Numerical manifold method for thermo-mechanical coupling simulation of fractured rock mass

As a calculation method based on the Galerkin variation,the numerical manifold method(NMM)adopts a double covering system,which can easily deal with discontinuous deformation problems and has a high calculation accuracy.Aiming at the thermo-mechanical(TM)coupling problem of fractured rock masses,this study uses the NMM to simulate the processes of crack initiation and propagation in a rock mass under the influence of temperature field,deduces related system equations,and proposes a penalty function method to deal with boundary conditions.Numerical examples are employed to confirm the effectiveness and high accuracy of this method.By the thermal stress analysis of a thick-walled cylinder(TWC),the simulation of cracking in the TWC under heating and cooling conditions,and the simulation of thermal cracking of the SwedishÄspöPillar Stability Experiment(APSE)rock column,the thermal stress,and TM coupling are obtained.The numerical simulation results are in good agreement with the test data and other numerical results,thus verifying the effectiveness of the NMM in dealing with thermal stress and crack propagation problems of fractured rock masses.

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.

Multiparameter Numerical Investigation of Two Types of Moving Interactions Between the Deep-Sea Mining Vehicle Track Plate and Seabed Soil:Digging and Rotating Motions

To ensure the safe performance of deep-sea mining vehicles(DSMVs),it is necessary to study the mechanical characteristics of the interaction between the seabed soil and the track plate.The rotation and digging motions of the track plate are important links in the contact between the driving mechanism of the DSMV and seabed soil.In this study,a numerical simulation is conducted using the coupled Eulerian–Lagrangian(CEL)large deformation numerical method to investigate the interaction between the track plate of the DSMV and the seabed soil under two working conditions:rotating condition and digging condition.First,a soil numerical model is established based on the elastoplastic mechanical characterization using the basic physical and mechanical properties of the seabed soil obtained by in situ sampling.Subsequently,the soil disturbance mechanism and the dynamic mechanical response of the track plate under rotating and digging conditions are obtained through the analysis of the sensitivity of the motion parameters,the grouser structure,the layered soil features and the soil heterogeneity.The results indicate that the above parameters remarkably influence the interaction between the DSMV and the seabed soil.Therefore,it is important to consider the rotating and digging motion of the DSMV in practical engineering to develop a detailed optimization design of the track plate.

Efficient method to calculate the eigenvalues of the Zakharov–Shabat system

A numerical method is proposed to calculate the eigenvalues of the Zakharov–Shabat system based on Chebyshev polynomials. A mapping in the form of tanh(ax) is constructed according to the asymptotic of the potential function for the Zakharov–Shabat eigenvalue problem. The mapping can distribute Chebyshev nodes very well considering the gradient for the potential function. Using Chebyshev polynomials, tanh(ax) mapping, and Chebyshev nodes, the Zakharov–Shabat eigenvalue problem is transformed into a matrix eigenvalue problem. This method has good convergence for the Satsuma–Yajima potential and the convergence rate is faster than the Fourier collocation method. This method is not only suitable for simple potential functions but also converges quickly for a complex Y-shape potential. It can also be further extended to other linear eigenvalue problems.

A Calculation Method of Double Strength Reduction for Layered Slope Based on the Reduction of Water Content Intensity

The calculation of the factor of safety(FOS)is an important means of slope evaluation.This paper proposed an improved double strength reductionmethod(DRM)to analyze the safety of layered slopes.The physical properties of different soil layers of the slopes are different,so the single coefficient strength reduction method(SRM)is not enough to reflect the actual critical state of the slopes.Considering that the water content of the soil in the natural state is the main factor for the strength of the soil,the attenuation law of shear strength of clayey soil changing with water content is fitted.This paper also establishes the functional relationship between different reduction coefficients.Then,a USDFLD subroutine is programmed using the secondary development function of finite element software.Controlling the relationship between field variables and calculation time realizes double strength reduction applicable to the layered slope.Finally,by comparing the calculation results of different examples,it is proved that the stress and displacement distribution of the critical slope state obtained by the improved method is more realistic,and the calculated safety factor is more reliable.The newly proposedmethod considers the difference of intensity attenuation between different soil layers under natural conditions and avoids the disadvantage of the strength reduction method with uniform parameters,which provides a new idea and method for stability analysis of layered and complex slopes.

Numerical investigation of geostress influence on the grouting reinforcement effectiveness of tunnel surrounding rock mass in fault fracture zones

Grouting is a widely used approach to reinforce broken surrounding rock mass during the construction of underground tunnels in fault fracture zones,and its reinforcement effectiveness is highly affected by geostress.In this study,a numerical manifold method(NMM)based simulator has been developed to examine the impact of geostress conditions on grouting reinforcement during tunnel excavation.To develop this simulator,a detection technique for identifying slurry migration channels and an improved fluid-solid coupling(FeS)framework,which considers the influence of fracture properties and geostress states,is developed and incorporated into a zero-thickness cohesive element(ZE)based NMM(Co-NMM)for simulating tunnel excavation.Additionally,to simulate coagulation of injected slurry,a bonding repair algorithm is further proposed based on the ZE model.To verify the accuracy of the proposed simulator,a series of simulations about slurry migration in single fractures and fracture networks are numerically reproduced,and the results align well with analytical and laboratory test results.Furthermore,these numerical results show that neglecting the influence of geostress condition can lead to a serious over-estimation of slurry migration range and reinforcement effectiveness.After validations,a series of simulations about tunnel grouting reinforcement and tunnel excavation in fault fracture zones with varying fracture densities under different geostress conditions are conducted.Based on these simula-tions,the influence of geostress conditions and the optimization of grouting schemes are discussed.

