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.

A Hybrid Dung Beetle Optimization Algorithm with Simulated Annealing for the Numerical Modeling of Asymmetric Wave Equations

In the generalized continuum mechanics(GCM)theory framework,asymmetric wave equations encompass the characteristic scale parameters of the medium,accounting for microstructure interactions.This study integrates two theoretical branches of the GCM,the modified couple stress theory(M-CST)and the one-parameter second-strain-gradient theory,to form a novel asymmetric wave equation in a unified framework.Numerical modeling of the asymmetric wave equation in a unified framework accurately describes subsurface structures with vital implications for subsequent seismic wave inversion and imaging endeavors.However,employing finite-difference(FD)methods for numerical modeling may introduce numerical dispersion,adversely affecting the accuracy of numerical modeling.The design of an optimal FD operator is crucial for enhancing the accuracy of numerical modeling and emphasizing the scale effects.Therefore,this study devises a hybrid scheme called the dung beetle optimization(DBO)algorithm with a simulated annealing(SA)algorithm,denoted as the SA-based hybrid DBO(SDBO)algorithm.An FD operator optimization method under the SDBO algorithm was developed and applied to the numerical modeling of asymmetric wave equations in a unified framework.Integrating the DBO and SA algorithms mitigates the risk of convergence to a local extreme.The numerical dispersion outcomes underscore that the proposed SDBO algorithm yields FD operators with precision errors constrained to 0.5‱while encompassing a broader spectrum coverage.This result confirms the efficacy of the SDBO algorithm.Ultimately,the numerical modeling results demonstrate that the new FD method based on the SDBO algorithm effectively suppresses numerical dispersion and enhances the accuracy of elastic wave numerical modeling,thereby accentuating scale effects.This result is significant for extracting wavefield perturbations induced by complex microstructures in the medium and the analysis of scale effects.

Progressive fragmentation of granular assemblies within rockslides: Insights from discrete-continuous numerical modeling

Rock fragmentation plays a critical role in rock avalanches,yet conventional approaches such as classical granular flow models or the bonded particle model have limitations in accurately characterizing the progressive disintegration and kinematics of multi-deformable rock blocks during rockslides.The present study proposes a discrete-continuous numerical model,based on a cohesive zone model,to explicitly incorporate the progressive fragmentation and intricate interparticle interactions inherent in rockslides.Breakable rock granular assemblies are released along an inclined plane and flow onto a horizontal plane.The numerical scenarios are established to incorporate variations in slope angle,initial height,friction coefficient,and particle number.The evolutions of fragmentation,kinematic,runout and depositional characteristics are quantitatively analyzed and compared with experimental and field data.A positive linear relationship between the equivalent friction coefficient and the apparent friction coefficient is identified.In general,the granular mass predominantly exhibits characteristics of a dense granular flow,with the Savage number exhibiting a decreasing trend as the volume of mass increases.The process of particle breakage gradually occurs in a bottom-up manner,leading to a significant increase in the angular velocities of the rock blocks with increasing depth.The simulation results reproduce the field observations of inverse grading and source stratigraphy preservation in the deposit.We propose a disintegration index that incorporates factors such as drop height,rock mass volume,and rock strength.Our findings demonstrate a consistent linear relationship between this index and the fragmentation degree in all tested scenarios.

Numerical modeling of destress blasting for strata separation

Destress blasting(DB)implemented along the perimeter of safety pillars is a special application of destressing in coal longwall mining.The goal is to separate relatively more deformed mined areas from safety pillars,such as shaft pillars or cross-cut pillars,to reduce the transfer of high stresses to the protective pillar.This case study aims to numerically simulate selected destress blasts in the Czech part of the Upper Silesian Coal Basin and examine its impact on stress transfer to the safety pillar area.To separate the area between the protective pillar and the longwall(LW),two fans of five 93-mm blast holes(length of 93e100 m)were drilled from the gate roads into the overburden strata.Each set of blast holes was fired separately in two stages without time delay.The explosive charge(gelatin-type of explosive)of each stage is 3450 kg.The two DB stages were fired when the longwall face was approximately 158 m and 152 m away from the blast.A 3D mine-wide model is built and validated with in situ stress measured with hydrofracturing.Mining and destressing in three 5-m thick coal seams are simulated in the region.Numerical modeling of DB is successfully conducted using a rock fragmentation factor a of 0.05 and a stress reduction/dissipation factor β of 0.95.Buffering of transfer of additional stress from the mining area into the safety pillar is evaluated by comparison of yielding volume before and after DB.It is shown that yielding volume drops after DB by nearly 80%in the area of the destressing panel and near the safety shaft pillar.

