Non-uniform Convergence Research Articles

Simple approach to assessing excess pore water pressure induced by shield tunneling in saturated-unsaturated clay soil

Excess pore water pressure (EPWP) induced by shield tunneling has a significant influence on the stability of the tunnel face, post-construction settlement, and the mechanical behavior of the tunnel lining. However, the three-dimensional and unsaturated property of the soil field is seldom considered in current research. Considering the non-uniform radial convergence model, a modified three-dimensional displacement solution induced by shield tunneling was established first. Then based on unsaturated EPWP elastic theory, a reliable and efficient method is developed to expedite the evaluation of EPWP distribution in three-dimensional saturated and unsaturated clay soil. The validity of the method is confirmed through comparison with field test and numerical outcomes. The analysis examples demonstrate that negative and positive EPWP are generated above the tunnel crown and beneath the tunnel invert, respectively. In the vertical direction, the negative EPWP exhibits a decreasing trend ahead of the heading face and an increasing trend behind it. Along the longitudinal direction, the influence zone of EPWP extends to 1D ahead of and 6D behind the heading face. With the decrease of soil saturation, the EPWP values tend to diminish. The maximum EPWP values observed in saturated conditions can be 16.13 times higher than those under unsaturated conditions.

Complex variable solution for ground deformation responses considering the grouting pressure of shallow shield

During shield tunneling in shallow strata, ground deformation is highly sensitive to grouting pressure response. Excessive grouting pressure can cause ground uplift, and an excessively low grouting pressure can cause excessive ground settlement, affecting the surrounding environment. Current studies seldom analyze the influence of shield excavation on ground deformation when the grouting pressure is considered. To ensure micro-disturbance of the surrounding environment during shield tunneling, this study presents a complex variable analysis method for analyzing the influence of grouting pressure on surface settlement. This solution maps the semi-infinite strata outside the segment into circular domains by introducing a complex function conformal mapping method. The nonuniform convergence deformation pattern BC-4 was introduced as the boundary condition for the displacement around the tunnel. The Laurent series expansion of the analytical function considering the grouting pressure in the circular domain was substituted into each boundary condition to solve the analytical function. Subsequently, the accuracy of the analytical method was verified by combining it with a finite element numerical method. Finally, the influence of grouting pressure on ground deformation under different working conditions was analyzed through parameterization. This analytical method can provide a theoretical basis for controlling grouting pressure parameters in shield tunnels.

Non-uniform convergence of solution for the Camassa–Holm equation in the zero-filter limit

Non-uniform convergence of solution for the Camassa–Holm equation in the zero-filter limit

On sequences which are not uniformly converging on any open subset

We consider the property of nonuniform convergence to 0 of a sequence of functions on any open subset of a metric space. We consider three examples with respect to three different characteristics. Next we show that the three characteristics cannot be present simultaneously. For this purpose we introduce the so-called height function, which we use to quantify how far is a sequence of functions from satisfying any of the third characteristic. Moreover, we study properties of the height function and its relation to uniform convergence. Finally, we show that this quantification is precise.

Coupling Dependence on Chaos Synchronization Process in a Network of Rulkov Neurons

This paper investigates the impact of network conformation on chaos synchronization in driven complex networks in a master-slave setting. We analyze the control and prediction of complex networks under perturbations. Although chaotic dynamics suppression is well understood in low-dimensional systems, it is not well described in more complex ones. To assess a network’s ability to amplify or suppress disturbances, we track ensembles of possible trajectories in the driven subsystem which give insights into nonlocal convergence properties in a master-slave setting. We demonstrate how the stability of forced networks of Rulkov neurons is dependent on network coupling architecture and how the duration of chaotic transients depends on initial conditions. Furthermore, we find that each network generator algorithm has specific sets of parameters that can lead to stable or nonstable states. Our results indicate that network types are more critical than isolated connections for suppressing chaotic dynamics in driven networks. Finally, we identify that under certain circumstances, the transient period before synchronization correlates with high convergence rates for certain initial conditions. This suggests nonuniform convergence rates in the phase space of driven neural networks.

