应用数学和力学

2017 Vol. 38, No. 2

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An EEP Adaptive Strategy of the Galerkin FEM for Dynamic Equations of Discrete Systems

XING Qin-yan, YANG Xing, YUAN Si

2017, 38(2): 133-143. doi: 10.21656/1000-0887.370288

Abstract(1377) PDF(659)

Abstract:
For the solution of structural dynamic equations, generally the accuracy of results and the efficiency of computation both depend on the selection of the time step lengths, which makes the key difficulty for efficient solution of time-dependent problems. With the element energy projection (EEP) super-convergent solution computed at the post-processing stage of the finite element method (FEM) to replace the unknown true solution and then to estimate the error of the conventional FEM solution, the so-called EEP adaptive method can automatically refine the solution mesh and has achieved success in various boundary-value problems with spatial coordinates as the arguments. Based on the Galerkin FEM solution of the weak form, the EEP self-adaptive strategy was introduced and applied to the dynamic equations of discrete systems. As a result, an adaptive mesh was automatically produced in the time domain, and a dynamic displacement solution satisfying the pre-specified error tolerance at any moment was obtained, which leads to a new adaptive computation approach for time-dependent problems.

An Elastoplastic Damage Constitutive Model for Concrete Considering Unilateral Effects

Lü Cong-cong, LI Zong-li

2017, 38(2): 144-152. doi: 10.21656/1000-0887.370278

Abstract(1639) PDF(780)

Abstract:
In order to describe the nonlinear mechanical properties of concrete effectively and accurately, a new elastoplastic damage constitutive model for concrete was established with the explicit integration algorithm, under the theoretical framework of continuum damage mechanics and irreversible thermodynamics. First, a yield failure criterion based on the unified strength theory was employed for this model, and then 2 scalar damage variables were introduced to better describe the quite different tensile and compressive mechanisms of concrete, respectively. Besides, the influence factors of reverse loading were adopted in view of the unilateral effects under alternating cyclic loading conditions and the interaction effects of tension and compression damages in the multiaxial stress state. Several numerical simulations were presented, and good agreement was achieved between the numerical results and the test results for concrete specimens under uniaxial loading, multiaxial loading and alternating cyclic loading, which verifies the validity and accuracy of the proposed model.

Convergence and Precision of the Dual-Variable Brick Mixed Element and Its Displacement Element

QING Guang-hui, LIU Yan-hong

2017, 38(2): 153-162. doi: 10.21656/1000-0887.370089

Abstract(1123) PDF(477)

Abstract:
The symplectic characteristics of the equivalent stiffness coefficient matrix for the Hamiltonian canonical equations of elasticity and the Hamiltonian mixed element were intuitive, and the symplectic characteristics of the equivalent stiffness coefficient matrix for the dual-variable brick mixed element (DVBME), which was derived based on the Hellinger-Reissner (H-R) variational principle and the symplectic-conservative theory of elasticity, were similarly intuitive. The governing equations of elasticity were established immediately through the DVBME formulation, and the solution of the governing equations was obtained with the mixed method. Meanwhile, the dual-variable brick displacement element (DVBDE) formulation was deduced from the DVBME formulation, which was only related to displacement variables. The solution of the governing equations based on the DVBDE formulation was got with the displacement method. The numerical examples show that the convergence rates of displacement and stress variables of the 8-node DVBDE with reduced integration are balanced and stable with high precision. The convergence rate of stress of the DVBDE is almost equal to that of the translational 20-node displacement element with reduced integration. The DVBDE is universal.

Analysis of Diamond-Patterned Origami Tubes Under Axial Crushing Forces

LIU Xiang, LI Dong-heng

2017, 38(2): 163-169. doi: 10.21656/1000-0887.370173

Abstract(1616) PDF(931)

Abstract:
The diamond pattern was introduced into thin-wall tubes to reduce the initial peak forces during axial crushing processes. The finite element method was used to analyze the performance of diamond-patterned origami tubes with square cross sections. The results show that the diamond-patterned origami tubes have lower initial peak forces but more steady crushing processes, in comparison with the corresponding straight-wall tubes. The critical conditions were obtained for the origami tubes with buckling modes agreeing with the diamond pattern creases. The relationships between the sector angle of the origami tube and the initial peak force as well as the mean crushing force under axial impact were also studied.

