Simple Approximate Method Research Articles

On the Approximation of the UWB Dipole Elliptical Arms With Stepped-Edge Polygon

A simple method of approximation of the elliptical patch with stepped-edge polygon is proposed as an introduction to wider studies over the planar ultrawideband (UWB) antennas. The general idea is to replace the elliptical patch with an equivalent polygonal patch, with minimum loss in the performance. The principles of the proposed method are presented in this letter, as well as the results of performed numerical studies and its experimental verification. A satisfying level of agreement of the parameters of the original antenna and its stepped-edge polygon version indicates that the proposed approximation can be applied without important loss of antenna performance.

REDUCED TRANSFER MATRIX APPROACH FOR ISING MODEL

In this work, we present a simple approximate transfer matrix method for 2D and 3D Ising ferromagnet to calculate spontaneous magnetization of the system. The critical coupling strength Kc of 2D and 3D Ising models in reduced transfer matrix approximation is obtained quite accurately by simple improvements over the mean field theory. The important physical effect we include is the some of the fluctuations effects of the systems directly with help of this method. We predict from the spontaneous magnetization curve that the critical coupling strength Kc=J/kBT = 0.401 and 0.245 for two-dimensional (2D) and three-dimensional (3D) systems, respectively.

Efficient Approximation Using Probabilistically Improved Combinatorial Structure of Bernstein's Polynomial Operator's Weights through the Fusion of Dual-Perspectives

A new polynomial approximation operator has been proposed which uses weight-functions of the well-known Bernstein’s Polynomial operator in its probabilistically improved combinatorial structure, achieved through a rather-ingenious ‘Fusion’ of two dual perspectives. These weights are functions of the impugned variable of the unknown function being approximated, and are not mere constants. The new approximation formula has been compared empirically with the simple classical method of polynomial approximation using the well-known “Bernstein Operator”. The percentage absolute relative errors for the proposed approximation formula and that with the “Bernstein Operator” have been computed for certain selected functions and with different number of node points in the interval of approximation. It has been observed that the proposed approximation formula produces exceedingly-significantly better results.

Is computational complexity a barrier to manipulation?

When agents are acting together, they may need a simple mechanism to decide on joint actions. One possibility is to have the agents express their preferences in the form of a ballot and use a voting rule to decide the winning action(s). Unfortunately, agents may try to manipulate such an election by mis-reporting their preferences. Fortunately, it has been shown that it is NP-hard to compute how to manipulate a number of different voting rules. However, NP-hardness only bounds the worst-case complexity. In this survey article, we summarize the evidence for and against computational complexity being a barrier to manipulation. We look both at techniques identified to increase complexity (for example, hybridizing together two or more voting rules), as well as other features that may change the computational complexity of computing a manipulation (for example, if votes are restricted to be single peaked then some of the complexity barriers fall away). We discuss recent theoretical results that consider the average case, as well as simple greedy and approximate methods. We also describe how computational “phase transitions”, which have been fruitful in identifying hard instances of propositional satisfiability and other NP-hard problems, have provided insight into the hardness of manipulating voting rules in practice. Finally, we consider manipulation of other related problems like stable marriage and tournament problems.

An Integer Approximation Method for Discrete Sinusoidal Transforms

Approximate methods have been considered as a means to the evaluation of discrete transforms. In this work, we propose and analyze a class of integer transforms for the discrete Fourier, Hartley, and cosine transforms (DFT, DHT, and DCT), based on simple dyadic rational approximation methods. The introduced method is general, applicable to several block-lengths, whereas existing approaches are usually dedicated to specific transform sizes. The suggested approximate transforms enjoy low multiplicative complexity and the orthogonality property is achievable via matrix polar decomposition. We show that the obtained transforms are competitive with archived methods in literature. New 8-point square wave approximate transforms for the DFT, DHT, and DCT are also introduced as particular cases of the introduced methodology.

Approximate up-scaling of geo-spatial variables applied to deep foundation design

We present a series of simple approximate methods for up-scaling the cumulative distribution function of spatially correlated variables by using an effective number n e of independent variables. Methods are based on the property of distribution permanence of the gamma and inverse Gaussian distributions under averaging, bootstrap sampling and expansions about the normal and gamma distributions. A stochastic simulation study is used to validate each method, and simple parameters are defined to identify respective ranges of applicability. A practical example is presented where core sample rock strength data are up-scaled to shaft size for probabilistic (risk-based) deep foundation design. Supplemental material is available online.

