THE NUMERICAL STABILITY OF THE BLOCK θ-METHODS FOR DELAY DIFFERENTIAL EQUATIONS

This paper focuses on the numerical stability of the block θ methods adapted to differential equations with a delay argument. For the block θ methods, an interpolation procedure is introduced which leads to the numerical processes that satisfy an important asymptotic stability condition related to the class of test problems y′(t)=ay(t)+by(t-τ) with a,b∈C, Re(a)<-|b| and τ>0. We prove that the block θ method is GP stable if and only if the method is A stable for ordinary differential equations. Furthermore, it is proved that the P and GP stability are equivalent for the block θ method.

ON THE ARBITRARY DIFFERENCE PRECISE INTEGRATION METHOD AND ITS NUMERICAL STABILITY

Based on the subdomain precise integration method, the arbitrary difference precise integration method (ADPIM) is presented to solve PDEs. While retaining all the merits of the former method, ADPIM further demonstrates advantages such as the abilities of better description of physical properties of inhomogeneous media and convenient treatment of various boundary conditions. The explicit integration schemes derived by ADPIM are proved unconditionally stable.

On the Arbitrary Difference Precise Integration Method and Its Numerical Stability

Based on the subdomain precise integration method, the arbitrary difference precise integration method (ADPIM) is presented to solve PDEs. While retaining all the merits of the former method, ADPIM also demonstrates advantages such as the abilities of better description of physical properties of inhomogeneous media and convenient treatment of various boundary conditions. The explicit integration schemes derived by ADPIM are proved unconditionally stable.

Numerical Stability Analysis for a Stationary and Translating Droplet at Extremely Low Viscosity Values Using the Lattice Boltzmann Method Color-GradientMulti-Component Model with Central Moments Formulation

Multicomponent models based on the Lattice Boltzmann Method(LBM)have clear advantages with respect to other approaches,such as good parallel performances and scalability and the automatic resolution of breakup and coalescence events.Multicomponent flow simulations are useful for a wide range of applications,yet many multicomponent models for LBMare limited in their numerical stability and therefore do not allow exploration of physically relevant low viscosity regimes.Here we performa quantitative study and validations,varying parameters such as viscosity,droplet radius,domain size and acceleration for stationary and translating droplet simulations for the color-gradientmethod with centralmoments(CG-CM)formulation,as this method promises increased numerical stability with respect to the non-CMformulation.We focus on numerical stability and on the effect of decreasing grid-spacing,i.e.increasing resolution,in the extremely low viscosity regime for stationary droplet simulations.The effects of small-and large-scale anisotropy,due to grid-spacing and domain-size,respectively,are investigated for a stationary droplet.The effects on numerical stability of applying a uniform acceleration in one direction on the domain,i.e.on both the droplet and the ambient,is explored into the low viscosity regime,to probe the numerical stability of the method under dynamical conditions.

PLANE WAVES NUMERICAL STABILITY OF SOME EXPLICIT EXPONENTIAL METHODS FOR CUBIC SCHRODINGER EQUATION

Numerical stability when integrating plane waves of cubic SchrSdinger equation is thor- oughly analysed for some explicit exponential methods. We center on the following second- order methods： Strang splitting and Lawson method based on a one-parameter family of 2-stage 2nd-order explicit Runge-Kutta methods. Regions of stability are plotted and numerical results are shown which corroborate the theoretical results. Besides, a tech- nique is suggested to avoid the possible numerical instabilities which do not correspond to continuous ones.

Research of the CICC Stability by the Numerical Code Gandalf

The basic approach to computer analysis of the CICC in superconducting Tokamak HT-7U is given and discussed. We will apply a 1-D mathematical model (Gandalf) to investigate the stability of CICC at real operating modes of Tokamak. 1-D model can be directly adopted to follow the evolution of the zone when the energy input is large enough and the coil quenches. In this report, we will analyze the stability of typical CICC (including pure copper) and discuss effect of copper on the stability of CICC.

