Global Multi-resolution Half-Honeycomb Trapezoid Grid

In order to break through the limitationof thelatitude/longitudegrid and hexagon grid, a new subdivision unit, Half-honeycomb Trapezoid, is proposed. Based on the summarization of the geometric properties and subdivision performance of Half-honeycomb Trapezoid, a new discrete global topographic grid system is established, and its compatibility with hexagonal grid is analyzed. At last, the visualization of multi-resolution global grid is achieved.

A Generalized Array Factor for Time-Modulated Hexagonal Based Antenna Array Geometry With Novel Trapezoidal Switching

The concept of the time-modulated array has been emerging as an alternative to the complex phase shifters,which lowers the cost of the array feeding network due to the utilization of radio frequency(RF)switches.The various forms of hexagonal antenna array geometries can be used for applications like surveillance tracking in phased array radar and wireless communication systems.This work proposes the generalized array factor(AF)for the hexagonal antenna array geometry based on time modulation.The time modulation in generalized hexagonal geometry can maintain the fixed static amplitude excitation,giving more flexibility over time.Furthermore,a novel trapezoidal switching function is also proposed and applied to the generalized array factor to enable future researchers to use this array factor in the field of advancement to observe how switching schemes like trapezoidal and rectangular affect the array pattern's side lobe level(SLL).The generalized equation can be utilized for the analysis and synthesis of radiation characteristics of the time-modulated hexagonal array(TMHA),time-modulated concentric hexagonal array(TMCHA),time-modulated hexagonal cylindrical array(TMHCA),and time-modulated hexagonal concentric cylindrical array(TMHCCA).The numerical result illustrates the generation of AF of time-modulated hexagonal structures and also shows that the trapezoidal switching sequence outperforms the rectangular switch using the cat swarm optimization(CSO)approach.

Composite Fractional Trapezoidal Rule with Romberg Integration

The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator.This would be achieved by first establishing a fractional-order version of the 2-point Trapezoidal rule and then by proposing another fractional-order version of the(n+1)-composite Trapezoidal rule.In particular,the so-called divided-difference formula is typically employed to derive the 2-point Trapezoidal rule,which has accordingly been used to derive a more accurate fractional-order formula called the(n+1)-composite Trapezoidal rule.Additionally,in order to increase the accuracy of the proposed approximations by reducing the true errors,we incorporate the so-called Romberg integration,which is an extrapolation formula of the Trapezoidal rule for integration,into our proposed approaches.Several numerical examples are provided and compared with a modern definition of the Riemann-Liouville fractional integral operator to illustrate the efficacy of our scheme.

Study of a Hydraulic Jump in an Asymmetric Trapezoidal Channel with Different Sluice Gates

In this study,the main properties of the hydraulic jump in an asymmetric trapezoidal flume are analyzed experimentally,including the so-called sequent depths,characteristic lengths,and efficiency.In particular,an asymmetric trapezoidal flume with a length of 7 m and a width of 0.304 m is considered,with the bottom of the flume transversely inclined at an angle of m=0.296 and vertical lateral sides.The corresponding inflow Froude number is allowed to range in the interval(1.40<F1<6.11).The properties of this jump are compared to those of hydraulic jumps in channels with other types of cross-sections.A relationship for calculating hydraulic jump efficiency is proposed for the considered flume.For F1>5,the hydraulic jump is found to be more effective than that occurring in triangular and symmetric trapezoidal channels.Also,when■mes>8 and■>5,the hydraulic jump in the asymmetrical trapezoidal channel downstream of a parallelogram sluice gate is completely formed as opposed to the situation where a triangular sluice is considered.