Flow-Induced Clogging in Microfiltration Membranes: Numerical Modeling and Parametric Study

Microfiltration membrane technology has been widely used in various industries for solid-liquid separation. However, pore clogging remains a persistent challenge. This study employs (CFD) and discrete element method (DEM) models to enhance our understanding of microfiltration membrane clogging. The models were validated by comparing them to experimental data, demonstrating reasonable consistency. Subsequently, a parametric study was conducted on a cross-flow model, exploring the influence of key parameters on clogging. Findings show that clogging is a complex phenomenon affected by various factors. The mean inlet velocity and transmembrane flux were found to directly impact clogging, while the confinement ratio and cosine of the membrane pore entrance angle had an inverse relationship with it. Two clog types were identified: internal (inside the pore) and external (arching at the pore entrance), with the confinement ratio determining the type. This study introduced a dimensionless number as a quantitative clogging indicator based on transmembrane flux, Reynolds number, filtration time, entrance angle cosine, and confinement ratio. While this hypothesis held true in simulations, future studies should explore variations in clogging indicators, and improved modeling of clogging characteristics. Calibration between numerical and physical times and consideration of particle volume fraction will enhance understanding.

Numerical modeling of the 2D time-domain transient electromagnetic secondary field of the line source of the current excitation

To effectively minimize the electromagnetic field response in the total field solution, we propose a numerical modeling method for the two-dimensional （2D） time- domain transient electromagnetic secondary field of the line source based on the DuFort- Frankel finite-difference method. In the proposed method, we included the treatment of the earth-air boundary conductivity, calculated the normalized partial derivative of the induced electromotive force （Emf）, and determined the forward time step. By extending upward the earth-air interface to the air grid nodes and the zero-value boundary conditions, not only we have a method that is more efficient but also simpler than the total field solution. We computed and analyzed the homogeneous half-space model and the fiat layered model with high precision--the maximum relative error is less than 0.01% between our method and the analytical method--and the solution speed is roughly three times faster than the total-field solution. Lastly, we used the model of a thin body embedded in a homogeneous half-space at different delay times to depict the downward and upward spreading characteristics of the induced eddy current, and the physical interaction processes between the electromagnetic field and the underground low-resistivity body.

Finite-difference numerical modeling with even-order accuracy in two-phase anisotropic media

To improve the accuracy of the conventional finite-difference method, finitedifference numerical modeling methods of any even-order accuracy are recommended. We introduce any even-order accuracy difference schemes of any-order derivatives derived from Taylor series expansion. Then, a finite-difference numerical modeling method with any evenorder accuracy is utilized to simulate seismic wave propagation in two-phase anisotropic media. Results indicate that modeling accuracy improves with the increase of difference accuracy order number. It is essential to find the optimal order number, grid size, and time step to balance modeling precision and computational complexity. Four kinds of waves, static mode in the source point, SV wave cusps, reflection and transmission waves are observed in two-phase anisotropic media through modeling.

Numerical modeling of PP- and PS-wave azimuthal anisotropy in HTI media

This study focuses on the factors that may affect the feasibility of performing elliptical anisotropy analysis on azimuthal PP and PS-wave data in HTI media, with the aim of using the modeling results as guidance in real seismic data application. Our results reveal that there is an offset limitation for both PP- and PS-waves in elliptical anisotropy fitting, and that PS-waves show a wider applicable offset range and larger observable azimuthal anisotropy than PP-waves. The major axis of the elliptical fit to the amplitudes of the R-component is perpendicular to the fracture strike, which is opposite to that in PP-wave analysis. The azimuthal interval travel time of PS-waves shows a nearly elliptical distribution and the major axis of the fit ellipse is perpendicular to the fracture strike, which is same as that in PP-wave analysis. For data within the applicable offset range, the anisotropic magnitude obtained from amplitude and travel time attributes of PP- and PS-waves exhibits a dependence on fracture density, and the major to minor axis ratio of the fit ellipse may be used to infer the relative distribution of fracture densities.