Physical Model Tests and Discrete-Element Simulation of Pile and Soil Displacement Response Induced by DOT Shield Tunneling Based on Transparent Soil Technology

With the advantages of saving underground space, small construction impact, and saving cost, the double-O-tube (DOT) shield tunneling construction technology has been used many times in the tunnel construction of particular areas with dense buildings. This paper describes a device developed for modeling DOT shield tunneling based on transparent soil technology and investigating the displacement response of pile and soil during the construction process of a DOT shield tunnel adjacent to an existing pile, which is considered in the uniform convergence mode (UCM) of soil loss. Meanwhile, the displacement response of pile and soil in the nonuniform convergence mode (NCM) was analyzed contrastively using the discrete-element method (DEM). The results showed that the displacements of soils around the pile are limited and the shapes of the soil settlement curves above the DOT shield tunnel change approximatively from “V” to “W” with the depth of the soil layer increasing. When the buried depth of the pile was 1.0D, the settlements of the pile and soils were obviously affected by the convergence mode. The maximum surface settlements were 1.52, 1.34, and 1.21 times larger under NCM than those under UCM, while the maximum horizontal displacements of the vertical soil layer beside the pile were 1.92, 1.83, and 1.80 times larger under NCM than those under UCM, respectively, corresponding to H/D = 1.0, 1.5, and 2.0. With the buried depth of the DOT tunnel increasing, the deflection and settlement increased first and then decreased. When H/D = 2.0, the excavation of the DOT tunnel could barely cause pile deflection under NCM.

Influence of Tunnel Excavation on the Deformation of a Frame Building

Tunnel excavation inevitably causes surface deformation. In urban areas, surface deformation could lead to the deformation of surrounding buildings, which may cause damage to communities when accumulated to a certain extent. However, the current construction organization and management mainly rely on on-site deformation measurements, and there is still a lack of reliable prediction methods. Here, we proposed an effective evaluation method for frame building deformation based on the stochastic medium theory and the equivalent beam theory. This method could effectively evaluate the surface and building deformation after a horseshoe tunnel excavation by considering the non-uniform convergence. Furthermore, its accuracy and practicability were verified using the Nanyan Fourth Circuit Transmission Reconstruction located in Dalian, China. The results show that the spatial distribution and characteristic values of the maximum tensile strains were closely related to the ratio of Young’s modulus to the shear modulus (E/G), the building height (h), the tunnel depth (H), the tangent of the tunnel influence angle (tan β) and the convergence radius (ΔR). These achievements can provide a theoretical basis and analytical ideas for investigating the influence of shallow buried tunnel excavation on frame structure buildings in cities.

Semi-analytical solutions of three-dimensional ground movements due to shallow tunnelling

This paper presents the semi-analytical solutions to predict the three-dimensional ground movements of shallowly buried tunnels in clayey and sandy soils based on the virtual image technique. Both the ground nonuniform convergence and the tunnel ovalization are taken into account in the proposed method. The accuracy of the proposed method is verified by comparing with the results from field monitoring, numerical simulation and existing solutions. The results indicate that the transverse surface settlements mainly occur within H/tan(45° + φ/2) + R from the tunnel axis on both sides of the tunnel, and the variations of the longitudinal surface settlements are mainly produced within ± 2H from the tunnel face. The internal friction angle, Poisson’s ratio, cover depth and tunnel diameter have significant effects on the surface and subsurface movements according to the parametric analysis. The proposed method can serve as a practical tool in the preliminary design of tunnels and provide a theoretical basis for ensuring the construction safety of actual engineering.

General criteria for the study of quasi-stationarity

For Markov processes with absorption, we provide general criteria ensuring the existence and the exponential non-uniform convergence in weighted total variation norm to a quasi-stationary distribution. We also characterize a subset of its domain of attraction by an integrability condition, prove the existence of a right eigenvector for the semigroup of the process and the existence and exponential ergodicity of the Q-process. These results are applied to one-dimensional and multi-dimensional diffusion processes, to pure jump continuous time processes, to reducible processes with several communication classes, to perturbed dynamical systems and discrete time processes evolving in discrete state spaces.

Double-O-Tube Shield Tunneling-Induced Soil Displacement Considering Burial Depth and Convergence Mode: Transparent Soil Experiment and DEM Simulation

Double-O-tube shield tunneling technology has gained extensive attention due to saving space and the high cost-effectiveness of the underground construction method. Estimating and predicting DOT shield tunneling-induced soil displacements is essential to prevent damage to nearby aboveground and underground structures in densely populated urban areas. This study develops a device for modeling DOT shield tunneling based on transparent soil technology and image processing technique and investigates soil displacement during the construction process of the DOT shield tunnel, which is considered in the uniform convergence mode (UCM) of soil loss. Meanwhile, the soil displacement under the non-uniform convergence mode (NCM) is analyzed contrastively using a two-dimension particle flow code (PFC2D). The results show that horizontal displacements increase gradually when the shield tail passes the monitoring face, while settlements increase rapidly. The maximum horizontal displacement of the surface and the maximum surface settlement under NCM are slightly larger than those under UCM. A trapezoid-shaped failure pattern of soil is experienced at three tunnel depths. The maximum soil displacement under NCM is 1.93, 2.10, and 2.05 times, respectively, as much as that under UCM, corresponding to H/D (the ratio of the tunnel depth to the tunnel diameter) = 1.0, 1.5, and 2.0. The soil arch effect above DOT tunnels arises as the DOT tunnel depth increases. Moreover, the experimental and numerical results are adopted to assess the validity of the proposed model, which indicates that the proposed model is close to the test results of the disturbance zone on DOT tunnels.

