Finite Element Difference Methods Research Articles

Investigation on the failure mechanism for surrounding rock in small interval tunnels using finite element limit analysis method

This paper presents a case study to evaluate the failure mechanism of surrounding rock in mountain tunnels caused by small interval parallelism and the reinforcement effect of intermediate rock columns. The characteristics and reasons of surrounding rock collapse were analyzed through field investigation, taking Yaoxi Tunnel as a case study. The critical collapse and blowout pressure results obtained by the finite element difference method (FEDM) software FLAC and the finite element limit method (FELAM) software OPTUM were compared based on Mollon’s theoretical solution. The critical support pressure errors for collapse are 18.803% and 8.129%, indicating that OPTUM has a slight advantage in accuracy for solving tunnel collapse. Therefore, the models of small interval tunnels were established by OPTUM to investigate the collapse failure mechanism and progressive failure process of surrounding rock. The effects of tunnel depth ratio (H/D), spacing ratio (S/D), and surrounding rock parameters (γ, φ, c) on critical support pressure, failure mode, and surrounding rock settlement laws were studied through simulation models. It shows that the gradual decrease in mechanical parameters of rock is the main cause of the increase in critical support pressure. The design parameters (H/D, S/D) have a mutagenic effect on the pressure arch of the double tunnel surrounding rock. As the distance between the two holes increases, the joint pressure point in the intermediate rock column gradually rises. Finally, the effectiveness of the reinforcement measures for the intermediate rock column was verified through on-site monitoring. The research results of this study have reference price for similar projects.

Theoretical study of electron mobility in AlGaN back-barrier Al2O3/InAlN/GaN heterostructures under optical phonon scattering

Considering the strong built-in electric field caused by polarization in wurtzite nitride heterostructures, the energy levels and wave functions of the two-dimensional electrons in AlyGa1−yN back-barrier Al2O3/InxAl1−xN/GaN heterostructures are calculated using the finite element difference method. The dispersion relations and electrostatic potentials of optical phonons are obtained by the transfer matrix method. The electron mobility under optical phonon scattering is studied based on the theory of Lei–Ting force balance equation. The influences of AlyGa1−yN back barrier, ternary mixed crystal effect, and size effect are also analyzed by comparing with the Al2O3/InxAl1−xN/GaN heterostructure without back barrier. It is found that the introduction of a back barrier can attenuate the effect of gate-insulating layer, but enhance the effect of barrier and channel layers on electronic states. Then, the interaction between electrons and optical phonons is weakened, so the electron mobility in AlyGa1−yN back-barrier Al2O3/InxAl1−xN/GaN heterostructure is higher than that in Al2O3/InxAl1−xN/GaN structure under the same conditions. It is also found that using a thinner AlyGa1−yN film with y = 0.25 as the back-barrier layer of Al2O3/InxAl1−xN/GaN heterostructure is more conducive to improving 2DEG mobility. These conclusions can provide references for the preparation of InxAl1−xN/GaN heterojunction devices.

Direct Shear Tests on Soft Clay Reinforced with Single Ordinary Granular Column: Discrete Element-Finite Difference Method

Direct Shear Tests on Soft Clay Reinforced with Single Ordinary Granular Column: Discrete Element-Finite Difference Method

Influence of different axis-translation techniques using ceramic disks/microporous membrane filters on mechanical/hydraulic behavior of unsaturated soil

Element tests, such as water retention tests and triaxial tests, are efficient and widely used methods for investigating the elementary hydro-mechanical behavior of unsaturated soil. In recent research, discrepancies observed in unsaturated test results using axis-translation techniques (ATTs) with ceramic disks and microporous membrane filters (MM filters) have indicated that misunderstandings might be hiding behind the perceived elementary behavior seen in element tests. In this study, triaxial tests with ceramic disks and MM filters were firstly conducted on specimens of unsaturated completely decomposed granite sand, referred to as Masado in Japan, to compare the influence of different techniques on the test results. Then a soil–water-air coupled finite element-finite difference (FE-FD) method, based on a newly proposed unsaturated/saturated constitutive model coupled with a deformation-dependent water retention curve (WRC), was utilized to simulate triaxial tests as boundary value problems (BVPs) because of the non-uniform deformation of the specimens. Drained/vented triaxial tests showed that the stress–strain relation of Masado is basically the same regardless of the applied technique, while the amount of drainage discharge and degree of saturation during the shearing stage are quite different. The calculated average stress–strain relation basically reproduced a tendency similar to that of the tests. It was also shown from the calculation that the ATT with ceramic disks or MM filters significantly influences the distributions of saturation and stress within the specimens during shearing. It should be emphasized that unsaturated triaxial tests with the ATT are basically BVPs instead of so-called element tests.