Numerical modeling of seabed morphological effects from the construction of artificial islands in Danzhou, Hainan

Haihua Islands is a large artificial island in Danzhou, Hainan. The construction of Haihua Islands changes the hYdrodynamic environment of Yangpu waters, and further affects its morphological change. Delft3D is used to set up a two dimensional nested hydrodynamic and sediment model for Yangpu waters in this paper, and this paper focuses on simulating the velocity and morphological change due to the construction of Haihua Islands after the verification of the model. The seabed deposition is small because of low suspended sediment concentration and less sand source near Yangpu waters. The bed level erodes in the south area of Xiaochan Reef and the Yangpu channel due to the velocity increase in the area.

A Heavy Sea Fog Event over the Yellow Sea in March 2005: Analysis and Numerical Modeling

In this paper, a heavy sea fog episode that occurred over the Yellow Sea on 9 March 2005 is investigated. The sea fog patch, with a spatial scale of several hundred kilometers at its mature stage, reduced visibility along the Shandong Peninsula coast to 100 m or much less at some sites. Satellite images, surface observations and soundings at islands and coasts, and analyses from the Japan Meteorology Agency （JMA） axe used to describe and analyze this event. The analysis indicates that this sea fog can be categorized as advection cooling fog. The main features of this sea fog including fog area and its movement axe reasonably reproduced by the Fifth-generation Pennsylvania State University/National Center for Atmospheric Research Mesoscale Model （MM5）. Model results suggest that the formation and evolution of this event can be outlined as： （1） southerly warm/moist advection of low-level air resulted in a strong sea-surface-based inversion with a thickness of about 600 m; （2） when the inversion moved from the warmer East Sea to the colder Yellow Sea, a thermal internal boundary layer （TIBL） gradually formed at the base of the inversion while the sea fog grew in response to cooling and moistening by turbulence mixing; （3） the sea fog developed as the TIBL moved northward and （4） strong northerly cold and dry wind destroyed the TIBL and dissipated the sea fog. The principal findings of this study axe that sea fog forms in response to relatively persistent southerly waxm/moist wind and a cold sea surface, and that turbulence mixing by wind shear is the primary mechanism for the cooling and moistening the marine layer. In addition, the study of sensitivity experiments indicates that deterministic numerical modeling offers a promising approach to the prediction of sea fog over the Yellow Sea but it may be more efficient to consider ensemble numerical modeling because of the extreme sensitivity to model input.

Examining the effect of adverse geological conditions on jamming of a single shielded TBM in Uluabat tunnel using numerical modeling

Severe shield jamming events have been reported during excavation of Uluabat tunnel through adverse geological conditions, which resulted in several stoppages at advancing a single shielded tunnel boring machine(TBM). To study the jamming mechanism, three-dimensional(3D) simulation of the machine and surrounding ground was implemented using the finite difference code FLAC3D. Numerical analyses were performed for three sections along the tunnel with a higher risk for entrapment due to the combination of overburden and geological conditions. The computational results including longitudinal displacement contours and ground pressure profiles around the shield allow a better understanding of ground behavior within the excavation. Furthermore, they allow realistically assessing the impact of adverse geological conditions on shield jamming. The calculated thrust forces, which are required to move the machine forward, are in good agreement with field observations and measurements. It also proves that the numerical analysis can effectively be used for evaluating the effect of adverse geological environment on TBM entrapments and can be applied to prediction of loads on the shield and preestimating of the required thrust force during excavation through adverse ground conditions.

