应用数学和力学

2019 Vol. 40, No. 2

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On Solutions to Several Classes of Differential-Algebraic Equations Based on Artificial Neural Networks

YANG Zhao, LAN Jun, WU Yongjun

2019, 40(2): 115-126. doi: 10.21656/1000-0887.390122

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Abstract:
In nonlinear science, it is always an important subject and research focus to find the approximate analytical solutions to differential equations. The artificial neural network and the optimization method were combined to solve 2 special classes of differentialalgebraic equations (DAEs). The 1st 3 numerical examples, namely, the Hessenberg DAEs with indices 1, 2, 3, fell into a category of pure mathematical problems. Then the 2nd example related to EulerLagrange DAEs with indices 3, i.e. a pendulum without external force, arising from the background of nonholonomic mechanics. The approximate analytical solutions to the above 4 examples were obtained and compared with the exact solutions and the results from the RungeKutta method. High accuracy of the proposed method was demonstrated.

Neurodynamic Analysis of Mutual Antagonism Between Default Mode Networks and Task-Positive Networks

CHENG Xianjun, WANG Yihong, WANG Rubin

2019, 40(2): 127-138. doi: 10.21656/1000-0887.390027

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Abstract:
The working mechanism of task-positive activation and task-negative activation is the fundamental element of cognitive function. The imbalance or impairment of this antagonism may induce a series of severe degenerative neurological diseases. However, the neural mechanism of this antagonism is unclear. Based on the mutual synaptic inhibitory assumption for the default mode network and the task-positive network, the numerical simulation of a working memory model was performed under multiple stimuli conditions. The results show that: 1) neural activities of task-positive and task-negative networks appear to antagonize each other; 2) the neural activity decay of the task-negative network will intensify as the count of stimulus directions of working memory increases; 3) the activity of the task-negative network will drop when the neural activity of the brain area related to working memory increases; 4) as the difficulty of working memory rises, the neural activity of the task-negative network will quickly decrease. These computational results match well with the experimental data. Since task-negative activation is the primary property of the default mode network, the mutual synaptic inhibition of default mode and task-positive networks makes the fundamental reason why the antagonism is generated between these 2 networks.

Neurodynamic Analysis of Cochlear Hair Cell Activity

RONG Weifeng, WANG Rubin

2019, 40(2): 139-149. doi: 10.21656/1000-0887.390184

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Abstract:
To understand the neurodynamic mechanism of cochlear hair cell activity more profoundly, a hair cell model based on the Hodgkin-Huxley equation was established. Through numerical simulation, neurodynamic analysis of hair cell membrane potential, power, and energy consumption was performed. The results show that, when the sound frequency is in the range of 0.1~20 kHz, the attenuation of outer hair cells’ (OHCs) membrane potential will be lower than that of inner hair cells (IHCs), while the gains in power and energy consumption of OHCs will be much larger than those of IHCs. The low attenuation of OHC membrane potential and the high gains in power and energy consumption support the view that the OHC amplification is driven by electromotility. The study on membrane potential, power and energy consumption of cochlear hair cell contributes to the profound understanding of the neurodynamic properties of hair cell activity.

A Robust and Low-Dissipation Flux Splitting Scheme

HU Lijun, YUAN Li, ZHAI Jian

2019, 40(2): 150-166. doi: 10.21656/1000-0887.390132

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Abstract:
With the rapid development of computational fluid dynamics, it is particularly important to design accurate, efficient and robust numerical schemes. Through the characteristics analyses of 3 popular flux splitting methods (AUSM, Zha-Bilgen and Toro-Vázquez), a simple, low-dissipation and robust flux splitting scheme (named as R-ZB) was constructed. The flux of Euler equations was split into a convection flux and a pressure flux with the Zha-Bilgen splitting procedure. The convection flux was computed with a simple upwinding scheme, and the pressure flux was evaluated with a low-dissipation HLL scheme to overcome the flaw of failing to capture contact discontinuities. Numerical experiments show that, the proposed R-ZB scheme not only retains the merits of the original Zha-Bilgen scheme, such as simpleness, efficiency and capturing contact discontinuities accurately, etc., but also has better robustness, which eliminates the numerical shock instabilities in the calculation of 2D problems.

