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Tunnel excavation was a typical three-dimensional space problem. As a two-dimensional numerical method, the combined finite-discrete element method did not demonstrate the spatial effect during the tunnel excavation simulation. A new excavation path for the combined finite-discrete element method was proposed based on the spatial evolution characteristics. Then, the tunnel excavation simulation was carried out under different excavation paths (linear softening curve, exponential softening curve, power softening curve, and new softening curve). The results showed that the advantages of the exponential softening curve and power softening curve were inherited by the new softening curve. The new softening curve demonstrated the spatial effect during the tunnel excavation simulation. Moreover, both the maximum softening rate and peak value of model kinetic energy occurred near the intermediate moment, which was an advantage over other softening curves. The discussion indicated that both the softening rate and elastic modulus range should be considered when determining new softening curve parameters. And the optimal total softening steps can be estimated by the proposed equation with an error tolerance. Finally, the new softening curve was well applied in the numerical analysis of tunnel excavation and support design. [ABSTRACT FROM AUTHOR]

• A new hysteretic damping model is proposed. • The critical hysteretic damping coefficient acquisition method is proposed. • The effects of different hysteretic damping ratios on the simulation results of quasi-static tunnel excavation and quasi-dynamic rock slope slip are investigated. In the field of mechanical numerical simulation, damping is always a vital parameter. However, in the combined finite-discrete element method (FDEM), the current application of damping is insufficient. Therefore, in this paper, a new hysteretic damping model and its corresponding critical damping coefficient acquisition method are proposed. Furthermore, simulations of quasi-static tunnel excavation and quasi-dynamic rock slope slip are employed to investigate the influence of different hysteretic damping ratios on the numerical results. The study results show that the new hysteretic damping model, which is independent on the calculation time step, considers the effects of the element size, Young's modulus and material density and can obtain the critical damping coefficient by comprehensively using cantilever beam vibration numerical modelling and theoretical equations. In addition, the sensitivity of quasi-static tunnel excavation to the hysteretic damping ratio is very weak; however, for quasi-dynamic rock slope slip simulation, the damping ratio η has a substantial influence on the simulation results. The new hysteretic damping model proposed in this paper is important for improving the accuracy of FDEM numerical simulation results, especially for quasi-dynamics simulation. [ABSTRACT FROM AUTHOR]

Grouting is widely used to reduce hydraulic conductivity and enhance the strength of rock masses in geotechnical engineering. With the depletion of shallow resources and expansion of human activities, more underground projects are operated in deeper underground under high in-situ stress conditions. Understanding and controlling the grout penetration process under high in-situ stress conditions are critical to optimizing the grouting process in deep underground projects. In this work, we develop a FDEM-grouting model to study the effect of in-situ stress conditions on the grout penetration process by combining the combined finite-discrete element method (FDEM) with a grout simulator. The ability of FDEM-grouting to model grout penetration in fracture networks is confirmed by numerical tests compared with analytical solutions. The necessity of considering the effects of the fracture roughness, in-situ stress and HM coupling on the grout penetration process are discussed. A persistent fracture network is designed to investigate the grout penetration process under various in-situ stress conditions. The results show that the in-situ stress conditions can greatly affect the grout penetration in fracture networks. The anisotropic grout penetration under in-situ stress conditions is potentially useful for improving the grouting design. [ABSTRACT FROM AUTHOR]

