## 2022 Vol. 43, No. 3

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2022, 43(3): 237-254. doi: 10.21656/1000-0887.420123
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Abstract:

The Bragg resonant reflection excited by a finite periodic array of parabolic trenches was analytically studied. First, the modified mild-slope equation (MMSE) with implicit coefficients was transformed into an ordinary differential equation with explicit coefficients through variable substitution. Second, an analytical solution to the MMSE was established in terms of the Frobenius series, and the convergence condition for the series solution was given. Finally, by means of the mass-conservation matching conditions, an analytical formula for the reflection coefficient was built. With the analytical formula, the effects of the number, the depth and the width of trenches on the peak value, the phase and the band width of the resonance, were investigated. The results show that, when the depth and width of trenches keep constant, and the number of trenches increases, the Bragg resonance peak value will increase up to 1, while the resonance bandwidth will narrow down and approach a fixed value. When the number and width of trenches keep constant, the Bragg resonance peak value will increase with the depth of trenches. When the number and depth of trenches keep constant, the Bragg resonance peak value will increase at first and then decrease with the width of trenches, which implies that there exists a certain width of trenches to make the Bragg resonance peak value reach the maximum, laying a theoretical base for the optimization of Bragg resonance vs. the trench width. Particularly, the phase upshift of the Bragg resonance wave reflection peak value recently observed over finite periodically arranged cycloidal trenches, was confirmed again over the parabolic trenches. That implies that, the phase upshift of the Bragg resonance reflection peak value is a common phenomenon excited by finite periodic trenches with arbitrary cross sections. Consequently, for sinusoidal ripples and periodic artificial bars, the phase of the Bragg resonance reflection will shift downward, while for an array of periodic trenches, regardless of the shape of the trench cross section, the phase of the Bragg resonance reflection will shift upward. In addition, starting from the initial definition of the Bragg resonance, the mathematical mechanism of the phenomenon of phase upshift is well explained.

2022, 43(3): 255-271. doi: 10.21656/1000-0887.420182
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Abstract:

The movement mechanisms of Newtonian droplets in power-law fluids in T-junction microchannels were studied with the lattice Boltzmann method. The effects of power-law index n , capillary number Ca , flow ratio Q , viscosity ratio M, and surface wettability θ on the droplet formation size, the formation time and the deformation index (DI) were investigated in detail. The results show that, first, with the increase of power-law index n from 0.4 to 1.6, the droplet formation size decreases almost linearly, and both the droplet formation time and the deformation index decrease quickly at first and then much more slowly. Second, the influences of the viscosity ratio on the droplet formation size, the formation time and the deformation index are basically the same as those of the power law index. In addition, with the increases of Ca and θ of the main channel, the droplet formation size decreases almost linearly, while the droplet formation time and deformation index decrease rapidly at first and then slowly, and the decreasing rates weaken with the increase of the power-law index. At last, with the increase of flow ratio Q of the continuous phase over the dispersed phase, the droplet formation size increases, and the droplet formation time as well as the deformation index decrease.

2022, 43(3): 272-280. doi: 10.21656/1000-0887.420197
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Abstract:

The unsteady flow of the upper-convected Oldroyd-B fluid over the heated wedge in the presence of velocity slip was discussed. The process of heat transfer and the effect of the thermal retardation time on heat transfer were simulated with the relaxation-retardation heat flux model. The buoyancy, the thermal radiation and the convective heat transfer boundary condition were considered to further elucidate the flow and heat transfer characteristics. The homotopy analysis method was used to obtain the approximate analytical solutions to ordinary differential equations. It is found that the magnification of the slip parameter can promote the flow of fluid, and the fluid temperature rises with the thermal radiation parameter. In addition, the temperature field shows opposite trends in the thermal relaxation time and the thermal retardation time.

2022, 43(3): 281-289. doi: 10.21656/1000-0887.420115
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Abstract:

To design reasonable main dimensions of catenary anchor leg mooring (CALM) buoys, 5 main design requirements were summarized. Based on practical engineering experiences, the mother model estimation method was established against the buoy weight. The reserve buoyancy was discussed, and the equal freeboard scheme set was proposed. The free floating stability of buoys was discussed based on the statics principle. Dynamic responses of buoys were discussed base on hydrodynamics, with the heave and roll characteristic vs. CALM buoy diameters presented. With the upper and lower limits on buoy diameters under 25 m and 100 m water depths as the boundary, a feasible region was built up. Its accuracy was verified in the actual engineer case under the 3 environmental conditions and by reference to the theoretical CALM buoy diameter feasible region under the 45 m water depth. The results show that, different factors, such as the environmental conditions, the mooring chain weight and others can influence the feasible region of buoy diameters, even under the same water depth. A deviation margin of the feasible region should be further considered in practical application. The study of the feasible region has a positive guiding significance to the determination of the main dimensions of CALM buoys.