Sliding behaviors of the trapezoidal roof rock block under a lateral dynamic disturbance

The surrounding rock of underground space is always affected by external dynamic disturbance from the side position,such as blasting vibration from a stope at the same level or seismic waves from adjacent strata.A series of laboratory tests,numerical simulations and theoretical analyses were carried out in this study to disclose the sliding mechanism of roof rock blocks under lateral disturbance.Firstly,the experiments on trapezoidal key block under various clamping loads and disturbance were conducted,followed by numerical simulations using the fast Lagrangian analysis of continua(FLAC3D).Then,based on the conventional wave propagation model and the classical shear-slip constitutive model,a theoretical model was proposed to capture the relative displacement between blocks and the sliding displacement of the key block.The results indicate that the sliding displacement of the key block increased linearly with the disturbance energy and decreased exponentially with the clamping load when the key block was disturbed to slide(without instability).Meanwhile,when the key block was disturbed to fall,two types of instability process may appear as immediate type or delayed type.In addition,the propagation of stress waves in the block system exhibited obvious low-velocity and lowfrequency characteristics,resulting in the friction reduction effect appearing at the contact interface,which is the essential reason for the sliding of rock blocks.The results can be applied to practical underground engineering and provide valuable guidance for the early detection and prevention of rockfalling disasters.

Multiple attribute decision making method based on trapezoid fuzzy linguistic variables

The problem of multiple attribute decision making under fuzzy linguistic environments, in which decision makers can only provide their preferences （attribute values）in the form of trapezoid fuzzy linguistic variables（TFLV）, is studied. The formula of the degree of possibility between two TFLVs is defined, and some of its characteristics are studied. Based on the degree of possibility of fuzzy linguistic variables, an approach to ranking the decision alternatives in multiple attribute decision making with TFLV is developed. The trapezoid fuzzy linguistic weighted averaging （TFLWA） operator method is utilized to aggregate the decision information, and then all the alternatives are ranked by comparing the degree of possibility of TFLV. The method can carry out linguistic computation processes easily without loss of linguistic information, and thus makes the decision results reasonable and effective. Finally, the implementation process of the proposed method is illustrated and analyzed by a practical example.

Aggregation operators on intuitionistic trapezoidal fuzzy number and its application to multi-criteria decision making problems

Intuitionistic trapezoidal fuzzy numbers and their operational laws are defined. Based on these operational laws, some aggregation operators, including intuitionistic trapezoidal fuzzy weighted arithmetic averaging operator and weighted geometric averaging operator are proposed. Expected values, score function, and accuracy function of intuitionitsic trapezoidal fuzzy numbers are defined. Based on these, a kind of intuitionistic trapezoidal fuzzy multi-criteria decision making method is proposed. By using these aggregation operators, criteria values are aggregated and integrated intuitionistic trapezoidal fuzzy numbers of alternatives are attained. By comparing score function and accuracy function values of integrated fuzzy numbers, a ranking of the whole alternative set can be attained. An example is given to show the feasibility and availability of the method.

Experimental Study of Wave Attenuation in Trapezoidal Floating Breakwaters

A comprehensive experimental study was carried out on the regular wave attenuation with a trapezoidal pontoontype floating breakwater(FB) in deep water. The functionalities of two simple FB geometries consist of a rectangle and a trapezoid with the slope of 60° were investigated under the wave attack. A two-dimensional wave flume was used in the experiment; the incident, transmitted waves, mooring line forces and motion responses of the floating breakwaters were measured. Also the influence of the sea state conditions(incident wave height and wave period)and structural parameters(draught of the structure) were investigated using the trapezoidal FB. Our experimental results indicated that the trapezoidal FB significantly reduced the wave transmission and mooring line force when compared with rectangular FBs. A new formula was developed in order to predict the value of the transmission coefficient in trapezoidal FBs with the slope of 60°. Experimental data showed to be consistent with the results of the formula.

Effect of trapezoidal collars as a scour countermeasure around wing-wall abutments

Local scour around bridge abutments is a widespread problem that can result in structural failure. Collars can be used as a countermeasure to reduce the scour depth. In this study, the temporal scour development around a wing-wall abutment was investigated with and without collars. The tests were carried out under clear-water conditions for different abutment lengths, with collars of different sizes placed at the bed level. When no collar was used in the experiments, 70% of the maximum scour depth occurred in less than 2 h. However, when a collar with a width greater than the length of the abutment was used, no scour was observed for up to 200 min from the beginning of the experiments. The results show that an increase in the collar width not only led to a lag time for the onset of scouring but also reduced the maximum scour depth. Moreover, an increased collar width led to a better performance in mitigating scouring around smaller abutments. Generally, the scour depth decreased by 9%-37% with different collar widths.

