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.

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.

Mobility and dynamic erosion process of granular flow:insights from numerical investigation using material point method

In order to understand the dynamics of granular flow on an erodible base soil,in this paper,a series of material point method-based granular column collapse tests were conducted to investigate numerically the mobility and dynamic erosion process of granular flow subjected to the complex settings,i.e.,the aspect ratio,granular mass,friction and dilatancy resistance,gravity and presence of water.A set of power scaling laws were proposed to describe the final deposit characteristics of granular flow by the relations of the normalized run-out distance and the normalized final height of granular flow against the aspect ratio,being greatly affected by the complex geological settings,e.g.,granular mass,the friction and dilatancy resistance of granular soil,and presence of water in granular flow.An index of the coefficient of friction of granular soil was defined as a ratio of the target coefficient of friction over the initial coefficient of friction to quantify the scaling extent of friction change(i.e.,friction strengthening or weakening).There is a characteristic aspect ratio of granular column corresponding to the maximum mobility of granular flow with the minimum index of the apparent coefficient of friction.The index of the repose coefficient of friction of granular flow decreased gradually with the increase in aspect ratio because higher potential energy of granular column at a larger aspect ratio causes a larger kinetic energy of granular soil to weaken the friction of granular soil as a kind of velocity-related friction weakening.An increase in granular mass reduces gradually the indexes of the apparent and repose coefficients of friction of granular soil to enhance the mobility of granular flow.The mobility of granular flow increases gradually with the decrease in friction angle or increase in dilatancy angle of granular soil.However,the increase of gravity accelerates granular flow but showing the same final deposit profile without any dependence on gravity.The mobility of granular flow increases gradually by lowering the indexes of the apparent and repose coefficients of friction of granular flow while changing the surroundings,in turn,the dry soil,submerged soil and saturated soil,implying a gradually increased excessive mobility of granular flow with the friction weakening of granular soil.Presence of water in granular flow may be a potential catalyzer to yield a long run-out granular flow,as revealed in comparison of water-absent and water-present granular flows.In addition,the dynamic erosion and entrainment of based soil induced by granular flow subjected to the complex geological settings,i.e.,the aspect ratio,granular mass,gravity,friction and dilatancy resistance,and presence of water,were comprehensively investigated as well.

Numerical Simulation of Oil-Water Two-Phase Flow in Low Permeability Tight Reservoirs Based on Weighted Least Squares Meshless Method

In response to the complex characteristics of actual low-permeability tight reservoirs,this study develops a meshless-based numerical simulation method for oil-water two-phase flow in these reservoirs,considering complex boundary shapes.Utilizing radial basis function point interpolation,the method approximates shape functions for unknown functions within the nodal influence domain.The shape functions constructed by the aforementioned meshless interpolation method haveδ-function properties,which facilitate the handling of essential aspects like the controlled bottom-hole flow pressure in horizontal wells.Moreover,the meshless method offers greater flexibility and freedom compared to grid cell discretization,making it simpler to discretize complex geometries.A variational principle for the flow control equation group is introduced using a weighted least squares meshless method,and the pressure distribution is solved implicitly.Example results demonstrate that the computational outcomes of the meshless point cloud model,which has a relatively small degree of freedom,are in close agreement with those of the Discrete Fracture Model(DFM)employing refined grid partitioning,with pressure calculation accuracy exceeding 98.2%.Compared to high-resolution grid-based computational methods,the meshless method can achieve a better balance between computational efficiency and accuracy.Additionally,the impact of fracture half-length on the productivity of horizontal wells is discussed.The results indicate that increasing the fracture half-length is an effective strategy for enhancing production from the perspective of cumulative oil production.

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 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.

Numerical Methods for a Class of Quadratic Matrix Equations

Quadratic matrix equations arise in many elds of scienti c computing and engineering applications.In this paper,we consider a class of quadratic matrix equations.Under a certain condition,we rst prove the existence of minimal nonnegative solution for this quadratic matrix equation,and then propose some numerical methods for solving it.Convergence analysis and numerical examples are given to verify the theories and the numerical methods of this paper.

