应用数学和力学

2007 Vol. 28, No. 4

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Basic Solution of Two Parallel Non-Symmetric Permeable Cracks in Piezoelectric Materials

ZHOU Zhen-gong, WANG Biao

2007, 28(4): 379-390.

Abstract(2802) PDF(697)

Abstract:
The behavior of two parallel non-symmetric cracks in piezoelectric materials subjected to the anti-plane shear loading is studied by the Schmidt method for the permeable crack electric boundary conditions. Through the Fourier transform, the present problem can be solved with two pairs of dual integral equations in which the unknown variables are the jumps of displacements across crack surfaces. To solve the dual integral equations, the jumps of displacements across crack surfaces were directly expanded in a series of Jacobi polynomials. Finally, the relations between electric displacement intensity factors and stress intensity factors at crack tips can be obtained. Numerical examples are provided to show the effect of the distance between two cracks upon stress and electric displacement intensity factors at crack tips. Contrary to the impermeable crack surface condition solution, it is found that electric displacement intensity factors for the permeable crack surface conditions are much smaller than those for the impermeable crack surface conditions. At the same time, it can be found that the crack shielding effect is also present in the piezoelectric materials.

Analytical Solutions for the Thermal Forcing Vortices in the Boundary Layer and Its Applications

LIU Xiao-ran, LI Guo-ping

2007, 28(4): 391-400.

Abstract(2733) PDF(597)

Abstract:
Using the Boussinesq approximation, the vortex in the boundary layer was assumed to be axisymmetrical and thermal-wind balanced system forced by diabatic heating and friction, and was solved as an initial-value problem of linearized vortex equation set in cylindrical coordinates. The impacts of thermal forcing on the flow field structure of vortex were analyzed. It is found that thermal forcing has significant impacts on the flow field structure, and the material representative forms of these impacts are closely related to the radial distribution of heating. The discussion for the analytical solutions for the vortex in the boundary layer can explain some main structures of the vortex over the Tibetan Plateau.

Mathematical Analysis of EEP Method for One-Dimensional Finite Element Postprocessing

ZHAO Qing-hua, ZHOU Shu-zi, ZHU Qi-ding

2007, 28(4): 401-405.

Abstract(2820) PDF(847)

Abstract:
Fora class of two-point boundary value problems, by virtue of one-dimensional projection interpolation and finite element superconvergence fundamental estimations, it was proved that the nodal recovery derivative obtained by Yuan's element energy projection (EEP) method had the optimal order superconvergence on condition that the degree of finite element space is no more than 4. The theoretical analysis coincides with the reported numerical results.

HUAI Wen-xin, ZENG Yu-hong

2007, 28(4): 406-412.

Abstract(2822) PDF(586)

Abstract:
The turbulent flow of vertical plane wall plume with concentration variation was studied with the finite analytical method. The k-epsilon model with the effect of buoyancy on turbulent kinetic energy and its dissipation rate was adopted. There were similarity solutions in the uniform environment for the system of equations including the equation of continuity, the equation of momentum along the flow direction and concentration, and equations of k, epsilon. The finite analytic method was applied to obtain the similarity solution. The calculated data of velocity, relative density difference, the kinetic energy of turbulence and its dissipation rate distribution for vertical plane plumes are in good agreement with the experimental data at the turbulent Schmidt number equal to 1.0. The variations of their maximum value along the direction of main flow were also given. It shows that the present model with numerical method is good, i. e., the effect of buoyancy on turbulent kinetic energy and its dissipation rate should be taken into account. The finite analytic method is effective.

Reconstruction of the Probabilistic S-N Curves Under Fatigue Life Following Lognormal Distribution With a Given Confidence

ZHAO Yong-xiang, YANG Bing, PENG Jia-chun

2007, 28(4): 413-418.

Abstract(3258) PDF(680)

Abstract:
When the historic probabilistic S-N curves were given under special survival probability and confidence levels and there is no possibility to retest, fatigue reliability analysis at other levels can not be done except for the special levels. Therefore, the widely applied curves are expected. Monte Carlo reconstruction methods of the test data and the curves are investigated under fatigue life following lognormal distribution. To overcome the non-conservative assessment of existent man-made enlarging the sample size up to thousands, a simulation policy is employed to address the true production which the sample size is controlled less than 20 for material specimens, 10 for structural component specimens and the errors matching the statistical parameters less than 5%. Availability and feasibility of the present methods have been indicated by the reconstruction practice of the test data and curves for 60Si2Mn high strength spring steel of railway industry.

