应用数学和力学

2019 Vol. 40, No. 1

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A Note on Symplectic Water Wave Dynamics

WU Feng, ZHONG Wanxie

2019, 40(1): 1-7. doi: 10.21656/1000-0887.390254

Abstract(1283) HTML (279) PDF(548)

Abstract:
The numerical method for the simulations of fully nonlinear water waves was discussed. The symplectic perturbation method, developed in Symplectic Water Wave Dynamics,was extended to compute the pressure of the nonlinear water wave. Numerical examples show that the proposed method can be used to analyze the nonlinear evolutions of the nonlinear water waves, simulate the nonlinear water waves such as the solitary wave and the swell wave with sharp peaks, and find out the pressure distribution of the water wave.

Nonlinear Rheology of Glassy Materials: a Simplified Maxwell Model

XUE Haifeng, NI Yong

2019, 40(1): 8-19. doi: 10.21656/1000-0887.390149

Abstract(1304) HTML (202) PDF(749)

Abstract:
The mechanical behaviors of glassy materials show great differences under different loading conditions. A simplified Maxwell model combined with rate equations was proposed to study the influences of the strain rate, the temperature, and the aging time on the nonlinear mechanical responses of glassy materials controlled by the evolution of free volume defects. With this model, it is revealed that, within a certain range the higher the strain rate is, the lower the temperature, the longer the aging time and the higher the peak stress will be. In addition, the model also indicates that the peak stress and the critical strain of the stressstrain curve exhibit logarithmic dependence on the aging time. These conclusions are consistent with the results previously reported out of molecular dynamics simulations.

Modified Particle Approximation to Pressure Gradients in the SPH Algorithm for Interfacial Flows With High Density Ratios

XU Chengjun, XU Shengli, LIU Qingyuan

2019, 40(1): 20-35. doi: 10.21656/1000-0887.390126

Abstract(1780) HTML (347) PDF(531)

Abstract:
Interfacial flows with high density ratios ranging from 1 to 1 000 were numerically investigated. Modified particle approximation was proposed for the pressure gradient term in the momentum equation and the repulsive force was imposed for virtual particles outside the interfaces. The Rayleigh-Taylor instability, non-Boussinesq lock exchange, dam-break flow and bubble buoyancy were numerically tested for validation of accuracy and robustness of the new SPH algorithm for multi-fluid flows. The particle distributions, pressure contours and pressure-time distributions at specified points were obtained from the computations. The results are in good agreement with those from references and experimental measurements. The captured interfaces are more smooth in comparison with those from previous literatures and no obvious oscillations are observed in the vicinity of the interfaces.

Longitudinal Vibration and Wave Propagation of Viscoelastic Nanorods Based on the Nonlocal Theory

TANG Guangze, YAO Linquan, LI Cheng, JI Changjian

2019, 40(1): 36-46. doi: 10.21656/1000-0887.390166

Abstract(1284) HTML (236) PDF(630)

Abstract:
The longitudinal dynamics of viscoelastic nanorods was investigated based on the nonlocal theory and the Kelvin viscoelastic theory, including axial free vibration and wave propagation. Firstly, the partial differential governing equations were derived and then the 1st 3 vibration properties were discussed under 3 kinds of typical boundary conditions with the dimensionless method. Finally, the relationships between the circular frequency, the wave speed and the wave number were obtained in the problem of wave propagation. The numerical results show that, the small-scale effect makes the 1st and 2nd frequencies decrease persistently and the 3rd frequency increase first and decrease later, which indicates that the nanostructural stiffness is weakened or strengthened. In particular, for a concentrated mass at the free end of the nanorod, the 2nd frequency has multiple values when the viscoelastic coefficient increases, which may cause instability. The numerical examples also prove that stronger nonlocal effect brings lower damping effect of viscoelastic materials. The longitudinal wave can propagate at high wave numbers due to occurrence of the escape frequency. The effects of viscoelastic coefficients on the damping ratio may be ignored at low wave numbers, however, be significant at high wave numbers.

An Average Vector Field Method for Nonlinear Vibration Analysis

BAO Siyuan, DENG Zichen

2019, 40(1): 47-57. doi: 10.21656/1000-0887.390178

Abstract(984) HTML (140) PDF(676)

Abstract:
Through construction of differential equations in the vector form, the differential iteration form of the vibration response was obtained according to the average vector field (AVF) method. This discrete form is energy-preserving for the Hamiltonian system, and has the characteristics of 2ndorder accuracy. The detailed steps of the AVF method were given. To establish the AVF scheme, the mapping forms were deduced directly for several common items in the differential equations. The pendulum problem and the Kepler problem were studied with the AVF method. The numerical results demonstrate the advantages of the AVF method in solving nonlinear vibration problems, i.e. the conservation of energy and the longterm solution stability.

