Analysis of the interaction between bolt-reinforced rock and surface support in tunnels based on convergence-confinement method

To investigate the interaction of the bolt-reinforced rock and the surface support,an analytical model of the convergence-confinement type is proposed,considering the sequential installation of the fully grouted rockbolts and the surface support.The rock mass is assumed to be elastic-brittle-plastic material,obeying the linear Mohr-Coulomb criterion or the non-linear Hoek-Brown criterion.According to the strain states of the tunnel wall at bolt and surface support installation and the relative magnitude between the bolt length and the plastic depth during the whole process,six cases are categorized upon solving the problem.Each case is divided into three stages due to the different effects of the active rockbolts and the passive surface support.The fictitious pressure is introduced to quantify the threedimensional(3D)effect of the tunnel face,and thus,the actual physical location along the tunnel axis of the analytical section can be considered.By using the bolt-rock strain compatibility and the rocksurface support displacement compatibility conditions,the solutions of longitudinal tunnel displacement and the reaction pressure of surface support along the tunnel axis are obtained.The proposed analytical solutions are validated by a series of 3D numerical simulations.Extensive parametric studies are conducted to examine the effect of the typical parameters of rockbolts and surface support on the tunnel displacement and the reaction pressure of the surface support under different rock conditions.The results show that the rockbolts are more effective in controlling the tunnel displacement than the surface support,which should be installed as soon as possible with a suitable length.For tunnels excavated in weak rocks or with restricted displacement control requirements,the surface support should also be installed or closed timely with a certain stiffness.The proposed method provides a convenient alternative approach for the optimization of rockbolts and surface support at the preliminary stage of tunnel design.

Optimal Shape Factor and Fictitious Radius in the MQ-RBF:Solving Ill-Posed Laplacian Problems

To solve the Laplacian problems,we adopt a meshless method with the multiquadric radial basis function(MQRBF)as a basis whose center is distributed inside a circle with a fictitious radius.A maximal projection technique is developed to identify the optimal shape factor and fictitious radius by minimizing a merit function.A sample function is interpolated by theMQ-RBF to provide a trial coefficient vector to compute the merit function.We can quickly determine the optimal values of the parameters within a preferred rage using the golden section search algorithm.The novel method provides the optimal values of parameters and,hence,an optimal MQ-RBF;the performance of the method is validated in numerical examples.Moreover,nonharmonic problems are transformed to the Poisson equation endowed with a homogeneous boundary condition;this can overcome the problem of these problems being ill-posed.The optimal MQ-RBF is extremely accurate.We further propose a novel optimal polynomial method to solve the nonharmonic problems,which achieves high precision up to an order of 10^(−11).

Highly Accurate Golden Section Search Algorithms and Fictitious Time Integration Method for Solving Nonlinear Eigenvalue Problems

This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.

Numerical modelling of stress analysis around rectangular tunnels with large discontinuities(fault) by a hybridized indirect BEM

Tunnels are one of the major transportation routes to pass mountains and difficult geological conditions. The behavior of these structures is significantly influenced by rock mass and discontinuities. Orientation of discontinuities is one of the most important geometrical parameters affecting discontinuities behavior. The effect of large discontinuities(faults) behavior on a jointed medium around rectangular tunnels is studied. A hybridized indirect boundary element code named TFSDDM(fictitious stress displacement discontinuity method) is used to study the stress distribution around the tunnels excavated in jointed rock masses. The code uses advantages of both fictitious stress and displacement discontinuity methods to analyze discontinuity effects more accurately. Results show that the dip angle of discontinuities has significant effect on stress distribution around the tunnel. It is also shown that increase in the discontinuities dip angle located in the roof will result in decrease in tensile stress of the roof. Stresses reaches to 8 MPa in the roof while due to dilation effect they reach up to 13 MPa.

Cooperative and Competitive Multi-Agent Systems:From Optimization to Games

Multi-agent systems can solve scientific issues related to complex systems that are difficult or impossible for a single agent to solve through mutual collaboration and cooperation optimization.In a multi-agent system,agents with a certain degree of autonomy generate complex interactions due to the correlation and coordination,which is manifested as cooperative/competitive behavior.This survey focuses on multi-agent cooperative optimization and cooperative/non-cooperative games.Starting from cooperative optimization,the studies on distributed optimization and federated optimization are summarized.The survey mainly focuses on distributed online optimization and its application in privacy protection,and overviews federated optimization from the perspective of privacy protection me-chanisms.Then,cooperative games and non-cooperative games are introduced to expand the cooperative optimization problems from two aspects of minimizing global costs and minimizing individual costs,respectively.Multi-agent cooperative and non-cooperative behaviors are modeled by games from both static and dynamic aspects,according to whether each player can make decisions based on the information of other players.Finally,future directions for cooperative optimization,cooperative/non-cooperative games,and their applications are discussed.

