应用数学和力学

Analysis and Modelling of Optimal Dynamical Systems of Incompressible NavierStokes Equations

WANG Jincheng, QI Jin, WU Chuijie

2020, 41(1): 1-15. doi: 10.21656/1000-0887.400279

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Abstract:
The modeling method for the optimal dynamical systems of NavierStokes equations satisfying both arbitrary velocity boundary conditions and velocity incompressible conditions was studied. Through the modeling and analysis of the optimal dynamical systems of the flow around the square column, it is found that the dynamics characteristics of the optimal dynamical systems are limit cycles. At the same time, the optimal dynamical system with only 3 optimal basis functions could well describe all the main flow field characteristics and the dynamics characteristics of the problem, so the proposed method is applicable to complex flow problems and their dynamics with minimal basis functions.

Simulation of 2-Phase Flow in the Nozzle of the ArcHeated Wind Tunnel Based on the Eulerian-Lagrangian Model

LUO Wanqing, LI Haiyan, LIANG Jianhan

2020, 41(1): 16-26. doi: 10.21656/1000-0887.400214

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Abstract:
Due to the ablation of the electrode during the heating of test gas in the arc-heated wind tunnel, some tiny metal particles enter the nozzle to form gas-particle 2-phase flow, of which the effect is worthy of attention on the flow field in the nozzle and the heat load on the model of the test section. The computation efficiency of the particle tracking algorithm was improved through modification of the time step, with the gas-phase flow and the particle-phase flow coupled based on the conservation law of momentum and energy. By means of the Eulerian-Lagrangian model, a numerical simulation method for the gas-particle 2-phase flow inside the nozzle was established under specific simplification, and some typical examples were simulated and analyzed. The research shows that, under the same particle mass fraction, the smaller the particle size is, the more uniform the flow field downstream of the nozzle will be. If the mass fraction of the particle phase is small, it will has little effect on the flow field parameters of the nozzle outlet. This work lays a foundation for further study of the 2-phase flow field characteristics inside the arc-heated wind tunnel.

A 2D Continuous Dynamic Traffic Assignment Model and Traffic-Related Emission Estimation Considering Housing Distribution

LI Xiaoyang, LIN Zhiyang, Lü Yupei, ZHANG Peng

2020, 41(1): 27-41. doi: 10.21656/1000-0887.400213

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Abstract:
A 2D continuous dynamic traffic assignment model considering housing distribution was proposed, and then the traffic-related emissions were estimated. In the model, the traffic demand is influenced by housing distribution, and travelers choose departure times and routes according to the dynamic user-optimal principle. The model can be used to describe the dynamic traffic and obtain the traffic density, velocity and flow. A speed and acceleration-based microscopic emission model VT-micro was applied to estimate the traffic-related CO₂ emissions. The finite volume method, the projection method and the successive average method were constructed based on triangular meshes. A numerical example of a city with a single CBD was presented to demonstrate the effectiveness of the model and the numerical algorithm.

Solution of 2D Shallow Water Wave Equations With the Moving-Grid Rotating-Invariance Method

ZHENG Supei, WANG Ling, WANG Miaomiao

2020, 41(1): 42-53. doi: 10.21656/1000-0887.400124

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Abstract:
In order to improve the resolution of the numerical algorithm for solving the 2D shallow water wave equation, a new algorithm was proposed based on the moving-grid method, with the entropy stable numerical flux function and by means of the mixed numerical flux obtained through the rotating invariance. The numerical solution of the shallow water wave equation and the grid computation process based on the characteristics of the solution were interleaved. The variational principle was used to reconstruct the mesh, and the physical quantity on the new mesh was computed with the 2nd-order precision conservation interpolation formula. The 3rd-order strongly stable Runge-Kutta method and the entropy stable format satisfying the 2nd law of thermodynamics were used to numerically solve the shallow water wave equation. The numerical results show that, the new algorithm has good discontinuity capture ability and high resolution.

Accuracy of the Mathematical Homogenization Method for Thermomechanical Problems

LI Hongpeng, LING Song, QI Zhenbiao, JIANG Keru, CHEN Lei

2020, 41(1): 54-69. doi: 10.21656/1000-0887.400119

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Abstract:
For thermo-mechanical problems of periodical composite structures, the full decoupled scheme of each order perturbation and the governing equation of each order influence function for the mathematical homogenization method (MHM) were derived, then the weighted residual method was utilized to transform them into the conveniently programmable finite element matrix form. The perturbation displacements in the uncoupled form were defined as the products of influence functions and the macro field derivatives, and the calculating accuracy of the perturbation displacements were determined by the accuracy of influence functions and the macro field derivatives, in turn the accuracy of influence functions depended mainly on the applicability of unit cell boundary conditions. For the static problems of 2D periodical composite structures, the super unit cell periodical boundary condition and the differential quadrature finite element method were applied to guarantee the calculating accuracy of the influence function and the macro field derivatives respectively. On this basis, the influence of the high-order perturbations on the true displacement of the MHM was studied, and the necessity of the 2nd-order perturbation was emphasized. Finally, the potential energy functional was used to evaluate the accuracy of the MHM. Numerical comparisons validate the conclusions.

