Applied Mathematics and Mechanics (English Edition)

江增荣段鹏飞郭杏林丁桦

2010, 31(11): 1337-1346. doi:10.1007/s10483-010-1366-x

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The discretization size is limited by the sampling theorem, and the limit is one half of the wavelength of the highest frequency of the problem. However, one half of the wavelength is an ideal value. In general, the discretization size that can ensure the accuracy of the simulation is much smaller than this value in the traditional finite element method. The possible reason of this phenomenon is analyzed in this paper, and an efficient method is given to improve the simulation accuracy.

Nonlinear flexural waves and chaos behavior in finite-deflection Timoshenko beam

张善元刘志芳

2010, 31(11): 1347-1358. doi:10.1007/s10483-010-1367-9

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Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling wave method and integration skills, the nonlinear partial differential equations can be converted into an ordinary differential equation. The qualitative analysis indicates that the corresponding dynamic system has a heteroclinic orbit under a certain condition. An exact periodic solution of the nonlinear wave equation is obtained using the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function tends to one in the degenerate case, a shock wave solution is given. The small perturbations are further introduced, arising from the damping and the external load to an original Hamilton system, and the threshold condition of the existence of the transverse heteroclinic point is obtained using Melnikov’s method. It is shown that the perturbed system has a chaotic property under the Smale horseshoe transform.

Approximate analytical solutions and experimental analysis for transient response of constrained damping cantilever beam

胡明勇王安稳章向明

2010, 31(11): 1359-1370. doi:10.1007/s10483-010-1368-9

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Vibration mode of the constrained damping cantilever is built up according to the mode superposition of the elastic cantilever beam. The control equation of the constrained damping cantilever beam is then derived using Lagrange’s equation. Dynamic response of the constrained damping cantilever beam is obtained according to the principle of virtual work, when the concentrated force is suddenly unloaded. Frequencies and transient response of a series of constrained damping cantilever beams are calculated and tested. Influence of parameters of the damping layer on the response time is analyzed. Analyitcal and experimental approaches are used for verification. The results show that the method is reliable.

Symplectic analysis for wave propagation in one-dimensional nonlinear periodic structures

侯秀慧邓子辰周加喜

2010, 31(11): 1371-1382. doi:10.1007/s10483-010-1369-7

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The wave propagation problem in the nonlinear periodic mass-spring structure chain is analyzed using the symplectic mathematical method. The energy method is used to construct the dynamic equation, and the nonlinear dynamic equation is linearized using the small parameter perturbation method. Eigen-solutions of the symplectic matrix are used to analyze the wave propagation problem in nonlinear periodic lattices. Nonlinearity in the mass-spring chain, arising from the nonlinear spring stiffness effect, has profound effects on the overall transmission of the chain. The wave propagation characteristics are altered due to nonlinearity, and related to the incident wave intensity, which is a genuine nonlinear effect not present in the corresponding linear model. Numerical results show how the increase of nonlinearity or incident wave amplitude leads to closing of transmitting gaps. Comparison with the normal recursive approach shows effectiveness and superiority of the symplectic method for the wave propagation problem in nonlinear periodic structures.

A class of coupled nonlinear Schrödinger equations: Painlev´e property, exact solutions, and application to atmospheric gravity waves

刘萍李子良楼森岳

2010, 31(11): 1383-1404. doi:10.1007/s10483-010-1370-6

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The Painlev´e integrability and exact solutions to a coupled nonlinear Schrödinger (CNLS) equation applied in atmospheric dynamics are discussed. Some parametric restrictions of the CNLS equation are given to pass the Painlev´e test. Twenty periodic cnoidal wave solutions are obtained by applying the rational expansions of fundamental Jacobi elliptic functions. The exact solutions to the CNLS equation are used to explain the generation and propagation of atmospheric gravity waves.