Jointed rock masses as metamaterials – Implications for railway tunnel vibrations

Vibrations from underground railways are known to affect receptors located close to the tunnel. Understanding the transmission pathway between these tunnels and receptors is important in determining the magnitude of vibrations which can be transmitted. Rail tunnels can be excavated in a range of geological conditions, including jointed rock masses. Jointed rock masses have been found to display certain resonant characteristics, namely spring resonance. Therefore, this paper studies the resonant characteristics of jointed rock masses using a series of models solved using the combined discrete element-finite difference method and the finite difference method. Modelling assumptions are tested as well as different equivalent material models. It is found that spring resonances occur at the same frequencies as predicted by analytical functions when different modelling assumptions are used. This indicates that the spring resonance effect will prevail in complex rock masses under a range of geological settings. The spring resonance mechanism is found to cause jointed rock masses to behave like periodic metamaterials in respect to the transmission of stress waves, which can operate as a band-pass or low pass filter, depending on the number of joints within the material. New evidence is presented showing that periodic metamaterials exhibit spring resonance. Results for metamaterials in laboratory scale frequency sweep tests are shown to feature high transmission zones occurring at the predicted spring resonant frequency for that material. Finally, the effects of the spring resonance mechanism operating within the jointed rock masses are appraised in the context of vibrations from railway tunnels.

Numerical modelling of resonance mechanisms in jointed rocks using transfer functions

Resonance effects in parallel jointed rocks subject to stress waves are investigated using transfer functions, derived from signals generated through numerical modelling. Resonance is important for a range of engineering situations as it identifies the frequency of waves which will be favourably transmitted. Two different numerical methods are used for this study, adopting the finite difference method and the combined discrete element-finite difference method. The numerical models are validated by replicating results from previous studies. The two methods are found to behave similarly and show the same resonance effects; one operating at low frequency and the other operating at relatively high frequency. These resonance effects are interpreted in terms of simple physical systems and analytical equations are derived to predict the resonant frequencies of complex rock masses. Low frequency resonance is shown to be generated by a system synonymous with masses between springs, described as spring resonance, with an equal number of resonant frequencies as the number of blocks. High frequency resonance is generated through superposition of multiple reflected waves developing standing waves within intact blocks, described as superposition resonance. While resonance through superposition has previously been identified, resonance based on masses between springs has not been previously identified in jointed rocks. The findings of this study have implications for future analysis of multiple jointed rock masses, showing that a wave travelling through such materials can induce other modes of propagation of waves, i.e. spring resonance.

Numerical Simulation of Electrode Structure on High Voltage Pulse Discharge Characteristics

For the study of the different electrode structures of the high voltage pulse discharge breakdown process of the insulating medium, the numerical simulation software is compiled to establish two kinds of the electrode structure of high voltage pulse discharge three-dimensional model, finite element difference method to simulate the expansion of plasma channel, the discharge characteristics of electrode structure internal voltage, and current of plasma channel and the impact of energy consumption. The simulation results show that, compared with the needle-needle electrode structure, the needle-plate electrode structure has a shorter time to reach the critical field strength of distilled water smaller breakdown voltage, shorter time to establish the discharge channel, smaller equivalent resistance, faster extension speed of the discharge channel, and less energy consumption for breakdown. And other conditions are the same. The results of this study can provide a theoretical reference for the structural design of high-voltage pulse-crushing electrodes.

