A new similarity function for generalized trapezoidal fuzzy numbers

Numerous authors have proposed functions to quantify the degree of similarity between two fuzzy numbers using various descriptive parameters, such as the geometric distance, the distance between the centers of gravity or the perimeter. However, these similarity functions have drawback for specific situations. We propose a new similarity measure for generalized trapezoidal fuzzy numbers aimed at overcoming such drawbacks. This new measure accounts for the distance between the centers of gravity and the geometric distance but also incorporates a new term based on the shared area between the fuzzy numbers. The proposed measure is compared against other measures in the literature.

Searching for a consensus similarity function for generalized trapezoidal fuzzy numbers

There is controversy regarding the use of the similarity functions proposed in the literature to compare generalized trapezoidal fuzzy numbers since conflicting similarity values are sometimes output for the same pair of fuzzy numbers. In this paper we propose a similarity function aimed at establishing a consensus. It accounts for the different approaches of all the similarity functions. It also has better properties and can easily incorporate new parameters for future improvements. The analysis is carried out on the basis of a large and representative set of pairs of trapezoidal fuzzy numbers.

Cast-in-place 2-foot concrete trapezoidal flow-measuring flumes /

Issued Nov. 1977.

Determination of discharge coefficient for flow over one cycle compound trapezoidal plan form labyrinth weir

Weirs are common structure to regulate discharge and flow control inwater conveyance channels and hydraulic structures. Labyrinth weirsconsidered one of economical and effective methods to increase the efficiency of weirs that crest length of weirs increase without a related increase in structure width. Therefore, flow discharge will be increased. Compared to use compound weir, there are some benefits including the simultaneous passage of floating materials such as wood, ice, etc. Also, sedimentations are pass through compound weir. The trapezoidal Labyrinth weir is one of the combined models. In present study 15 physical models that discussion effect changes sidewall angle of labyrinth weir on discharge coefficient of flow over and through the compound trapezoidal one cycle Labyrinth weir. Also, it is developed design curves with various shapes and configurations. The researchshowed here mainly objectives at determining the coefficient of discharge for flow-over trapezoidal labyrinth weir by performing tests at wide range of values of side wall angles (α) from 6° to 35° and compound linear weir to be compared.

Computationally efficient numerical methods for time- and space-fractional Fokker–Planck equations

Fractional Fokker–Planck equations have been used to model several physical situations that present anomalous diffusion. In this paper, a class of time- and space-fractional Fokker–Planck equations (TSFFPE), which involve the Riemann–Liouville time-fractional derivative of order 1-α (α(0, 1)) and the Riesz space-fractional derivative (RSFD) of order μ(1, 2), are considered. The solution of TSFFPE is important for describing the competition between subdiffusion and Lévy flights. However, effective numerical methods for solving TSFFPE are still in their infancy. We present three computationally efficient numerical methods to deal with the RSFD, and approximate the Riemann–Liouville time-fractional derivative using the Grünwald method. The TSFFPE is then transformed into a system of ordinary differential equations (ODE), which is solved by the fractional implicit trapezoidal method (FITM). Finally, numerical results are given to demonstrate the effectiveness of these methods. These techniques can also be applied to solve other types of fractional partial differential equations.

Natural balancing of flying capacitor voltages in multicell inverter under PD carrier-based PWM

The flying capacitor multicell inverter (FCMI) possesses natural balancing property. With the phase-shifted (PS) carrier-based scheme, natural balancing can be achieved in a straightforward manner. However, to achieve natural balancing with the harmonically optimal phase-disposition (PD) carrierbased scheme, the conventional approaches require (n-1) x (n-1) trapezoidal carrier signals for an n-level inverter, which is (n-1) x (n-2) times more than that in the standard PD scheme. This paper proposes two improved natural balancing strategies for FMI under PD scheme, which use the same (n-1) carrier signals as used in the standard PD scheme. In the first scheme, on-line detection is performed of the band in which the modulation signal is located, corresponding period number of the carrier, and rising or falling half cycle of the carrier waveform to generate the switching signals based on certain rules. In the second strategy, the output voltage level selection is first processed and the switching signals are then generated according to a rule based on preferential cell selection algorithm. These methods are easy to use and can be simply implemented as compared to the other available methods. Simulation and experimental results are presented for a five-level inverter to verify these proposed schemes.

Effect of overload cycles on thin steel roof claddings during cyclonic winds

During an investigation on thin steel roof claddings under simulated cyclonic wind loading, it was found that trapezoidal roof claddings behaved quite differently to corrugated (arc and tangent type) roof claddings due to the presence of overload cycles. The overload cycles caused a reduction in fatigue life for corrugated roofing whereas the reverse occurred for trapezoidal roofing. This contrasting behavior of the two crest-fixed roof claddings was investigated using small scale roofing models instead of the commonly used large scale two-span roof claddings. It was found that overload cycles formed a weaker locally dimpled mechanism around the fastener holes of corrugated roofing and thus accelerated the fatigue-caused pull-through failure. In contrast, a stronger deformed shape was formed in trapezoidal roofing which delayed the pull-through failure. Both laboratory testing and finite element analysis of small scale models were used to study the contrasting behavior of roof claddings.