Replacement of annular domain with trapezoidal domain in computational modeling of nonaqueous-phase-liquid dissolution-front propagation problems

In order to simulate the instability phenomenon of a nonaqueous phase liquid(NAPL) dissolution front in a computational model, the intrinsic characteristic length is commonly used to determine the length scale at which the instability of the NAPL dissolution front can be initiated. This will require a huge number of finite elements if a whole NAPL dissolution system is simulated in the computational model. Even though modern supercomputers might be used to tackle this kind of NAPL dissolution problem, it can become prohibitive for commonly-used personal computers to do so. The main purpose of this work is to investigate whether or not the whole NAPL dissolution system of an annular domain can be replaced by a trapezoidal domain, so as to greatly reduce the requirements for computer efforts. The related simulation results have demonstrated that when the NAPL dissolution system under consideration is in a subcritical state, if the dissolution pattern around the entrance of an annulus domain is of interest, then a trapezoidal domain cannot be used to replace an annular domain in the computational simulation of the NAPL dissolution system.However, if the dissolution pattern away from the vicinity of the entrance of an annulus domain is of interest, then a trapezoidal domain can be used to replace an annular domain in the computational simulation of the NAPL dissolution system. When the NAPL dissolution system under consideration is in a supercritical state, a trapezoidal domain cannot be used to replace an annular domain in the computational simulation of the NAPL dissolution system.

Characteristics from a hydrodynamic model of a trapezoidal artificial reef

Flume experiments and numerical simulation were conducted to characterize the hydrodynamics of a trapezoid artificial reef.Measurements in particle image velocimetry were conducted to observe the formation of upwelling and vortices;and forces for the reef model were measured by load cell.The results of flume experiments agree well with the numerical data.In addition,the flow structure around a reef combining trapezoidal and cubic blocks was simulated numerically under two deployment schemes,showing a more complicated flow structure than that of a stand-alone reef.Relationship between drag coefficient and Reynolds number suggest that the degree of turbulence can be assessed from the value of drag coefficient downstream from the reef.The role of the reef in water flow is to reduce flow velocity and generate turbulence.

The Application of Trapezoidal Fuzzy AHP on Evaluation Item Weight Calculation of Highway Network Programming Schemes

The fuzzy theory is invited in this paper to calculate weights of evaluation items, which is a very important stage in highway network evaluation. Highway network evaluation is a composite of various attributes that lack of pertinency and comparability, and their commonality is that they are more or less tainted with uncertainty, with different daily decision making problems of diverse intensity, the results can be misleading if the fuzziness of human decision-making is not taken into account. So we introduce fuzzy theory into the measurement of evaluating weights to overcome the issue and Analytic Hierarchy Process in scheme ranking. It is testified to be a more reliable way through the case study of Yangtze River Delta highway network evaluation.

Microstructure and Mechanical Strength Predictive Modeling in Al 5052-Trapezoidal Grooved SS 304 Explosive Cladding

Aluminum alloy plates were explosively cladded to stainless steel plates with trapezoidal grooves on the mating surface.The process parameters viz,loading ratio,standoff distance and flyer plate thickness were varied based on the Taguchi analogy.The variation in the process parameters alters the kinetic energy dissipation and the deformation work performed at the interface,and dictates the interfacial wave amplitude and the mechanical strength of the dissimilar explosive clad.The optimum level of process parameters for attaining higher tensile and shear strength is computed by signal-to-noise ratio.Further,a mathematical model is developed for calculating tensile and shear strength of the clad,based on the regression analysis using statistical software Minitab-16,and the level of fit is determined by analysis of variance.