A New Layerwise Theory for Vibration Analysis of Laminated Structures Based on Modified Chebyshev Polynomials

YE Tiangui, JIN Guoyong, LIU Zhigang

2019, 40(1): 58-74. doi: 10.21656/1000-0887.390098

Abstract(1093) HTML (148) PDF(473)

Abstract:
A new layerwise theory for vibration analysis of laminated structures based on modified Chebyshev polynomials was proposed. The displacement field in each discrete layer was composed of a global linear component introduced under the layerwise strategy and a local highorder counterpart considered to improve the accuracy of the theory. In each discrete layer, the highorder displacement field distribution through the laminate thickness was determined with the modified Chebyshev polynomials. Therefore, the proposed theory offers an easy analysis operation to realize different modeling precision requirements only by changing the truncation order without the need for reprogramming from case to case. The theory also has the ability of achieving arbitrary modeling precision according to practical requirements. Based on the proposed theory, the general spectral method was combined to formulate the vibration equations of laminated beams, plates and shells. To test the efficiency and accuracy of the present theory, dynamic properties of laminated beams, plates and shells with different dimensions, boundary conditions and lamination schemes were studied. The numerical results obtained from the present theory are in good agreement with exact elasticity solutions published previously.

Numerical Solution of Schwarz-Christoffel Transformation From Rectangles to Arbitrary Polygonal Domains

WANG Yufeng, JI Anzhao, CUI Jianbin

2019, 40(1): 75-88. doi: 10.21656/1000-0887.390050

Abstract(1923) HTML (369) PDF(620)

Abstract:
With the Schwarz-Christoffel transformation method, a mathematical model of conformal mapping from polygonal domains to strip domains was established. For constraint conditions and singular integral problems in the model, the reciprocal transformation between complex parameters and real parameters was conducted based on the Riemann principle, which eliminates constraint conditions of the nonlinear system. By means of reasonable integration paths, the singular integral in the model was transformed into the Gauss-Jacobi integral, and the nonlinear system model was solved with the Levenberg-Marquardt algorithm. According to the first class elliptic function characteristics, the mathematical model of conformal mapping from rectangular domains to strip domains was built, and the relationship between the rectangular boundary and the strip boundary was obtained through calculation of the complex parameter elliptic function. At last, an 8-point polygonal domain and a 27-point irregular strip domain were calculated to map the irregular closed domain boundary to the rectangular domain boundary. The orthogonal grid in the rectangular domain still meets orthogonality in the polygonal domain after mapping. This study provides a foundation for numerical calculation of mapping from irregular domains to regular ones.

Solitary Travelling Wave Solutions to Strongly Nonlinear Wave Equations

FENG Yihu1, 2

2019, 40(1): 89-96. doi: 10.21656/1000-0887.390054

Abstract(1171) HTML (185) PDF(691)

Abstract:
A strongly nonlinear wave equation was studied. With the functional analytic variational iteration method, firstly, a variational iteration was constructed, and the corresponding Lagrangian multiplicator was solved. Secondly, the initial solitary wave was selected and the iteration method was used to obtain the approximate solution of arbitrarydegree accuracy for the solitary wave. This method is easy and feasible for getting approximate solutions to nonlinear wave equations.

Almost Sure Asymptotic Stability of the Euler-Maruyama Method With Random Variable Stepsizes for Stochastic Functional Differential Equations

MA Li, MA Ruinan

2019, 40(1): 97-107. doi: 10.21656/1000-0887.390057

Abstract(1003) HTML (177) PDF(406)

Abstract:
The Euler-Maruyama (EM) approximation to a class of stochastic functional differential equations was studied. First, a numerical approximation with the EM method with random variable stepsizes was defined, then two characteristics of the random variable stepsizes were got: the summation of finite stepsizes is a stopping time and the summation of countably infinite stepsizes diverges. Finally, with the theory of non-negative semi-martingale convergence in discrete time, it was proved that the numerical approximation converges to zero almost surely if the coefficients satisfy the local Lipschitz condition and the monotonic condition. The results generalize the corresponding results of MAO Xuerong in a previous literature, where the EM approximation to a class of stochastic differential equations was studied and the numerical solution was proved to converge to zero almost surely.

Strong Convergence of CQ Algorithms for Split Feasibility Problems in the Hilbert Spaces

ZHAO Shilian

2019, 40(1): 108-114. doi: 10.21656/1000-0887.390012

Abstract(1174) HTML (225) PDF(412)

Abstract:
To study the strong convergence of split feasibility problems, a new CQ algorithm was proposed in the Hilbert spaces. Firstly, the modified Halpern iterative sequence was obtained with the CQ method. Furthermore, the split feasibility problem was transformed into the fixed point for operators, and it was proved that the sequence converges strongly to a solution of the split feasibility problem under some weak conditions. The findings generalize the corresponding results of Wang and Xu.