Global Analysis of Ivlev’s Type Predator-Prey Dynamical System

XIAO Hai-bin

2007, 28(4): 419-427.

Abstract(2811) PDF(810)

Abstract:
A class of Ivlev. s type predator-prey dynamic systems with prey and predator both having linear density restricts is considered. By using the qualitative methods of ODE, the positive equilibrium's global stability and existence and uniqueness of non-small amplitude stable limit cycle were obtained. Especially under certain conditions, it shows that existence and uniqueness of non-small amplitude stable limit cycle is equivalent to the positive equilibrium's local unstability and the positive equilibrium's local stability implies its global stability. That is to say, the global dynamic of the system is entirely determined by the local stability of the positive equilibrium.

Discussion of Stability in a Class of Models on Recurrent Wavelet Neural Networks

DENG Ren, LI Zhu-xin, FAN You-hong

2007, 28(4): 428-432.

Abstract(2977) PDF(592)

Abstract:
Based on wavelet neural networks (WNNs) and recurrent neural networks (RNNs), a class of models on recurrent wavelet neural networks (RWNNs) was proposed. The new networks possess the advantages of WNNs and RNNs. Asymptotic stability of RWNNs was researched according to the Liapunov theorem. Some theorems and formulae were given. The simulation results show the excellent performance of the networks in nonlinear dynamic system recognition.

Developing Shock-Capturing Difference Methods

TU Guo-hua, YUAN Xiang-jiang, LU Li-peng

2007, 28(4): 433-440.

Abstract(2823) PDF(744)

Abstract:
A new shock-capturing method was proposed which is based on upwind schemes and flux-vector splittings. Firstly, original upwind schemes were projected along characteristic directions. Secondly, the amplitudes of the characteristic decompositions were carefully controlled by limiters to prevent non-physical oscillations. Lastly, the schemes were converted into conservative forms and the oscillation-free shock-capturing schemes were acquired. Two explicit upwind schemes (2nd-order and 3rd-order) and three compact upwind schemes (3rd-order, 5th-order and 7th-order) were modified by the method for hyperbolic systems and the modified schemes were checked on several oneOdimensional and two-dimensional test cases. Some numerical solutions of the schemes were compared with those of a WENO scheme and an MP scheme as well as a compact-WENO scheme. The results show that the method with high order accuracy and high resolutions can capture shock waves smoothly.

Active Vibration Control of Nonlinear Benchmark Buildings

ZHOU Xing-de, CHEN Dao-zheng

2007, 28(4): 441-446.

Abstract(3002) PDF(600)

Abstract:
The present nonlinear model reduction methods unfit for nonlinear benchmark buildings as their vibration equations belong to non affine system. Meanwhile, the controllers designed directly by nonlinear control strategy have a high order and are the difficult to be applied actually. Therefore, a new active vibration control way which fits nonlinear buildings was proposed. The idea of the proposed way was based on model identification and structural model linearization, exerting the control force to the built model according to the force action principle. The proposed way has a better practicability as the built model can be reduced by balance reduction method based on the empirical Grammian matrix. At last, a 3 storey benchmark structure was presented. Simulation results illustrate that the proposed method is viable for civil engineering structures.

Perfect Elastic-Viscoplastic Field at Mode Ⅰ Dynamic Propagating Crack-Tip

WANG Zhen-qing, LIANG Wen-yan, ZHOU Bo, SU Juan

2007, 28(4): 447-452.

Abstract(2328) PDF(679)

Abstract:
The viscosity of material is considered at propagating crack-tip. Under the assumption that the artificial viscosity coefficient is in inverse proportion to power law of the plastic strain rate, an elastic-viscoplastic asymptotic analysis was carried out for moving crack-tip fields in power-hardening materials under plane-strain condition. A continuous solution was obtained containing no discontinuities. The variations of numerical solution were discussed for mode Ⅰ crack according to each parameter. It is shown that stress and strain both possess exponential singularity. The elasticity, plasticity and viscosity of material at crack-tip only can be matched reasonably under linear-hardening condition. And the tip field contains no elastic unloading zone for mode Ⅰ crack. It approaches the limiting case, crack-tip is under ultra-viscose situation and energy accumulates, crackOtip begins to propagate under different compression situations.