Control over Surrounding Rocks Deformation of Soft Floor and Whole-Coal Gateways with Trapezoidal Supports

In Gengcun Colliery, Yima Coal Group Co. Ltd.the characteristics of the gateways of thick coal seam and the coal seam is with fully mechanized sublevel caving mining are that the thickness of roof coal seam of gateways is larger, their surrounding rocks are the whole-coal mass and the coal seam is prone to Spontaneous Combustion. With the natural equilibrium arch theory, the reasonable adjacent distance of No.11 mine-type metal supports was calculated in trapezoidal gateways based on these characteristics. Then, in-situ supporting experiments were carried out. The results indicate that under the action of virgin rock stress, the width of broken rocks zone of surrounding rocks is 1.7–2.0 m in return heading and 1.1–1.3 m in going headway. And their surrounding rocks belong to the Ⅳ-type soften rock and the Ⅲ-type common surrounding rock respectively. Therefore, under the movable abutment pressure, the gateways deformation is serious. It is suggested that the designed gateways have to use pre-broadened cross section to suit their deformation. At the same time, the accumulated water on gateway floor must be drained in time. These measures were taken in the 1302 and 1304 coal faces in Gengcun Colliery, and the satisfactory results have been obtained.

Effect of loading rates on crack propagating speed,fracture toughness and energy release rate using single-cleavage trapezoidal open specimen under impact loads

The former studies indicate that loading rates significantly affect dynamic behavior of brittle materials,for instance,the dynamic compressive and tensile strength increase with loading rates.However,there still are many unknown or partially unknown aspects.For example,whether loading rates have effect on crack dynamic propagating behavior(propagation toughness,velocity and arrest,etc).To further explore the effect of loading rates on crack dynamic responses,a large-size single-cleavage trapezoidal open(SCTO)specimen was proposed,and impacting tests using the SCTO specimen under drop plate impact were conducted.Crack propagation gauges(CPGs)were employed in measuring impact loads,crack propagation time and velocities.In order to verify the testing result,the corresponding numerical model was established using explicit dynamic software AUTODYN,and the simulation result is basically consistent with the experimental results.The ABAQUS software was used to calculate the dynamic SIFs.The universal function was calculated by fractal method.The experimental-numerical method was employed in determining initiation toughness and propagation toughness.The results indicate that crack propagating velocities,dynamic fracture toughness and energy release rates increase with loading rates;crack delayed initiation time decreases with loading rates.

Positioning-control Based on Trapezoidal Velocity Curve for High-precision Basis Weight Control Valve

Traditionally, basis weight control valve is driven by a constant frequency pulse signal. Therefore, it is difficult for the valve to match the control precision of basis weight. Dynamic simulation research using Matlab/Simulink indicates that there is much more overshoot and fluctuating during the valve-positioning process. In order to improve the valve-positioning precision, the control method of trapezoidal velocity curve was studied. The simulation result showed that the positioning steady-state error was less than 0.0056%, whereas the peak error was less than 0.016% by using trapezoidal velocity curve at 10 positioning steps. A valve-positioning precision experimental device for the stepper motor of basis weight control valve was developed. The experiment results showed that the error ratio of 1/10000 positioning steps was 4% by using trapezoidal velocity curve. Furthermore, the error ratio of 10/10000 positioning steps was 0.5%. It proved that the valve-positioning precision of trapezoidal velocity curve was much higher than that of the constant frequency pulse signal control strategy. The new control method of trapezoidal velocity curve can satisfy the precision requirement of 10000 steps.