A Long-Time-Step-Permitting Tracer Transport Model on the Regular Latitude–Longitude Grid

If an explicit time scheme is used in a numerical model, the size of the integration time step is typically limited by the spatial resolution. This study develops a regular latitude–longitude grid-based global three-dimensional tracer transport model that is computationally stable at large time-step sizes. The tracer model employs a finite-volume flux-form semiLagrangian transport scheme in the horizontal and an adaptively implicit algorithm in the vertical. The horizontal and vertical solvers are coupled via a straightforward operator-splitting technique. Both the finite-volume scheme's onedimensional slope-limiter and the adaptively implicit vertical solver's first-order upwind scheme enforce monotonicity. The tracer model permits a large time-step size and is inherently conservative and monotonic. Idealized advection test cases demonstrate that the three-dimensional transport model performs very well in terms of accuracy, stability, and efficiency. It is possible to use this robust transport model in a global atmospheric dynamical core.

Science Letters:On numerical calculation in symplectic approach for elasticity problems

The symplectic approach proposed and developed by Zhong et al. in 1990s for elasticity problems is a rational analytical method, in which ample experience is not needed as in the conventional semi-inverse method. In the symplectic space, elasticity problems can be solved using the method of separation of variables along with the eigenfunction expansion technique, as in traditional Fourier analysis. The eigensolutions include those corresponding to zero and nonzero eigenvalues. The latter group can be further divided into α-and β-sets. This paper reformulates the form of β-set eigensolutions to achieve the stability of numerical calculation, which is very important to obtain accurate results within the symplectic frame. An example is finally given and numerical results are compared and discussed.

Stability conditions of explicit integration algorithms when using 3D viscoelastic artificial boundaries

Viscoelastic artificial boundaries are widely adopted in numerical simulations of wave propagation problems.When explicit time-domain integration algorithms are used,the stability condition of the boundary domain is stricter than that of the internal region due to the influence of the damping and stiffness of an viscoelastic artificial boundary.The lack of a clear and practical stability criterion for this problem,however,affects the reasonable selection of an integral time step when using viscoelastic artificial boundaries.In this study,we investigate the stability conditions of explicit integration algorithms when using three-dimensional(3D)viscoelastic artificial boundaries through an analysis method based on a local subsystem.Several boundary subsystems that can represent localized characteristics of a complete numerical model are established,and their analytical stability conditions are derived from and further compared to one another.The stability of the complete model is controlled by the corner regions,and thus,the global stability criterion for the numerical model with viscoelastic artificial boundaries is obtained.Next,by analyzing the impact of different factors on stability conditions,we recommend a stability coefficient for practically estimating the maximum stable integral time step in the dynamic analysis when using 3D viscoelastic artificial boundaries.

Sensitivity analysis of influencing parameters in cavern stability

In order to analyze the stability of the underground rock structures,knowing the sensitivity of geomechanical parameters is important.To investigate the priority of these geomechanical properties in the stability of cavern,a sensitivity analysis has been performed on a single cavern in various rock mass qualities according to RMR using Phase 2.The stability of cavern has been studied by investigating the side wall deformation.Results showed that most sensitive properties are coefficient of lateral stress and modulus of deformation.Also parameters of Hoek-Brown criterion and r c have no sensitivity when cavern is in a perfect elastic state.But in an elasto-plastic state,parameters of Hoek-Brown criterion and r c affect the deformability;such effect becomes more remarkable with increasing plastic area.Other parameters have different sensitivities concerning rock mass quality(RMR).Results have been used to propose the best set of parameters for study on prediction of sidewall displacement.

Stability analysis of backflling in subsiding area and optimization of the stoping sequence

In underground mining by sublevel caving method, the deformation and damage of the surface induced by subsidence are the major challenging issues. The dynamic and soft backflling body increases the safety risks in the subsiding area. In this paper, taking Zhangfushan iron mine as an example, the ore body and the general layout are focused on the safety of backflling of mined-out area. Then, we use the ANSYS software to construct a three-dimensional(3D) model for the mining area in the Zhangfushan iron mine. According to the simulation results of the initial mining stages, the ore body is stoped step by step as suggested in the design. The stability of the backflling is back analyzed based on the monitored displacements, considering the stress distribution to optimize the stoping sequence. The simulations show that a reasonable stoping sequence can minimize the concentration of high compressive stress and ensure the safety of stoping of the ore body.

