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.

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 study on the mold strength of magnetic mold casting based on a coupled electromagnetic-structural method

The properties of the magnetic mold in magnetic mold casting directly determine the quality of the final cast parts.In this study,the magnetic mold properties in magnetic mold casting,were studied utilizing a coupled electromagnetic-structural method through numerical simulation.This study investigated key factors including equivalent stress,the distribution of tensile and compressive stresses,and the area ratio of tensile stress.It compared molds made entirely of magnetic materials with those made partially of magnetic materials.Simulation results indicate that as current increases from 4 A to 8 A,both the initial magnetic mold and the material-replaced magnetic mold initially show an increasing trend in equivalent stress,tensile-compressive stress,and the area ratio of tensile stress,peaking at 6 A before declining.After material replacement,the area ratio of tensile stress at 6 A decreases to 19.84%,representing a reduction of 29.72%.Magnetic molds comprising a combination of magnetic and non-magnetic materials exhibit sufficient strength and a reduced area ratio of tensile stress compared to those made entirely from magnetic materials.This study provides valuable insights for optimizing magnetic mold casting processes and offers practical guidance for advancing the application of magnetic molds.

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.

Calculation of effective temperature for pavement rutting using numerical simulation methods

In order to predict the long-term rutting of asphalt pavement, the effective temperature for pavement rutting is calculated using the numerical simulation method. The transient temperature field of asphalt pavement was simulated based on actual meteorological data of Nanjing. 24-hour rutting development under a transient temperature field was calculated in each month. The rutting depth accumulated under the static temperature field was also estimated and the relationship between constant temperature parameters was analyzed. Then the effective temperature for pavement rutting was determined based on the rutting equivalence principle. The results show that the monthly effective temperature is above 40 t in July and August, while in June and September it ranges from 30 to 40 Rutting development can be ignored when the monthly effective temperature is less than 30 t. The yearly effective temperature for rutting in Nanjing is around 38. 5 t. The long-term rutting prediction model based on the effective temperature can reflect the influences of meteorological factors and traffic time distribution.

Application of ALE Method on the Numerical Simulation of Reinforced Concrete Penetration

LS-DYNA program and the principle of ALE method were introduced, and the target features of the reinforced concrete penetration were analyzed by using the D material model and the ALE method. A numerical simulation has been done to show the penetration visually and veritably. The simulation results are analyzed carefully and explicitly prove their significance to the research of reinforced concrete penetration.

A numerical study of two-and three-dimensional detonation dynamics of pulse detonation engine by the CE/SE method

In this paper, the CE/SE method is developed to simulate the two- and three-dimensional flow-field of Pulse Detonation Engine (PDE). The conservation equations with stiff source terms for chemical reaction are solved in two steps. The detailed analysis of computational results of a PDE with a single detonation tube and a PDE with five detonation tubes are given in this paper. Complex wave systems are observed inside and outside a PDE. For a PDE with 5 detonation tubes, there is a big bow shock produced from a number of little shocks near the open ends of tubes. A lot of vortexes interact with shocks and a large expansion wave propagates forward and backward with respect to the PDE in a semi-oval shape.

Numerical simulation of the welding deformation for the side sill of the bogie frame based on local-global method

Considering the limitation of computational capacity, a new finite element solution is used to simulate the welding deformation of the side sill of railroad car＇ s bogie frame based on the local-global method. Firstly, a volumetric heat source defined by a double ellipsoid is adopted to simulate the thermal distributions of the arc welding process. And then, the local models extracted from the global model are computed with refined meshes. On these bases, the global distortions of the subject studied are ascertained by transferring the inner forces of computed local models to the global model. It indicates that the local-global method is feasible for simulating the large welded structures by comparing the computed results with the corresponding actual measured values. The work provides basis for optimizing the welding sequence and clamping conditions, and has theoretical values and engineering significance in the integral design, manufacturing technique selection of the bogie frame, as well as other kinds of large welded structures.