Effect of sediment on vertical dynamic impedance of rock-socketed pile with large diameter

Based on the fictitious soil pile model, the effect of sediment on the vertical dynamic impedance of rock-socketed pile with large diameter was theoretically studied by means of Laplace transform technique and impedance function transfer method. Firstly, the sediment under rock-socketed pile was assumed to be fictitious soil pile with the same sectional area. The Rayleigh-Love rode model was used to simulate the rock-socketed pile and the fictitious soil pile with the consideration of the lateral inertial effect of large-diameter pile. The layered surrounding soils and bedrock were modeled by the plane strain model. Then, by virtue of the initial conditions and boundary conditions of the soil pile system, the analytical solution of the vertical dynamic impedance at the head of rock-socketed pile was derived for the arbitrary excitation acting on the pile head. Lastly, based on the presented analytical solution, the effect of sediment properties, bedrock property and lateral inertial effect on the vertical dynamic impedance at rock-socketed pile head were investigated in detail. It is shown that the sediment properties have significant effect on the vertical dynamic impedance at the rock-socketed pile head. The ability of soil-pile system to resist dynamic vertical deformation is weakened with the increase of sediment thickness, but amplified with the increase of shear wave velocity of sediment. The ability of soil pile system to resist dynamic vertical deformation is amplified with the bedrock property improving, but the ability of soil-pile system to resist vertical vibration is weakened with the improvement of bedrock property.

Longitudinal vibration of pile in layered soil based on Rayleigh-Love rod theory and fictitious soil-pile model

The dynamic response of pile in layered soil is theoretically investigated when considering the transverse inertia effect.Firstly, the fictitious soil-pile model is employed to simulate the dynamic interaction between the pile and the soil layers beneath pile toe. The dynamic interactions of adjacent soil layers along the vertical direction are simplified as distributed Voigt models.Meanwhile, the pile and fictitious soil-pile are assumed to be viscoelastic Rayleigh-Love rods, and both the radial and vertical displacement continuity conditions at the soil-pile interface are taken into consideration. On this basis, the analytical solution for dynamic response at the pile head is derived in the frequency domain and the corresponding quasi-analytical solution in the time domain is then obtained by means of the convolution theorem. Following this, the accuracy and parameter value of the hypothetical boundaries for soil-layer interfaces are discussed. Comparisons with published solution and measured data are carried out to verify the rationality of the present solution. Parametric analyses are further conducted by using the present solution to investigate the relationships between the transverse inertia effects and soil-pile parameters.

Improvement of Aircraft Rolling Power by Use of Piezoelectric Actuators

Piezoelectric actuators are distributed on both side of a rectangular wing model,and the possibility of improvement of aircraft rolling power is investigated. The difference between the model with aileron deflection and the model without aileron (fictitious control surface, FCS) is studied. The analytical results show that these two cases are substantial different. In aileron deflection case, the aeroelastic effect is disadvantageous, so the structural stiffness should be high until the electrical voltage is not necessary. But in the case of FCS,the aeroelastic effect is advantageous and it means that lower structural stiffness can lead to lower voltage. Compared with aileron project, the FCS project can save structure weight.

A Matrix Approach to the Modeling and Analysis of Networked Evolutionary Games With Time Delays

Using the semi-tensor product method, this paper investigates the modeling and analysis of networked evolutionary games(NEGs) with finite memories, and presents a number of new results. Firstly, a kind of algebraic expression is formulated for the networked evolutionary games with finite memories, based on which the behavior of the corresponding evolutionary game is analyzed. Secondly, under a proper assumption, the existence of Nash equilibrium of the given networked evolutionary games is proved and a free-type strategy sequence is designed for the convergence to the Nash equilibrium. Finally, an illustrative example is worked out to support the obtained new results.