Pavement Rutting Analysis Based on Vehicle-Road Interaction Under Thermal Effects

LIU Jun-qing1, LIU Hong1, LI Qian2

2017, 38(2): 170-180. doi: 10.21656/1000-0887.370129

Abstract(1108) PDF(724)

Abstract:
To study the rutting of asphalt pavement, first, based on the vehicle-road interaction, a simplified model for the dynamic vehicle load caused by road roughness was established; then according to the data from creep tests of asphalt mixture, the parameters for the calculation of asphalt pavement rutting with the modified Burgers’ model, were obtained through fitting; finally, in combination with the observed temperature data and with the ABAQUS finite element software, the pavement temperature field was introduced and the calculation model was established for the daily rutting prediction in view of consecutive temperature variation. The rutting growth of asphalt pavement based on vehicle-road interaction under thermal effects was simulated, and the effects of the pavement temperature, the vehicle load and the vehicle speed on the rutting were parametrically analyzed. The results show that, the rutting depth increases by 6.5% when the vehicle-road interaction is considered, indicating this factor can’t be ignored in the asphalt pavement rutting prediction. Both the axle load and the pavement roughness are related to the rutting linearly. Under the same axle load, the higher the temperature is, the faster the rutting grows and the bigger the final rutting depth gets, and the rutting decreases with the vehicle speed. Moreover, at a vehicle speed where the resonant vehicle-road interaction occurs, the resulting rutting depth is 32% bigger than that in the case without this interaction.

Parametric Vibration Stability of Controlled Stay Cables With Time Delays

PENG Jian, LI Lu-xin, HU Xia, WANG Xiu-yong

2017, 38(2): 181-188. doi: 10.21656/1000-0887.370110

Abstract(1577) PDF(615)

Abstract:
The effects of time delays on the primary parametric vibration of controlled stay cables under axial excitation were studied. In view of cable sag and geometric nonlinearity, the nonlinear parametric vibration equation for the controlled stay cable system under axial excitation was built based on the Hamiltonian principle. Then the dynamic system with time delay was formulated by means of the Galerkin method. The multiscale method was used to analyze the primary parametric resonance of the controlled stay cable system and obtain the effects of different time delays and control gains on the time histories of the parametric vibration and the stability region of the controlled stay cable. The study shows that time delay weakens the vibration controlling effects on the stay cable, and the stability region of the parametric vibration is shifted. The larger the time delay is, the worse the controlling effects will be. The work plays a guiding role in the parametric design of the control system for stay cables.

Generalized Hydrodynamics for Second 2D Soft-Matter Quasicrystals

FAN Tian-you

2017, 38(2): 189-199. doi: 10.21656/1000-0887.370198

Abstract(1476) PDF(462)

Abstract:
The concept on the first and second kinds of 2D quasicrystals was put forward, and through extension of ref.[1], the generalized hydrodynamics for possible soft-matter quasicrystals with 7-, 9- and 14-fold symmetries was suggested based on the Langevin equation and with the derivation method of the Poisson bracket. The derivation referred to the previous work of the author. The soft-matter quasicrystals observed so far were 2D ones. From this point of view, all the soft-matter quasicrystals both discovered and possibly discovered were considered in ref.[1] and the present work from the angle of symmetry and generalized hydrodynamics, where the equation of state as a key, was built by ref.[1]. Final governing equations (7), (9) and (11) were similar to those given in ref.[1], and the solution was also similar, so that the way of solution for the first kind of 2D soft-matter quasicrystals is beneficial to the second kind.