Analysis of deformation ＆ destruction mechanism and stability of F0 fault crush zone in Fangezhuang Coal Mine

Under the mining influence, carried on the electron microscope, the thin section analysis and creep tests to the fault matter＇s original sample and five groups of duplication samples, which have three kinds of moisture. The results of those tests indicate that confining pressure effect, structure effect and moisture effect all have influence to fault matter＇s nature. Meanwhile, the distortion destruction and stability variation of fault crush zone are the main reason which causes water-inrush lag-effect. Simultaneously, the stimulation computation we made by the numerical simulation software FLAC 3D also describes the mining influence to floor strata, fault crush zone＇s range and size, the influence of confined water on overburden and the fault zones, also it describes the time effect of bearing influenced by confined water and the rock dank height of confined water along the fault zones influenced by the specific water head.

Slope stability analysis of Southern slope of Chengmenshan copper mine,China

The engineering geology and hydrogeology in the southern slope of Chengmenshan copper mine are very complicated,because there is a soft-weak layer between two kinds of sandstones.Field investigations demonstrate that some instability problems might occur in the slope.In this research,the southern slope,which is divided into six sections(I-0,I-1,I-2,II-0,II-1 and II-2),is selected for slope stability analysis using limit equilibrium and numerical method.Stability results show that the values of factor of safety(FOS) of sections I-0,I-1 and I-2 are very low and slope failure is likely to happen.Therefore reinforcement subjected to seismic,water and weak layer according to sections were carried out to increase the factor of safety of the three sections,two methods were used;grouting with hydration of cement and water to increase the cohesion(c) and pre-stressed anchor.Results of reinforcement showed that factor of safety increased more than 1.15.

Investigation of stability of a tunnel in a deep coal mine in China

Stability level of tunnels that exist in an underground mine has a great influence on the safety,production and economic performance of mines.Ensuring of stability for soft-rock tunnels is an important task for deep coal mines located in high in situ stress conditions.Using the available information on stratigraphy,geological structures,in situ stress measurements and geo-mechanical properties of intact rock and discontinuity interfaces,a three-dimensional numerical model was built by using 3DEC software to simulate the stress conditions around a tunnel located under high in situ stress conditions in a coal rock mass in China.Analyses were conducted for several tunnel shapes and rock support patterns.Results obtained for the distribution of failure zones,and stress and displacement felds around the tunnel were compared to select the best tunnel shape and support pattern to achieve the optimum stability conditions.

Numerical simulation of vegetated mine dump slope with reference to small plants

The present paper discusses the effects of small plants on the dump mass reinforcement and slope stability.The roots of smaller plants(such as grasses and shrubs)do not go deep.However,they stabilize the slope by binding the upper layer of dump slope.Shear strength of the dump mass with and without root reinforcement is determined by laboratory shear box instrument.The increased cohesion(apparent cohesion)of upper layer of the dump mass due to plants is determined by fabricated shear box.The kinetic behavior of the dump has been investigated using the FLAC software.The factor of safety has been calculated in order to determine the possible effect of small plants on the stability of the dump slope.It is observed that the small plants do not significantly improve the factor of safety(FOS)of slope.However,it could be useful for early stabilization.The grasses quickly bind the upper surface,whereas shrubs too immensely strengthen the stability of the dump in the initial stage.

Crosswind stability of high-speed trains in special cuts

Analysis of the aerodynamic performance of high-speed trains in special cuts would provide references for the critical overturning velocity and complement the operation safety management under strong winds.This work was conducted to investigate the flow structure around trains under different cut depths,slope angles using computational fluid dynamics(CFD).The high-speed train was considered with bogies and inter-carriage gaps.And the accuracy of the numerical method was validated by combining with the experimental data of wind tunnel tests.Then,the variations of aerodynamic forces and surface pressure distribution of the train were mainly analyzed.The results show that the surroundings of cuts along the railway line have a great effect on the crosswind stability of trains.With the slope angle and depth of the cut increasing,the coefficients of aerodynamic forces tend to reduce.An angle of 75°is chosen as the optimum one for the follow-up research.Under different depth conditions,the reasonable cut depth for high-speed trains to run safely is 3 m lower than that of the conventional cut whose slope ratio is 1:1.5.Furthermore,the windward slope angle is more important than the leeward one for the train aerodynamic performance.Due to the shield of appropriate cuts,the train body is in a minor positive pressure environment.Thus,designing a suitable cut can contribute to improving the operation safety of high-speed trains.

