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.

Numerical simulations for radon migration and exhalation behavior during measuring radon exhalation rate with closed-loop method

Accurate measurements of the radon exhalation rate help identify and evaluate radon risk regions in the environment.Among these measurement methods,the closed-loop method is frequently used.However,traditional experiments are insufficient or cannot analyze the radon migration and exhalation patterns at the gas–solid interface in the accumulation chamber.The CFD-based technique was applied to predict the radon concentration distribution in a limited space,allowing radon accumulation and exhalation inside the chamber intuitively and visually.In this study,three radon exhalation rates were defined,and two structural ventilation tubes were designed for the chamber.The consistency of the simulated results with the variation in the radon exhalation rate in a previous experiment or analytical solution was verified.The effects of the vent tube structure and flow rate on the radon uniformity in the chamber;permeability,insertion depth,and flow rate on the radon exhalation rate and the effective diffusion coefficient on back-diffusion were investigated.Based on the results,increasing the inser-tion depth from 1 to 5 cm decreased the effective decay constant by 19.55%,whereas the curve-fitted radon exhalation rate decreased(lower than the initial value)as the deviation from the initial value increased by approximately 7%.Increasing the effective diffusion coefficient from 2.77×10^(-7) to 7.77×10^(-6) m^(2) s^(-1) made the deviation expand from 2.14 to 15.96%.The conclusion is that an increased insertion depth helps reduce leakage in the chamber,subject to notable back-diffusion,and that the closed-loop method is reasonably used for porous media with a low effective diffusion coefficient in view of the back-diffusion effect.The CFD-based simulation is expected to provide guidance for the optimization of the radon exhalation rate measurement method and,thus,the accurate measurement of the radon exhalation rate.

Comparative Studies between Picard’s and Taylor’s Methods of Numerical Solutions of First Ordinary Order Differential Equations Arising from Real-Life Problems

To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section.

Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation

In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al.in J Sci Comput 66:321–345,2016;Dong and Wang in J Comput Appl Math 380:1–11,2020)for a one-dimensional stationary Schrödinger equation.Previous work showed that penalty parameters were required to be positive in error analysis,but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes.In this work,by performing extensive numerical experiments,we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods,and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.

An intelligent control method based on artificial neural network for numerical flight simulation of the basic finner projectile with pitching maneuver

In this paper,an intelligent control method applying on numerical virtual flight is proposed.The proposed algorithm is verified and evaluated by combining with the case of the basic finner projectile model and shows a good application prospect.Firstly,a numerical virtual flight simulation model based on overlapping dynamic mesh technology is constructed.In order to verify the accuracy of the dynamic grid technology and the calculation of unsteady flow,a numerical simulation of the basic finner projectile without control is carried out.The simulation results are in good agreement with the experiment data which shows that the algorithm used in this paper can also be used in the design and evaluation of the intelligent controller in the numerical virtual flight simulation.Secondly,combined with the real-time control requirements of aerodynamic,attitude and displacement parameters of the projectile during the flight process,the numerical simulations of the basic finner projectile’s pitch channel are carried out under the traditional PID(Proportional-Integral-Derivative)control strategy and the intelligent PID control strategy respectively.The intelligent PID controller based on BP(Back Propagation)neural network can realize online learning and self-optimization of control parameters according to the acquired real-time flight parameters.Compared with the traditional PID controller,the concerned control variable overshoot,rise time,transition time and steady state error and other performance indicators have been greatly improved,and the higher the learning efficiency or the inertia coefficient,the faster the system,the larger the overshoot,and the smaller the stability error.The intelligent control method applying on numerical virtual flight is capable of solving the complicated unsteady motion and flow with the intelligent PID control strategy and has a strong promotion to engineering application.

A methodology for damage evaluation of underground tunnels subjected to static loading using numerical modeling

We have proposed a methodology to assess the robustness of underground tunnels against potential failure.This involves developing vulnerability functions for various qualities of rock mass and static loading intensities.To account for these variations,we utilized a Monte Carlo Simulation(MCS)technique coupled with the finite difference code FLAC^(3D),to conduct two thousand seven hundred numerical simulations of a horseshoe tunnel located within a rock mass with different geological strength index system(GSIs)and subjected to different states of static loading.To quantify the severity of damage within the rock mass,we selected one stress-based(brittle shear ratio(BSR))and one strain-based failure criterion(plastic damage index(PDI)).Based on these criteria,we then developed fragility curves.Additionally,we used mathematical approximation techniques to produce vulnerability functions that relate the probabilities of various damage states to loading intensities for different quality classes of blocky rock mass.The results indicated that the fragility curves we obtained could accurately depict the evolution of the inner and outer shell damage around the tunnel.Therefore,we have provided engineers with a tool that can predict levels of damages associated with different failure mechanisms based on variations in rock mass quality and in situ stress state.Our method is a numerically developed,multi-variate approach that can aid engineers in making informed decisions about the robustness of underground tunnels.

