应用数学和力学

2016 Vol. 37, No. 8

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Only symplectic conservation is characteristic of discrete dynamics

ZHONG Wan-xie

2016, 37(8): 775-777. doi: 10.21656/1000-0887.370186

Abstract(1675) PDF(1066)

Abstract:
Symplectic conservation is altered to "(geometric) structurepreserving" in some foreign work. "Symplectic" is a terminology, but what "structure" means here is still a fuzzy concept that can’t be stated clearly. This alteration seems to make a foreign original achievement. Hence the foreign fuzzified concept will displace the definite Chinese invention. It is very important to distinguish between Chinese original work and foreign distorted ones. The positively correct concept that symplectic conservation is characteristic of discrete dynamics, is a Chinese original one, shall not be obscured behind a foreign camouflage. "People respect those who respect themselves", which deserves enough attention of the Chinese people.

Solutions of Symmetries for Piezoelectric Stack Actuators

XIE Yu, FU Jing-li, CHEN Ben-yong

2016, 37(8): 778-790. doi: 10.21656/1000-0887.370048

Abstract(1210) PDF(774)

Abstract:
The symmetries of piezoelectric stack actuators were investigated, and the solutions of conserved quantities and symmetries were given. The piezoelectric stack actuator of axial movement was considered and its structural characteristics were analyzed, accordingly the displacement and the flux linkage were selected as the generalized coordinates, then the electromechanical coupling Lagrangian equations were established with the energy method. Through the infinitesimal transformation of the displacement and flux linkage coordinates, the Noether symmetries and Lie symmetries were studied respectively, in turn the generalized Noether identity, the generalized Killing equations, the generalized Noether theorm and the Lie theorm were presented. The generators of the Noether symmetries and the Lie symmetries for the piezoelectric stack actuator were calculated, and the corresponding conserved quantities were derived. At last, with the obtained conserved quantities the solutions of symmetries were got, and the dynamic response curves of the actuator’s displacement and speed were calculated under the changing control voltage.

Analysis on Shear Deformation and Shear-lag Effects on Twin-Cell Box Girders

ZHANG Hui, ZHANG Yu-yuan, ZHANG Yuan-hai, LI Wei

2016, 37(8): 791-803. doi: 10.21656/1000-0887.370056

Abstract(1732) PDF(847)

Abstract:
Based on the flanges’ different maximum shear angle differences selected as the shear-lag generalized displacements, the energy variational principle was applied to deduce the shear-lag differential equations for twin-cell box girders with and without shear deformation in consideration, and the initial parametric solutions of the box girder’s longitudinal stress and vertical deflection were given according to the boundary conditions. From the angle of mechanics and mathematics it was proved that the shear deformation and the shear lag were 2 relatively independent mechanical behaviors, of which the impacts on the box girder mechanics were expounded: the shear deformation had no effect on the longitudinal stress but had great influence on the vertical deflection. The example of a simply-supported box girder shows that the longitudinal stress at the middle cross section calculated with the presented analytic method and with the numerical method are in good agreement, of which the lateral distribution is similar to that in the case of single-cell box girders; however, the shear-lag effect around the side webs is slightly greater than that around the middle web. The shear deformation adds to the middle deflection of the box girder under concentrated and uniformly distributed loads by 4.6% and 2.7%, respectively.

A Partial Functional Differential Method for Shear Lag Analysis of Composite Double-Cell Box Girders

YU Jiang, HU Shao-wei

2016, 37(8): 804-819. doi: 10.21656/1000-0887.370112

Abstract(1258) PDF(762)

Abstract:
The shear lag effects caused large deformation and significant damage of some structural members in engineering, and had unfavorable influences on engineering safety. To deepen the understanding of the shear lag mechanism, the partial functional differential system for steel-concrete composite double-cell box girders was proposed through introduction of different parabolic partial functional modes. In turn, the particular solutions of the differential system were obtained under a specific load (the span-center concentrated load), and the shear lag coefficients under 4 modes of partial functional expressions were also calculated. As for different concrete material parameters and steel material parameters, the shear lag characteristics of the double-cell box girder examples were comparatively studied. This work provides some reference for the design and calculation of thin-wall box girders.