Surplus Space Method: A New Numerical Model for Prediction of Shallow-seated Magmatic Bodies

Based on the data of field measurement and drilling in the Tongling area, a series of numerical simulations are carried out by using the 'Surplus Space Method' (SSM), which is first put forward in this paper and applied to predict the shallow-seated magmatic bodies. The results of the numerical simulations show the existence and the 3-D shape of a conical magmatic structure at a depth of-1000 m beneath the center of the area: its top offsets southwards and bifurcates to several branches, while its lower part stretches northeastwards and contracts rapidly to a point at about -1000 m depth. This point is reckoned to be a 'sink' of magma system, transferring ore materials and heat energy from the deep magma chamber to the sub-surface apophyses. The preliminary application of the SSM proves that it may be developed as a new detection means for determining the existence of shallow-seated magmatic bodies and analyzing their three-dimensional features.

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

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

Stress initialization methods for dynamic numerical simulation of rock mass with high in-situ stress

In the context of deep rock engineering,the in-situ stress state is of major importance as it plays an important role in rock dynamic response behavior.Thus,stress initialization becomes crucial and is the first step for the dynamic response simulation of rock mass in a high in-situ stress field.In this paper,stress initialization methods,including their principles and operating procedures for reproducing steady in-situ stress state in LS-DYNA,are first introduced.Then the most popular four methods,i.e.,explicit dynamic relaxation(DR)method,implicit-explicit sequence method,Dynain file method and quasi-static method,are exemplified through a case analysis by using the RHT and plastic hardening rock material models to simulate rock blasting under in-situ stress condition.Based on the simulations,it is concluded that the stress initialization results obtained by implicit-explicit sequence method and dynain file method are closely related to the rock material model,and the explicit DR method has an obvious advantage in solution time when compared to other methods.Besides that,it is recommended to adopt two separate analyses for the whole numerical simulation of rock mass under the combined action of in-situ stress and dynamic disturbance.

Analogue correction method of errors and its application to numerical weather prediction

In this paper, an analogue correction method of errors （ACE） based on a complicated atmospheric model is further developed and applied to numerical weather prediction （NWP）. The analysis shows that the ACE can effectively reduce model errors by combining the statistical analogue method with the dynamical model together in order that the information of plenty of historical data is utilized in the current complicated NWP model, Furthermore, in the ACE, the differences of the similarities between different historical analogues and the current initial state are considered as the weights for estimating model errors. The results of daily, decad and monthly prediction experiments on a complicated T63 atmospheric model show that the performance of the ACE by correcting model errors based on the estimation of the errors of 4 historical analogue predictions is not only better than that of the scheme of only introducing the correction of the errors of every single analogue prediction, but is also better than that of the T63 model.

Computing Streamfunction and Velocity Potential in a Limited Domain of Arbitrary Shape.Part II:Numerical Methods and Test Experiments

Built on the integral formulas in Part I,numerical methods are developed for computing velocity potential and streamfunction in a limited domain.When there is no inner boundary（around a data hole） inside the domain,the total solution is the sum of the internally and externally induced parts.For the internally induced part,three numerical schemes（grid-staggering,local-nesting and piecewise continuous integration） are designed to deal with the singularity of the Green＇s function encountered in numerical calculations.For the externally induced part,by setting the velocity potential（or streamfunction） component to zero,the other component of the solution can be computed in two ways：（1） Solve for the density function from its boundary integral equation and then construct the solution from the boundary integral of the density function.（2） Use the Cauchy integral to construct the solution directly.The boundary integral can be discretized on a uniform grid along the boundary.By using local-nesting（or piecewise continuous integration）,the scheme is refined to enhance the discretization accuracy of the boundary integral around each corner point（or along the entire boundary）.When the domain is not free of data holes,the total solution contains a data-hole-induced part,and the Cauchy integral method is extended to construct the externally induced solution with irregular external and internal boundaries.An automated algorithm is designed to facilitate the integrations along the irregular external and internal boundaries.Numerical experiments are performed to evaluate the accuracy and efficiency of each scheme relative to others.

