应用数学和力学

2019 Vol. 40, No. 3

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Some Preconditioning Iterative Algorithms for Non-Hermitian Linear Equations

ZHANG Yingchun, LI Yin, XIAO Manyu, XIE Gongnan

2019, 40(3): 237-249. doi: 10.21656/1000-0887.390222

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Abstract:
Non-Hermitian linear equations have extensive application in scientific and engineering calculations and are expected to be solved with high efficiency. To accelerate the convergence rate of original algorithms, a preconditioning technique was developed and applied to some iterative methods chosen to solve the nonHermitian linear equations and complex linear systems with multiple righthand sides. Several numerical experiments show that the preconditioned iterative methods are superior to the original methods in terms of both the convergence rate and the number of iterations. In addition, the preconditioned generalized conjugate A-orthogonal residual squared method (GCORS2) has better convergent behavior and stability than other preconditioned methods.

Optimization of RBF Parameterized Airfoils With the Aerodynamic ROM

ZHANG Jun, LI Lizhou, YUAN Meini

2019, 40(3): 250-258. doi: 10.21656/1000-0887.390187

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Abstract:
Under the assumption of small perturbations and weak nonlinearity, a new airfoil optimization method was proposed based on the aerodynamic reduction order model (ROM) and the radial basis function (RBF) parameterization. The RBF was used to parameterize the airfoil shape perturbations, the ROM kernels of the airfoil aerodynamics corresponding to shape perturbations were identified with the computational fluid dynamics (CFD), the aerodynamic ROM was built through superposition, and the airfoil liftdrag ratio was calculated and optimized with the ROM. The optimized results of the NACA0012 airfoil show that, the proposed optimization method based on the ROM is feasible and can greatly accelerate the airfoil optimization procedure.

MATLAB Implementation of a Singular Boundary Method for Transient Heat Conduction

LI Yudong, WANG Fajie, CHEN Wen

2019, 40(3): 259-268. doi: 10.21656/1000-0887.390225

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Abstract:
The singular boundary method (SBM) based on the timedependent fundamental solutions to dynamic problems is a meshless boundary collocation technique with the merits of easy implementation and mathematical simplicity. This method effectively avoids the singularity of source points through introduction of the origin intensity factor. The SBM was implemented to simulate the transient heat conduction problem via the MATLAB programming, and the MATLAB toolbox was created to provide a simple and efficient tool for the numerical analysis of transient heat conduction problems and actual operating problems. In numerical experiments, 2D and 3D problems were investigated in regular geometrical regions. The method was applied to the solution of the temperature field in the low temperature transient state. The results indicate that the MATLAB toolbox for the SBM with the inverse interpolation technique and the empirical formula is simple, accurate and efficient.

Numerical Simulation of the Whole Instability and Destruction Process for Fully Weathered Slopes Based on the FEMLIP

WANG Manling, WANG Aitao, LI Yang, PENG Junqiang, LI Shuchen

2019, 40(3): 269-281. doi: 10.21656/1000-0887.390206

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Abstract:
Slope angles and strengths are important factors influencing slope stability, while slope instability is often accompanied with large deformation, ranging from tens to thousands of meters. For traditional finite element methods, calculation often terminates due to grid distortion in the large deformation cases. The finite element method with Lagrangian integral points (FEMLIP) was adopted to simulate the large deformation landslide process of slopes and study the effects of slope angles and strengths on slope stability. The C language was used for the Ellipsis programming to simulate the whole process of slope instability and collapse, which was verified with a typical case. The stability and landslide processes of the slope under different slope angles and strengths were analyzed with this method. The results show that, the FEMLIP can accurately find out the potential slip surface of the slope and simulate the landslide development process after the slope instability, making a new numerical way for the large deformation analysis of the slope landslide.

Analysis on Nonlinear Dynamics of Circular Truss Antennae in 6D Systems

SUN Ying, ZHANG Wei, WU Ruiqin

2019, 40(3): 282-301. doi: 10.21656/1000-0887.390058

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Abstract:
The main tendency of circular truss antennae will be large scale, light weight and high flexibility in future. The circular truss antenna keeps in a folded state during the time of launching. After blastoff, the circular truss antenna unfolds in sequence according to the instruction, saving much space for the satellite. In addition, the caliber of the circular truss antenna can be designed as an ideal value according to requirement. Due to the structural characteristics and the complex spatial environment, the antenna may suffer large-amplitude vibrations, which severely affect the stability of the satellite. The circular truss antenna was simplified as an equivalent cylindrical shell model and the dynamic equations were established. The theoretical analysis and numerical simulation were used to investigate the nonlinear dynamic behaviors of the circular truss antenna in the 6D system. The normal form theory was adopted to simplify the averaged equations. The dynamics of the unperturbed system and the perturbed system was studied. The Shilnikov-type multi-pulse chaotic motion was proved with the energy-phase method, and the effects of the thermal excitation on the nonlinear vibrations of the circular truss antenna system was verified through numerical simulation.