2022, 43(3): 290-299. doi: 10.21656/1000-0887.420136
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Abstract:

Generally, the macro-scopic mechanical properties of heterogeneous composites depend on meso-components’ distribution and mechanical properties, but it is extremely difficult to establish a clear macro-meso relationship expression. To cope with this challenge, for concrete, a strategy based on deep learning was proposed to obtain the stress-strain curves through meso-model image information. First, the GoogLeNet model based on convolutional neural networks was used for image information recognition and extraction. According to the complexity of the stress-strain curve, data preprocessing operations were performed and the corresponding multi-task loss function was designed. The meso-model images in the data set were generated with the random aggregate model based on the Monte Carlo method, and numerical simulation experiments were conducted to obtain the uniaxial compressive stress-strain curve of the corresponding meso-model. Finally, the feasibility of the proposed method was evaluated through training and testing. The training efficiency and prediction accuracy of the GoogLeNet model are better than the AlexNet and ResNet models, and have good generalization ability and robustness.

2022, 43(3): 300-311. doi: 10.21656/1000-0887.420104
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Abstract:

With the continual increase of mining depths, the mechanical behaviors of deep rock masses present new forms and new characteristics. The whipping effect widely used in the construction industry is very similar to the partial dynamic responses of deep rock masses. Based on the structural characteristics, with the sandstone blocks as the research object, and the horizontal displacement and acceleration of the working block (horizontally impacted block) as the reference indicators, by experiments and through FLAC-3D numerical simulations, the influential mechanisms of the working block positions and sizes on the ultralow friction whipping effects were investigated. The work shows that, the intensity of the system’s ultralow friction whipping effect is closely related to the size of the working block. In the simulation, when the side length of the working block is 2/5 of that of the standard block (with a cube side length of 100 mm), the system structure will induce particularly severe ultralow friction whipping effects; within a certain range, the farther the working block position is from the disturbance source, the greater the intensity of the ultralow friction whipping effect will be. Beyond this range, a decreasing trend will occur, that is, the ultralow friction whipping effect intensity will increase first and then decrease with the distance between the working block and the source block.

2022, 43(3): 312-321. doi: 10.21656/1000-0887.420178
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Abstract:

The displacement pattern of wall footing rotation occurs for the backfill wall constrained at the bottom. The non-limit states of the soil layers at different depths are different in this pattern, and present difficulties for soil stress calculation. Based on the existing research, the functional relationship between soil strength parameters of the wall footing rotation pattern and wall displacements were detruded. Under the assumption that the backfill forms a circular arch and the slip surface is uncertain, the backfill was divided into long horizontal slices, the numerical iteration scheme for the nonlimit-state active earth pressure in the wall footing rotation pattern was constructed, and the numerical calculation method for the active earth pressure was given. The numerical method not only determines the slip surface shape, but calculates the intensity, the resultant force and the action point of the nonlimit-state active earth pressure. The backfill slip surface is a curved one and the new numerical solution is more consistent with the existing full-scale test results than the existing analytical results. This work provides more accurate numerical solutions of the nonlimit-state active earth pressure on the rigid retaining wall in the footing rotation pattern, and makes a practical guide to design of such retaining walls.

2022, 43(3): 322-329. doi: 10.21656/1000-0887.420275
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Abstract:

A new type of approximate subdifferential was proposed for quasiconvex functions. Their properties were studied, and the approximate subdifferential was applied to the characterization of approximate solutions to quasiconvex multiobjective optimization problems. Firstly, the existing approximate subdifferentials were improved to get a new approximate subdifferential of the quasiconvex function, and their relationships and properties were given. Then, the optimality conditions for approximate efficient solutions and approximate properly efficient solutions to quasiconvex multiobjective optimization problems were obtained by means of the new approximate subdifferential.

2022, 43(3): 330-340. doi: 10.21656/1000-0887.420210
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Abstract:

The PageRank algorithm has become the core technology for web search engines. For the linear equations derived from the PageRank problem, firstly, the restarted GMRES (generalized minimal residual) method of the Krylov subspace methods was combined with the multi-splitting iterative method, and a modified multi-splitting iterative method with the restarted GMRES was proposed. Then, the detailed calculating process and the convergence analysis of this new algorithm were given. Finally, the effectiveness of the algorithm was demonstrated through some numerical experiments.

2022, 43(3): 341-352. doi: 10.21656/1000-0887.420168
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Abstract:

The nonlinear time-fractional diffusion equations were considered in the 2D domain, and the physical information in initial state

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of the material was recovered from the measured data in the final state. This problem is seriously ill-posed, that is, the solution to this problem does not continuously depend on the measured data. Therefore, a variational regularization method was proposed to construct the approximate solution to the problem, and the convergence error estimates of the exact and approximate solutions were obtained under the assumption of the priori bounds on the exact solutions. Finally, a numerical example was given to verify the effectiveness of the proposed method.

2022, 43(3): 353-362. doi: 10.21656/1000-0887.420094
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Blow-up of solutions to semilinear Moore-Gibson-Thompson (MGT) equations with space-dependent coefficients and source terms was studied. Under subcritical conditions, through selection of suitable energy functionals and test functions, and with an iteration method and some differential inequality techniques, the nonexistence of global solutions to the Cauchy problem was obtained. Furthermore, the upper bound estimate of the solutions of the lifespan was derived.