Preconditioned Conjugate Gradient Methods for the 3D Wilson Nonconforming FEM Discretizations

XIAO Ying-xiong, LI Zhen-you

2016, 37(8): 820-831. doi: 10.21656/1000-0887.370037

Abstract(1632) PDF(550)

Abstract:
The nonconforming finite element method (FEM) is an efficient method to overcome the volume locking trouble in 3D elasticity problems. This method has the advantages of a few degrees of freedom and high accuracy. In order to improve the overall efficiency of the FEM analysis, it is necessary to design some faster solvers for the corresponding system of discretization equations. The faster solvers for the Wilson nonconforming FEM discretizations were considered. When Poisson’s ratio ν was close to 0.5, the resulting system of equations was symmetric positive definite and highly illconditioned, and the preconditioned conjugate gradient (PCG) method was one of the most efficient methods for solving such FEM equations. Moreover, in practical applications, anisotropic meshes are often obtained due to the specificity of the structure considered, which will greatly decrease the convergence rate of the PCG method. A type of PCG method based on the DAMG was presented and then applied to the solution of the Wilson FEM discretizations. This DAMG was an algebraic multi grid (AMG) method based on the distance matrix and can be used to solve the system of equations discretized on anisotropic meshes. The numerical results show that, in combination with the effective smoothing operators, the proposed PCG method has high efficiency and robustness for nearly incompressible problems.

Global Synchronization of Complex-Valued Neural Networks With Leakage Time Delays

YAN Huan, ZHAO Zhen-jiang, SONG Qian-kun

2016, 37(8): 832-841. doi: 10.21656/1000-0887.370168

Abstract(1586) PDF(1050)

Abstract:
The global synchronization of complex-valued neural networks with leakage time delays was investigated. For the considered complex-valued neural networks, the activation functions need not be separated into real parts and imaginary parts. Through the construction of appropriate Lyapunov-Krasovskii functionals, with the drive-response synchronization method, the free-weighting matrix method and the matrix inequality technique, a delay-dependent criterion for checking the global synchronization of complex-valued neural networks with leakage time delays was established in the form of complex-valued linear matrix inequality (LMI), meanwhile the design method for the synchronization controllers was given. Finally, a simulation example shows the effectiveness of the present work.

A Local Stabilized Nonconforming Finite Element Method for the Optimal Control of Navier-Stokes Equations

QIN Yan-mei, LI Hui, FENG Min-fu

2016, 37(8): 842-855. doi: 10.21656/1000-0887.370137

Abstract(1487) PDF(529)

Abstract:
For the optimal control of Navier-Stokes equations, a new local stabilized nonconforming finite element method was proposed. The time-dependent problem was fully discretized with lowest-equal-order nonconforming finite element NCP₁-Psub>1in the velocity and pressure spaces and the reduced Crank-Nicolson scheme in the time domain. The scheme was stable for the equal-order combination of discrete velocity and pressure spaces through the addition of a local L² projection term. Specially, based on an extrapolation formula, the method requires only the solution of one linear system per time step. Stability of the method was proved. For the state, adjoint state and control variables, the a priori error estimates were obtained. The error estimation results show that the method has 2nd-order accuracy.

