method) to numerically evaluate the derivatives of the functions.Its high efficiency and accuracy attract many engineers to apply the method to solve most of the numerical problems in engineering.However,difficulties can still be found in some particular problems.In the following study,the LDQ was applied to solve the Sod shock tube problem.This problem is a very particular kind of problem,which challenges many common numerical methods.Three different examples were given for testing the robustness and accuracy of the LDQ.In the first example,in which common initial conditions and solving methods were given,the numerical oscillations could be found dramatically;in the second example,the initial conditions were adjusted appropriately and the numerical oscillations were less dramatic than that in the first example;in the third example,the momentum equation of the Sod shock tube problem was corrected by adding artificial viscosity,causing the numerical oscillations to nearly disappear in the process of calculation.The numerical results presented demonstrate the detailed difficulties encountered in the calculations,which need to be improved in future work.However,in summary,the localized differential quadrature is shown to be a trustworthy method for solving most of the nonlinear problems in engineering.展开更多
This study investigated a water supply recovery problem involving municipal water service piping. The problem consisted in recovering full service after network failure, in order to rapidly satisfy all urgent citywide...This study investigated a water supply recovery problem involving municipal water service piping. The problem consisted in recovering full service after network failure, in order to rapidly satisfy all urgent citywide demands. The optimal recovery solution was achieved through the application of so-called network design problems （NDPs）, which are a form of combinatorial optimization problem. However, a conventional NDP is not suitable for addressing urgent situations because （1） it does not utilize the non-failure arcs in the network, and （2） it is solely concerned with stable costs such as flow costs. Therefore, to adapt the technique to such urgent situations, the conventional NDP is here modified to deal with the specified water supply problem. In addition, a numerical illustration using the Sendai water network is presented.展开更多
This paper presents a combination of the hybrid spectral collocation technique and the spectral homotopy analysis method(SHAM for short) for solving the nonlinear boundary value problem(BVP for short) for the electroh...This paper presents a combination of the hybrid spectral collocation technique and the spectral homotopy analysis method(SHAM for short) for solving the nonlinear boundary value problem(BVP for short) for the electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylindrical conduit. The accuracy of the present solution is found to be in excellent agreement with the previously published solution. The authors use an averaged residual error to find the optimal convergence-control parameters. Comparisons are made between SHAM generated results, results from literature and Matlab ode45 generated results, and good agreement is observed.展开更多
The necessary condition established in Part I of this paper for the global maximizers of the maximization problem max V tr（VTAV）/tr（VTBV）＋tr（VTCV）over the Stiefel manifold{V∈Rm×l ｜VTV=Il}（l〈m）,natural...The necessary condition established in Part I of this paper for the global maximizers of the maximization problem max V tr（VTAV）/tr（VTBV）＋tr（VTCV）over the Stiefel manifold{V∈Rm×l ｜VTV=Il}（l〈m）,naturally leads to a self-consistent-field（SCF）iteration for computing a maximizer.In this part,we analyze the global and local convergence of the SCF iteration,and show that the necessary condition for the global maximizers is fulfilled at any convergent point of the sequences of approximations generated by the SCF iteration.This is one of the advantages of the SCF iteration over optimization-based methods.Preliminary numerical tests are reported and show that the SCF iteration is very efficient by comparing with some manifold-based optimization methods.展开更多
文摘The localized differential quadrature (LDQ) method is a numerical technique with high accuracy for solving most kinds of nonlinear problems in engineering and can overcome the difficulties of other methods (such as difference method) to numerically evaluate the derivatives of the functions.Its high efficiency and accuracy attract many engineers to apply the method to solve most of the numerical problems in engineering.However,difficulties can still be found in some particular problems.In the following study,the LDQ was applied to solve the Sod shock tube problem.This problem is a very particular kind of problem,which challenges many common numerical methods.Three different examples were given for testing the robustness and accuracy of the LDQ.In the first example,in which common initial conditions and solving methods were given,the numerical oscillations could be found dramatically;in the second example,the initial conditions were adjusted appropriately and the numerical oscillations were less dramatic than that in the first example;in the third example,the momentum equation of the Sod shock tube problem was corrected by adding artificial viscosity,causing the numerical oscillations to nearly disappear in the process of calculation.The numerical results presented demonstrate the detailed difficulties encountered in the calculations,which need to be improved in future work.However,in summary,the localized differential quadrature is shown to be a trustworthy method for solving most of the nonlinear problems in engineering.
文摘This study investigated a water supply recovery problem involving municipal water service piping. The problem consisted in recovering full service after network failure, in order to rapidly satisfy all urgent citywide demands. The optimal recovery solution was achieved through the application of so-called network design problems （NDPs）, which are a form of combinatorial optimization problem. However, a conventional NDP is not suitable for addressing urgent situations because （1） it does not utilize the non-failure arcs in the network, and （2） it is solely concerned with stable costs such as flow costs. Therefore, to adapt the technique to such urgent situations, the conventional NDP is here modified to deal with the specified water supply problem. In addition, a numerical illustration using the Sendai water network is presented.
文摘This paper presents a combination of the hybrid spectral collocation technique and the spectral homotopy analysis method(SHAM for short) for solving the nonlinear boundary value problem(BVP for short) for the electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylindrical conduit. The accuracy of the present solution is found to be in excellent agreement with the previously published solution. The authors use an averaged residual error to find the optimal convergence-control parameters. Comparisons are made between SHAM generated results, results from literature and Matlab ode45 generated results, and good agreement is observed.
基金Acknowledgements The first author was supported by National Natural Science Foundation of China(Grant Nos.11101257 and 11371102)the Basic Academic Discipline Program,the 11th Five Year Plan of 211 Project for Shanghai University of Finance and Economics+1 种基金supported by National Science Foundation of USA(Grant Nos.1115834and 1317330)a Research Gift Grant from Intel Corporation
文摘The necessary condition established in Part I of this paper for the global maximizers of the maximization problem max V tr（VTAV）/tr（VTBV）＋tr（VTCV）over the Stiefel manifold{V∈Rm×l ｜VTV=Il}（l〈m）,naturally leads to a self-consistent-field（SCF）iteration for computing a maximizer.In this part,we analyze the global and local convergence of the SCF iteration,and show that the necessary condition for the global maximizers is fulfilled at any convergent point of the sequences of approximations generated by the SCF iteration.This is one of the advantages of the SCF iteration over optimization-based methods.Preliminary numerical tests are reported and show that the SCF iteration is very efficient by comparing with some manifold-based optimization methods.