• An nonlinear Schrödinger equation is obtained in simulating the evolution of planetary Rossby solitary waves while considering the generalized β (y). • Generalized β (y) is denoted to modify the linear phase speed and growth/decay characteristic of the planetary scale solitary wave packet. And asymmetry, intensity and persistence of each kind of flow fields depend strongly upon β (y). • The generalized β (y) and weak shear background current are found to share qualitative similarity in influencing the evolution of Rossby solitary wave packet. The interaction between planetary-scale wave and synoptic-scale wave in atmospheres is important in understanding the physical mechanism of short or long term weather or climate events, such as the blocking phenomena. Kinds of physical factors are disclosed to affect the interaction processes, such as the topography, background current. The effect of beta parameter is investigated in the present paper, it is called the generalized beta effect. By using methods of multiple scales and perturbation expansions, a new nonlinear forced Schrödinger equation is obtained in describing the evolution of planetary-scale envelope Rossby solitary waves, and a modified equation for synoptic-scale waves is derived. By constructing the numerical solution for the nonlinear Schrödinger equation, it reveals that the generalized beta can shift phase and modify the magnitude of planetary-scale envelope solitary waves. An analytical expression for synoptic-scale waves, including the generalized beta effect, is also obtained. It shows that the asymmetry, intensity and persistence of both planetary-scale wave and synoptic-scale wave depend strongly upon the generalized beta. The results provide new theoretical explanations for our understanding of wave-wave interaction. [ABSTRACT FROM AUTHOR]

Abstract The geothermal production from enhance geothermal system (EGS) involves complex thermal–hydraulic (TH) coupling process, which may affect the production efficiency and performance of EGS reservoir. In the study, a numerical manifold method (NMM) for TH coupling in fractured EGS is proposed. The EGS reservoir is considered to be formed from intact rock matrix blocks and discrete fractures networks. TH coupling control equations are firstly established using discrete fracture networks model. Then, the weight residual method was used to establish the discretized governing equation for transient heat conduction. Furthermore, the element sub-matrixes for heat conduction were presented. NMM approach simulating the fluid flow, heat conduction in fracture and rock matrix and TH coupling process was established. Finally, a single fracture heat extraction example was used to validate the effectiveness of the proposed method for solving a simplified TH coupling problem. Furthermore, a 2D fracture networks case was conducted to predict the performance of an EGS reservoir. The results in here validate the capability and accuracy of the proposed method and extend the application of NMM. [ABSTRACT FROM AUTHOR]

Abstract For Mandelbrot's cascade in a random environment, we find the critical value for the existence of harmonic moments of the limit variable of Mandelbrot's martingale, and establish large and moderate deviation principles for the free energy. As applications, we show the corresponding limit theorems for branching random walks in random environments. [ABSTRACT FROM AUTHOR]

In this work, a discontinuous hybrid “FE-Meshfree” method is developed to simulate three-dimensional (3D) cracking. Then, an algorithm of Local Adaptive Background Sub-element (LABS-element) is developed in the context of hybrid “FE-Meshfree” method to simulate intersecting cracks. Compared to the extended finite element method (XFEM), the present hybrid “FE-Meshfree” method is not required to increase the size of global stiffness matrix, introduce extra unknowns, and construct special enrichment functions. The present algorithms of LABS-element and “FE-Meshfree” method are validated by intensive numerical tests, which achieves the balance between accuracy and flexibility. Additionally, a hydraulic and mechanical (H-M) coupling model is generated, in which the deformation of rocks and the propagation of cracks are solved by the hybrid “FE-Meshfree” method, meanwhile the fluid flow in cracks is solved by a fluid simulator based on the principle of parallel-plate flow model. This H-M coupling model is then used to investigate the propagation of fluid-driven cracks in rock mass with multiple pre-existing cracks, and the observed fluid pressure feedback is potentially useful during shale oil/gas well simulation. [ABSTRACT FROM AUTHOR]

Modelling heat conduction is of significant importance for evaluating temperature effects of granular materials. Since the randomness of the grain structure and the heat resistance characteristics of grain interfaces, to realistically modelling the heat conduction of granular materials, the heat interactive among these random-shaped grains should be correctly reflected. In this study, the numerical manifold method (NMM) is extended to model the transient heat conduction of granular materials. The random-shaped grain structure of granular materials is represented with Voronoi polygons. The heat interactive among grains is realistically simulated by inserting heat conductive cohesive elements between grain boundaries. Besides, an interfacial heat conductivity is defined for the cohesive element to better represent the heat conduct capacity of grain interfaces. As a result, the temperature jumps at grain interfaces are naturally captured due to the dual cover systems of the NMM, while the heat fluxes across the interfaces are assumed to be continuous. To validate the developed numerical method, a benchmark test is carried out. At last, effects of the grain characteristics and interfacial heat resistance on the temperature field as well as the effective heat conductivity (EHC) of a plane consists of granular material are investigated by the developed NMM. [ABSTRACT FROM AUTHOR]

