应用数学和力学

2020 Vol. 41, No. 8

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Modelling and Analysis of Optimal Dynamical Systems of Incompressible Navier-Stokes Equations With Pressure Base Functions

WANG Jincheng, QI Jin, WU Chuijie

2020, 41(8): 817-833. doi: 10.21656/1000-0887.400276

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Abstract:
The modelling theory for optimal truncated low-dimensional dynamical systems of Navier-Stokes equations with pressure base functions and velocity base functions was studied. In the viscous incompressible fluid, the flow field around 3 square columns was simulated. According to that flow problem, the optimal dynamical systems of the Navier-Stokes equations with pressure base functions and velocity base functions were modelled and studied. The results show that, the dynamics behavior of the optimal dynamical systems around the 3 square columns is chaos, which is essentially different from the limit cycle dynamics behavior of the flow field around 2 square columns, so the complexity of the wake increases in the multi-column flow, which thereby means promotion of fluid mixing.

Error Estimates of Mixed Space-Time Finite Element Solutions to Sobolev Equations

PANG Naihong, LI Hong

2020, 41(8): 834-843. doi: 10.21656/1000-0887.410053

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Abstract:
The mixed space-time finite element scheme for Sobolev equations was constructed through introduction of auxiliary variables. The scheme can not only reduce the order of the equation with the mixed method, but discretize both space and time variables by means of the finite element technique. A numerical model of high-order accuracy in space and time was obtained. The existence, stability and uniqueness of the mixed space-time finite element solution were proved. The error estimates were derived with the space-time projection operator.

A Discontinuous Galerkin FEM for 2D Navier-Stokes Equations of Incompressible Viscous Fluids

CHEN Yafei, ZHENG Yunying

2020, 41(8): 844-852. doi: 10.21656/1000-0887.400379

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Abstract:
The incompressible Navier-Stokes equations are composed of the conservation law and the diffusion and constrained development equations. To test the numerical method, based on the unstructured grid, a discontinuous Galerkin scheme was established. The numerical results of the eddy current problem for different viscosity coefficients υ were discussed. The simulation results show that, the method has high precision and can solve the incompressible viscous fluid problem with moving interface, which makes the simulation boundary layer, the shear layer and the complex vortex solution be very effective, and the shock structure can be successfully extended to the numerical simulation of complex phenomena.

Vibration of Piezoelectric Nanobeams With Surface Effects

ZHOU Qiang, ZHANG Zhichun, LONG Zhilin, WU Jingxiang, HUANG Bin, JIN Hua

2020, 41(8): 853-865. doi: 10.21656/1000-0887.400330

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Abstract:
One-dimensional piezoelectric nanomaterials are widely used in MEMS/NEMS systems, and it is important to effectively characterize their mechanical properties. Based on the Gurtin-Murdoch surface theory, a new model for one-dimensional nanomaterials with surface effects was established. Based on the Timoshenko beam theory, the governing equations for piezoelectric nanowires with surface effects were derived, and the exact solutions of frequency equations and mode equations for piezoelectric nanowires under different boundary conditions were obtained. A method to simulate the surface effects with the finite element software was proposed, and the numerical simulation of piezoelectric nanobeams with surface effects was realized in ABAQUS. The theoretical results are in good agreement with the finite element simulation results, which verifies the correctness and validity of the theoretical model. The surface effect was very significant to the vibration frequencies of nanobeams and somewhat influences the mode shapes.

Model Updating for Bolted Structures Based on the Bayesian FFT Method

ZHANG Yong, ZHAO Yan, OUYANG Huajiang

2020, 41(8): 866-876. doi: 10.21656/1000-0887.400373

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Abstract:
A model updating method for bolted joints based on the Bayesian fast Fourier transform (FFT) method was proposed. In this method, the bolted joint was simulated with spring and thin-layer elements, and the dynamic equations for the composite structure were established with the sub-structure technique. The asymptotic distribution of the scaled FFT of the measured data in the time domain was used to formulate the posterior probability distribution function of the bolted parameters, and its negative log function was taken as the objective to conduct the parameter updating. The maximum posterior estimation generates the optimal estimation, and the uncertainty of the parameters was quantified with the asymptotic property of the posterior probability distribution. The developed method was validated in the model updating of a composite cantilever beam under stochastic excitation, where two kinds of jointed modeling methods were given. The comparison between the measured power spectrum and the updated power spectrum demonstrates the effectiveness of the developed method.

