应用数学和力学

Experimental Investigation of Evaporating Sessile Droplets on PDMS Surface

XIA Xue-lian, ZHENG Xu, HUANG Xian-fu, ZHOU Jin-zhi, YU Ying-song

2017, 38(5): 495-502. doi: 10.21656/1000-0887.370358

Abstract(1094) PDF(663)

Abstract:
Evaporation of sessile water droplets on polydimethylsiloxane (PDMS) surface was experimentally studied with the particle tracking velocimetry (PTV) technique. The fluorescent microspheres at the solid-liquid interface first moved towards the center and then back to the contact line. Because the evaporative flux near the contact line is less than that far from the line, there will be a capillary flow towards the center when the contact line is pinned. Such a flow will carry microspheres towards the center. Moving characteristics of the contact line were also investigated. It is found that the theoretical values of the moving velocity at different moments in the CCA stage is of the same order with the experimental values. However, the experimental moving accelerations were much larger than the theoretical ones because the microspheres at the contact line weaken the interaction between the PDMS surface and the contact line.

Threshold Selection for the Extreme Value Estimation of Bridge Strain Under Vehicle Load

YANG Xia, ZHANG Jing, REN Wei-xin, YUAN Ping-ping.

2017, 38(5): 503-512. doi: 10.21656/1000-0887.370395

Abstract(1462) PDF(868)

Abstract:
The selection of a reasonable threshold is critical to estimate the extreme strain under vehicle load on bridges with the peak-over-threshold method. Little information can be used if the threshold is too high, while the bias of parameters of the general Pareto distribution will be large if the threshold is too low. Common threshold selection methods are not suitable to be applied in estimation of the extreme strain under vehicle load. Based on 1-year strain data of the Taiping Lake Bridge, 3 types of mixed distributions for the strain peaks induced by vehicle load were chosen to generate a large number of samples with the Monte-Carlo method. The estimated extreme values of the samples based on the generalized Pareto distributions with different thresholds were compared and analyzed. Then, an empirical threshold selection method was proposed for the strain data induced by vehicle load. Finally, the Taiping Lake Bridge was chosen as the case verification. It is demonstrated that the estimated weekly extreme strain based on the threshold selected with the proposed method is more close to the measured results than those with the common methods.

Global Stability of Clifford-Valued Recurrent Neural Networks With Mixed Time-Varying Delays

SHU Han-qi, SONG Qian-kun

2017, 38(5): 513-525. doi: 10.21656/1000-0887.370319

Abstract(1412) PDF(939)

Abstract:
The global exponential stability of Clifford-valued recurrent neural networks (RNNs) with both asynchronous time-varying and continuously distributed delays was studied. First, the existence and uniqueness of the equilibrium points of delayed Clifford-valued RNNs were proved with the inequality technique and the M-matrix properties. Then, based on the mathematical analysis method, some determinant conditions ensuring the global exponential stability of such systems were obtained. The simulation results of a numerical example substantiate the effectiveness of the theoretical analysis.

Sufficient Optimality Conditions for Nonsmooth Semi-Infinite Multiobjective Optimization Problems

YANG Yu-hong, LI Fei

2017, 38(5): 526-538. doi: 10.21656/1000-0887.380012

Abstract(1398) PDF(875)

Abstract:
The nonsmooth semi-infinite multiobjective optimization problem (SIMOP) was addressed and its optimality conditions were discussed. First, the Clarke F-convexity hypothesis was imposed on some combinations of the objective functions and the constraint functions, the sufficient optimality conditions for the (weakly) efficient solution to the SIMOP were established. Next, the sufficient optimality conditions for the optimal solution to its scalar problem were obtained with the ChankongHaimes method.

Solutions to the Nonlinear Schrödinger Equation and Coupled Nonlinear Schrödinger Equations With a New G′/（G+G′）-Expansion Method

SHI Lan-fang, NIE Zi-wen

2017, 38(5): 539-552. doi: 10.21656/1000-0887.370269

Abstract(1455) PDF(698)

Abstract:
A new G′/（G+G′）-expansion method was proposed. Exact solutions to a class of Schrödinger equations and coupled nonlinear Schrödinger equations were obtained with this new method. The solutions can be expressed with the hyperbolic cotangent functions, the cotangent functions and the rational functions. This new G′/（G+G′）-expansion method not only help gets new exact solutions to the equations directly and effectively, but also expands the scope of the solutions. This new method promises a very wide range of application for the study of related partial differential equations.

New 2-Soliton Solutions to the Arbitrary Order Nonlinear Camassa-Holm Equation

Taogetusang

2017, 38(5): 553-560. doi: 10.21656/1000-0887.370211

Abstract(1184) PDF(604)

Abstract:
The method combining the auxiliary equation, the function transformation and the variable separation solutions was proposed to construct the new 2-soliton and 2-period solutions to the arbitrary order nonlinear Camassa-Holm equation. Step 1, with 2 auxiliary equations, the function transformation and the variable separation solutions, the problem of solving the arbitrary order nonlinear Camassa-Holm equation was transformed to the problem of solving the nonlinear algebraic equations. Step 2, by means of symbolic computation system Mathematica, the solutions to the algebraic equations were obtained, and with the help of the relative conclusions on the auxiliary equation, the new 2-soliton and 2-period solutions were constructed.