A Control Method for Underactuated Cranes Based on Virtual Holonomic Constraints

ZHAO Chen, GE Xinsheng

2019, 40(3): 302-310. doi: 10.21656/1000-0887.390163

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Abstract:
The control of underactuated systems was an important field of nonlinear control. The underactuated system refers to a nonlinear system with an input control variable number less than the degree of freedom number. At present, the main methods of dynamics and control research of underactuated nonlinear systems include the linear quadraticform optimal control method and the partial feedback linearization method, and how to make the system stabilize in the equilibrium position is always a difficult point. For the virtual constraint method, the periodic motion of the system is designed with a selected cyclic independent variable. Based on the typical underactuated model crane, the virtual constraint method was adopted to make the system stabilize or oscillate in equilibrium position. First, through establishment of the virtual constraints, the system’s degrees of freedom were reduced. Then, the system state equations were derived according to the partial feedback linearization theory. Finally, the feedback controller was designed with the linear quadraticform regulator. The simulation results show that, the weight can reach a stable state near the vertical position under the feedback control, which reflects the effectiveness of the virtual constraint method for underactuated systems.

Dynamic Analysis of the Avian Influenza A (H7N9) Transmission Model

ZHOU Piaopiao, ZHU Guanghu

2019, 40(3): 311-320. doi: 10.21656/1000-0887.390175

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Abstract:
Avian influenza A(H7N9) is always a big threat to human health and safety. Aimed at the transmission patterns of A(H7N9), a new SI-V-SEIR epidemic model was put forward, which incorporated the viral interactions among humans, poultry and environment. Through dynamic analysis, the expression of basic reproduction number R₀ was given, and the stability of disease-free and endemic equilibrium points was proved. The proposed model was further applied to study the 2016—2017 A(H7N9) outbreaks in Guangdong province. It is found thatR₀=18.8 in the early outbreak, which indicates 94.7% of poultry to be vaccinated for the control of the virus transmission in poultry and environment. After control,R₀ will fall down to 0.14. The results show that, reduction of the viral load in environment and the infection ratios among poultry and from poultry to humans could effectively lower human infections.

Influence of Diffusion on an InvasionDiffusion Prey-Predator Model With Disease Infection in Both Populations

LIU Wenqing, CHEN Qingwan

2019, 40(3): 321-331. doi: 10.21656/1000-0887.390100

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Abstract:
An invasiondiffusion preypredator epidemic system with disease infection in both populations was studied. The influence of invasion diffusion on the equilibrium solutions of positive constants was obtained through analysis of the eigenvalue and construction of the Lyapunov function. Furthermore, with the topological method, it was proved that the coefficient of invasion diffusion will be big enough while the selfdiffusion coefficient is sufficiently small, then there exists a positive nonconstant equilibrium solution.

Hopf Bifurcation Analysis of a Model for Spruce Budworm Populations With Delays

CAO Jianzhi, TAN Jun, WANG Peiguang

2019, 40(3): 332-342. doi: 10.21656/1000-0887.390111

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Abstract:
The dynamic behavior of a population model with stage structure for spruce budworms with time delay was investigated. Firstly, existence of a unique positive equilibrium of the model was discussed and sufficient conditions for local stability of the positive equilibrium and Hopf bifurcation occurrence were obtained. Next, the direction of the Hopf bifurcation and the stability of the periodic bifurcation solutions were analyzed with the normal form method combined with the center manifold theorem. Finally, some numerical simulations to verify the theoretical results were also conducted. The work provides an applicable reference for control of spruce budworms.

D-η-E-Semi-Preinvex Mapping and D-η-E-Properly Semi-Prequasi-Invex Mapping

WANG Haiying, FU Zufeng

2019, 40(3): 343-354. doi: 10.21656/1000-0887.390143

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Abstract:
Some properties of the D-η-E-semi-preinvex mapping and the D-η-E-properly semi-prequasi-invex mapping were studied. Firstly, relationships among the D-η-E-semi-preinvex mapping, the semistrictly semi-preinvex mapping, the strictly D-η-E-semi-preinvex mapping and the D-η-E-properly semi-prequasi-invex mapping were discussed. Three important theorems of properties were obtained based on the middle point D-η-E-semi-preinvex and some suitable conditions. Secondly, relationships among the D-η-E-properly semi-prequasi-invex mapping, the semistrictly D-η-E-properly semi-prequasi-invex mapping and the strictly D-η-E-properly semi-prequasi-invex mapping also were addressed. Finally, an important application in terms of the D-η-E-properly semi-prequasi-invex mapping was demonstrated in optimization.