应用数学和力学

2016 Vol. 37, No. 11

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LES Discretization Methods for Unstructured Meshes Based on the Finite Volume Method

XIONG Ying, GUAN Hui, WU Chui-jie

2016, 37(11): 1129-1144. doi: 10.21656/1000-0887.370228

Abstract(1889) PDF(1404)

Abstract:
The LES of unstructured meshes is an effective way to solve the high Reynolds number flow around complex geometries. Firstly, based on the finite volume method, the discretization methods for the convection term and the diffusion term were analyzed. For the convection term, the central difference scheme with a TVD limiter ensured 2nd-order accuracy and inhibited the nonphysical oscillations. Dissipation of the linear upwind scheme was large and can not guarantee boundedness. The central difference scheme caused a period of nonphysical oscillations. For the diffusion term, the method of over relaxation nonorthogonal correction reduced the discretization error caused by the nonorthogonal mesh. The correction coefficients were chosen according to the nonorthogonal degree of the mesh. Secondly, numerical simulation of unsteady flow around a sphere with high Reynolds number was conducted based on the improved delayed detached eddy simulation (IDDES) model and the tetrahedral mesh. The limited central difference scheme was used for the convection term, and the over relaxation correction was used for the diffusion term. The least squares method was used for the interpolation scheme. The 2nd-order backward difference scheme was used for the time term. The calculation results show that, the proposed discretization methods are stable and in good agreement with the experimental data.

Solution of the Oseen Flow With the Multigrid Method Based on Alternating-Line Relaxation

ZHU Xing-wen, ZHANG Li-xiang

2016, 37(11): 1145-1155. doi: 10.21656/1000-0887.370062

Abstract(916) PDF(514)

Abstract:
The 1st-order upwind discretization form of the Oseen flow was obtained through the Godunov-type flux-difference splitting approach based on the Riemann solver. The convergence analysis of 2 kinds of cycling algorithms, i.e., the V-cycle and the W-cycle in the multigrid method for the solution of the discretized equations, was performed. Furthermore, the smooth properties of the collective symmetrical alternating-line Gauss-Seidel relaxation was investigated by means of the local Fourier analysis. The numerical results show that the collective symmetrical alternating-line Gauss-Seidel relaxation has sound smooth properties, and the convergence of the W-cycle algorithm is better than that of the V-cycle one in the multigrid method for the solution of the Oseen flow with different Reynolds numbers.

Numerical Study of Local Receptivity in the Boundary Layer

SHEN Lu-yu, LU Chang-gen

2016, 37(11): 1156-1168. doi: 10.21656/1000-0887.370076

Abstract(974) PDF(714)

Abstract:
The research on local receptivity in the boundary layer is very important for the prediction and control of the laminar-turbulent transition, and especially the study on the formation mechanism of 3D Tollmien-Schlichting (T-S) waves is meaningful in theory. The high-order high-resolution non-uniform compact finite difference schemes were utilized to study the local receptivity under the interaction of free-stream turbulence and 2D localized wall roughness. The numerical results verify the exsistance of the local receptivity under the interaction of free-stream turbulence and 2D localized wall roughness, and the streamwise vorticity forms and gets stronger downstream as the excited 3D T-S wave packets evolve in the streamwise direction. The numerical results also show that the propagation direction of the excited 3D T-S wave packets is influenced by the propagation direction of the free-stream turbulence, and the propagation speed is close to 1/3 of the free-stream velocity; the wave-length conversion mechanism only changes streamwise wave number α, whereas spanwise wave number β keeps unchanged. In addition, the relation between the free-stream amplitude and incident angle, the localized wall roughness height and length, and the local receptivity, is confirmed. The in-depth research on this subject is helpful for further understanding of the laminar-turbulent transition and tubulence formation.