Flow of a biomagnetic viscoelastic fluid: application to estimation of blood flow in arteries during electromagnetic hyperthermia, a therapeutic procedure for cancer treatment

J.C.MISRA A.SINHA G.C.SHIT

2010, 31(11): 1405-1420. doi:10.1007/s10483-010-1371-6

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The paper deals with the theoretical investigation of a fundamental problem of biomagnetic fluid flow through a porous medium subject to a magnetic field by using the principles of biomagnetic fluid dynamics (BFD). The study pertains to a situation where magnetization of the fluid varies with temperature. The fluid is considered to be non-Newtonian, whose flow is governed by the equation of a second-grade viscoelastic fluid. The walls of the channel are assumed to be stretchable, where the surface velocity is proportional to the longitudinal distance from the origin of coordinates. The problem is first reduced to solving a system of coupled nonlinear differential equations involving seven parameters. Considering blood as a biomagnetic fluid and using the present analysis, an attempt is made to compute some parameters of the blood flow by developing a suitable numerical method and by devising an appropriate finite difference scheme. The computational results are presented in graphical form, and thereby some theoretical predictions are made with respect to the hemodynamical flow of the blood in a hyperthermal state under the action of a magnetic field. The results clearly indicate that the presence of a magnetic dipole bears the potential so as to affect the characteristics of the blood flow in arteries to a significant extent during the therapeutic procedure of electromagnetic hyperthermia. The study will attract the attention of clinicians, to whom the results would be useful in the treatment of cancer patients by the method of electromagnetic hyperthermia.

Unsteady three-dimensional boundary layer flow due to a permeable shrinking sheet

N.BACHOK A.ISHAK I.POP

2010, 31(11): 1421-1428. doi:10.1007/s10483-010-1372-6

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The unsteady viscous flow over a continuously permeable shrinking surface is studied. Similarity equations are obtained through the application of similar transformation techniques. Numerical techniques are used to solve the similarity equations for different values of the unsteadiness parameter, the mass suction parameter, the shrinking parameter and the Prandtl number on the velocity and temperature profiles as well as the skin friction coefficient and the Nusselt number. It is found that, different from an unsteady stretching sheet, dual solutions exist in a certain range of mass suction and unsteadiness parameters.

A parallel two-level finite element method for the Navier-Stokes equations

尚月强罗振东

2010, 31(11): 1429-1438. doi:10.1007/s10483-010-1373-7

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Based on domain decomposition, a parallel two-level finite element method for the stationary Navier-Stokes equations is proposed and analyzed. The basic idea of the method is first to solve the Navier-Stokes equations on a coarse grid, then to solve the resulted residual equations in parallel on a fine grid. This method has low communication complexity. It can be implemented easily. By local a priori error estimate for finite element discretizations, error bounds of the approximate solution are derived. Numerical results are also given to illustrate the high efficiency of the method.

Nonconforming local projection stabilization for generalized Oseen equations

白艳红冯民富王川龙

2010, 31(11): 1439-1452. doi:10.1007/s10483-010-1374-x

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A new method of nonconforming local projection stabilization for the generalized Oseen equations is proposed by a nonconforming inf-sup stable element pair for approximating the velocity and the pressure. The method has several attractive features. It adds a local projection term only on the sub-scale (H > h). The stabilized term is simple compared with the residual-free bubble element method. The method can handle the influence of strong convection. The numerical results agree with the theoretical expectations very well.

Regularity and finite dimensionality of attractor for plate equation on Rⁿ

肖海滨

2010, 31(11): 1453-1462. doi:10.1007/s10483-010-1375-9

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This paper addresses the regularity and finite dimensionality of the global attractor for the plate equation on the unbounded domain. The existence of the attractor in the phase space has been established in an earlier work of the author. It is shown that the attractor is actually a bounded set of the phase space and has finite fractal dimensionality.

Precise integration method for solving singular perturbation problems

富明慧张文志 S.V.SHESHENIN

2010, 31(11): 1463-1472. doi:10.1007/s10483-010-1376-x

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This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.

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