Stability analysis of slope in strain-softening soils using local arc-length solution scheme

Soils with strain-softening behavior — manifesting as a reduction of strength with increasing plastic strain — are commonly found in the natural environment. For slopes in these soils,a progressive failure mechanism can occur due to a reduction of strength with increasing strain. Finite element method based numerical approaches have been widely performed for simulating such failure mechanism,owning to their ability for tracing the formation and development of the localized shear strain. However,the reliability of the currently used approaches are often affected by poor convergence or significant mesh-dependency,and their applicability is limited by the use of complicated soil models. This paper aims to overcome these limitations by developing a finite element approach using a local arc-length controlled iterative algorithm as the solution strategy. In the proposed finite element approach,the soils are simulated with an elastoplastic constitutive model in conjunction with the Mohr-Coulomb yield function. The strain-softening behavior is represented by a piece-wise linearrelationship between the Mohr-Coulomb strength parameters and the deviatoric plastic strain. To assess the reliability of the proposed finite element approach,comparisons of the numerical solutions obtained by different finite element methods and meshes with various qualities are presented. Moreover,a landslide triggered by excavation in a real expressway construction project is analyzed by the presented finite element approach to demonstrate its applicability for practical engineering problems.

Multiple solutions and stability of the steady transonic small-disturbance equation

Numerical solutions of the steady transonic small-disturbance（TSD） potential equation are computed using the conservative Murman-Cole scheme. Multiple solutions are discovered and mapped out for the Mach number range at zero angle of attack and the angle of attack range at Mach number 0.85 for the NACA 0012 airfoil. We present a linear stability analysis method by directly assembling and evaluating the Jacobian matrix of the nonlinear finite-difference equation of the TSD equation. The stability of all the discovered multiple solutions are then determined by the proposed eigen analysis. The relation of stability to convergence of the iterative method for solving the TSD equation is discussed. Computations and the stability analysis demonstrate the possibility of eliminating the multiple solutions and stabilizing the remaining unique solution by adding a sufficiently long splitter plate downstream the airfoil trailing edge. Finally, instability of the solution of the TSD equation is shown to be closely connected to the onset of transonic buffet by comparing with experimental data.

Stability analysis and determination of rock pillar between two adjacent caverns in different regions of Asmari formation in Iran

Large underground caverns are commonly used in variety of applications. In many cases, because of the geomechanical limitations of dimensions and requirement of high volume, several parallel caverns are used. Plastic zone integration requires a larger rock pillar distance of theses adjacent caverns while eco- nomic and access reasons require smaller distance. In lran many underground projects are located in West and South West, Asmari formation covers a large part of these regions. The stability of underground spaces that are constructed or will be constructed in this formation has been investigated. A proper cross section based on plastic analysis and a stability criterion has been proposed for each region. Finally, in each case, allowable rock pillar between adjacent caverns with similar dimension was determined with two methods （numerical analysis and fire service law）. Results show that Fire Service Law uses a very con- servative safety factor and it was proposed to use a correction factor for allowable distance based on application of underground space.

Efficient Mass Transport and Electrochemistry Coupling Scheme for Reliable Multiphysics Modeling of Planar Solid Oxide Fuel Cell Stack

A multiphysics model for a production scale planar solid oxide fuel cell （SOFC） stack is important for the SOFC technology, but usually requires an unpractical amount of computing resource. The major cause for the huge computing resource requirement is identified as the need to solve the cathode O2 transport and the associated electrochemistry. To overcome the technical obstacle, an analytical model for solving the O2 transport and its coupling with the electrochemistry is derived. The analytical model is used to greatly reduce the numerical mesh complexity of a multiphysics model. Numerical test shows that the analytical approximation is highly accurate and stable. A multiphysics numerical modeling tool taking advantage of the analytical solution is then developed through Fluent@. The numerical efficiency and stability of this modeling tool are further demonstrated by simulating a 30- cell stack with a production scale cell size. Detailed information about the stack performance is revealed and briefly discussed. The multiphysics modeling tool can be used to guide the stack design and select the operating parameters.