High-Precision Approximate Analytical Solutions for Free Bending Vibrations of Thin Plates and an Improvement

BAO Si-yuan, DENG Zi-chen

2016, 37(11): 1169-1180. doi: 10.21656/1000-0887.370005

Abstract(1402) PDF(932)

Abstract:
For free bending vibrations of thin plates, based on the variable separation method, the analytical solutions of deflection functions were obtained in the Hamiltonian system, and the eigenvalue equations about the 2 coordinate axes were established. Then the vibration frequencies were solved as parametrical variables, and the mode shapes of different orders were got. The analytical form of the deflection function was in fact a high-precision approximate solution satisfying the displacement boundary conditions. From the approximate frequency values calculated with the Rayleigh-Ritz method, it is found that the previous analytical method had great precision, thus the effectiveness was demonstrated. In addition, for clamped or simply supported boundary conditions, different points were properly selected as the coordinate origins. The unified forms of the frequency equations were given. These forms were used to discuss the rectangular plates with 4 edges clamped, or 4 edges simply supported, or some clamped and the other(s) simply supported, and so on. The correctness of the frequency equation forms was verified with the symmetry of the plates’ deformations. The linkage and transformation between the frequency equations under different boundary conditions were obtained.

Principal Resonance Bifurcation and Chaos of Rotating Annular Plates in Magnetic Fields

PIAO Jiang-min, HU Yu-da

2016, 37(11): 1181-1197. doi: 10.21656/1000-0887.370141

Abstract(1344) PDF(842)

Abstract:
The magneto-elastic principal resonance bifurcation and chaos of rotating annular plates in magnetic fields were studied. Based on the expressions of kinetic energy, strain energy and virtual work done by external forces and electromagnetic forces, the nonlinear vibration equations of a rotating annular plate in magnetic field were deduced with the Hamiltonian principle. The Galerkin method with the Bessel mode shape functions was used to achieve the ordinary differential vibration equations. The static bifurcation equations and corresponding transition sets with the physical parameters as the bifurcation control parameters were achieved by means of the method of multiple scales. Finally, the critical conditions for the break of the heteroclinic orbits were obtained under the conditions of fixed outer boundary and free inner boundary with the Mel’nikov method. Moreover, the global bifurcation diagrams under the external forces as the control parameters and other response diagrams with specified control parameters were drawn. The results show that the magnetic field deters the occurence of multi-value phenomena. With the decreasing of the external force frequency, the rotating speed and the magnetic induction, and with the increasing of the external force, the system’s heteroclinic orbits break more easily, meanwhile chaos or almost periodic motion of the system is induced.

Stochastic Nonlinear Dynamics Analysis of Double-Well Duffing Systems Under Random Parametric Excitations

ZHANG Ying, DU Lin, YUE Xiao-le, HU Jian, FANG Tong

2016, 37(11): 1198-1207. doi: 10.21656/1000-0887.370257

Abstract(1066) PDF(634)

Abstract:
Based on the orthogonal polynomial approximation theory, the stochastic dynamical behaviors of double-well Duffing systems under random parametric excitations were investigated. Firstly, the complex dynamical behaviors of deterministic Duffing systems were studied by means of the Poincaré sections. Then, the Duffing systems with random stiffnesses and damping parameters were reduced to equivalent deterministic expanded-order systems, and the effectiveness of this approximation method was proved. Thus, the ensemble-mean responses of the equivalent systems were applied to reveal the stochastic dynamical properties and the effects of the random variable intensity on the double-well Duffing systems. The numerical simulation results indicate that, in the case of coexistent attractors, the double-well Duffing system with random stiffness parameters has the similar stable dynamical behaviors to those in deterministic cases. However, for the Duffing system with random damping parameters, during the increase of the random variable intensity, the bifurcation phenomena occur to some coexistent attractors.

Primary Resonance of Elastic Beams Under Time-Delay Velocity Feedback Control

PENG Jian, ZHANG Gai, SUN Ce-shi

2016, 37(11): 1208-1216. doi: 10.21656/1000-0887.370083

Abstract(1023) PDF(531)

Abstract:
The vibration control of the elastic beam subjected to axial force was investigated with the time-delay velocity feedback control method. A nonlinear control model for the piezoelectric coupling elastic beam was established according to Newton’s second law. The direct method was employed to obtain the 1st-order approximate response of the primary resonance of the elastic beam with time-delay velocity feedback. Then the relationship between the system response and the control parameters was given. The results show that, multiple solutions and jumping phenomena exist in the primary resonance response, and the large-amplitude vibration can be effectively suppressed through regulation of the control gain and the time delay.