The Adjustment Analysis Method of the Active Surface Antenna Based on Convolutional Neural Network

Active surface technique is one of the key technologies to ensure the reflector accuracy of the millimeter/submillimeter wave large reflector antenna.The antenna is complex,large-scale,and high-precision equipment,and its active surfaces are affected by various factors that are difficult to comprehensively deal with.In this paper,based on the advantage of the deep learning method that can be improved through data learning,we propose the active adjustment value analysis method of large reflector antenna based on deep learning.This method constructs a neural network model for antenna active adjustment analysis in view of the fact that a large reflector antenna consists of multiple panels spliced together.Based on the constraint that a single actuator has to support multiple panels(usually 4),an autonomously learned neural network emphasis layer module is designed to enhance the adaptability of the active adjustment neural network model.The classical 8-meter antenna is used as a case study,the actuators have a mean adjustment error of 0.00252 mm,and the corresponding antenna surface error is0.00523 mm.This active adjustment result shows the effectiveness of the method in this paper.

Numerical Solutions of the Classical and Modified Buckley-Leverett Equations Applied to Two-Phase Fluid Flow

Climate change is a reality. The burning of fossil fuels from oil, natural gas and coal is responsible for much of the pollution and the increase in the planet’s average temperature, which has raised discussions on the subject, given the emergencies related to climate. An energy transition to clean and renewable sources is necessary and urgent, but it will not be quick. In this sense, increasing the efficiency of oil extraction from existing sources is crucial, to avoid waste and the drilling of new wells. The purpose of this work was to add diffusive and dispersive terms to the Buckley-Leverett equation in order to incorporate extra phenomena in the temporal evolution between the water-oil and oil-water transitions in the pipeline. For this, the modified Buckley-Leverett equation was discretized via essentially weighted non-oscillatory schemes, coupled with a three-stage Runge-Kutta and a fourth-order centered finite difference methods. Then, computational simulations were performed and the results showed that new features emerge in the transitions, when compared to classical simulations. For instance, the dispersive term inhibits the diffusive term, adding oscillations, which indicates that the absorption of the fluid by the porous medium occurs in a non-homogeneous manner. Therefore, based on research such as this, decisions can be made regarding the replacement of the porous medium or the insertion of new components to delay the replacement.

Comprehensive analysis of slope stability and determination of stable slopes in the Chador-Malu iron ore mine using numerical and limit equilibrium methods

One of the critical aspects in mine design is slope stability analysis and the determination of stable slopes. In the Chador- Malu iron ore mine, one of the most important iron ore mines in central Iran, it was considered vital to perform a comprehensive slope stability analysis. At first, we divided the existing rock hosting pit into six zones and a geotechnical map was prepared. Then, the value of MRMR （Mining Rock Mass Rating） was determined for each zone. Owing to the fact that the Chador-Malu iron ore mine is located in a highly tectonic area and the rock mass completely crushed, the Hoek-Brown failure criterion was found suitable to estimate geo-mechanical parameters. After that, the value of cohesion （c） and friction angle （tp） were calculated for different geotechnical zones and relative graphs and equations were derived as a function of slope height. The stability analyses using numerical and limit equilibrium methods showed that some instability problems might occur by increasing the slope height. Therefore, stable slopes for each geotechnical zone and prepared sections were calculated and presented as a function of slope height.

2D numerical manifold method based on quartic uniform B-spline interpolation and its application in thin plate bending

A new numerical manifold （NMM） method is derived on the basis of quartic uniform B-spline interpolation. The analysis shows that the new interpolation function possesses higher-order continuity and polynomial consistency compared with the conven- tional NMM. The stiffness matrix of the new element is well-conditioned. The proposed method is applied for the numerical example of thin plate bending. Based on the prin- ciple of minimum potential energy, the manifold matrices and equilibrium equation are deduced. Numerical results reveal that the NMM has high interpolation accuracy and rapid convergence for the global cover function and its higher-order partial derivatives.

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.