Properties of Positive Solutions to Sturm-Liouville Boundary Value Problems With p-Laplace Operators

YANG Jing-bao

2016, 37(8): 856-862. doi: 10.21656/1000-0887.370047

Abstract(1503) PDF(648)

Abstract:
The properties of positive solutions were investigated for a class of SturmLiouville boundary value problems with p-Laplace operators. Based on the properties of p-Laplace operators, and according to the L’H?pital’s rule and the extreme value theorem for continuous functions on closed intervals, the SturmLiouville boundary value problems with p-Laplace operators were studied. The 2 necessary conditions for the existence of positive solutions were obtained. In the last part, the application of the main findings was given. The work enriches the content in the field of boundary value problems, and provides a new channel of using computer and iterative techniques to find approximate solutions to boundary value problems, meanwhile extending some conclusions in previous literatures.

Instability of Aircraft Multiple Trailing Vortex Pairs

CHEN Li-fang, WANG Zhi-bo, SUN Gang

2016, 37(8): 863-872. doi: 10.21656/1000-0887.370059

Abstract(1555) PDF(806)

Abstract:
For aircrafts the wake vortex separation and the disturbance-induced vortex instability draw more and more concerns. A 1st-order approximation of the Biot-Savart law was used to build the induced motion model for arbitrarily multiple vortex pairs. Symmetric and antisymmetric modes of the induced motion of vortex pairs were deduced with the linear combination method. The properties of the modal matrix eigenvalues were obtained to describe instability of the multiple vortex pairs. The modes of 2 vortex pairs were used for validation of the proposed model, which was thereafter extended to 3 vortex pairs to give the instable modes corresponding to the modal matrices. The theoretical analysis and eigenvalue calculation show that, with the increase of the number of vortex filaments, the instability of the 3 vortex pairs will rise, and the amplification effects of the vortex system on the disturbance will highten.

Effects of Pore Structure on Hydrodynamic Lubrication of Multi-Layer Oil Bearings

ZHANG Guo-tao, YIN Yan-guo, LI Ji-ning, LIU Zhen-ming, TIAN Ming

2016, 37(8): 873-879. doi: 10.21656/1000-0887.360296

Abstract(1093) PDF(685)

Abstract:
The hydrodynamic lubrication model for multi-layer oil bearings was established based on the Darcy’s law. The pores’effects depicted with the Kozeny-Carman equation were considered in the analysis. The effects of the bearing structure and the pore structure on the lubrication properties were simulated and discussed with the finite difference method. Results show that the lubrication properties get worse with the increase of the layer heights and the porosity; a lower-permeability surface will be more beneficial to improve the lubrication performance when the total porosity is fixed. Therefore, in the design of double-layer oil bearings, the surface porosity should be reduced as far as possible as long as the oil content in pores is guaranteed. This work is referential to the analysis of the tribological properties and the structural design of multi-layer oil bearings.

Study on MHD Stagnation-Point Flow Over Permeable Stretching Sheets With 2nd-Order Slip Boundaries

XU Xiao-qin, CHEN Shu-mei

2016, 37(8): 880-888. doi: 10.21656/1000-0887.370032

Abstract(1594) PDF(573)

Abstract:
The incompressible MHD viscous flow with 2nd-order slip about stagnation points over permeable exponentially stretching sheets in porous media was studied. The governing equations describing the stagnation point flow were reduced to nonlinear differential equations through the similarity transformations. Then the bvp5c function in MATLAB was employed to solve the nonlinear problem. Finally, the effects of the 1st- and 2nd-order slip parameters, the suction/injection parameter and the permeability parameter on the velocity and the skin friction were analyzed and discussed. The results show that the velocity increases but the skin friction decreases and is positive with the 1st-order slip parameter and the absolute value of the 2nd-order slip parameter, the suction/injection parameter and the permeability parameter, when the sheet’s stretching velocity is smaller than the external mainstream velocity; however, an anti-boundary layer forms, the velocity decreases and the absolute value of the skin friction also decreases but is negative, when the sheet’s stretching velocity is larger than the external mainstream velocity. The effect of the 2nd-order slip parameter is slightly greater than that of the 1st-order slip parameter on the velocity and the skin friction, and the effect of the suction/injection parameter is significantly greater than that of the permeability parameter.