Study on Equivalent Stiffnesses of Orthotropic Hemi-Spherical Convex Plates

LIU Hang, DU Guojun, FENG Yan

2020, 41(1): 70-80. doi: 10.21656/1000-0887.400181

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Abstract:
The hemispherical convex plate was periodically divided into representative unit structures. Firstly, the stiffness characteristics of representative units were studied, and the equivalent stiffness of the hemispherical convex plate was obtained by means of the deformation equivalence principle, the homogenization procedure and the stiffness combination method. Then the 3 principal stiffnesses were brought into the theoretical solution of the 4-side simple plate to solve the plate center deflection. The finite element numerical simulation solution and the theoretical solution were compared and analyzed to verify the accuracy of the theoretical principal stiffnesses. The effect of the material dimensions of the representative units on the equivalent stiffness was then discussed. As the ratio of the length of the representative unit to the convex radius increases, the accuracy of the theoretical results will improve, and the equivalent stiffness formula is applicable to hemispherical convex plates of different thicknesses. Finally, a relatively simple engineering application formula was given with the approximate range of the convex radius based on several examples.

Research on Transient Temperature in the Work Room of a Rotary Arm Type Centrifuge

YIN Yihui, LI Qisheng, HAO Yu, LI Yifei, FAN Zhigeng, LUO Zhaoyu

2020, 41(1): 81-97. doi: 10.21656/1000-0887.400047

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Abstract:
Firstly, in view of the transient heat conduction in the top, bottom and side walls of the work room of a rotary arm type centrifuge, the governing equations of transient temperature for the walls were developed. Through the Laplace transform and solution, the total heat flux amount permeating the interior surfaces of the walls in terms of air temperature in the work room was obtained. Then, the governing equation of transient air temperature in the work room was established according to the energy conservation principle. In the equation, the input energy from the drive system was balanced with the total energy absorbed by the air and solid parts in the work room, absorbed and transferred out by the walls, and taken away by the outflow air over inflow air. Finally, the explicit series expressions of the transient temperature in the work room were deduced by means of the expansion theorem of the inverse Laplace transform. As an example, the work room transient temperature of one constructed geotechnical centrifuge was computed theoretically and the results were compared with the experimentally measured values. The comparison indicates that, the theoretical solution is in good agreement with the experimental one. The established transient temperature formula can increase the forecast accuracy for the work room temperature and improve the temperature control design of the work rooms of rotary arm type centrifuges.

A Nonlinear Boussinesq Equation With External Source and Dissipation Forcing Under Generalized β Plane Approximation and Its Solitary Wave Solutions

CHEN Liguo, YANG Liangui

2020, 41(1): 98-106. doi: 10.21656/1000-0887.400067

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Abstract:
Under generalized β plane approximation, based on the quasigeostrophic potential vorticity equation, and by means of the Gardner-Morikawa transform and the weak nonlinear perturbation expansion method, a Boussinesq equation with external source and dissipation forcing was derived to describe the generation and evolution of the Rossby wave amplitude. The periodic wave solutions and solitary wave solutions for the Boussinesq equation were presented with the modified Jacobi elliptic function expansion method. The solution structure shows that, the generalized β effect, the shear basic flow, the external source and the dissipation are extremely important factors influencing the nonlinear Rossby wave.

Asymptotic Solutions to a Class of Catalytic Reaction Robin Problems

XU Jianzhong, MO Jiaqi

2020, 41(1): 107-114. doi: 10.21656/1000-0887.400185

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Abstract:
A class of Robin problems of nonlinear catalytic reaction differential equations were studied. Firstly, under the suitable conditions, the outer solution to the original Robin problem was obtained with the perturbation method. Then by means of the stretched variable and the power series, the 1st and 2nd boundary layer corrective terms were constructed respectively, and the formal asymptotic expansion was structured. Finally, based on the theory of differential inequalities the formal asymptotic expression of the solution to the Robin problem was given. Finally, the uniform validity of the asymptotic expression of the solution to problem was proved.

A Splitting Iterative Algorithm for Solving Continuous Sylvester Matrix Equations

LI Ying

2020, 41(1): 115-124. doi: 10.21656/1000-0887.400133

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Abstract:
The solution of continuous Sylvester matrix equations has significant application value in scientific and engineering calculations, hence, a splitting iterative algorithm was proposed. The core idea of the algorithm is to split the coefficient matrix of the continuous Sylvester matrix equation into a symmetric matrix and an antisymmetric matrix with an outer iterative scheme, and to solve the complex symmetric matrix equation with the inner iterative scheme. Compared with the traditional splitting algorithms, the proposed splitting algorithm effectively avoids the selection of optimal iterative parameters and takes advantages of the efficient solution of complex symmetric equations, which improves the easy implementation and easy operation of the algorithm. In addition, the convergence of the splitting iterative algorithm was further proved theoretically. Numerical examples show that, the splitting iterative algorithm has good convergence and robustness, and the convergence of the splitting iterative algorithm depends on the selection of the inner iterative schemes.

2020 Vol. 41, No. 1