FFT-Based Numerical Method for Nonlinear Elastic

In theoretical research pertaining to sealing, a contact model must be used to obtain the leakage channel. However, for elastoplastic contact, current numerical methods require a long calculation time. Hyperelastic contact is typically simplifed to a linear elastic contact problem, which must be improved in terms of calculation accuracy. Based on the fast Fourier transform, a numerical method suitable for elastoplastic and hyperelastic frictionless contact that can be used for solving two-dimensional and three-dimensional (3D) contact problems is proposed herein. The nonlinear elastic contact problem is converted into a linear elastic contact problem considering residual deformation (or the equivalent residual deformation). Results from numerical simulations for elastic, elastoplastic, and hyperelastic contact between a hemisphere and a rigid plane are compared with those obtained using the fnite element method to verify the accuracy of the numerical method. Compared with the existing elastoplastic contact numerical methods, the proposed method achieves a higher calculation efciency while ensuring a certain calculation accuracy (i.e., the pressure error does not exceed 15%, whereas the calculation time does not exceed 10 min in a 64 × 64 grid). For hyperelastic contact, the proposed method reduces the dependence of the approximation result on the load, as in a linear elastic approximation. Finally, using the sealing application as an example, the contact and leakage rates between complicated 3D rough surfaces are calculated. Despite a certain error, the simplifed numerical method yields a better approximation result than the linear elastic contact approximation. Additionally, the result can be used as fast solutions in engineering applications.

Comparative Analysis for Evaluating Wind Energy Resources Using Intelligent Optimization Algorithms and Numerical Methods

Statistical distributions are used to model wind speed,and the twoparameters Weibull distribution has proven its effectiveness at characterizing wind speed.Accurate estimation of Weibull parameters,the scale(c)and shape(k),is crucial in describing the actual wind speed data and evaluating the wind energy potential.Therefore,this study compares the most common conventional numerical(CN)estimation methods and the recent intelligent optimization algorithms(IOA)to show how precise estimation of c and k affects the wind energy resource assessments.In addition,this study conducts technical and economic feasibility studies for five sites in the northern part of Saudi Arabia,namely Aljouf,Rafha,Tabuk,Turaif,and Yanbo.Results exhibit that IOAs have better performance in attaining optimal Weibull parameters and provided an adequate description of the observed wind speed data.Also,with six wind turbine technologies rating between 1 and 3MW,the technical and economic assessment results reveal that the CN methods tend to overestimate the energy output and underestimate the cost of energy($/kWh)compared to the assessments by IOAs.The energy cost analyses show that Turaif is the windiest site,with an electricity cost of$0.016906/kWh.The highest wind energy output is obtained with the wind turbine having a rated power of 2.5 MW at all considered sites with electricity costs not exceeding$0.02739/kWh.Finally,the outcomes of this study exhibit the potential of wind energy in Saudi Arabia,and its environmental goals can be acquired by harvesting wind energy.

Computing Streamfunction and Velocity Potential in a Limited Domain of Arbitrary Shape.Part II:Numerical Methods and Test Experiments

Built on the integral formulas in Part I,numerical methods are developed for computing velocity potential and streamfunction in a limited domain.When there is no inner boundary（around a data hole） inside the domain,the total solution is the sum of the internally and externally induced parts.For the internally induced part,three numerical schemes（grid-staggering,local-nesting and piecewise continuous integration） are designed to deal with the singularity of the Green＇s function encountered in numerical calculations.For the externally induced part,by setting the velocity potential（or streamfunction） component to zero,the other component of the solution can be computed in two ways：（1） Solve for the density function from its boundary integral equation and then construct the solution from the boundary integral of the density function.（2） Use the Cauchy integral to construct the solution directly.The boundary integral can be discretized on a uniform grid along the boundary.By using local-nesting（or piecewise continuous integration）,the scheme is refined to enhance the discretization accuracy of the boundary integral around each corner point（or along the entire boundary）.When the domain is not free of data holes,the total solution contains a data-hole-induced part,and the Cauchy integral method is extended to construct the externally induced solution with irregular external and internal boundaries.An automated algorithm is designed to facilitate the integrations along the irregular external and internal boundaries.Numerical experiments are performed to evaluate the accuracy and efficiency of each scheme relative to others.

Numerical Method for Modeling the Constitutive Relationship of Sand under Different Stress Paths

A numerical method was used in order to establish the constitutive relationship of sands under different stress paths, Firstly, based on the numerical method modeling the constitutive law of sands, the elastoplastic constitutive relationship of sand was established for three paths： the constant proportion of principle stress path, the conventional triaxial compression （CTC） path, and the p=constant （TC） path. The yield lines of plastic volumetric strain and plastic generalized shear strain were given. Through visualization, the three dimensional surface of the stress-strain relationship in the whole stress field （p, q） obtained under the three paths was plotted. Also, by comparing the stress-strain surfaces and yield locus of the three stress paths, the differences were found to be obvious, which demonstrates that the influence of the stress paths on constitutive law was not neglected. The numerical modeling method overcame the difficulty of finding an analytical expression for plastic potential. The results simulated the experimental data with an accuracy of 90% on average, so the constitutive model established in this paper provides an effective constitutive equation for this kind of engineering, reflecting the effect of practical stress paths that occur in sands.