A Prediction Model of Effective Thermal Conductivity for Metal Powder Bed in Additive Manufacturing

In current research, many researchers propose analytical expressions for calculating the packing structure of spherical particles such as DN Model, Compact Model and NLS criterion et al. However, there is still a question that has not been well explained yet. That is: What is the core factors affecting the thermal conductivity of particles? In this paper, based on the coupled discrete element-finite difference (DE-FD) method and spherical aluminum powder, the relationship between the parameters and the thermal conductivity of the powder (ETCp) is studied. It is found that the key factor that can described the change trend of ETCp more accurately is not the materials of the powder but the average contact area between particles (aave) which also have a close nonlinear relationship with the average particle size d50. Based on this results, the expression for calculating the ETCp of the sphere metal powder is successfully reduced to only one main parameter d50 and an efficient calculation model is proposed which can applicate both in room and high temperature and the corresponding error is less than 20.9% in room temperature. Therefore, in this study, based on the core factors analyzation, a fast calculation model of ETCp is proposed, which has a certain guiding significance in the field of thermal field simulation.

A nodal immersed finite element-finite difference method

The immersed finite element-finite difference (IFED) method is a computational approach to modeling interactions between a fluid and an immersed structure. The IFED method uses a finite element (FE) method to approximate the stresses, forces, and structural deformations on a structural mesh and a finite difference (FD) method to approximate the momentum and enforce the incompressibility of the entire fluid-structure system on a Cartesian grid. The fundamental approach used by this method follows the immersed boundary framework for modeling fluid-structure interaction (FSI), in which a force spreading operator prolongs structural forces to a Cartesian grid, and a velocity interpolation operator restricts a velocity field defined on that grid back onto the structural mesh. With an FE structural mechanics framework, force spreading first requires that the force itself be projected onto the finite element space. Similarly, velocity interpolation requires projecting velocity data onto the FE basis functions. Consequently, evaluating either coupling operator requires solving a matrix equation at every time step. Mass lumping, in which the projection matrices are replaced by diagonal approximations, has the potential to accelerate this method considerably. This paper provides both numerical and computational analyses of the effects of this replacement for evaluating the force projection and for the IFED coupling operators. Constructing the coupling operators also requires determining the locations on the structure mesh where the forces and velocities are sampled. Here we show that sampling the forces and velocities at the nodes of the structural mesh is equivalent to using lumped mass matrices in the IFED coupling operators. A key theoretical result of our analysis is that if both of these approaches are used together, the IFED method permits the use of lumped mass matrices derived from nodal quadrature rules for any standard interpolatory element. This is different from standard FE methods, which require specialized treatments to accommodate mass lumping with higher-order shape functions. Our theoretical results are confirmed by numerical benchmarks, including standard solid mechanics tests and examination of a dynamic model of a bioprosthetic heart valve.

Study on the Evolution of a Flooded Tailings Pond and Its Post-Failure Effects

In order to avoid the risk of tailing pond failures and to minimize the post-failure losses, it is necessary to analyze the current operation status of tailings ponds, to explore the evolution law of their failure process, to grasp their post-failure impact range, and to propose corresponding effective prevention and control measures. Based on a tailings pond in China, this paper establishes a 1:200 scale indoor model to explore the evolution law of post-failure tailings discharge in a tailings pond under flooded roof conditions; secondly, the finite element difference method and smooth particle fluid dynamics are combined to compare and analyze the post-failure impact area and to delineate the risk prevention and control area. The results of the study show that, during the dam break, the lower tailing sand in the breach is the first to slip, and after forming a steep can, the upper tailing sand in the steep can is pulled to slip, so that the erosion trench mainly develops vertically first, and then laterally. The velocity of the discharged tailing sand will quickly reach its peak value in a short period of time and then decrease to the creeping stage; the front edge of the sand flow is the first to stop moving, and the trailing edge of the tailing sand accumulation depth continues to increase until the end of the dam failure, at which point the initial bottom dam area of the discharge tailing sand flow velocity is the largest. The further the tailings are released from the initial dam, the smaller the accumulation depth and the larger the particle size, and the larger the elevation of the foundation in the same section, the smaller the accumulation depth and the larger the particle size; further, the presence of blocking materials will increase the local tailings accumulation depth. Based on the maximum flow velocity of the discharged tailings and the accumulation depth, the risk area downstream of the tailings pond is divided, so that relocation measures can be formulated. The test results can provide an important reference for the operation and management of similar tailings ponds.