A Power Law Equation of State for Actual Gases

CHEN Wen, LIANG Ying-jie

2017, 38(2): 200-205. doi: 10.21656/1000-0887.370140

Abstract(1455) PDF(698)

Abstract:
A power law equation of state for actual gases was proposed with fewer parameters compared with the Onnes equation of state. The new model requires only two parameters, of which the order of the power function characterizes the deviation extent of actual gas from the ideal gas. The gas whose behavior obeys the power law equation of state is called the power law gas. The power law equation of state was applied to describe the behaviors of nitrogen (N₂) and tetrafluoromethane (CF₄) gases. Compared with the Onnes equation, the proposed power law equation can accurately capture the power law relation between the gas pressure and volume in a simpler fashion with fewer parameters. The results show that the power law order tends to be smaller under a lower temperature, reflecting a larger deviation of the actual gas state from that of the ideal gas.

Analysis of High Speed Flow in Circular and Annular Ducts Occupied by Bidisperse Porous Media

WANG Ke-yong, LI Pei-chao

2017, 38(2): 206-215. doi: 10.21656/1000-0887.370105

Abstract(1118) PDF(487)

Abstract:
Based on the two-velocity Brinkman-extended Darcy flow model, the characteristics of high speed flow in circular and annular ducts occupied by bidisperse porous media were analyzed. The flow fields of the fracture (f) and porous (p) phases were inherently governed by the 4th-order system of coupled differential equations. The original governing equations were simplified to a 2nd-order system of decoupled differential equations with the normal mode reduction method. Furthermore, the analytical solutions of velocity distributions were readily derived for the f- and p-phases. Results from both the circular and the annular ducts show that an increase in the Darcy number leads to a reduction in not only the flow velocities of the two phases but their difference. However, the flow velocities of the two phases exhibit an opposite trend with the increase of the momentum transfer between the two phases, resulting in a decrease in the velocity difference.

Rossby Waves Excited by Large Topography and Beta Change in Barotropic Atmosphere

SONG Jian, LIU Quan-sheng, YANG Lian-gui

2017, 38(2): 216-223. doi: 10.21656/1000-0887.370135

Abstract(1374) PDF(473)

Abstract:
Based on the potential vorticity equation, the large topography and the change of Rossby wave parameter β with the latitude were considered and parameter δ was introduced. With the normal mode method, the Rossby wave phase velocity formula was obtained in the high latitude regions with the large topography, the Froude number and parameter δ. The research points out that the large topography and the Froude number under the change of β influence the stability of Rossby waves, and these factors usually play a stabilizing part in the Rossby waves.

Chaos Control for the Duopoly Cournot-Puu Model

DU Lin, ZHANG Ying, HU Gao-ge, LEI You-ming

2017, 38(2): 224-232. doi: 10.21656/1000-0887.370256

Abstract(1822) PDF(540)

Abstract:
Based on the linearization method for nonlinear dynamics and the linear stability theorem, the duopoly Cournot-Puu model and the associated chaos control methods were investigated. In view of the essential features of the model, the delayed feedback control (DFC) method and the adaptive control method were applied to address the chaotic behavior of this system and to control chaos during the output adjustment process in the actual economic sense. The theoretical formulations were numerically simulated. Furthermore, the rational value ranges of the control parameters were given and the economic meanings of both the introduced control methods were discussed.

LI Shun-chu1, REN Li1, ZHENG Peng-she1, GUI Qin-min2

2017, 38(2): 233-242. doi: 10.21656/1000-0887.370190

Abstract(1150) PDF(434)

Abstract:
For naturally fractured shale gas reservoirs, the problem about fixed output was addressed in 3 outer boundary conditions (infinite boundary, constant pressure and closed boundary) and inner boundary conditions, to build a well test analysis model for unsteady seepage flow in view of stress sensitivity and desorption & adsorption. Firstly, the model was linearized. Secondly, the perturbation method and the Laplace transform were used to get the exact solution of dimensionless reservoir pressure in the Laplace space with the linearized model. Finally, according to the similar structure theory, the steps leading to the similar structure of the solution to this model were presented. In addition, the kernel functions in the 3 outer boundary conditions were defined and it was found that a similar structure existed among the exact solutions of this model in the 3 outer boundary conditions. The work not only facilitates the development of well test analysis software with higher calculation efficiency, but also helps a lot in the investigation of shale gas seepage mechanisms, making a new method for the solution of the shale gas seepage model.