The Numerical Modeling of 3-D Elastic Wave Equation Using a High-Order,Staggered-Grid, Finite Difference Scheme

This article provides the application of the high-order, staggered-grid, finite-difference scheme to model elastic wave propagation in 3-D isotropic media. Here, we use second-order, tempo- ral- and high-order spatial finite-difference formulations with a staggered grid for discretization of the 3-D elastic wave equations of motion. The set of absorbing boundary conditions based on paraxial approximations of 3-D elastic wave equations are applied to the numerical boundaries. The trial re- sults for the salt model show that the numerical dispersion is decreased to a minimum extent, the accuracy high and diffracted waves abundant. It also shows that this method can be used for modeling wave propagation in complex media with the lateral variation of velocity.

Numerical Modeling of Response and Damage of Masonry Walls to Blast Loading

To model the damage process of masonry walls under blast loading, a dynamic continuum damage material model is constructed for brick and mortar separately. The degradation of both the stiffness and strength are governed by a damage variable. By using the proposed material model, damage and fragmentation of a typical masonry wall under blast loading at different scaled distances is calculated. The hazard level of the masonry wall to blast loading is evaluated by analyzing the numerical results.

Application of numerical modeling and genetic programming to estimate rock mass modulus of deformation

Estimation of the rock mass modulus of deformation(Em)is one of the most important design parameters in designing many structures in and on rock.This parameter can be obtained by in situ tests,empirical relations between deformation modulus and rock mass classifcation,and estimating from laboratory tests results.In this paper,a back analysis calculation is performed to present an equation for estimation of the rock mass modulus of deformation using genetic programming(GP)and numerical modeling.A database of 40,960 datasets,including vertical stress(rz),horizontal to vertical stresses ratio(k),Poisson’s ratio(m),radius of circular tunnel(r)and wall displacement of circular tunnel on the horizontal diameter(d)for input parameters and modulus of deformation for output,was established.The selected parameters are easy to determine and rock mass modulus of deformation can be obtained from instrumentation data of any size circular galleries.The resulting RMSE of 0.86 and correlation coeffcient of97%of the proposed equation demonstrated the capability of the computer program(CP)generated by GP.

Numerical modeling for rockbursts:A state-of-the-art review

As the depth of excavation increases,rockburst becomes one of the most serious geological hazards damaging equipment and facilities and even causing fatalities in mining and civil engineering.This has forced researchers worldwide to identify different methods to investigate rockburst-related problems.However,some problems,such as the mechanisms and the prediction of rockbursts,continue to be studied because rockburst is a very complicated phenomenon influenced by the uncertainty and complexity in geological conditions,in situ stresses,induced stresses,etc.Numerical modeling is a widely used method for investigating rockbursts.To date,great achievements have been made owing to the rapid development of information technology(IT)and computer equipment.Hence,it is necessary and meaningful to conduct a review of the current state of the studies for rockburst numerical modeling.In this paper,the categories and the origin of different numerical approaches employed in modeling rockbursts are reviewed and the current usage of various numerical modeling approaches is investigated by a literature research.Later,a state-of-the-art review is implemented to investigate the application of numerical modeling in the mechanism study,and prediction and prevention of rockbursts.The main achievements and problems are highlighted.Finally,this paper discusses the limitations and the future research of numerical modeling for rockbursts.An approach is proposed to provide researchers with a systematic and reasonable numerical modeling framework.

Numerical modeling of metamorphic core complex formation:Implications for the destruction of the North China Craton

Widespread magmatism, metamorphic core complexes(MCCs), and significant lithospheric thinning occurred during the Mesozoic in the North China Craton(NCC). It has been suggested that the coeval exhumation of MCCs with uniform northwest-southeast shear senses and magmatism probably resulted from a decratonization event during the retreat of the paleo-Pacific Plate. Here we used two-dimensional finite element thermomechanical numerical models to investigate critical parameters controlling the formation of MCCs under far-field extensional stress. We observed three end-member deformation modes: the MCC mode, the symmetric-dome mode, and the pure-shear mode. The MCC mode requires a Moho temperature of ≥700 ℃ and an extensional strain rate of ≥5 × 10^(-16)s^(-1), implying that the lithosphere had already thinned when the MCC was formed in the Mesozoic. Considering that the widespread MCCs have the same northwest-southeast extension direction in the NCC, we suggest that the MCCs are surface expressions of both large-scale extension and craton destruction and that rollback of the paleo-Pacific slab might be the common driving force.