Thermo-Mechanical Buckling Analysis of Thin Plates

LI Ruoyu, WANG Tianhong

2020, 41(8): 877-886. doi: 10.21656/1000-0887.400308

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Abstract:
Based on the RayleighRitz theory and the finite element method, the expressions of critical buckling loads on thin plates under thermomechanical coupling loads was derived. Under simultaneously imposed mechanical and thermal loads, the finite element program was compiled in the MATLAB environment to solve the buckling critical loads on thin plates. In the buckling analysis, the thermal load was applied at the nodes in the form of temperature field. The effects of mechanical load components and thermal load components on the instability of thin plates were analyzed through nonuniform temperature field loading. The results show that, with the increase or decrease of given thermomechanical loads, the critical load increases or decreases almost linearly.

Event-Based State Estimation for Neural Network With Time-Varying Delay and Infinite-Distributed Delay

DU Yuwei, LI Bing, SONG Qiankun

2020, 41(8): 887-898. doi: 10.21656/1000-0887.400377

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Abstract:
The event-based state estimation problem was investigated for a class of neural networks with mixed delays. A novel event-triggering scheme depending on both the output and exponential decay function was designed to reduce the frequency of updating. In view of both the mixed delays and the event-triggering properties, a new state estimation error system was built. The exponential stability of the error system was derived with the Lyapunov function and the inequality technique. The Zeno phenomenon was analyzed and excluded. Finally, a numerical example and its simulations were presented to illustrate the effectiveness of the proposed approach.

Stability Analysis of Uncertain Singularly Perturbed Filter Error Dynamic Systems With Time Delays

SUN Fengqi

2020, 41(8): 899-911. doi: 10.21656/1000-0887.400368

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Abstract:
With the Lyapunov stability theory, the linear matrix inequality method, the time delay piecewise analysis method, the matrix analysis method and the free weight matrix method, the stability of singularly perturbed filter error dynamic systems with time-varying delays and uncertain structures was studied on the basis of the previous filter design theory. Through construction of a new Lyapunov function, with a new cross-term definition method and according to the system characteristics, a new criterion for stability of filter error dynamic systems in 2 cases of time delay dependence and time delay independence was derived. Finally, the selected numerical example illustrates the effectiveness and feasibility of the obtained results.

Berge Lower Semi-Continuity of Parametric Generalized Vector Quasi-Equilibrium Problems Under Improvement Set Mappings

SHAO Chongyang, PENG Zaiyun, LIU Fuping, WANG Jingjing

2020, 41(8): 912-920. doi: 10.21656/1000-0887.400307

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Abstract:
The Berge lower semi-continuity of solution mapping for a new class of parametric generalized vector quasi-equilibrium problems was discussed. Firstly, the improvement set mapping was defined, based on which the order structure was generalized and applied to the study of vector quasi-equilibrium problems, to lead to parametric generalized vector quasi-equilibrium problems under improvement set mappings (IPGVQEP). Then, a nonlinear scalarization function Ψ associated with the improvement set mapping was introduced, the scalar problem (IPGVQEP)_Ψ corresponding to the above problem (IPGVQEP) was given, and the relation between solution sets of (IPGVQEP) and (IPGVQEP)_Ψ was obtained. Finally, by virtue of a key hypothesis H_Ψ and the relation between solution sets, the sufficient and necessary conditions for Berge lower semi-continuity of the solution mapping for (IPGVQEP) were established, and an example was given to verify the results.

Finite-Time Stability of Fractional-Oder Linear Differential Systems With Delays

TIAN Hongqiao, ZHANG Zhixin, JIANG Wei

2020, 41(8): 921-930. doi: 10.21656/1000-0887.400365

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Abstract:
The finite-time stability of fractional-order linear differential systems with time delays was studied. Firstly, with a new Lyapunov function and the linear matrix inequality, some sufficient conditions for the finite-time stability of fractional linear differential systems with time delays were derived. Then, under the action of a state feedback controller, some conditions for the finite-time stability of fractional differential closed-loop systems with time delays were given, and the design method for the controller was given. In the end, the effectiveness of the theoretical results was illustrated with two examples.