应用数学和力学

2014 Vol. 35, No. 6

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20-Node Rational Elements for 3D Anisotropic Elastic Problems

MAO Ling, YAO Wei-an, GAO Qiang, ZHONG Wan-xie

2014, 35(6): 589-597. doi: 10.3879/j.issn.1000-0887.2014.06.001

Abstract(1299) PDF(985)

Abstract:
For the conventional finite element method, only the geometry and node locations of elements were considered in the interpolation functions, while the physical parameters which reflect the key features of the physical problems were ignored, so its numerical performance may be not satisfying in some cases. The construction of the rational finite element method was different from that of the conventional finite element method. The linear combinations of the fundamental solutions to the problem’s controlling differential equations were used as the interpolation functions, so the stress and strain fields were interpolated directly in the physical domain at the same time. The transfer matrix was modified at the element level to pass the patch test, and the resulting element stiffness matrix was related closely to the physical parameters of the problem. The rational finite element avoids the separation between the mathematical and physical aspects of a problem, so the stability and accuracy of numerical analysis could be improved significantly. Two kinds of 20-node rational brick elements based on the principle of minimum potential energy and satisfying the requirements of the patch test, were constructed according to the fundamental solutions to general 3D anisotropic problems. Numerical examples show that the rational elements give numerical results with not only high accuracy, but also good numerical stability.

DEM Simulation and Experimental Investigation of Burden Distribution in the Parallel-Hopper Bell-Less Top Blast Furnace

QIU Jia-yong, ZHANG Jian-liang, SUN Hui, YAN Bing-ji, LI Feng-guang, GUO Hong-wei

2014, 35(6): 598-609. doi: 10.3879/j.issn.1000-0887.2014.06.002

Abstract(860) PDF(722)

Abstract:
The flow behavior of particles during the burden distribution process in the parallel-hopper bell-less top blast furnace was investigated with the discrete element method (DEM) as well as a model experiment. It is shown that the DEM simulation results agree with the experimental results well. The contact force distribution in the hopper is nonuniform, i.e., the strong force chains mainly locate in the lower part of the parallel-hopper and in the vicinity of the inclined wall. The flow pattern in the parallel-hopper resembles a deflective funnel flow which comprises the quasi-stagnant zone, the central accelerated flow zone and the wall shear layer. Both the particle falling trajectory and distribution are related to the discharging sequence which is affected by the flow pattern in the hopper. As the flow trajectory is influenced by the opening of the flowgate, it should be controlled within a reasonable range for the stability of burden distribution. The heap peak position varies with the falling point during the heaping process, and the peak radius is larger than that of the falling point.

An Implicit Piecewise Dogleg Algorithm for Solving Trust-Region Subproblems

WANG Xi-yun, LI Liang, YU Hai-bo

2014, 35(6): 610-619. doi: 10.3879/j.issn.1000-0887.2014.06.003

Abstract(1334) PDF(669)

Abstract:
Based on the premise that Hessian matrix was positive definite, a differential equation model was established for the optimal curve. Then an implicit piecewise dogleg was constructed according to the differential equation. In turn, the implicit piecewise dogleg algorithm for solving trust-region subproblems was presented. And the rationality of the implicit piecewise dogleg path was analyzed and demonstrated. Numerical results indicate that the new algorithm is effective and practicable.

A Dual Wavelet Shrinkage Procedure for Suppressing Numerical Oscillation in Shock Wave Calculation

ZHAO Yong, ZONG Zhi>, WANG Tian-lin

2014, 35(6): 620-629. doi: 10.3879/j.issn.1000-0887.2014.06.004

Abstract(1196) PDF(1349)

Abstract:
In the numerical calculation of shock waves, numerical oscillation often occurred and contaminated the real solution in serious cases. For the purpose of suppressing the numerical oscillation, various complicated numerical schemes or artificial viscosity methods had been applied. From the view of signal processing, a dual wavelet shrinkage procedure was formulated to extract the real solution hidden in the numerical solution with oscillation. The localized differential quadrature (LDQ) method was firstly used to solve the shock wave problems governed by the shallow water equations and Euler equations for ideal fluid flow, and heavy oscillation emerged in these cases, then the dual wavelet shrinkage procedure was employed to supplement the LDQ method and the results without numerical oscillation were obtained, in which not only the position of shock/rarefaction wave was captured but the shock wave structure well kept. Compared with the previous complicated schemes, the present procedure enables some relatively simple scheme such as the LDQ method to effectively solve the shock wave problems.