A Meshless Local Petrov-Galerkin Method Based on theMoving Kriging Interpolation for Structural Uncoupled Thermal Stress Analysis

WANG Feng, ZHOU Yi-hong, ZHENG Bao-jing, LIN Gao

2016, 37(11): 1217-1227. doi: 10.21656/1000-0887.370189

Abstract(1035) PDF(602)

Abstract:
A meshless local PetrovGalerkin (MLPG) method based on the moving Kriging interpolation was employed for the solution of 2D structural uncoupled thermal stress problems. The transient heat conduction problem was solved firstly and then the thermal solutions were imposed as body loads with the sequential coupledfield method in the stress analysis. The local weak forms were developed with the weighted residual method locally from the partial differential equations of transient heat conduction and structural dynamics, where the Heaviside step function was used as the weighted function in each subdomain. The essential boundary conditions can be implemented directly since the shape functions constructed from the moving Kriging interpolation possess the Kronecker δ property. This method does not involve the subdomain integral during generation of the global stiffness matrix except for the boundary integral, so the computational costs are reduced largely. The results of 2 numerical examples show the effectiveness of this method.

Whole-Process Simulation of Complex-Structure Buildings After Aircraft Impacts

CUI Tie-jun, LI Sha-sha, MA Yun-dong, WANG Lai-gui

2016, 37(11): 1228-1238. doi: 10.21656/1000-0887.360336

Abstract(1062) PDF(857)

Abstract:
In order to study the collapse processes of complexstructure highrise buildings after aircraft impacts, with the particle flow theory the whole process including impact, explosion, combustion and structure collapse was simulated. The particle flow models were introduced into different steps in the forms of the crash model, the explosion model and the combustion model. A building model of a 216 m height, a large atrium and a complex structure composed of 4 core tubes was established. The impact heights were 200 m and 100 m, respectively. The results show that, mainly due to the complexity of the structure and different impact heights, distinctly different failure processes after aircraft impacts are to come, especially in the aspects of the collapse cause, the collapse mode and the collapse time.

Stress Concentration in Finite Plates With Functionally Graded Rings Around Circular Holes

YANG Quan-quan, ZHU Wei-guo, LIU Fei

2016, 37(11): 1239-1246. doi: 10.21656/1000-0887.370131

Abstract(1325) PDF(423)

Abstract:
Based on the complex variable method and combined with the least squares boundary collocation technique, the problem of stress concentration in finite plates with functionally graded rings around circular holes was studied when the plates were subjected to arbitrary uniform distributed loads at the outer boundaries. With the method of piecewise homogeneous layers, the semi-analytic solutions of stresses around the hole were first given through formulation of the complex potentials in the functionally graded ring with arbitrary radial elastic properties. The numerical results of stress distributions around the holes were then obtained for plates with different material properties, ring thicknesses, plate sizes and hole eccentricities, respectively. It is found that the stress concentration in the finite plate can be significantly reduced on the condition of a proper distribution of the radial elastic property and a proper thickness of the ring.

Stability Analysis of an SEIS Epidemic ModelWith a Nonlinear Incidence Rate

XING Wei, YAN Qi-sheng, YANG Zhi-hui, GAO Jin-fang

2016, 37(11): 1247-1254. doi: 10.21656/1000-0887.370166

Abstract(1319) PDF(673)

Abstract:
An SEIS epidemic model with a nonlinear incidence rate and involving a constant input rate, a natural mortality rate and a mortality rate due to disease, was investigated. Firstly, the basic reproduction number for the model was defined. Then the disease-free equilibrium point was proved to be globally asymptotically stable when R₀<1. Finally, the conditions for the theorem that the unique endemic equilibrium point was globally asymptotically stable, were obtained when R₀>1.