Nonuniform convergence in moment expansions of integral work relations.

Exponential averages that appear in integral fluctuation theorems can be recast as a sum over moments of thermodynamic observables. We use two examples to show that such moment series can exhibit nonuniform convergence in certain singular limits. The first example is a simple model of a process with measurement and feedback. In this example, the limit of interest is that of error-free measurements. The second system we study is an ideal gas particle inside an (infinitely) fast expanding piston. Both examples show qualitative similarities; the low-order moments are close to their limiting value, while high-order moments strongly deviate from their limit. As the limit is approached the transition between the two groups of moments is pushed toward higher and higher moments. Our findings highlight the importance of the ordering of limits in certain nonequilibrium-related calculations.

Time-dependent analyses for ground movement and stress field induced by tunnelling considering rainfall infiltration mechanics

The evaluation of influence of heavy rainfall on tunnel structural construction is a problem being faced by engineers around the world. Current analytical solutions on the tunnelling-induced ground movements are generally based on the normal sunny weather and provide little attention on the rainfall intensity and duration risk. In this paper, a time-dependent complex variable approach is proposed to estimate the deformation and stress of surrounding soils caused by tunnelling considering the rainfall infiltration mechanisms. The modified Green-Ampt model under the assumption of stratification is used to simulate the rainfall infiltration process and the complex function theory is employed to calculate the subregion mapping for the saturated, transitional, and natural layers. The non-uniform convergence deformation boundary is adopted at the tunnel structure opening so as to consider the displacement controlled excavation effects due to tunnelling. The analytical presented solution is then verified by comparisons with a total of three engineering case records. It is revealed that the presented complex variable solution considering the rainfall infiltration mechanism can provide a prediction of surface settlement, soil subsidence and horizontal displacement in good agreements with the field measurements. Finally, the parameters that affect the ground deformation and stress induced by tunneling in rainy day, including the rainfall intensity, saturated permeability, matric suction, initial water content, are studied. The paper contributes to provides a relatively quick and easy way of accurately evaluating the time-dependent feedback in tunnel-soil interaction system involving heavy rainfall as a potential risk in the preliminary design of tunnel structures considering the adverse weather condition.

Truncated sum-of-squares estimation of fractional time series models with generalized power law trend

We consider truncated (or conditional) sum-of-squares estimation of a parametric fractional time series model with an additive deterministic structure. The latter consists of both a drift term and a generalized power law trend. The memory parameter of the stochastic component and the power parameter of the deterministic trend component are both considered unknown real numbers to be estimated and belonging to arbitrarily large compact sets. Thus, our model captures different forms of nonstationarity and noninvertibility as well as a very flexible deterministic specification. As in related settings, the proof of consistency (which is a prerequisite for proving asymptotic normality) is challenging due to non-uniform convergence of the objective function over a large admissible parameter space and due to the competition between stochastic and deterministic components. As expected, parameter estimates related to the deterministic component are shown to be consistent and asymptotically normal only for parts of the parameter space depending on the relative strength of the stochastic and deterministic components. In contrast, we establish consistency and asymptotic normality of parameter estimates related to the stochastic component for the entire parameter space. Furthermore, the asymptotic distribution of the latter estimates is unaffected by the presence of the deterministic component, even when this is not consistently estimable. We also include Monte Carlo simulations to illustrate our results.

Robust Hybrid Control for Demand Side Management in Islanded Microgrids

This paper focuses on designing a robust control law to manage the demand response of islanded microgrids composed of shifting and adjusting loads. On one side, Hybrid Dynamical System theory is suited here, because the microgrid model is composed of continuous-time dynamics (the energy stored in the battery and the adjustable loads), and discrete-time dynamics (the shifting loads). On the other side, Multi Agent System theory is used to control the adjusting loads in order to guarantee a consensus between them. Hence, non-uniform convergence of the State of Charge of the battery to a given reference is ensured. Robustness with respect to plug and play of any load and parameter variations is also ensured. Experimental results from a laboratory-scale microgrid validate the approach.

Non-uniform ISS small-gain theorem for infinite networks

Abstract We introduce the concept of non-uniform input-to-state stability for networks. It combines the uniform global stability with the uniform attractivity of any subnetwork while it allows for non-uniform convergence of all components. For an infinite network consisting of input-to-state stable subsystems, which do not necessarily have a uniform $\mathscr{K}\mathscr{L}$-bound on the transient behaviour, we show the following: if the gain operator satisfies the uniform small-gain condition, then the whole network is non-uniformly input-to-state stable and all its finite subnetworks are input-to-state stable.