Wave Attenuation Properties of Double Trapezoidal Submerged Breakwaters on Flat-Bed

This paper investigates the wave attenuation properties of the double trapezoidal submerged breakwaters on the flat-bed by conducting physical experiments subjected to linear and cnoidal incident waves.The method of Goda's two points is used to separate the heights of incident,reflected and transmitted waves based on the experimental data.The possible factors affecting the wave attenuation properties of the double trapezoidal submerged breakwaters(i.e.,the relative submerged water depth,relative breakwater spacing,wave steepness and relative wave height) are investigated with respect to the reflection and transmission coefficients.The results show that there is a range,within which the breakwater spacing has little impact on the reflection coefficient,and the transmission coefficient tends to be a constant.The influence of the wave steepness is reduced while the breakwater spacing is too large or too small.Within the range of the relative wave height tested in this study,the reflection and transmission coefficients increase and decrease with the relative wave height,respectively.The double trapezoidal submerged breakwaters model indicates a good attenuation effect for larger wave steepness,big relative wave height and within the range of the relative breakwater spacing between 12.5 and 14 according to linear and cnoidal waves.The changes of wave energy spectra between the double submerged breakwaters on the flat-bed are investigated by the fast Fourier transform(FFT) method,showing that wave energy dissipation can be reached more effectively when the relative breakwater spacing is 12.5.

Spectral redshift of high-order harmonics by adding a weak pulse in the falling part of the trapezoidal laser pulse

We investigate the spectral redshift of high-order harmonics of the H_2~＋（D_2~＋） molecule by numerically solving the non-Born–Oppenheimer time-dependent Schr ¨odinger equation（TDSE）. The results show that the spectral redshift of highorder harmonics can be observed by adding a weak pulse in the falling part of the trapezoidal laser pulses. Comparing with the H_2~＋ molecule, the shift of high-order harmonic generation（HHG） spectrum for the D_2~＋ molecule is more obvious.We employ the spatial distribution in HHG and time-frequency analysis to illustrate the physical mechanism of the spectral redshift of high-order harmonics.

Scattering loss in optical waveguide with trapezoidal cross section

To get the scattering loss of the trapezoidal core waveguide,a new analysis model is presented based on the perturbation equivalent method and modified effective-index method.Firstly,the trapezoidal core waveguide is successfully equivalent to the rectangular one with both restricting the same optical field energy by adopting the perturbation method,Then,the equivalent rectangular core waveguide is decomposed into two slab waveguides by employing the modified effective-index method,The trapezoidal core waveguide scattering theory model is established based on the slab waveguide scattering theory.With the sidewalls surface roughness in the range from 0 to 100 nm in the single model trapezodial core waveguide,optical simulation shows excellent agreement with the results from the scattering loss model presented.The relationship between the dimension and side-wall roughness with the scattering loss can be determined in the trapezoidal core waveguide by the scattering loss model.

Finite Element Analysis of Magnetohydrodynamic Mixed Convection in a Lid-Driven Trapezoidal Enclosure Having Heated Triangular Block

A numerical research on magnetohydrodynamic mixed convection flow in a lid-driven trapezoidal enclosure at non-uniform heating of bottom wall has been studied numerically. The enclosure consists of insulated top wall and cold side walls, too. It also contains a heated triangular block (Rot = 0° - 90°) located somewhere inside the enclosure. The boundary top wall of the enclosure is moving through uniform speed U0. The geometry of the model has been represented mathematically by coupled governing equations in accordance with proper boundary conditions and then a two-dimensional Galerkin finite element based numerical approach has been adopted to solve this paper. The numerical computations have been carried out for the wide range of parameters Prandtl number (0.5 ≤ Pr ≤ 2), Reynolds number (60 ≤ Re ≤ 120), Rayleigh number (Ra = 103) and Hartmann number (Ha = 20) taking with different rotations of heated triangular block. The results have been shown in the form of streamlines, temperature patterns or isotherms, average Nusselt number and average bulk temperature of the fluid in the enclosure at non-uniform heating of bottom wall. It is also indicated that both the streamlines, isotherm patterns strongly depend on the aforesaid governing parameters and location of the triangular block but the thermal conductivity of the triangular block has a noteworthy role on the isotherm pattern lines. Moreover, the variation of Nuav of hot bottom wall and θav in the enclosure is demonstrated here to show the characteristics of heat transfer in the enclosure.