应用数学和力学

2020 Vol. 41, No. 6

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A StructurePreserving Algorithm for Hamiltonian Systems With Nonholonomic Constraints

MAN Shumin, GAO Qiang, ZHONG Wanxie

2020, 41(6): 581-590. doi: 10.21656/1000-0887.400375

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Abstract:
Based on the concept of variational integrator and the Lagrange-d’Alembert principle with dual variables, a high-order structure-preserving algorithm for Hamiltonian systems with nonholonomic constraints was proposed. Based on the variational integrator, a discretization form of the Lagrange-d’Alembert principle with dual variables was obtained by means of appropriate polynomials and quadrature rules. On the basis of this discretization form, a numerical integration method was given with displacements at both ends of the integral interval as independent variables. Meanwhile, the nonholonomic constraints were strictly met at the endpoints of the integral interval and the control points within the interval. The symmetric property of the proposed algorithm was proved. Numerical examples show that, the proposed algorithm has a high convergence order, strictly meets the nonholonomic constraints and has good long-time behaviors.

Solutions of Continuous and Discontinuous Anisotropic Heat Conduction Problems With the Numerical Manifold Method

LIU Simin, ZHANG Huihua, HAN Shangyu, LIU Qiang

2020, 41(6): 591-603. doi: 10.21656/1000-0887.400289

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Abstract:
The heat conduction is a common problem in engineering practice. Compared with those of isotropic materials, the heat conduction problem of anisotropic materials is more complicated, so it is of great significance to accurately predict the internal temperature distribution. The numerical manifold method (NMM) was developed to solve typical continuous and discontinuous heat conduction problems in anisotropic materials. According to the governing differential equation, boundary conditions and variational principles, the NMM discrete equations for such problems were derived. Several representative examples involving continuous and discontinuous situations were analyzed with the uniform mathematical cover independent of all physical boundaries. The results prove the feasibility and accuracy of the method and indicate that the NMM can simulate the heat conduction problem of anisotropic materials well. Besides, the influences of the material properties and crack configurations on the temperatures were also investigated.

Optimal Design of Flexible Skin on the Leading Edge of a 3D Variable-Camber Wing

Lü Shuaishuai, WANG Binwen, YANG Yu

2020, 41(6): 604-614. doi: 10.21656/1000-0887.400384

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Abstract:
The smooth continuous wing leading edge with variable cambers has the advantages of reduced noise and improved aerodynamic efficiency. Based on the 2D airfoil flexible skin design method, a design method for flexible skin on the variable-curvature leading edge of a swept-back airfoil was proposed. The main improvement lies in the synchronous optimization of multiple airfoil profiles along the wingspan direction, the modification of the objective function to solve the deformation problem, and the modification of NSGA-II with the elite strategy to adapt to the multi-objective optimization of 3D skin. The research shows that, compared with the existing design method, this method can improve the deformation accuracy of the flexible skin by 27% and realize the accurate shape of the flexible skin in the drooping state.

A 2D Flux Splitting Scheme Based on the AUSM Splitting

HU Lijun, WU Shifeng, ZHAI Jian

2020, 41(6): 615-626. doi: 10.21656/1000-0887.400264

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Abstract:
The AUSM-type schemes based on the advection upstream splitting method have the advantages of simpleness, high efficiency and high resolution, and are widely applied in computational fluid dynamics. The traditional AUSM-type schemes only consider the normal waves to the cell interface while ignoring the influence of tangential waves to the interface in the computation of the interfacial numerical flux. The flux of 2D Euler equations was split into the convective flux and the pressure flux by means of the AUSM splitting method, and they were both computed with the modified AUSM scheme. In the solution of the numerical flux at the corners where the influence of tangential waves was considered, a genuinely 2D AUSM flux splitting scheme was constructed. In the computation of 1D numerical examples, the proposed scheme keeps the merits of capturing shocks and contact discontinuities accurately. In the computation of the 2D numerical examples, the scheme has higher resolution and better robustness, while eliminating the instability behind the strong shock waves. In addition, with the scheme the stable CFL number greatly improves and the computation efficiency rises in the simulation of multidimensional problems. Therefore, the proposed scheme makes an accurate, efficient and robust numerical method.

Networked Non-Clustering Phase Synchronization in Coupled Neuron Systems

XIE Yiding, WANG Zhengping, LIU Shuai

2020, 41(6): 627-635. doi: 10.21656/1000-0887.400297

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Abstract:
The phase synchronization of coupled neurons under different complex network environments (including classical small-world, scale-free and random networks) was studied. Differing from the clustering phase synchronization in coupled phase oscillators generally found and reported in previous literature, a novel non-clustering phase synchronization was uncovered. The global synchronization involves 2 different dynamical processes: the frequency increase and the frequency decrease, where the frequency increase is induced by the spike insertion, and the frequency decrease is induced by the spike merge. Therefore, the neuron’s frequency variation mainly depends on the change of spike numbers, and the usual phase clustering phenomenon cannot be found here. The findings could enrich the understanding of networked dynamical behaviors including the phase synchronization and the computational neuron dynamics.