Response Sensitivity Analysis of the Dynamic Milling Process Based on the Numerical Integration Method

As one of the bases of gradient-based optimization algorithms, sensitivity analysis is usually required to calculate the derivatives of the system response with respect to the machining parameters. The most widely used approaches for sensitivity analysis are based on time-consuming numerical methods, such as finite difference methods. This paper presents a semi-analytical method for calculation of the sensitivity of the stability boundary in milling. After transforming the delay-differential equation with time-periodic coefficients governing the dynamic milling process into the integral form, the Floquet transition matrix is constructed by using the numerical integration method. Then, the analytical expressions of derivatives of the Floquet transition matrix with respect to the machining parameters are obtained. Thereafter, the classical analytical expression of the sensitivity of matrix eigenvalues is employed to calculate the sensitivity of the stability lobe diagram. The two-degree-of-freedom milling example illustrates the accuracy and efficiency of the proposed method. Compared with the existing methods, the unique merit of the proposed method is that it can be used for analytically computing the sensitivity of the stability boundary in milling, without employing any finite difference methods. Therefore, the high accuracy and high efficiency are both achieved. The proposed method can serve as an effective tool for machining parameter optimization and uncertainty analysis in high-speed milling.

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.

Numerical manifold method modeling of coupled processes in fractured geological media at multiple scales

The greatest challenges of rigorously modeling coupled hydro-mechanical(HM)processes in fractured geological media at different scales are associated with computational geometry.These challenges include dynamic shearing and opening of intersecting fractures at discrete fracture scales as a result of coupled processes,and contact alteration along rough fracture surfaces that triggers structural and physical changes of fractures at micro-asperity scale.In this paper,these challenges are tackled by developing a comprehensive modeling approach for coupled processes in fractured geological media based on numerical manifold method(NMM)at multiple scales.Based on their distinct geometric features,fractures are categorized into three different scales:dominant fracture,discrete fracture,and discontinuum asperity scales.Here the scale is relative,that of the fracture relative to that of the research interest or domain.Different geometric representations of fractures at different scales are used,and different governing equations and constitutive relationships are applied.For dominant fractures,a finite thickness zone model is developed to treat a fracture as a porous nonlinear domain.Nonlinear fracture mechanical behavior is accurately modeled with an implicit approach based on strain energy.For discrete fractures,a zero-dimensional model was developed for analyzing fluid flow and mechanics in fractures that are geometrically treated as boundaries of the rock matrix.With the zero-dimensional model,these fractures can be modeled with arbitrary orientations and intersections.They can be fluid conduits or seals,and can be open,bonded or sliding.For the discontinuum asperity scale,the geometry of rough fracture surfaces is explicitly represented and contacts involving dynamic alteration of contacts among asperities are rigorously calculated.Using this approach,fracture alteration caused by deformation,re-arrangement and sliding of rough surfaces can be captured.Our comprehensive model is able to handle the computational challenges with accurate representation of intersections and shearing of fractures at the discrete fracture scale and rigorously treats contacts along rough fracture surfaces at the discontinuum asperity scale.With future development of three-dimensional(3D)geometric representation of discrete fracture networks in porous rock and contacts among multi-body systems,this model is promising as a basis of 3D fully coupled analysis of fractures at multiple scales,for advancing understanding and optimizing energy recovery and storage in fractured geological media.

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.

Searching for critical slip surfaces of slopes using stress fields by numerical manifold method

This study first reviews the numerical manifold method(NMM)which possesses some advantages over the traditional limit equilibrium methods(LEMs)in calculating the factors of safety(Fs)of the slopes.Then,with regard to a trial slip surface(TSS),associated stress fields reproduced by NMM as well as the enhanced limit equilibrium method are combined to compute Fs.In order to search for the potential critical slip surface(CSS),the MAX-MIN ant colony optimization algorithm(MMACOA),one of the best performing algorithms for some optimization problems,is adopted.Procedures to obtain Fs in conjunction with the potential CSS are described.Finally,the proposed numerical model and traditional methods are compared with stability analysis of three typical slopes.The numerical results show that Fs and CSSs of the slopes can be accurately calculated with the proposed model.

Determining Atmospheric Boundary Layer Height with the Numerical Differentiation Method Using Bending Angle Data from COSMIC

This paper presents a new method to estimate the height of the atmospheric boundary layer(ABL) by using COSMIC radio occultation bending angle(BA) data. Using the numerical differentiation method combined with the regularization technique, the first derivative of BA profiles is retrieved, and the height at which the first derivative of BA has the global minimum is defined to be the ABL height. To reflect the reliability of estimated ABL heights, the sharpness parameter is introduced, according to the relative minimum of the BA derivative. Then, it is applied to four months of COSMIC BA data(January, April, July, and October in 2008), and the ABL heights estimated are compared with two kinds of ABL heights from COSMIC products and with the heights determined by the finite difference method upon the refractivity data. For sharp ABL tops(large sharpness parameters), there is little difference between the ABL heights determined by different methods, i.e.,the uncertainties are small; whereas, for non-sharp ABL tops(small sharpness parameters), big differences exist in the ABL heights obtained by different methods, which means large uncertainties for different methods. In addition, the new method can detect thin ABLs and provide a reference ABL height in the cases eliminated by other methods. Thus, the application of the numerical differentiation method combined with the regularization technique to COSMIC BA data is an appropriate choice and has further application value.

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.