An Effective Numerical Method for the Solution of a Stochastic Coronavirus(2019-nCovid)Pandemic Model

Nonlinear stochastic modeling plays a significant role in disciplines such as psychology,finance,physical sciences,engineering,econometrics,and biological sciences.Dynamical consistency,positivity,and boundedness are fundamental properties of stochastic modeling.A stochastic coronavirus model is studied with techniques of transition probabilities and parametric perturbation.Well-known explicit methods such as Euler Maruyama,stochastic Euler,and stochastic Runge–Kutta are investigated for the stochastic model.Regrettably,the above essential properties are not restored by existing methods.Hence,there is a need to construct essential properties preserving the computational method.The non-standard approach of finite difference is examined to maintain the above basic features of the stochastic model.The comparison of the results of deterministic and stochastic models is also presented.Our proposed efficient computational method well preserves the essential properties of the model.Comparison and convergence analyses of the method are presented.

A numerical method for determining the stuck point in extended reach drilling

A stuck drill string results in a major non-productive cost in extended reach drilling engineering. The first step is to determine the depth at which the sticking has occurred. Methods of measurement have been proved useful for determining the stuck points, but these operations take considerable time. As a result of the limitation with the current operational practices, calculation methods are still preferred to estimate the stuck point depth. Current analytical methods do not consider friction and are only valid for vertical rather than extended reach wells. The numerical method is established to take full account of down hole friction, tool joint, upset end of drill pipe, combination drill strings and tubular materials so that it is valid to determine the stuck point in extended reach wells. The pull test, torsion test and combined test of rotation and pulling can be used to determine the stuck point. The results show that down hole friction, tool joint, upset end of drill pipe, tubular sizes and materials have significant effects on the pull length and/or the twist angle of the stuck drill string.

Analysis profile of the fully grouted rock bolt in jointed rock using analytical and numerical methods

The purpose of this study was to investigate the effect of bolt profile on load transfer mechanism of fully grouted bolts in jointed rocks using analytical and numerical methods. Based on the analytical method with development of methods, a new model is presented. To validate the analytical model, five different profiles modeled by ANSYS software. The profile of rock bolts T3 and T4with load transfer capacity,respectively 180 and 195 kN in the jointed rocks was selected as the optimum profiles. Finally, the selected profiles were examined in Tabas Coal Mine. FLAC analysis indicates that patterns 6+7 with2 NO flexi bolt 4 m better than other patterns within the faulted zone.

Numerical method of studying nonlinear interactions between long waves and multiple short waves

Although the nonlinear interactions between a single short gravity wave and a long wave can be solved analytically, the solution is less tractable in more general cases involving multiple short waves. In this work we present a numerical method of studying nonlinear interactions between a long wave and multiple short harmonic waves in infinitely deep water. Specifically, this method is applied to the calculation of the temporal and spatial evolutions of the surface elevations in which a given long wave interacts with several short harmonic waves. Another important application of our method is to quantitatively analyse the nonlinear interactions between an arbitrary short wave train and another short wave train. From simulation results, we obtain that the mechanism for the nonlinear interactions between one short wave train and another short wave train （expressed as wave train 2） leads to the energy focusing of the other short wave train （expressed as wave train 3）. This mechanism occurs on wave components with a narrow frequency bandwidth, whose frequencies are near that of wave train 3.

A NUMERICAL METHOD OF K-S ENTROPY CALCULATION FOR A STRANGE ATTRACTOR

Based directly on the original definition of K-S entropy, a new algorithm for calculating K-S entropy from chaotic time series is developed by using some techniques of coding and code operation.

Horizontal Slow Oscillation of a Moored Tanker and Numerical Method of Hopf-Bifurcation

In this paper, a six-order nonlinear dynamic model with three degrees of freedom is presented for the study of the 'fishtail' motion of a Single Point Mooring System. The effect of parameter variations on the equilibrium state of the system is analyzed. In order to study the stability of the equilibrium state, the mooring-line length l is chosen as a bifurcation parameter, so that all eigenvalues of the Jacobian matrix under different parameters can be worked out, and then the Hopf-bifurcation point can be found. Finally, the Hopf-bifurcation periodic solution of the system is computed.