Shear strength and stiffness of crest-fixed steel claddings

Diaphragm action of crest-fixed profiled steel claddings is present in low-rise buildings whether the designer acknowledges it or not. For the designers to take advantage of the diaphragm strength of the crest-fixed steel claddings in the design of low-rise buildings in a similar manner to valley-fixed claddings, and to design the buildings based on the true behaviour rather than the assumed behaviour, shear/racking behaviour of the three trapezoidal and corrugated steel claddings commonly used at present was investigated using large scale experiments. Crest-fixed claddings (up to a maximum size of 6 x 6.2m) with different aspect ratio and fastening systems were tested to failure, based on which suitable shear strength and stiffness values have been proposed for these claddings as they are used at present. A simple analytical model combined with basic connection data from small scale experiments was used to predict the shear strength of tested panels. Currently attempts are being made to develop general design formulae to determine shear strength and stiffness of crest-fixed steel claddings...

Stochastic linear multistep methods for the simulation of chemical kinetics

In this paper, we introduce the Stochastic Adams-Bashforth (SAB) and Stochastic Adams-Moulton (SAM) methods as an extension of the tau-leaping framework to past information. Using the theta-trapezoidal tau-leap method of weak order two as a starting procedure, we show that the k-step SAB method with k >= 3 is order three in the mean and correlation, while a predictor-corrector implementation of the SAM method is weak order three in the mean but only order one in the correlation. These convergence results have been derived analytically for linear problems and successfully tested numerically for both linear and non-linear systems. A series of additional examples have been implemented in order to demonstrate the efficacy of this approach.

Influence of External Light Waves on the Thermoelectric Power Under Strong Magnetic Field in Ultrathin Films, Quantum Wires and Quantum Dots of Optoelectronic Materials

In this paper, an attempt is made to study the influence of external light waves on the thermoelectric power under strong magnetic field (TPSM) in ultrathin films (UFs), quantum wires (QWs) and quantum dots (QDs) of optoelectronic materials whose unperturbed dispersion relation of the conduction electrons are defined by three and two band models of Kane together with parabolic energy bands on the basis of newly formulated electron dispersion laws in each case. We have plotted the TPSM as functions of film thickness, electron concentration, light intensity and wavelength for UFs, QWs and ODs of InSb, GaAs, Hg1-xCdxTe and In1-xGaxAsyP1-y respectively. It appears from the figures that for UFs, the TPSM increases with increasing thickness in quantum steps, decreases with increasing electron degeneracy exhibiting entirely different types of oscillations and changes with both light intensity and wavelength and these two latter types of plots are the direct signature of light waves on opto-TPSM. For QWs, the opto-TPSM exhibits rectangular oscillations with increasing thickness and shows enhanced spiky oscillations with electron concentration per unit length. For QDs, the opto-TPSM increases with increasing film thickness exhibiting trapezoidal variations which occurs during quantum jumps and the length and breadth of the trapezoids are totally dependent on energy band constants. Under the condition of non-degeneracy, the results of opto-TPSM gets simplified into the well-known form of classical TPSM equation which the function of three constants only and being invariant of the signature of band structure.

Practical constant-accuracy linear weir

This paper is concerned with the modifications of the Extended Bellmouth Weir (EBM weir) earlier designed by Keshava Murthy. It is shown that by providing inclined sides (equivalent to providing an inward-trapezoidal weir) over a sector of a circle of radius R, separated by a distance 2t, and depth d, the measurable range of EBM can be considerably enhanced (over 375%). Simultaneously, the other parameters of the weir are optimized such that the reference plane of the weir coincides with its crest making it a constant-accuracy linear weir. Discharge through the aforementioned weir is proportional to the depths of flow measured above the crest of the weir for all heads in the range of 0.5R less-than-or-equal-to h less-than-or-equal-to 7.9R, within a maximum deviation of +/-1% from the theoretical discharge. Experiments with two typical weirs show excellent agreement with the theory by giving a constant-average coefficient of discharge of 0.619

On the dual head-discharge characteristics of a modified chimney weir

This paper is concerned with the dual head-discharge characteristics of a modified chimney weir. It is shown by an optimization procedure that the modified chimney weir having an inward trapezoidal weir over an inverted V-notch can produce discharges proportional to both the logarithm of the head as well as the linear power of the head reckoned independently over two different reference planes, within a prefixed maximum allowable percentage error from the theoretical discharge. A new technique is adopted to optimize the weir parameters, in order to obtain the maximum ranges of measurement under logarithmic, linear as well as combined characteristics. In the case of linear weir it is shown that it is possible to enhance the linearity range of the chimney weir by more than 540% and for a weir with constant indication accuracy by more than 350%. In addition, about 86% of the overall depth of the designed linear weir and over 90% of the logarithmic weir is converted as the corresponding measurable ranges. Experiments with four typical weirs give consistent constant average coefficient of discharge for each weir confirming the theory. The practical application of the weir in minor irrigation, hydraulic and other engineering fields is highlighted.

Geometrically Simple Logarithmic Weir

This paper discusses the design and experimental verification of a geometrically simple logarithmic weir. The weir consists of an inward trapezoidal weir of slope 1 horizontal to n vertical, or 1 in n, over two sectors of a circle of radius R and depth d, separated by a distance 2t. The weir parameters are optimized using a numerical optimization algorithm. The discharge through this weir is proportional to the logarithm of head measured above a fixed reference plane for all heads in the range 0.23R less than or equal to h less than or equal to 3.65R within a maximum deviation of +/-2% from the theoretical discharge. Experiments with two weirs show excellent agreement with the theory by giving a constant average coefficient of discharge of 0.62. The application of this weir to the field of irrigation, environmental, and chemical engineering is highlighted.