Improvement of well drainage radius calculation

The calculation of well drainage radius is improved on the example of Chervonozayarske gas field. Theoretical research methods are applied, which include a systematic analysis of the information used, numerical modeling based on the combined finite element-difference method, methods of visual representation of the information obtained, analytical methods To increase the accuracy of the calculation of the well drainage radius, the heterogeneity of the porous medium structure was taken into account. It is established that the drainage radius of a gas well expands with time. This allows to determine the optimal grid of production wells, the number and location of injection wells to ensure their interaction, and hence ¬ reduce material costs in the gas industry

Approval of the combined reserve calculation method (Chervonozayarske Gas Field as an example)

The object of the study is the calculation of the producing reserves of the gas-bearing reservoir. Increasingly, published studies provide justification for the possibility of restoring reserves in old depleted gas fields by gas flow from deep horizons. Given the possibility of resuming producing gas reserves, the issue of clarifying their volume in the reservoirs of fields at a late stage of development is promising. In the course of the study, theoretical research methods were used: a system analysis of the inreservoir used, numerical modeling based on a combined finite element-difference method, methods for visualizing the inreservoir obtained, and analytical methods. The method proposed in the paper for refining gas reserves combines the volumetric method and modeling of filtration processes using a combined finite element-difference method. The latter makes it possible to take into account the reservoir structure that is heterogeneous in terms of permeability and adequately describe the distribution of non-stationary reservoir pressure around the production well on a quantitative level. By applying an analytical formula based on the values of average reservoir and bottomhole pressures, the radii of the well feed contour were calculated for different periods of reservoir development. Thus, the active area (and volume) of the reservoir is determined, according to which the calculations of the producing reserves of the field are carried out. The study was carried out on the example of the Chervonozayarske Gas Field (Ukraine) for the reservoir V-26-T-1a, discovered by one production well 468-B(D). The recoverable reserves of the V-26-T-1a reservoir calculated in this way are 597.69 million m3 of gas. At the same time, the error relative to the value indicated in the Atlas of Ukrainian Fields is 4.63 %. The method of calculating reserves proposed in this study is useful for refining the reserves of depleted fields. The combination of the volumetric method with the results of simulation of filtration processes is an operational method for calculating the reserves of a reservoir discovered by one production well. At the same time, the use of a combined finite element-difference method makes it possible to take into account the complex heterogeneous structure of the reservoir and predict the distribution of reservoir non-stationary pressures around the production well.

Dynamic Response of a Four-Pile Group Foundation in Liquefiable Soil Considering Nonlinear Soil-Pile Interaction

Piles, which are always exposed to dynamic loads, are widely used in offshore structures. The dynamic response of the pile-soil-superstructure system in liquefiable soils is complicated, and the interaction between the pile and soil and the pile volume effect are the key influencing factors. In this study, a water-soil fully coupled dynamic finite element-finite difference (FE-FD) method was used to numerically simulate the centrifuge shaking table (CST) test of a four-pile group in saturated sand soil. An interface contact model was proposed to simulate the pile-soil interaction, and a solid element was used to consider the volume effect of the pile. The acceleration responses of the soil and pile, settlement deformation, excess pore water pressure, and bending moment were examined. The results show that the bending moment response of the two piles parallel to the shaking direction show minor differences, while the two piles perpendicular to the shaking direction show almost the same distribution. The values of excess pore water pressure at the same depth but different azimuth angles around the pile are also different. The numerical simulation can accurately reproduce soil deformation and pile internal force during and after dynamic loading.

Magnetic Field Effect on Heat and Momentum of Fractional Maxwell Nanofluid within a Channel by Power Law Kernel Using Finite Difference Method

The mathematical model of physical problems interprets physical phenomena closely. This research work is focused on numerical solution of a nonlinear mathematical model of fractional Maxwell nanofluid with the finite difference element method. Addition of nanoparticles in base fluids such as water, sodium alginate, kerosene oil, and engine oil is observed, and velocity profile and heat transfer energy profile of solutions are investigated. The finite difference method involving the discretization of time and distance parameters is applied for numerical results by using the Caputo time fractional operator. These results are plotted against different physical parameters under the effects of magnetic field. These results depicts that a slight decrease occurs for velocity for a high value of Reynolds number, while a small value of Re provides more dominant effects on velocity and temperature profile. It is observed that fractional parameters α and β show inverse behavior against u y , t and θ y , t . An increase in volumetric fraction of nanoparticles in base fluids decreases the temperature profile of fractional Maxwell nanofluids. Using mathematical software of MAPLE, codes are developed and executed to obtain these results.