Numerical method of Rayleigh-Stokes problem for heated generalized second grade fluid with fractional derivative

In this paper,we consider the Rayleigh-Stokes problem for a heated generalized second grade fluid（RSP-HGSGF）with fractional derivative.An effective numerical method for approximating RSP-HGSGF in a bounded domain is presented.The stability and convergence of the method are analyzed.Numerical examples are presented to show the application of the present technique.

DETERMINATION OF CLOSURE EFFECT IN FATIGUE CRACKING BY MEANS OF COMPLIANCE TECHNIQUE AND NUMERICAL METHOD

The crack tip strain gauge method in the compliance technique was used to determine the opening load of notched crack of axle steel,and the nonlinear finite element ADINA program, to which the cyclic stress-strain curve of axle steel was applied,was used to analyze the stress-strain field ahead of the crack tip and the opening load of notched crack.The results of both the compliance technique and the numerical method were in good agreement.In this pa- per,the concept of the sensitive point is proposed and the key to the determination of the crack opening load in the experiment is to place a strain gauge at sensitive point.It is certified by both experimental and numerical methods that the sensitive point has the best linear relation- ship character and the value of strain is much greater.

A baroclinic typhoon model with a moving multi-nested grid and variational adjustment initialisationI. Numerical method

A baroclinic typhoon model with a moving multi--nested grid is applied in marine environmental forecasts. This paper describes the numerical methods of the model including governing equations, finite differencing, split scheme and time integration.

Numerical Method for Determination of Composition of Weak Acid Mixtures by Potentiometric Titration

An equation describing the state of weak acid mixtures was derived from the relationships between mole balance and charge balance. The equation was solved with numerical method and the compositions of the acid mixtures were determined. The advantages of this treatment were demonstrated by analyzing binary mixtures of chloroacetic, formic and acetic acids.

A NUMERICAL METHOD FOR FRACTIONAL INTEGRAL WITH APPLICATIONS

A new numerical method for the fractional integral that only stores part history data is presented, and its discretization error is estimated. The method can be used to solve the integro_differential equation including fractional integral or fractional derivative in a long history. The difficulty of storing all history data is overcome and the error can be controlled. As application,motion equations governing the dynamical behavior of a viscoelastic Timoshenko beam with fractional derivative constitutive relation are given. The dynamical response of the beam subjected to a periodic excitation is studied by using the separation variables method. Then the new numerical method is used to solve a class of weakly singular Volterra integro_differential equations which are applied to describe the dynamical behavior of viscoelastic beams with fractional derivative constitutive relations. The analytical and unmerical results are compared. It is found that they are very close.

A two scaled numerical method for global analysis of high dimensional nonlinear systems

This paper first analyzes the features of two classes of numerical methods for global analysis of nonlinear dynamical systems, which regard state space respectively as continuous and discrete ones. On basis of this understanding it then points out that the previously proposed method of point mapping under cell reference (PMUCR), has laid a frame work for the development of a two scaled numerical method suitable for the global analysis of high dimensional nonlinear systems, which may take the advantages of both classes of single scaled methods but will release the difficulties induced by the disadvantages of them. The basic ideas and main steps of implementation of the two scaled method, namely extended PMUCR, are elaborated. Finally, two examples are presented to demonstrate the capabilities of the proposed method.

The Application of Barr Numerical Method to Realize Abel Inversion

An 8-channel HCN laser interferometer will be installed on HL-2A in near term. In order to get the spatial profile of the electron density Barr numerical method is adopted to realize the Abel inversion. In this article the result of the Abel inversion by Matlab GUI is given which can be updated to process the measured data of the 8-channel laser interferometer and provide the spatial distribution of the electron density.

NUMERICAL METHOD FOR SOLVING THE EULER EQUATION FOR UNSTEADY TRANSONIC FLOWS OVER OSCILLATING AIRFOILS

Through transformations, the time-dependent boundary condition on the airfoil contour and the boundary condition at infinity are brought fixed to the boundaries of a finite domain. The boundary conditions can thus be satisfied exactly without increasing the computational time. The novel scheme is useful for computing transonic, strong disturbance, unsteady flows with high reduced frequencies. The scheme makes use of curvefitted orthogonal meshes and the lattice control technique to obtain the optimal grid distribution. The numerical results are satisfactory.

RECONSTRUCTION OF PART OF AN ACTUAL BLAST-WAVE FLOW FIELD TO AGREE WITH EXPERIMENTAL DATA BY USING NUMERICAL METHOD WITH HIGH IDENTIFICATION

In this paper, on the basis of experimental data of two kinds of chemical explosions, the piston-pushing model of spherical blast-waves and the second-order Godunov-type scheme of finite difference methods with high identification to discontinuity are used to the numerical reconstruction of part of an actual hemispherical blast-wave flow field by properly adjusting the moving bounary conditions of a piston. This method is simple and reliable. It is suitable to the evaluation of effects of the blast-wave flow field away from the explosion center.

Numerical method for wave height distribution within the artificial harbor with water depth of steep variation

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