Quantum search in a possible three-dimensional complex subspace

Suppose we are given an unsorted database with N items and N is sufficiently large. By using a simpler approximate method, we re-derive the approximate formula cos2 ?, which represents the maximum success probability of Grover's algorithm corresponding to the case of identical rotation angles $${\phi=\theta}$$ for any fixed deflection angle $${\Phi \in\left[0,\pi/2\right)}$$ . We further show that for any fixed $${\Phi \in\left[0,\pi/2\right)}$$ , the case of identical rotation angles $${\phi=\theta}$$ is energetically favorable compared to the case $${\left|{\theta - \phi}\right|\gg 0}$$ for enhancing the probability of measuring a unique desired state.

COUNTRY PORTFOLIOS IN OPEN ECONOMY MACRO-MODELS

This paper develops a simple approximation method for computing equilibrium portfolios in dynamic general equilibrium open economy macro models. The method is widely applicable, simple to implement, and gives analytical solutions for equilibrium portfolio positions in any combination or types of asset. It can be used in models with any number of assets, whether markets are complete or incomplete, and can be applied to stochastic dynamic general equilibrium models of any dimension, so long as the model is amenable to a solution using standard approximation methods. We first illustrate the approach using a simple two-asset endowment economy model, and then show how the results extend to the case of any number of assets and general economic structure.

A simple approximate method for free vibration analysis of framed tube structures

SUMMARYA simple approximate method is developed for determining natural frequency of tall buildings. Timoshenko's beam model, which considers the influence of shear and flexural deformation, was used in modelling framed tube structures. In this paper, natural frequency and mode shape of framed tube structures were calculated based on the flexural and shear rigidities along with the effects of rotational inertia. Dynamic model of Timoshenko's beam can be obtained by writing equilibrium equations of forces acting on an infinite element. The solution of the dynamic model was obtained by first applying the separation of variables to time and space, followed by the assumption of harmonic motion in time, the steady state eigen system was obtained. Natural frequencies of framed tube structures were calculated by solving the eigenproblem. A numerical example has been presented to demonstrate the ease of application and accuracy of the proposed method. The structure's fundamental frequency was computed using ETABS V9.0.0 (Computer and Structures, Berkeley, California, USA) and compared with the result obtained from proposed method, which shows that the percentage of error was low and acceptable. The proposed method can be adopted as an alternative procedure to evaluate the natural frequency of framed tube in the preliminary stages of structural design. Copyright © 2010 John Wiley & Sons, Ltd.

Tolerance Intervals for the Distribution of the Difference Between Two Independent Normal Random Variables

The problem of constructing tolerance interval for the distribution of the difference between two independent normal random variables possibly with different variances is considered. Assuming that the variance ratio is known, an exact method and a simple approximate method for constructing tolerance factor are presented. Also, approximate methods for obtaining tolerance interval are provided when the variances are unknown and arbitrary. The coverage probabilities of the approximate methods and an available approximate method are evaluated via Monte Carlo simulation study. Simulation study indicates that the approximate tolerance intervals are quite satisfactory even when the sample sizes are small whereas the existing method is conservative for small to moderate samples. The proposed tolerance interval can be used safely for practical applications. The methods are illustrated using an example.

Freeform surface design for a light-emitting diode–based collimating lens

A freeform collimating lens is designed to project light rays emitted from an LED light source to a far target plane. Generally, the projection distance is assumed to be more than 100 m, and the light beam to have negligible divergence. The lens consists of a total internal reflection (TIR) side surface, a spherical surface in the rear, a vertical plane surface in the outer part of the front, and a freeform refractive surface in the central part of the front. Light rays emitted from the LED source with large spread angles hit the TIR surface and are redirected parallel to the light axis, and those having small spread angles will be collimated by the freeform refractive surface, which is designed with a simple approximation method, and then travel parallel to the light axis. Computer simulation results show that an optical efficiency of 81.5% is achieved under a view angle of ±5 deg and for a 1 mm×1 mm LED source.