Optimization of radial flexible mechanism in scroll compressor based on oil film lubrication analysis

PurposeThe leakage problem caused by machining error, assembly error, wearing and thermal deformation has been the main factor hindering the development of scroll compressor. This paper aims to investigate the lubrication characteristics of radial clearance and further optimize the radial clearance, which can reduce the leakage in the tangential direction of the working chamber.Design/methodology/approachThis paper establishes a model of radial clearance oil film lubrication in scroll compressor. And, the method to solve the Reynolds and energy equations is presented, as well as the dimensionless and discretization by finite element difference method. To verify the established model, performance experiment of scroll compressor for electric vehicle air conditioning system is also carried out.FindingsBased on the presented model, the temperature field and distribution of the oil film in the radial clearance are analyzed. And the influence of the structural parameter on the radial clearance is further discussed. The optimum radial clearance could be achieved at β = 40°–42°, where the orbiting scroll is in the state of rotary balance. And, the simulation results coincide well with the experimental results.Originality/valueThis work provides an effective model to evaluate the lubrication characteristics of radial clearance in scroll compressor, which can provide guidance for the design of scroll compressor.

A Nodal Immersed Finite Element-Finite Difference Method

The immersed finite element-finite difference (IFED) method is a computational approach to modeling interactions between a fluid and an immersed structure. This method uses a finite element (FE) method to approximate the stresses and forces on a structural mesh and a finite difference (FD) method to approximate the momentum of the entire fluid-structure system on a Cartesian grid. The fundamental approach used by this method follows the immersed boundary framework for modeling fluid-structure interaction (FSI), in which a force spreading operator prolongs structural forces to a Cartesian grid, and a velocity interpolation operator restricts a velocity field defined on that grid back onto the structural mesh. Force spreading and velocity interpolation both require projecting data onto the finite element space. Consequently, evaluating either coupling operator requires solving a matrix equation at every time step. Mass lumping, in which the projection matrices are replaced by diagonal approximations, has the potential to accelerate this method considerably. Constructing the coupling operators also requires determining the locations on the structure mesh where the forces and velocities are sampled. Here we show that sampling the forces and velocities at the nodes of the structural mesh is equivalent to using lumped mass matrices in the coupling operators. A key theoretical result of our analysis is that if both of these approaches are used together, the IFED method permits the use of lumped mass matrices derived from nodal quadrature rules for any standard interpolatory element. This is different from standard FE methods, which require specialized treatments to accommodate mass lumping with higher-order shape functions. Our theoretical results are confirmed by numerical benchmarks, including standard solid mechanics tests and examination of a dynamic model of a bioprosthetic heart valve.

Difference finite element method for the 3D steady Stokes equations

In this paper, a difference finite element (DFE) method is presented for the 3D steady Stokes equations. This new method consists of transmitting the finite element solution (uh,ph) of the 3D steady Stokes equations in the direction of (x,y,z) into a series of the finite element solution (uhk,phk) of the 2D steady Stokes equations. Here the 2D steady Stokes equations are solved by the finite element space pair (P1b,P1b,P1)×P1, where the 2D finite element pair (P1b,P1b)×P1 satisfies the discrete inf-sup condition in a 2D domain ω. Here we design the weak formulation of the DFE method based on the 3D finite element pair ((P1b,P1b,P1)×P1)×(P1×P0) under the quasi-uniform mesh condition, prove that the 3D finite element pair satisfies the discrete inf-sup condition in a 3D domain Ω and provide the existence, uniqueness and stability of the DFE solution (uh,ph)=(∑k=0l3uhkϕk(z),∑k=1l3phkψk(z)) and deduce the first order convergence of the DFE solution (uh,ph) with respect to the exact solution (u,p) of the 3D steady Stokes equations. Finally, some numerical tests are presented to show the accuracy and efficiency for the proposed method.