Influence of undercrossing tunnel excavation on the settlement of a metro station in Dalian

Numerous tunnel excavation projects have been undertaken in China to address the ever-increasing demand for tunnels, which is driven by the rapid development of urban cities. Tunnel excavations generally induce the settlement of existing tunnels and sometimes result in the derailment of metro vehicles. In this study, the Nanyan Fourth Circuit Transmission Reconstruction, located in Dalian City in China, was selected to investigate the local settlement of a metro station caused by an underlying tunnel excavation project. The theoretical solution of the stochastic medium theory was calculated for a horseshoe-shaped tunnel based on the Gauss–Legendre integral in a non-uniform convergence situation. Results were compared with data measured at the metro station. A three-dimensional model was developed to further explore the variations in stress and deformation, which were difficult to measure during the excavation. The results showed the existence of a settlement gradient of the metro station in the direction of the excavation after the underlying tunnel was excavated. The numerical results were in good agreement with the data measured onsite. The non-uniformity of the settlement in the direction of the excavation was investigated. The findings of this study can provide a reference for predicting and controlling the impact of undercrossing tunnel excavation on the deformation of metro stations.

Fast and efficient finite difference/finite element method for the two-dimensional multi-term time-space fractional Bloch-Torrey equation

This paper proposes a numerical approximation combining finite difference method in time and finite element method in space to solve the two-dimensional time-space fractional Bloch-Torrey equation. Unlike the existing works which focus on the decoupled model, the proposed numerical scheme is established and discussed based on the coupled equations which is from the separation of real and imaginary parts of the original fractional Bloch-Torrey model. In addition, in order to get fast estimation to Caputo fractional derivative and reduce the storage of numerical scheme, an efficient sum-of-exponentials approximation for the kernel t−α,α∈(0,1) is adopted. Under the non-uniform time mesh, the stability and convergence are discussed for the semi-discrete scheme, and the error estimate is investigated for the fully discrete scheme in detail. Finally, several numerical tests are provided to verify the correctness of the obtained theoretical results and the effectiveness of our method.

Properties of the Null Distance and Spacetime Convergence

AbstractThe null distance for Lorentzian manifolds was recently introduced by Sormani and Vega. Under mild assumptions on the time function of the spacetime, the null distance gives rise to an intrinsic, conformally invariant metric that induces the manifold topology. We show when warped products of low regularity and globally hyperbolic spacetimes endowed with the null distance are (local) integral current spaces. This metric and integral current structure sets the stage for investigating convergence analogous to Riemannian geometry. Our main theorem is a general convergence result for warped product spacetimes relating uniform, Gromov–Hausdorff, and Sormani–Wenger intrinsic flat convergence of the corresponding null distances. In addition, we show that nonuniform convergence of warping functions in general leads to distinct limiting behavior, such as limits that disagree.

Analytical prediction of time-dependent behavior for tunneling-induced ground movements and stresses subjected to surcharge loading based on rheological mechanics

Current analytical solutions on the ground movements and stresses induced by shield tunneling in soft soils are generally based on the instantaneous excavation behavior. Furthermore, previous studies seldom took account of the influence of surcharge loading during tunnel excavation process. This paper introduces a complex variable analytical solution to estimate the deformation and stress of surrounding soils caused by shield tunneling subjected to surcharge loading considering the soil rheological mechanisms. The non-uniform convergence deformation boundary is adopted at the tunnel opening and the Boltzmann viscoelastic model is employed in the time-domain solution so as to consider the displacement controlled boundary and the rheological effects. The accuracy of complex variable time-dependent solution is then verified by comparisons with in-situ observed data, and good consistency is obtained. Finally, the parametric analyses are performed to evaluate the influence of correlative coefficients in Boltzmann viscoelastic model and paramount factors of surcharge loading on the feedback around tunneling. The ratios of the peak values of final surface settlements induced by tunneling in viscoelastic soils to those in elastic soils in the initial period vary approximately from 40% to 120%. The soil stresses around the tunnel increase about 50% significantly over time.

Consensus-Based Formation Control of Networked Nonholonomic Vehicles With Delayed Communications

For a network of nonholonomic vehicles communicating according to an undirected connected graph, a consensus-based formation control problem is solved via a smooth time-varying, proportional-derivative, δ-persistently-exciting, controller. It is assumed that the communication among agents is affected by time-varying, nondifferentiable, communication delays and uniform global asymptotic stability is demonstrated. This goes beyond the more-often encountered property of nonuniform convergence and, what is more, for the first time in the literature, a strict Lyapunov-Krasovskii functional is provided.