Overhanging rock slope by design:An integrated approach using rock mass strength characterisation,large-scale numerical modelling and limit equilibrium methods

Overhanging rock slopes(steeper than 90°) are typically avoided in rock engineering design, particularly where the scale of the slope exceeds the scale of fracturing present in the rock mass. This paper highlights an integrated approach of designing overhanging rock slopes where the relative dimensions of the slope exceed the scale of fracturing and the rock mass failure needs to be considered rather than kinematic release of individual blocks. The key to the method is a simplified limit equilibrium(LE) tool that was used for the support design and analysis of a multi-faceted overhanging rock slope. The overhanging slopes required complex geometries with constantly changing orientations. The overhanging rock varied in height from 30 m to 66 m. Geomechanical modelling combined with discrete fracture network(DFN)representation of the rock mass was used to validate the rock mass strength assumptions and the failure mechanism assumed in the LE model. The advantage of the simplified LE method is that buttress and support design iterations(along with sensitivity analysis of design parameters) can be completed for various cross-sections along the proposed overhanging rock sections in an efficient manner, compared to the more time-intensive, sophisticated methods that were used for the initial validation. The method described presents the development of this design tool and assumptions made for a specific overhanging rock slope design. Other locations will have different geological conditions that can control the potential behaviour of rock slopes, however, the approach presented can be applied as a general guiding design principle for overhanging rock cut slope.

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

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

A two-dimensional numerical wave flume based on SA-MPLS method

A spatially adaptive (SA) two-dimensional (2-D) numerical wave flume is presented based on the quadtree mesh system,in which a new multiple particle level set (MPLS) method is proposed to solve the problem of interface tracking,in which common intersection may be traversed by multiple interfaces.By using the adaptive mesh technique and the MPLS method,mesh resolution is updated automatically with time according to flow characteristics in the modeling process with higher resolution around the free surface and the solid boundary and lower resolution in less important area.The model has good performance in saving computer memory and CPU time and is validated by computational examples of small amplitude wave,second-order Stokes wave and cnoidal wave.Computational results also indicate that standing wave and wave overtopping are also reasonably simulated by the model.

An overview of particle-based numerical manifold method and its application to dynamic rock fracturing

This review summarizes the development of particle-based numerical manifold method(PNMM)and its applications to rock dynamics.The fundamental principle of numerical manifold method(NMM)is first briefly introduced.Then,the history of the newly developed PNMM is given.Basic idea of PNMM and its simulation procedure are presented.Considering that PNMM could be regarded as an NMM-based model,a comparison of PNMM and NMM is discussed from several points of view in this paper.Besides,accomplished applications of PNMM to the dynamic rock fracturing are also reviewed.Finally,some recommendations are provided for the future work of PNMM.

A linearly-independent higher-order extended numerical manifold method and its application to multiple crack growth simulation

The numerical manifold method(NMM)can be viewed as an inherent continuous-discontinuous numerical method,which is based on two cover systems including mathematical and physical covers.Higher-order NMM that adopts higher-order polynomials as its local approximations generally shows higher precision than zero-order NMM whose local approximations are constants.Therefore,higherorder NMM will be an excellent choice for crack propagation problem which requires higher stress accuracy.In addition,it is crucial to improve the stress accuracy around the crack tip for determining the direction of crack growth according to the maximum circumferential stress criterion in fracture mechanics.Thus,some other enriched local approximations are introduced to model the stress singularity at the crack tip.Generally,higher-order NMM,especially first-order NMM wherein local approximations are first-order polynomials,has the linear dependence problems as other partition of unit(PUM)based numerical methods does.To overcome this problem,an extended NMM is developed based on a new local approximation derived from the triangular plate element in the finite element method(FEM),which has no linear dependence issue.Meanwhile,the stresses at the nodes of mathematical mesh(the nodal stresses in FEM)are continuous and the degrees of freedom defined on the physical patches are physically meaningful.Next,the extended NMM is employed to solve multiple crack propagation problems.It shows that the fracture mechanics requirement and mechanical equilibrium can be satisfied by the trial-and-error method and the adjustment of the load multiplier in the process of crack propagation.Four numerical examples are illustrated to verify the feasibility of the proposed extended NMM.The numerical examples indicate that the crack growths simulated by the extended NMM are in good accordance with the reference solutions.Thus the effectiveness and correctness of the developed NMM have been validated.

A method to calculate design tide levels on the basis of numerical model of tidal current and its application

In order to determine the design tide levels in the areas without measured tide level data, especially in the areas where it is difficult to measure tidal levels, a calculation method based on a numerical model of tidal current is proposed. The essentials of the method are described, and its application is illustrated with an example. The results of the application show that the design tide levels calculated by the method are close to those determined by long-time measured tide level data, and its calculation precision is high, so it is feasible to use the method to determine the design tide levels in the areas.