Asymmetrical Dynamic Propagation Problems on Mode Ⅲ Interface Crack

Lü Nian-chun, YANG Ding-ning, CHENG Yun-hong, CHENG Jin

2007, 28(4): 453-461.

Abstract(2712) PDF(612)

Abstract:
By application of the theory of complex functions, asymmetrical dynamic propagation problems on mode Ⅲ interface crack are studied. The universal representations of analytical solutions are obtained by the approaches of self-similar functions. The problems researched can be facilely transformed into Riemann-Hilbert problems and analytical solutions to an asymmetrical propagation crack under the condition of point loads and unit-step loads, respectively, are acquired. After those solutions were used by superposition theorem, the solutions of arbitrarily complex problems could be attained.

Life-Span of Classical Solutions for One Dimensional Hydromagnetic Flow

LIU Fa-gui

2007, 28(4): 462-470.

Abstract(2983) PDF(752)

Abstract:
The Cauchy problem for one dimensional hydromagnetic dynamics with dissipative terms is concerned with. For the case of non-dissipation, it is shown that the smooth solutions will develop shocks in the finite time, if the initial amounts of entropy and themagnetic field. is smaller than that of sound waves. And for the case of dissipation, the initial amounts of entropy, dissipative effect and the-magnetic field. in each period is smaller than that of sound waves. Then the smooth solutions must blow up in the finite time. Moreover, the life-span of smooth solution is given.

Interval Finite Element Method and Its Application on Anti-Slide Stability Analysis

SHAO Guo-jian, SU Jing-bo

2007, 28(4): 471-478.

Abstract(2613) PDF(583)

Abstract:
The problem of interval correlation results in interval extension is discussed by the relationship of interva-lvalued functions and real-valued functions. The methods of reducing interval extension are given. Based on the ideas of the paper, the formulas of sub-interval perturbed finite element method based on the elements were given. The sub-interval amount is discussed and the approximate computation formula was given. At the same time, the computational precision was discussed and some measures of improving computational efficiency were given. Finally, based on sub-interval perturbed finite element method and anti-slide stability analysis method, the formula for computing the bounds of stability factor was given. Which will provide a basis for estimating and evaluating reasonably anti-slide stability of structures.

A Newton Type Iterative Method for Heat-Conduction Inverse Problems

HE Guo-qiang, MENG Ze-hong

2007, 28(4): 479-486.

Abstract(2686) PDF(812)

Abstract:
An inverse problem for identification of the coefficient in heat-conduction equation is considered. After reducing the problem to a nonlinear ill-posed operator equation, Newton type iterative methods were considered. The implicit iterative method was applied to the linearized Newton equation, and the key step in the process was that a new reasonable a posteriori stopping rule for the inner iteration was presented. Numerical experiments for the new method as well as for Tikhonov method and Bakushikskii method are given. And these results show the obvious advantages of the new method over the other ones.

Nonlinear Incidence Rate of a Pest Management S-I Model With Biological and Chemical Control Concern

JIAO Jian-jun, CHEN Lan-sun

2007, 28(4): 487-496.

Abstract(2883) PDF(763)

Abstract:
A pest management S-I model with impulsive releases of infective pests and spraying pesticides is proposed and investigated. It was proved that all solutions of the model are uniformly ultimately bounded. The sufficient conditions of globally asymptotic stability periodic solution of pest-extinction and permanence of the model were also obtained.The approach of combining impulsive releasing infective pests with impulsive spraying pesticides provides reliable tactical basis for the practical pest management.

Stability of Theoretical Model for Catastrophic Weather Prediction

SHI Wei-hui, WANG Yue-peng

2007, 28(4): 497-504.

Abstract(2502) PDF(689)

Abstract:
Stability related to theoretical model for catastrophic weather prediction that includes non-hydrostatic perfect elastic model, anelastic model was discussed and analyzed in detail. It was proved that in infinitely differentiable function class non-hydrostatic perfect elastic equations set is stable. However, for anelastic equations set, its continuity equation is changed in form because of the particular hypothesis for fluid, so/the matching consisting of both viscosity coefficient and incompressible assumption-appears, thereby the most important equations set of this class in practical prediction shows the same instability in topological property as Navier-Stokes equation, which should be avoided first in practical numerical prediction. In light of this, the referenced suggestions to amend applied model are finally presented.