Multistage Coexistence of Different Chaotic Routes in a Delayed Neural System

LI Xiaohu, ZHANG Dingyi, SONG Zigen

2020, 41(6): 636-645. doi: 10.21656/1000-0887.400130

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Abstract:
Chaos and its coexistence involve very important problems in dynamical analysis. A delayed inertial 2-neuron system with non-monotonic activation function was studied with the Poincaré section method. With system parameters fixed and time delay τ chosen as the parametric variable, 1D bifurcation diagrams, i. e. period-doubling and quasi-periodic bifurcations were given under different initial conditions. The results show that, the neural system exhibits multistage coexistence of many period-doubling and quasi-periodic bifurcation sequences along different routes to chaos and stable coexistence of many chaotic attractors and multi-periodic solutions.

Asymptotic Stability Analysis of Fractional Neural Networks With Discrete Delays and Distributed Delays

LIU Jian, ZHANG Zhixin, JIANG Wei

2020, 41(6): 646-657. doi: 10.21656/1000-0887.400286

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Abstract:
The asymptotic stability of fractional-order neural networks with discrete delays and distributed delays in the sense of Caputo derivatives was studied. Through construction of the Lyapunov function and with the fractional Razumikhin theorem, sufficient conditions for asymptotic stability of fractional-order neural networks with discrete and distributed delays were given, and 4 examples were given to illustrate the validity of the proposed theorem conditions.

Existence of Periodic Traveling Waves for Time-Periodic Lotka-Volterra Competition Systems With Delay

GU Yumeng, HUANG Mingdi

2020, 41(6): 658-668. doi: 10.21656/1000-0887.400275

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Abstract:
A time-periodic reaction-diffusion Lotka-Volterra competition model with delay was considered. Under certain conditions, with the method of super- and sub-solutions and monotone iterations, the existence of time-periodic traveling waves connecting 2 semi-trivial periodic solutions of the corresponding kinetic system was proved with wave speed c*.Furthermore, the traveling wave solutions for c* were proved to be monotone with the comparison principle, and the asymptotic behaviors of traveling wave solutions were obtained at minus/plus infinity. Finally, the existence of traveling wave solutions was proved at wave speed c=c^*.

Design of a Finite-Time State Estimator for Nonlinear Systems Under Event-Triggered Control

TONG Yinghao, TONG Dongbing, CHEN Qiaoyu, ZHOU Wuneng

2020, 41(6): 669-678. doi: 10.21656/1000-0887.400210

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Abstract:
The event-triggered state estimator for nonlinear systems with time delay was studied. Firstly, the state estimator for nonlinear systems was established by the event-triggered mechanism, and the Lyapunov function was used to make the system mean square bounded in finite time. Secondly, based on the H_∞ bounded condition, the system’s H_∞ finite time bounded criterion was obtained. Finally, a numerical example was given to illustrate the validity of the obtained result.

Asymptotic Solution for Fractional-Order 2-Parameter High-Order Nonlinear Perturbed Models

XU Jianzhong, MO Jiaqi

2020, 41(6): 679-686. doi: 10.21656/1000-0887.400238

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Abstract:
A class of nonlinear fractional-order perturbed higher-order differential models was considered. Firstly, under suitable conditions, the outer solution to the original problem was obtained with the perturbation method. Then by means of the stretched variable, the composite expansion method and the theory of power series, the first and second boundary layer correction terms were constructed and the formal asymptotic expansion was obtained. Finally, with the theory of differential inequalities the asymptotic behavior of the solution to the problem was studied and the uniform validity of the asymptotic estimate expression was proved.

First-Order Sufficient Conditions for Existence of Local Extremums of Multivariate Functions

HUANG Zhenggang

2020, 41(6): 687-694. doi: 10.21656/1000-0887.400237

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Abstract:
The unified 1st-order sufficient condition was proposed for existence of the local extremums of n-variable functions, in a case more general than classical unconstrained optimization ones. The difficulty of no such 1st-order sufficient condition in optimization theories was solved. Moreover, the 1st-order sufficient condition for 1-variable functions was proved to be a special case of the results. The work can eliminate the shortages of the 2nd-order sufficient conditions for existence of local extremums of classical multivariate functions, and the result is both necessary and sufficient under the assumption of quasiconvexity or quasiconcavity.