Application of the finite analytic numerical method to a flowdependent variational data assimilation

An anisotropic diffusion filter can be used to model a flow-dependent background error covariance matrix,which can be achieved by solving the advection-diffusion equation.Because of the directionality of the advection term,the discrete method needs to be chosen very carefully.The finite analytic method is an alternative scheme to solve the advection-diffusion equation.As a combination of analytical and numerical methods,it not only has high calculation accuracy but also holds the characteristic of the auto upwind.To demonstrate its ability,the one-dimensional steady and unsteady advection-diffusion equation numerical examples are respectively solved by the finite analytic method.The more widely used upwind difference method is used as a control approach.The result indicates that the finite analytic method has higher accuracy than the upwind difference method.For the two-dimensional case,the finite analytic method still has a better performance.In the three-dimensional variational assimilation experiment,the finite analytic method can effectively improve analysis field accuracy,and its effect is significantly better than the upwind difference and the central difference method.Moreover,it is still a more effective solution method in the strong flow region where the advective-diffusion filter performs most prominently.

Numerical Simulation of the Seismic Response of a Horizontal Storage Tank Based on a SPH-FEM Coupling Method

A coupled numerical calculation method combining smooth particle hydrodynamics(SPH)and the finite element method(FEM)was implemented to investigate the seismic response of horizontal storage tanks.Anumericalmodel of a horizontal storage tank featuring a free liquid surface under seismic action was constructed using the SPH–FEM coupling method.The stored liquid was discretized using SPH particles,while the tank and supports were discretized using the FEM.The interaction between the stored liquid and the tank was simulated by using the meshless particle contact method.Then,the numerical simulation results were compared and analyzed against seismic simulation shaking table test data to validate the method.Subsequently,a series of numerical models,considering different liquid storage volumes and seismic effects,were constructed to obtain time history data of base shear and top center displacement,which revealed the seismic performance of horizontal storage tanks.Numerical simulation results and experimental data showed good agreement,with an error rate of less than 18.85%.And this conformity signifies the rationality of the SPH-FEM coupling method.The base shear and top center displacement values obtained by the coupled SPH-FEM method were only 53.3% to 69.1% of those calculated by the equivalent mass method employed in the current code.As the stored liquid volume increased,the seismic response of the horizontal storage tank exhibited a gradual upward trend,with the seismic response increasing from 73% to 388% for every 35% increase in stored liquid volume.The maximum von Mises stress of the tank and the supports remained below the steel yield strength during the earthquake.The coupled SPH-FEM method holds certain advantages in studying the seismic problems of tanks with complex structural forms,particularly due to the representation of the flow field distribution during earthquakes by involving reservoir fluid participation.

Numerical Simulation of Surrounding Rock Deformation and Grouting Reinforcement of Cross-Fault Tunnel under Different Excavation Methods

Tunnel construction is susceptible to accidents such as loosening, deformation, collapse, and water inrush, especiallyunder complex geological conditions like dense fault areas. These accidents can cause instability and damageto the tunnel. As a result, it is essential to conduct research on tunnel construction and grouting reinforcementtechnology in fault fracture zones to address these issues and ensure the safety of tunnel excavation projects. Thisstudy utilized the Xianglushan cross-fault tunnel to conduct a comprehensive analysis on the construction, support,and reinforcement of a tunnel crossing a fault fracture zone using the three-dimensional finite element numericalmethod. The study yielded the following research conclusions: The excavation conditions of the cross-fault tunnelarray were analyzed to determine the optimal construction method for excavation while controlling deformationand stress in the surrounding rock. The middle partition method (CD method) was found to be the most suitable.Additionally, the effects of advanced reinforcement grouting on the cross-fault fracture zone tunnel were studied,and the optimal combination of grouting reinforcement range (140°) and grouting thickness (1m) was determined.The stress and deformation data obtained fromon-site monitoring of the surrounding rock was slightly lower thanthe numerical simulation results. However, the change trend of both sets of data was found to be consistent. Theseresearch findings provide technical analysis and data support for the construction and design of cross-fault tunnels.

Numerical Models and Methods of Atmospheric Parameters Originating in the Formation of the Earth’s Climatic Cycle

Atmospheric models are physical equations based on the ideal gas law. Applied to the atmosphere, this law yields equations for water, vapor (gas), ice, air, humidity, dryness, fire, and heat, thus defining the model of key atmospheric parameters. The distribution of these parameters across the entire planet Earth is the origin of the formation of the climatic cycle, which is a normal climatic variation. To do this, the Earth is divided into eight (8) parts according to the number of key parameters to be defined in a physical representation of the model. Following this distribution, numerical models calculate the constants for the formation of water, vapor, ice, dryness, thermal energy (fire), heat, air, and humidity. These models vary in complexity depending on the indirect trigonometric direction and simplicity in the sum of neighboring models. Note that the constants obtained from the equations yield 275.156˚K (2.006˚C) for water, 273.1596˚K (0.00963˚C) for vapor, 273.1633˚K (0.0133˚C) for ice, 0.00365 in/s for atmospheric dryness, 1.996 in2/s for humidity, 2.993 in2/s for air, 1 J for thermal energy of fire, and 0.9963 J for heat. In summary, this study aims to define the main parameters and natural phenomena contributing to the modification of planetary climate. .