An implicit-explicit local method for parabolic partial differential equations

PurposeThe purpose of this article is to derive an implicit-explicit local differential transform method (IELDTM) in dealing with the spatial approximation of the stiff advection-diffusion-reaction (ADR) equations.Design/methodology/approachA direction-free numerical approach based on local Taylor series representations is designed for the ADR equations. The differential equations are directly used for determining the local Taylor coefficients and the required degrees of freedom is minimized. The complete system of algebraic equations is constructed with explicit/implicit continuity relations with respect to direction parameter. Time integration of the ADR equations is continuously utilized with the Chebyshev spectral collocation method.FindingsThe IELDTM is proven to be a robust, high order, stability preserved and versatile numerical technique for spatial discretization of the stiff partial differential equations (PDEs). It is here theoretically and numerically shown that the order refinement (p-refinement) procedure of the IELDTM does not affect the degrees of freedom, and thus the IELDTM is an optimum numerical method. A priori error analysis of the proposed algorithm is done, and the order conditions are determined with respect to the direction parameter.Originality/valueThe IELDTM overcomes the known disadvantages of the differential transform-based methods by providing reliable convergence properties. The IELDTM is not only improving the existing Taylor series-based formulations but also provides several advantages over the finite element method (FEM) and finite difference method (FDM). The IELDTM offers better accuracy, even when using far less degrees of freedom, than the FEM and FDM. It is proven that the IELDTM produces solutions for the advection-dominated cases with the optimum degrees of freedom without producing an undesirable oscillation.

Study the effect of circular hole on dynamic fracture properties of cracked PMMA specimen under impact loads

Hole defects and crack defects usually coexist in a real engineering rock masses, they together affect the structural security of rock masses. With the purpose of investigating the effect of circular hole on crack initiation and propagation process under dynamic loads, a circular opening specimen with straight crack cavity (COSSCC) model was proposed in this study and polymethyl methacrylate was chosen to make the specimens. A series of dynamic experiments were carried out utilizing a drop hammer impact test system (DHITS), crack speeds and crack initiation time were determined utilizing crack propagation gauge (CPG). The maximum principal stress field evolution in the crack evolution process was simulated using a modified finite element difference method software, and dynamic fracture toughness was calculated by a traditional finite element method software. Some significantly conclusions can be obtained: As the inclination angle θ of the circular hole is less than 10o, crack propagation trajectories will propagate to the surface of the circular hole; As the inclination angle θ of circular hole is 20o and 30o, crack propagation trajectories will turn in the direction of circular hole but not reach the circular hole; As circular hole inclination angle θ is 40o and 50o, the crack shows a pure mode I crack propagation. As crack propagates near the circular hole, crack propagation speeds increase significantly. The dynamic propagation toughness during crack propagation is not constant and is less than dynamic initiation toughness (DIT) and the dynamic propagate toughness (DPT) is inversely proportional to the crack speeds.

Numerical Investigation of Multistage Fractured Horizontal Wells considering Multiphase Flow and Geomechanical Effects

Hydraulic fracturing is a key technology in unconventional reservoir production, yet many simulators only consider the single-phase flow of shale gas, ignoring the two-phase flow process caused by the retained fracturing fluid in the early stage of production. In this study, a three-dimensional fluid–gas–solid coupling reservoir model is proposed, and the governing equations which involve the early injection water phenomenon and stress-sensitive characteristics of shale gas reservoirs are established. The finite element–finite difference method was used for discretisation of stress and strain equations and the equations of flow balances. Further, a sensitivity analysis was conducted to analyse fracture deformation changes in the production. Fracture characteristics under different rock mechanics coefficients were simulated, and the influence of rock mechanics parameters on productivity was further characterised. The stimulated reservoir volume zone permeability could determine the retrofitting effect, the permeability increased from 0.02 to 0.1 mD, and cumulative gas production increased from 18.08 to 26.42 million m3, thus showing an increase of 8.34 million m3, or 46%. The effect of Young’s modulus on the yield was smaller than Poisson’s ratio and the width and length of the fractures. Production was most sensitive to the length of the fractures. The length of the fracture increased from 200 to 400 m, and the cumulative gas production increased from 26.44 to 38.34 million m3, showing an increase of 11.9 million m3, or 45%. This study deepens the understanding of the production process of shale gas reservoirs and has significance for the fluid–gas–solid coupling of shale gas reservoirs.