Numerical modeling of wave equation by a truncated high-order finite-difference method

Finite-difference methods with high-order accuracy have been utilized to improve the precision of numerical solution for partial differential equations. However, the computation cost generally increases linearly with increased order of accuracy. Upon examination of the finite-difference formulas for the first-order and second-order derivatives, and the staggered finite-difference formulas for the first-order derivative, we examine the variation of finite-difference coefficients with accuracy order and note that there exist some very small coefficients. With the order increasing, the number of these small coefficients increases, however, the values decrease sharply. An error analysis demonstrates that omitting these small coefficients not only maintain approximately the same level of accuracy of finite difference but also reduce computational cost significantly. Moreover, it is easier to truncate for the high-order finite-difference formulas than for the pseudospectral for- mulas. Thus this study proposes a truncated high-order finite-difference method, and then demonstrates the efficiency and applicability of the method with some numerical examples.

Numerical Modeling of the Hyperbolic Mild-Slope Equation in Curvilinear Coordinates

The mild-slope equation is familiar to coastal engineers as it can effectively describe wave propagation in nearshore regions. However, its computational method in Cartesian coordinates often renders the model inaccurate in areas with irregular shorelines, such as estuaries and harbors. Based on the hyperbolic mild-slope equation in Cartesian coordinates, the numerical model in orthogonal curvilinear coordinates is developed. The transformed model is discretized by the finite difference method and solved by the ADI method with space-staggered grids. The numerical predictions in curvilinear co- ordinates show good agreemenl with the data obtained in three typical physical expedments, which demonstrates that the present model can be used to simulate wave propagation, for normal incidence and oblique incidence, in domains with complicated topography and boundary conditions.

Numerical modeling of tidal current of LNG terminal in Caofeidian,Bohai Sea

In order to reasonably simulate tidal currents around small structures such as piles in a large-scale model domain, a 2-D hydrodynamic integrated model for Bohai Sea is established with the finite element method. The grid can be discretionarily refined as a non-structure triangle or quadrilateral so that piers can be treated as one or several impermeable elements with an area of 20 to 30 km^2 in a model domain over 85 700 km^2. The computational results of tidal levels and horizontal velocities are in good agreement with the field data. Based on the computed results by the model, the layout of an open 105 DWT liquefied natural gas （LNG）terminal in Caofeidian, Bohal Sea is effectively and reasonably optimized. It can be concluded that the model is suitable and reasonable for direct simulation of tidal currents around small structures in projects.

Numerical Modeling of a Spar Platform Tethered by a Mooring Cable

Virtual simulation is an economical and efficient method in mechanical system design. Numerical modeling of a spar platform, tethered by a mooring cable with a spherical joint is developed for the dynamic simulation of the floating structure in ocean. The geometry modeling of the spar is created using finite element methods. The submerged part of the spar bears the buoyancy, hydrodynamic drag force, and effect of the added mass and Froude-Krylov force. Strip theory is used to sum up the forces acting on the elements. The geometry modeling of the cable is established based on the lumped-mass-and-spring modeling through which the cable is divided into 10 elements. A new element-fixed local frame is used, which is created by the element orientation vector and relative velocity of the fluid, to express the loads acting on the cable. The bottom of the cable is fixed on the seabed by spring forces, while the top of the cable is connected to the bottom of the spar platform by a modified spherical joint. This system suffers the propagating wave and current in the X-direction and the linear wave theory is applied for setting of the propagating wave. Based on the numerical modeling, the displacement-load relationships are analyzed, and the simulation results of the numerical modeling are compared with those by the commercial simulation code, Proteus DS. The comparison indicates that the numerical modeling of the spar platform tethered by a mooring cable is well developed, which provides an instruction for the optimization of a floating structure tethered by a mooring cable system.