In order to gain an insight into creep induced cracking in viscoelastic materials, the creep crack phenomenon in viscoelastic materials are numerically investigated in this paper based on the numerical manifold method (NMM). To more realistically capture the mechanical behaviour in the vicinity of crack tips, the local displacements around crack tips are enriched with the viscoelastic crack-tip asymptotic displacement field. To better represent the viscoelastic materials, an incremental viscoelastic constitutive law based on the generalized Kelvin-Voigt model is incorporated into the NMM. To achieve these treatments in the NMM, an incremental viscoelastic NMM formulation is derived. In addition, an incremental formulation of the mixed mode viscoelastic energy release rate (ERR), which can be tackled in the same process with the incremental viscoelastic constitutive law, is derived and incorporated into the NMM for investigating the variations of the ERR in viscoelastic materials. Based on the developed method, three creep crack problems with different loading conditions and pre-existing crack configurations are simulated, and the crack opening and sliding displacements as well as corresponding mixed mode ERR are analysed. Simulation results show good agreement with the theoretical solutions, which indicate that the proposed NMM is suitable for complex creep crack problems. [ABSTRACT FROM AUTHOR]

The dynamics of tumor suppressor p53 is closely related to cell fate decisions in responses to various stress signals, especially p53 oscillation. In this paper, phase transition of p53 dynamics from stable steady state to oscillatory state induced by Lévy noise is investigated numerically in p53 gene regulatory network. First, we carry out bifurcation analysis of p53 protein expression levels with respect to the degradation rate of ataxia telangiectasia mutated (ATM) protein in deterministic model of the p53 gene regulatory network. The bifurcation diagram demonstrates that p53 activities can display a stable steady state, a stable limit cycle and the coexistence of a stable steady state with a stable limit cycle upon varying ATM degradation rate. Then, we investigate the influences of Lévy noise parameters on phase transition from stable steady state to stable limit cycle in stochastic model of the p53 gene regulatory network for control parameters near bifurcation points. Moreover, phase transitions are verified through autocorrelation at lag-1, variance, kurtosis, skewness and potential landscape. Our results reveal that phase transition may occur for an optimal interval of stability index α , and is positive and negative correlation with noise intensity D and skewness parameter β , respectively. Besides, phase transition from low stable steady state to stable limit cycle is sensitive to Lévy noise parameters, but it is the opposite for high one. These results may provide an insight to uncover p53 dynamics in p53 gene regulatory network upon noise. • Suitable ATM degradation rate induces the oscillation of p53 model with high initial conditions. • Phase transition easily occurs for stability index α around 1, large noise intensity D and small skewness parameter β of Lévy noise. • Phase transition from low stable steady state to stable limit cycle is sensitive to Lévy noise parameter. [ABSTRACT FROM AUTHOR]

Grouting is a typical HM (hydro-mechanical) coupling process between slurry and rock and widely used in underground engineering to reinforce the strength of fractured rock mass. In present work, based on numerical manifold method (NMM), an NMM-HM grouting model is proposed to investigate slurry flowing and analyze grouting efficiency. In this NMM-HM grouting model, the slurry flows in fractures and grouting pressure can cause fracture aperture variation and fracture propagation. Then, two numerical examples for central embedded crack and hydraulic fracturing are adopted to validate the presented model. Furthermore, grouting simulations are conducted on a multi-crack and a roadway with fracture networks. Results show that Grouting pressure in fractures can increase fracture aperture, which is beneficial for grouting efficiency. However, fluid pressure could also cause closure of neighboring fractures or even fracture propagation and failure of rock mass, which will weaken grouting effectiveness. Therefore, the coupled HM impacts on grouting efficiency and grouting pressure should be seriously considered and optimized to achieve the expected grouting performance in practice. [ABSTRACT FROM AUTHOR]