A Multigrid Preconditioned Conjugate Gradient Method for Isogeometric Analysis

LIU Shi, CHEN De-xiang, FENG Yong-xin, XU Zi-li, ZHENG Li-kun

2014, 35(6): 630-639. doi: 10.3879/j.issn.1000-0887.2014.06.005

Abstract(1936) PDF(1841)

Abstract:
Accuracy of the isogeometric analysis can be improved through increase of the order of the NURBS basis function, but convergence of the multigrid will be slowed down at the same time. A method which combined the multigrid technique and preconditioned conjugate gradient iteration was proposed to accelerate the multigrid convergence. In the proposed method, the conjugate gradient part serves as the primary iteration, while the multigrid part serves as the preconditioner. The Poisson’s equation was solved with the multigrid method and multigrid preconditioned conjugate gradient method repectively for comparison. The results show that the multigrid preconditioned conjugate gradient method converges faster than the multigrid method especially in the cases of high-order NURBS basis functions or 3-dimensional problems.

Generalized Thermoelastic Solutions to the Problems of Thermal Shock on Elastic Half Space

WANG Ying-ze, WANG Qian, LIU Dong, SONG Xin-nan

2014, 35(6): 640-651. doi: 10.3879/j.issn.1000-0887.2014.06.006

Abstract(1497) PDF(820)

Abstract:
Based on the Laplace transform technique and its limit theorem, the asymptotic solutions to the problems of thermal shock on elastic half space were derived according to different generalized thermoelasticity models with the fractional order calculus introduced. The wavelike properties of heat propagation in elastic media were revealed accurately by these asymptotic solutions, and the jumps at the elastic wave fronts induced by thermal shock were also captured. The elastic wave propagation and the thermoelastic responses of displacement, temperature and stress fields were studied. The predictive abilities of the different generalized thermoelasticity models for thermal behaviors under thermal shock were compared, and the influence of the fractional order parameter on thermal behaviors was also be analyzed. The results show that, the molecular diffusion of heat has notable influence on the heat wave propagation, the response zones of related physical fields and the jump peak values of the temperature and stress fields, but it has little effect on the thermoelastic wave propagation.

Bifurcation and Chaos Thresholds of Bistable Piezoelectric Vibration Energy Harvesting Systems

LI Hai-tao, QIN Wei-yang

2014, 35(6): 652-662. doi: 10.3879/j.issn.1000-0887.2014.06.007

Abstract(1499) PDF(914)

Abstract:
Nonlinear dynamic performances such as homoclinic bifurcation and chaos were modeled and analyzed for bistable nonlinear vibration energy harvesting systems. According to bistability of the beam under axial loading, a model of the bistable nonlinear vibration energy harvester was established. Based on the Melnikov theory, a qualitative method was proposed to address homoclinic bifurcation of the bistable energy harvester under harmonic excitation, and the criteria for homoclinic bifurcation and the high-energy solution were derived from the Melnikov function through parameter optimization. Numerical simulation shows that the singlewell-to-doublewell transitions occur at the critical thresholds, which verifies the theoretical analysis. Research on the Melnikov method for nonlinear energy harvesting systems promises effective tools for the parametric design of high-performance energy harvesters.

Study on Fractional Brownian Motion of Self-Propelled Janus Microspheres

LI Fei, ZHANG Hong-yan, WU Mei-ling, ZHENG Xu, CUI Hai-hang

2014, 35(6): 663-673. doi: 10.3879/j.issn.1000-0887.2014.06.008

Abstract(1206) PDF(1099)

Abstract:
Janus microspheres are a special class of particles with regular shape but irregular surface composition. On the PIV experimental platform, a selfpropulsion experiment about Pt-SiO₂ Janus microspheres with 1μm and 2μm diameters was carried out. The stochastic trajectories of the particles selfpropelled by the asymmetrical catalytic decomposition of H₂O₂ were obtained. The Hurst indexes related to different observation time intervals were calculated through statistic analysis of the particles suspended in different concentration (0%, 2.5%, 5%, 10% and 15%) solutions. From the experimental data, it is clear that the stochastic trajectory of a Janus microsphere is the superposition of a random motion and a directional motion, and the particle undergoes abnormal diffusion. Then, the current complex motion is deemed as the combined action of Brownian motion, selfpropulsion and random rotation. The characteristic time scales, within which different dynamic factors are dominating, are obtained, and the model presents a reasonable explanation about the observed phenomena.