Displacement-based seismic design of braced steel structures

This paper discusses concepts and presents procedures for the development of a displacement-based seismic design methodology for chevron bracing. Both theoretically rigorous and simplified approximate methods for calculating frame displacements corresponding to significant limit states are proposed. In other words, analytical formulations are proposed for the calculation of inter-storey drift angles and associated displacements corresponding to brace buckling and the achievement of a given brace ductility demand. The aim of the displacement-based design (DBD) methodology developed is to control both elastic (pre-yield) and inelastic (post-yield) deformation modes, thus permitting control of damage distribution within the structure. The results gained from the application of the proposed methodology to a case study are then given. The structure designed was studied using dynamic time-history response analyses. The numerical results illustrate the good performance of the DBD-compliant structure which matches the performance objectives set at the design stage.

A Probabilistic Lifetime Prediction Technique for Piping Under Creep Conditions

We outline here a simple approximate method of lifetime prediction for high-temperature, internally pressurized piping components that takes into account the very large scatter present in the creep data for commonly used piping materials. This scatter is so large as to make deterministic methods of life prediction quite problematic. The method presented here is based upon the well-known Monte Carlo technique, and makes use of the standard damage fraction as the basic measure of creep damage. The method yields predictions of the probability of failure after a fixed operating time, assumed to be large.

Three dimensional stress analysis of a composite patch using stress functions

A stress function-based analysis is proposed to provide a simple and efficient approximation method of three-dimensional (3D) state of stress that exists near the free edge of bonded composite patches. In order to apply plane strain assumption in a composite patch, a linear superposition of sliced section from a bonded patch is used. In addition, to describe the load transfer mechanism from the substrate to the composite patch, a simple shear lag model is introduced. The 3D stress behavior at the free edge of the composite patch is modeled by Lekhnitskii stress functions, and the governing equations of the given composite patch are obtained by applying the principle of complementary virtual work. After a suitable expansion of the functions, the governing equations are transformed into two coupled ordinary differential equations, and they are solved by a general eigenvalue solution procedure. As the number of base functions increases, the interlaminar stresses converge. The interlaminar stresses reach maximum at the free edge and decrease sharply at the inner part of the patch. The interlaminar stresses are concentrated at the interface between the layers because of the mismatch of material properties and the geometric singularity. Since the proposed method accurately predicts the 3D stresses in a composite patch bonded on the metal substrate, it can be used as a simple and efficient analytical tool for designing such structural components.

Nonequivalent Milling of Roller Gear Cams Based on Approximation Method of Offset Surface

The roller gear cam surface is manufactured with nonequivalent milling method in many cases, by which the machining error is unavoidable. In order to simplify the tool position, we can transform one problem approximating the designed surface with the tool envelope surface to another problem approaching the theoretical tool axis trajectory surface with the actual one. Furthermore, the theoretical tool axis trajectory surface, which is an offset surface of the designed cam surface, is reconstructed by the NURBS ruled surface. In order to find the best tool axis vector to minimize the machining error, a simple least square approximation method is established to figure out all the control points of the NURBS tool axis trajectory surface. A numerical calculation and simulation example is described to verify the effectiveness of the tool position method proposed in the paper.

An improvement to linear‐elastic procedures for seismic performance assessment

AbstractAn improved linear‐elastic analysis procedure is developed in this paper as a simple approximate method for displacement‐based seismic assessment of the existing buildings. The procedure is mainly based on reducing the stiffness of structural members that are expected to respond in the inelastic range in a single global iteration step. Modal spectral displacement demands are determined from the equal displacement rule. Response predictions obtained from the proposed procedure are evaluated comparatively by using the results of benchmark nonlinear response history analysis, and both the conventional and the multi‐mode pushover analyses. In comparative evaluations, a twelve‐story RC plane frame and a six‐story unsymmetrical‐plan RC frame are employed by using 91 ground motion components. It is observed that the proposed procedure estimates the flexural deformation demands in deformation‐controlled members and the shear forces in force‐controlled members with reasonable accuracy. Copyright © 2010 John Wiley & Sons, Ltd.