Grouting is a commonly used technique in rock engineering to enhance the joint strength and improve the stability of surrounding rock. Grout penetration characteristic is controlled by grouting parameters and has a significant role on practice. In the study, a numerical manifold method (NMM) for grout penetration process simulation in fractured rock mass is firstly proposed. The fluid flow behaviour of the grout is assumed to be a Bingham fluid and control equations are established using discrete fracture network model. The global discretization equation, element sub-matrixes and NMM simulation algorithm for grouting are presented. Then, numerical tests for grouting process in a single fracture and a regular fracture network are conducted firstly to verify the proposed NMM grouting model by comparing with analytical solutions and experimental results. Furthermore, the effects of mesh size, fracture and slurry parameters on the grouting performance are systematically investigated using a random fracture network example. The numerical results indicate that the grouted zone and propagation depth decrease as the mesh size of numerical model and yield strength increases, while it increases as initial fracture aperture and grouting pressure increases. [ABSTRACT FROM AUTHOR]

• A new model is derived to describe the envelope solitary waves propagation of the pulsating flow in the blood. • The influencing factors of stroke volume are discussed and analyzed which provide a strong theoretical basis for some blood diseases. In this paper, considering blood as an incompressible Newtonian fluid, we explore the propagation of blood flow. Based on the Navier–Stokes equations and the continuity equation, the vorticity equation is deduced. Then, using the multi-scale analysis and iterative perturbed method, we derive a new higher order nonlinear Schrödinger equation to describe the blood flow in the blood vessels. The traveling wave solution of the higher order equation is obtained with the help of the extended tanh-function method. By analyzing the higher-order term of the equation, we find that the width and velocity of the blood wave increase during the propagation process. To further study the numerical characteristic, the numerical solutions of the higher order nonlinear Schrödinger equation are obtained by the meshless radial basis function method. The absolute errors between the exact and numerical red solutions show that the meshless radial basis function method gives a more accurate solution. Finally, the influencing factors of stroke volume are discussed and analyzed which provide a strong theoretical basis for some blood diseases. [ABSTRACT FROM AUTHOR]

Abstract: For a supercritical branching process in a stationary and ergodic environment , we study the rate of convergence of the normalized population to its limit : we show a central limit theorem for with suitable normalization and derive a Berry–Esseen bound for the rate of convergence in the central limit theorem when the environment is independent and identically distributed. Similar results are also shown for for each fixed . [Copyright &y& Elsevier]

We consider a branching process ( Z n ) in a stationary and ergodic random environment ξ = ( ξ n ) . Athreya and Karlin (1971) proved the basic result about the concept of subcriticality and criticality, by showing that under the quenched law P ξ , the conditional distribution of Z n given the non-extinction at time n converges in law to a proper distribution on N + = { 1 , 2 , ⋯ } in the subcritical case, and to the null distribution in the critical case, under the condition that the environment sequence is exchangeable. In this paper we first improve this basic result by removing the exchangeability condition on the environment, and by establishing a more general result about the conditional law of Z n given the non-extinction at time n + k for each fixed k ≥ 0 . As a by-product of the proof we also remove the exchangeability condition in another result of Athreya and Karlin (1971) for the subcritical case about the decay rate of the survival probability given the environment. We then establish a convergence theorem about the ratio P ξ ( Z n = j ) / P ξ ( Z n = 1 ) , which can be applicable in each of the subcritical, critical, and supercritical cases. [ABSTRACT FROM AUTHOR]

By introducing the concept of mathematical cover and physical cover, the numerical manifold method (NMM) is able to solve continuous and discontinuous problems in a unified way. In this paper, the NMM is developed to analyze three dimensional (3D) fracture propagation. The maximum tensile stress criterion is implemented to determine whether the fracture will propagate and the direction of fracture propagation. Three benchmark problems are analyzed to validate the present algorithm and program. The numerical results replicate available experimental results and existing numerical results. The present algorithm and 3D NMM code are promising for 3D fracture propagation. They deserve to be further developed for the analysis of rock mechanic problems in which the initiation and propagation of multiple fractures, tensile and shear fractures, and fracture propagation under compressive loading are taken into account. [ABSTRACT FROM AUTHOR]