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.

Numerical Assessment of Nanofluid Natural Convection Using Local RBF Method Coupled with an Artificial Compressibility Model

In this paper,natural heat convection inside square and equilateral triangular cavities was studied using a meshless method based on collocation local radial basis function(RBF).The nanofluids used were Cu-water or Al_(2)O_(3)-water mixture with nanoparticle volume fractions range of 0≤φ≤0.2.A system of continuity,momentum,and energy partial differential equations was used in modeling the flow and temperature behavior of the fluids.Partial derivatives in the governing equations were approximated using the RBF method.The artificial compressibility model was implemented to overcome the pressure velocity coupling problem that occurs in such equations.Themain goal of this work was to present a simple and efficient method to deal with complex geometries for a variety of problem conditions.To assess the accuracy of the proposed method,several test cases of natural convection in square and triangular cavities were selected.For Rayleigh numbers ranging from 103 to 105,a validation test of natural convection of Cu-water in a square cavity was used.The numerical investigation was then extended to Rayleigh number 106,as well as Al_(2)O_(3)-water nanofluid with a volume fraction range of 0≤φ≤0.2.In a second investigation,the same nanofluids were used in a triangular cavitywith varying volume fractions to test the proposed meshless approach on non-rectangular geometries.The numerical results appear to be in agreement with those from earlier investigations.Furthermore,the suggested meshless method was found to be stable and accurate,demonstrating that it may be a viable alternative for solving natural heat transfer equations of nanofluids in enclosures with irregular geometries.

Numerical Simulation of a Viscoelastic Thinning Process for Preparing Flexible Glasses by Redrawing Method

The forming process of the flexible ultrathin glasses(UTG)prepared by the redrawing method was numerically simulated using ANSYS Polyflow software.In the forming process by the redrawing method,temperature,viscosity,transverse and longitudinal velocity distribution of the glasses with different compositions were studied.Furthermore,the influence of these factors on the width and thickness of the flexible glass plate was investigated.It is found that the internal and external heat exchange of glass has a dominant influence on the viscosity variation during the UTG forming process,which is inconsistent with the general viscosity-temperature dependence.The glass that first reaches the lower limit of forming viscosity can significantly resist the shrinking effect caused by surface tension,making the glass wider during the forming.If the original glass width remains unchanged,the glass thickness or feeding speed is reduced,wider and thinner flexible glasses can be produced.

A Numerical Investigation Based on Exponential Collocation Method for Nonlinear SITR Model of COVID-19

In this work,the exponential approximation is used for the numerical simulation of a nonlinear SITR model as a system of differential equations that shows the dynamics of the new coronavirus(COVID-19).The SITR mathematical model is divided into four classes using fractal parameters for COVID-19 dynamics,namely,susceptible(S),infected(I),treatment(T),and recovered(R).The main idea of the presented method is based on the matrix representations of the exponential functions and their derivatives using collocation points.To indicate the usefulness of this method,we employ it in some cases.For error analysis of the method,the residual of the solutions is reviewed.The reported examples show that the method is reasonably efficient and accurate.

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.

Numerical simulation on the seismic performance of retrofitted masonry walls based on the combined finite-discrete element method

Due to the long construction life,improper design methods,brittle material properties and poor construction techniques,most existing masonry structures do not perform well during earthquakes.The retrofitting method using an external steel-meshed mortar layer is widely used to retrofit existing masonry buildings.Assessing the seismic performance of masonry walls reinforced by an external steel-meshed mortar layer reasonably and effectively is a difficult subject in the research field of masonry structures.Based on the combined finite-discrete elements method,the numerical models of retrofitted brick walls with four different masonry mortar strengths by an external mortar layer are established.The shear strength of mortar and the contact between the retrofitted mortar layer and the brick blocks are discussed in detail.The failure patterns and load-displacement curves of the retrofitted brick walls were obtained by applying low cycle reciprocating loads to the numerical model,and the bearing capacity and the failure mechanism of the retrofitted walls were obtained by comparing the failure patterns,ultimate bearing capacity,deformability and other aspects with the tests.This study provides a basis for improving the seismic strengthening design method of masonry structures and helps to better assess the seismic performance of masonry structures after retrofitting.

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.

On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analyticalmeans.Thus,we need numerical inversionmethods to convert the obtained solution fromLaplace domain to a real domain.In this paper,we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with orderα,β.Our proposed numerical scheme is based on three main steps.First,we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense,and then into Caputo sense.Secondly,we transformthe fractional differential equation in Caputo sense to an equivalent equation in Laplace space.Then the solution of the transformed equation is obtained in Laplace domain.Finally,the solution is converted into the real domain using numerical inversion of Laplace transform.Three inversion methods are evaluated in this paper,and their convergence is also discussed.Three test problems are used to validate the inversion methods.We demonstrate our results with the help of tables and figures.The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.