In-situ measurement via the flow-through method and numerical simulations for radon exhalation during measurements of the radon exhalation rate

Small-scale measurements of the radon exhalation rate using the flow-through and closed-loop methods were conducted on the surface of a uranium tailing pond to better understand the differences between the two methods.An abnormal radon exhalation behavior was observed,leading to computational fluid dynamics(CFD)-based simulations in which dynamic radon migration in a porous medium and accumulation chamber was considered.Based on the in-situ experimental and numerical simulation results,variations in the radon exhalation rate subject to permeability,flow rate,and insertion depth were quantified and analyzed.The in-situ radon exhalation rates measured using the flow-through method were higher than those measured using the closed-loop method,which could be explained by the negative pressure difference between the inside and outside of the chamber during the measurements.The consistency of the variations in the radon exhalation rate between the experiments and simulations suggests the reliability of CFD-based techniques in obtaining the dynamic evolution of transient radon exhalation rates for diffusion and convection at the porous medium-air interface.The synergistic effects of the three factors(insertion depth,flow rate,and permeability)on the negative pressure difference and measured exhalation rate were quantified,and multivariate regression models were established,with positive correlations in most cases;the exhalation rate decreased with increasing insertion depth at a permeability of 1×10^(−11) m^(2).CFD-based simulations can provide theoretical guidance for improving the flow-through method and thus achieve accurate measurements.

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.

Determination method of mesh size for numerical simulation of blast load in near-ground detonation

In order to improve the overall resilience of the urban infrastructures, it is required to conduct blast resistant design for important building structures in the city. For complex terrain in the city, it is recommended to determine the blast load on the structures via numerical simulation. Since the mesh size of the numerical model highly depends on the explosion scenario, there is no generally applicable approach for the mesh size selection. An efficient method to determine the mesh size of the numerical model of near-ground detonation based on explosion scenarios is proposed in this study. The effect of mesh size on the propagation of blast wave under different explosive weights was studied, and the correlations between the mesh size effect and the charge weight or the scaled distance was described. Based on the principle of the finite element method and Hopkinson-Cranz scaling law, a mesh size measurement unit related to the explosive weight was proposed as the criterion for determining the mesh size in the numerical simulation. Finally, the applicability of the method proposed in this paper was verified by comparing the results from numerical simulation and the explosion tests and was verified in AUTODYN.

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.

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.

An Extended Numerical Method by Stancu Polynomials for Solution of Integro-Differential Equations Arising in Oscillating Magnetic Fields

In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method.

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 Analysis of Bacterial Meningitis Stochastic Delayed Epidemic Model through Computational Methods

Based on theWorld Health Organization(WHO),Meningitis is a severe infection of the meninges,the membranes covering the brain and spinal cord.It is a devastating disease and remains a significant public health challenge.This study investigates a bacterial meningitis model through deterministic and stochastic versions.Four-compartment population dynamics explain the concept,particularly the susceptible population,carrier,infected,and recovered.The model predicts the nonnegative equilibrium points and reproduction number,i.e.,the Meningitis-Free Equilibrium(MFE),and Meningitis-Existing Equilibrium(MEE).For the stochastic version of the existing deterministicmodel,the twomethodologies studied are transition probabilities and non-parametric perturbations.Also,positivity,boundedness,extinction,and disease persistence are studiedrigorouslywiththe helpofwell-known theorems.Standard and nonstandard techniques such as EulerMaruyama,stochastic Euler,stochastic Runge Kutta,and stochastic nonstandard finite difference in the sense of delay have been presented for computational analysis of the stochastic model.Unfortunately,standard methods fail to restore the biological properties of the model,so the stochastic nonstandard finite difference approximation is offered as an efficient,low-cost,and independent of time step size.In addition,the convergence,local,and global stability around the equilibria of the nonstandard computational method is studied by assuming the perturbation effect is zero.The simulations and comparison of the methods are presented to support the theoretical results and for the best visualization of results.

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.