Experimental evaluation of vertical response of rigid surface footing on layered soil

The study of the dynamic response of foundations is of significant importance in the machine foundation design and soil-foundation interaction problems, in general. This paper presents experimental investigations in the field studying the effect of layering of soil mass on the dynamic response of foundation under vertical mode of vibration. Model block vibration tests on a rigid surface footing were conducted on different layered soil systems. A concrete block of size 400×400×100 mm was used as the model block and a Lazan type mechanical oscillator was used for inducing vibration in vertical direction. In the investigation two different soils, namely, gravel and fly ash were used. In total 132 tests were conducted on different layered soil beds with different loading combinations. Dynamic response of foundation resting on layered soil system were also analyzed by two simple approximate methods namely, mass spring dashpot model and equivalent elastic half-space theory. Results obtained from the two methods were compared with that obtained from the experimental investigation. It was found from the comparisons that the simple mass-spring dashpot model was capable of predicting the resonance frequency and the resonance amplitude satisfactorily. Elastic half space theory with equivalent soil properties was found to underestimate the displacement amplitude.

THEORETICAL ESTABLISHMENT AND EVALUATION OF A NOVEL OPTIMAL PYRAMIDAL HORN DESIGN CRITERION

This paper proposes a novel design criterion for optimal pyramidal horns. According to the criterion, the optimal aperture phase error parameters of a pyramidal horn are determined from the minimization of the horn's lateral surface area. We present two families of curves that illustrate the optimal aperture phase error parameters for frequency and directivity values in the area of practical interest. We also discuss two simple approximate design methods for the calculation of the optimal horn parameters. Comparisons with well-known design methods demonstrate the e-cacy of our approach. The designed horns have also smaller aperture area and require less installation space than other optimal proposals in the published literature. Moreover, the proposed criterion produces, under certain conditions, the lightest horn for a given directivity; as a result its fabrication requires less material compared to other structures. The present approach is a useful design tool when the size and weight of a pyramidal horn are of concern.

Versatile Solution Phase Triangular Silver Nanoplates for Highly Sensitive Plasmon Resonance Sensing

Solution phase triangular silver nanoplates (TSNP) with versatile tunability throughout the visible-NIR wavelengths are presented as highly sensitive localized surface plasmon refractive index sensors. A range of 20 TSNP solutions with edge lengths ranging from 11 to 200 nm and aspect ratios from 2 to 13 have been studied comprehensively using AFM, TEM, and UV-vis-NIR spectroscopy. Studies of the localized surface plasmon resonance (LSPR) peak's sensitivity to refractive index changes are performed using a simple sucrose concentration method whereby the surrounding refractive index can solely be changed without variation in any other parameter. The dependence of the TSNP localized surface plasmon resonance (LSPR) peak wavelength lambda(max) and its bulk refractive index sensitivity on the nanoplate's structure is determined. LSPR sensitivities are observed to increase linearly with lambda(max) up to 800 nm, with the values lying within the upper limit theoretically predicted for optimal sensitivity, notwithstanding any diminution due to ensemble averaging. A nonlinear increase in sensitivity is apparent at wavelengths within the NIR region with values reaching 1096 nm.RIU(-1) at lambda(max) 1093 nm. Theoretical studies performed using a simple aspect ratio dependent approximation method and discrete dipole approximation methods confirm the dependence of the LSPR bulk refractive index sensitivity upon the TSNP aspect ratio measured experimentally. These studies highlight the importance of this key parameter in acquiring such high sensitivities and promote these TSNP sols for sensing applications at appropriate wavelengths for biological samples.

A new method for interval estimation of the mean of the Gamma distribution

The two parameter Gamma distribution is widely used for modeling lifetime distributions in reliability theory. There is much literature on the inference on the individual parameters of the Gamma distribution, namely the shape parameter k and the scale parameter theta when the other parameter is known. However, usually the reliability professionals have a major interest in making statistical inference about the mean lifetime mu, which equals the product thetak for the Gamma distribution. The problem of inference on the mean mu when both parameters theta and k are unknown has been less attended in the literature for the Gamma distribution. In this paper we review the existing methods for interval estimation of mu. A comparative study in this paper indicates that the existing methods are either too approximate and yield less reliable confidence intervals or are computationally quite complicated and need advanced computing facilities. We propose a new simple method for interval estimation of the Gamma mean and compare its performance with the existing methods. The comparative study showed that the newly proposed computationally simple optimum power normal approximation method works best even for small sample sizes.