Improved Conditions for Block-Sparse Signal Recovery via the Non-Convex Optimization Model

ZHOU Jun, HUANG Wei

2019, 40(2): 167-180. doi: 10.21656/1000-0887.390154

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Abstract:
Compressed sensing (CS) is a newly developed theoretical framework for information acquisition and processing, which shows that sparse signals can be recovered exactly from far less samples than those required by the classical ShannonNyquist theorem. The blocksparse signal recovery algorithm under the compressed sensing framework was mainly studied, and a class of improved exact recovery conditions based on the block restricted isometry property (RIP) were established in the noiseless cases via the mixed l₂/l_q(0

Periodicity of the Model for Electron Beam Focusing Systems

CHEN Lijuan, LU Shiping

2019, 40(2): 181-188. doi: 10.21656/1000-0887.390092

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Abstract:
The electron beam focusing system theories are widely applied in the fields of physics, biology, electronics and so on. To control the movement track of the electron beam to be focused on the object effectively, the existence of periodic solutions was discussed based on the electron beam focusing system model with Mawhin’s coincidence degree theory. The existence of positive periodic solutions to the model was obtained under specific conditions, and the feasibility of the ranges of the parameters in the model was analyzed. The study provides a theoretical basis for the design of traveling-wave tubes.

Effects of Circular Inhomogeneity on Crack Propagation

XING Shuaibing, WANG Qiangsheng, SHENG Yue, JIANG Xiaoyu

2019, 40(2): 189-199. doi: 10.21656/1000-0887.390136

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Abstract:
The solution of an infinite plane containing a crack and an arbitrarily oriented inhomogeneity under uniaxial tensile load was presented based on the distributed dislocation technique. The stress field and the strain energy density were obtained. The crack propagation direction was predicted according to the minimum strain energy density criterion. The results show that, the soft inhomogeneity has an amplifying effect on the stress intensity factor, the strain energy density and the stress field near the crack tip, while the hard inhomogeneity has a shielding effect. The effect of the inhomogeneity on the crack propagation direction increases with the decreasing distance, the increasing absolute value of lg(μ₂/μ₁), and the increasing inhomogeneity radius. The inhomogeneity has a little effect on the crack propagation direction for -30°＜θ＜30°.The soft inhomogeneity has an attracting effect, while the hard inhomogeneity has a repulsing effect on the crack propagation for -90°＜θ＜-30°and 30°＜θ＜90°.

Analysis of Multicrack Problems With Eigen COD Boundary Integral Equations

GUO Zhao, GUO Zitao, YI Lingyan

2019, 40(2): 200-209. doi: 10.21656/1000-0887.390183

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Abstract:
For multicrack problems, the conventional numerical solution techniques are of low efficiency. To realize large-scale numerical simulation of multicrack problems, the eigen crack opening displacement (COD) boundary integral equations and the pertinent iteration algorithm were established. To deal with the interactions between cracks, the local Eshelby matrix was introduced. In this way, the superposition technique was employed with all cracks divided into 2 groups, i.e. the adjacent group and the far-field group, according to a non-dimensional radial distance of a crack to the current crack. In comparison to the fast multipole boundary element method with a constant element as the discrete element, the proposed computational model and the iteration algorithm were numerically verified. The numerical results show that, the model for the eigen COD boundary integral equations gets great improvement in dealing with multicrack problems, and its computation efficiency is significantly higher than those of the traditional boundary element method and the fast multipole boundary element method.

Analytical Solutions to Problems of Elliptical Holes With 4 Edge Cracks in 1D Orthorhombic Quasicrystals

GAO Yuanyuan, LIU Guanting

2019, 40(2): 210-222. doi: 10.21656/1000-0887.390032

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Abstract:
The plane elastic problems of elliptical holes with 4 cracks in 1D orthorhombic quasicrystals were investigated through introduction of a new generalized conformal mapping with the generalized complex variable method. With the stress functions, the basic elasticity equations were reduced to 4th-order partial differential equations, the complex expression of stress components was derived in the image plane and the analytical solution of stress intensity factors (SIFs) at the crack tips was found out. With the change of parameters describing the defects, the results can not only reduce to the conclusions in previous literatures, but also give the SIFs of a variety of common defect configurations, which provides a theoretical basis for engineering mechanics analysis.

A Frictionless Contact Problem of 2D Decagonal Quasicrystal Plane Elasticity With Cracks

ZHANO Xuefen, LI Xing

2019, 40(2): 223-236. doi: 10.21656/1000-0887.390127

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Abstract:
With the classical complex function method, a frictionless contact problem of 2D decagonal quasicrystal semiplane elasticity with arbitraryform cracks was addressed under the action of a rigid convex basal punch. Based on complex expressions of stresses and displacements of 2D decagonal quasicrystals, the problem was converted into solvable boundary value problems with analytic functions, and then reduced to a class of Riemann boundary problems. Solutions to the Riemann boundary problems give the stress functions in closed form, the explicit expressions of the stress intensity factors at crack tips and the contact stress distribution under the punch. The expression of the contact stress shows that, it has singularity at the edge of the contact zone and the crack tips. Without the effect of the phason field, the obtained results match well with those classical conclusions for elastic materials. Numerical examples illustrated the solutions to the frictionless contact problem in 2D decagonal quasicrystal semiplane elasticity with a vertical crack and a horizontal straight crack under a rigid punch. The work provides a theoretical reference for the application of quasicrystalline materials.