Generalized Solution to a Class of Singularly Perturbed Problem of Nonlinear Reaction Diffusion Equation With Two Parameters

FENG Yi-hu, LIU Shu-de, MO Jia-qi

2017, 38(5): 561-569. doi: 10.21656/1000-0887.370177

Abstract(1260) PDF(594)

Abstract:
A class of generalized singularly perturbed problems of reaction diffusion equations with two parameters were considered with the singular perturbation method. Firstly, under suitable conditions, the outer solution to the problem was found. Next, the power series of the two small parameters were developed, and the first and second boundary layer corrective terms for the solution to the problem were constructed with the multiscale variable method, respectively. Finally, based on the composite expansion method, the asymptotic expression of the generalized solution to the problem was obtained, and according to the fixed point theory for functional analysis, the precision of the asymptotic expansion was estimated. Two corrective functions with different thicknesses were obtained for the generalized solution in the overlapping area, and they take effects on the boundary conditions respectively and expand the range of study; moreover, the work provides a costruction method for this kind of corrective terms with different thicknesses in the overlapping area, thus has a wide study foreground.

Dynamic Characteristics of Bistable Electromagnetic Vibration Energy Harvesters Under Colored Noise Excitation

WU Zi-ying, NIU Feng-qi, LIU Rui, YE Wen-teng

2017, 38(5): 570-580. doi: 10.21656/1000-0887.370215

Abstract(1062) PDF(1209)

Abstract:
With the progress of MEMS technology, it is possible to power the sensor system by means of the electricity produced by the vibration energy harvester. The vibration energy harvesting technology based on environment vibration has become a hot topic for the study of nonlinear dynamics. The bistable electromagnetic vibration energy harvester with an auxiliary linear oscillator was studied. The dynamic equations under colored noise were established. The dynamic characteristics of the bistable vibration energy harvester excited by colored noise were studied through numeric simulation for varying parameters including the noise intensity, the mass ratio and the tuning ratio. The influencing law of the noise intensity, the mass ratio and the tuning ratio was obtained. The research results provide a theoretical basis for the further study of bistable electromagnetic vibration energy harvesters.

Uncertainty Research of Natural Convection Heat Transfer Under Stochastic Boundary Condition Based on the Monte-Carlo Stochastic Finite Element Method

HE Yi-hai, JIANG Chang-wei, YAO Ming, ZHANG Bing-qing, ZHU Yan-he, ZHANG Zhong-qing

2017, 38(5): 581-593. doi: 10.21656/1000-0887.370224

Abstract(1446) PDF(714)

Abstract:
In order to study the effects of stochastic boundary conditions on natural convection heat transfer in square cavities, a Monte-Carlo stochastic finite element method was developed to solve uncertainty propagation of natural convection heat transfer under stochastic boundary condition. The input random parameters were expanded through the Karhunen-Loeve expansion and the random samples of boundary condition were generated with the Latin sampling method. The flow field and temperature field in the square cavity for different random samples of boundary condition were calculated numerically. The mathematical expectations and variances of stochastic output fields were calculated with the sampling statistical method. The stochastic finite element program with the MATLAB language was coded to solve the uncertainty propagation of natural convection heat transfer in cavity under stochastic boundary condition based on the computational framework. The effects of the correlation length and the variance of stochastic boundary condition on natural convection uncertainty were analyzed. The results show that the mean temperature field and flow field are basically the same as the deterministic temperature field and flow field, respectively. The probability distribution of the Nusselt number under stochastic boundary condition is a normal distribution. The mean Nusselt number increases with the correlation length and the variance, the variance has a greater influence on natural convection heat transfer than the correlation length.

2D Steady Heat Conduction Analysis With the Regular Hexagon Numerical Manifold Method

TAN Yu-xin, ZHANG Hui-hua, HU Guo-dong

2017, 38(5): 594-604. doi: 10.21656/1000-0887.370306

Abstract(1115) PDF(820)

Abstract:
The polygonal numerical manifold method (NMM) was developed to analyze two2dimensional (2D) steady heat conduction problems. Based on the governing equation, the boundary conditions and the NMM temperature approximation, the discrete NMM equations were deduced according to the modified variational principle. The domain integration schemes on the polygonal elements were presented. Due to the independence between the mathematical cover system and the physical domain and in virtue of the accuracy advantage of regular polygonal elements, the Wachspress regular hexagon mathematical elements were adopted in 2 typical examples, and the computed temperatures agreed well with the referential ones. The study shows that the regular hexagon NMM can well tackle 2D heat conduction problems.

A Meshless Natural Element Method for 2D Viscoelastic Problems

CHEN Shen-shen, ZHONG Bin

2017, 38(5): 605-612. doi: 10.21656/1000-0887.370300

Abstract(1376) PDF(601)

Abstract:
Based on the meshless natural element method, a new algorithm was proposed to solve 2D viscoelastic problems. According to the elasticviscoelastic correspondence principle and the Laplace transform technique, the viscoelastic problem was transformed into an elastic problem in the Laplace space and then the basic formula of the natural element method for the analysis of viscoelastic problems were derived. As a recently developed meshless method, the natural element method (NEM) is essentially a Galerkin method based on natural neighbour interpolation. Compared to most other meshless methods, the shape function employed in the NEM has interpolation property and its support domain is anisotropic. Some numerical examples verify the effectiveness of the developed method.

2017 Vol. 38, No. 5