Parametric Study on the Straight-Line Cruising Velocity of an Auto-Swimming Robotic Fish

HAO Dong-wei, WANG Wen-quan

2014, 35(6): 674-683. doi: 10.3879/j.issn.1000-0887.2014.06.009

Abstract(1155) PDF(953)

Abstract:
Study on the kinetic & kinematic mechanisms of fish-like swimming is of cardinal significance for increasing needs of bionic technologies. Therefore, an auto-swimming robotic fish model was established with a flexible body and muscular self-propelling force, involving the interactions between fish-body internal forces, fish-body motions and surrounding fluid dynamics. Then, respectively, the effects of different-length tail fins, different tail elasticities and different muscular self-propelling force densities on the robotic fish’s straight-line auto-swimming state were numerically simulated respectively. The hydrodynamics and kinematics of the auto-swimming robotic fish are analysed to reveal the key factors on the cruising velocity and clarify the related mechanisms. The key factors influence on swimming efficiency is shown by analysis of hydrodynamic performance.

On the Existence of Anti-Periodic Solutions in Time-Invariant Fractional Order Systems

YANG Xu-jun, SONG Qian-kun

2014, 35(6): 684-691. doi: 10.3879/j.issn.1000-0887.2014.06.010

Abstract(1138) PDF(900)

Abstract:
The anti-periodic solution problem makes an important characteristic of dynamics for nonlinear differential systems. In recent years, the anti-periodic solution problem in integer order nonlinear differential systems had been widely studied, while the anti-periodic solution problem in fractional order nonlinear differential systems had been preliminarily discussed. Other than the previous work, the existence of anti-periodic solutions in time-invariant fractional order systems was investigated. It is shown that although within a finite time interval the solutions do not show any anti-periodic behavior, when the lower limit of the fractional order derivative tends to infinity the anti-periodic orbits will be obtained.

Dependence of Equilibrium Stability of First Order Lagrange Systems on Parameters

SONG Duan

2014, 35(6): 692-696. doi: 10.3879/j.issn.1000-0887.2014.06.011

Abstract(1026) PDF(669)

Abstract:
Steady first order Lagrange systems with additive terms were considered as gradient systems under certain conditions. The characteristics of the gradient system were used to study the equilibrium stability and its dependence on the parameters of the system. With two examples, the first order Lagrange systems’ stability domains were given in the parameter plane. Further, the analytical results indicate that change of the parameters not only influence the systems’ stability, but also influence the quantity of the equilibrium points.

Bisymmetric Damping and Stiffness Matrices Calibration With Test Data of Vibration Systems

ZHOU Shuo, HAN Ming-hua, MENG Huan-huan

2014, 35(6): 697-711. doi: 10.3879/j.issn.1000-0887.2014.06.012

Abstract(1141) PDF(707)

Abstract:
The problem of bisymmetric damping and stiffness matrices calibration with test data of vibration systems was discussed. Based on the eigen equation as well as bisymmetry of the damping and stiffness matrices, existence and uniqueness of the solution to the problem was studied by means of the theory and method for the inverse algebraic quadratic eigenvalue problem. A new method for the calibration of damping and stiffness matrices was presented. According to the properties of bisymmetric matrices, the bisymmetric solution to the matrix equation was studied. The general expression of the bisymmetric solution was obtained. Moreover, the related optimal approximation problem of any related matrix was addressed and the solution given. The damping and stiffness matrices calibrated with the method not only satisfy the quadratic eigen equation, but also are the unique bisymmetric matrix solution. A numerical example proves efficiency of the present method.

Vibration and Acoustic Responses of Lightweight Sandwich Structures: Theoretical and Experimental Investigations

XIN Feng-xian

2014, 35(6): 1001-1007.

Abstract(1168) PDF(1213)

Abstract:
Lightweight sandwich structures have been increasingly used in a wide range of engineering applications (e.g., automobiles, express trains, ship/submarine hulls and aircraft fuselages), and hence their vibroacoustic characteristics are of paramount importance for interior noise reduction. In the pursuit of vibration and noise reduction in civil and military applications, this dissertation deals with the vibroacoustic problems of lightweight sandwich structures immersed in either static or convected fluid. Specifically, structural wave and sound wave propagation as well as dynamic responses and vibroacoustic performances of these structures are systematically investigated by incorporating theoretical modeling, experimental measurement and numerical simulation. An integrated optimal algorithm toward lightweight, high stiffness and superior sound insulation capability is subsequently proposed, based on which preliminary optimal design of prototype sandwich structures is performed. The contents and contributions of the dissertation are summarized in the full text.