Observer-Based Nonlinear Control of A Torque Motor with Perturbation Estimation

This paper presents an observer-based nonlinear control method that was developed and implemented to provide accurate tracking control of a limited angle torque motor following a 50Hz reference waveform. The method is based on a robust nonlinear observer, which is used to estimate system states and perturbations and then employ input-output feedbazk linearization to compensate for the system nonlinearities and uncertainties. The estimation of system states and perturbations allows input-output linearization of the nonlinear system without an accurate mathematical model of nominal plant. The simulation results show that the observer-based nonlinear control method is superior in comparison with the conventional model-based state feedback linearizing controller.

Road Pricing Design Based on Game Theory and Multi-agent Consensus

Consensus theory and noncooperative game theory respectively deal with cooperative and noncooperative interactions among multiple players/agents. They provide a natural framework for road pricing design, since each motorist may myopically optimize his or her own utility as a function of road price and collectively communicate with his or her friends and neighbors on traffic situation at the same time. This paper considers the road pricing design by using game theory and consensus theory. For the case where a system supervisor broadcasts information on the overall system to each agent, we present a variant of standard fictitious play called average strategy fictitious play(ASFP) for large-scale repeated congestion games.Only a weighted running average of all other players actions is assumed to be available to each player. The ASFP reduces the burden of both information gathering and information processing for each player. Compared to the joint strategy fictitious play(JSFP) studied in the literature, the updating process of utility functions for each player is avoided. We prove that there exists at least one pure strategy Nash equilibrium for the congestion game under investigation, and the players actions generated by the ASFP with inertia(players reluctance to change their previous actions) converge to a Nash equilibrium almost surely. For the case without broadcasting, a consensus protocol is introduced for individual agents to estimate the percentage of players choosing each resource, and the convergence property of players action profile is still ensured. The results are applied to road pricing design to achieve socially local optimal trip timing. Simulation results are provided based on the real traffic data for the Singapore case study.

Distributed Lagrange Multiplier/Fictitious Domain Finite Element Method for a Transient Stokes Interface Problem with Jump Coefficients

The distributed Lagrange multiplier/fictitious domain(DLM/FD)-mixed finite element method is developed and analyzed in this paper for a transient Stokes interface problem with jump coefficients.The semi-and fully discrete DLM/FD-mixed finite element scheme are developed for the first time for this problem with a moving interface,where the arbitrary Lagrangian-Eulerian(ALE)technique is employed to deal with the moving and immersed subdomain.Stability and optimal convergence properties are obtained for both schemes.Numerical experiments are carried out for different scenarios of jump coefficients,and all theoretical results are validated.

MULTI-MONTE-CARLO METHOD FOR GENERAL DYNAMIC EQUATION CONSIDERING PARTICLE COAGULATION

Monte-Carlo （MC） method is widely adopted to take into account general dynamic equation （GDE） for particle coagulation, however popular MC method has high computation cost and statistical fatigue. A new Multi-Monte-Carlo （MMC） method, which has characteristics of time-driven MC method, constant number method and constant volume method, was promoted to solve GDE for coagulation. Firstly MMC method was described in details, including the introduction of weighted fictitious particle, the scheme of MMC method, the setting of time step, the judgment of the occurrence of coagulation event, the choice of coagulation partner and the consequential treatment of coagulation event. Secondly MMC method was validated by five special coagulation cases in which analytical solutions exist. The good agreement between the simulation results of MMC method and analytical solutions shows MMC method conserves high computation precision and has low computation cost. Lastly the different influence of different kinds of coagulation kernel on the process of coagulation was analyzed： constant coagulation kernel and Brownian coagulation kernel in continuum regime affect small particles much more than linear and quadratic coagulation kernel,whereas affect big particles much less than linear and quadratic coagulation kernel.

Spline Fictitious Boundary Element Alternating Method for Edge Crack Problems with Mixed Boundary Conditions

The alternating method based on the fundamental solutions of the infinite domain containing a crack,namely Muskhelishvili’s solutions,divides the complex structure with a crack into a simple model without crack which can be solved by traditional numerical methods and an infinite domain with a crack which can be solved by Muskhelishvili’s solutions.However,this alternating method cannot be directly applied to the edge crack problems since partial crack surface of Muskhelishvili’s solutions is located outside the computational domain.In this paper,an improved alternating method,the spline fictitious boundary element alternating method(SFBEAM),based on infinite domain with the combination of spline fictitious boundary element method(SFBEM)and Muskhelishvili’s solutions is proposed to solve the edge crack problems.Since the SFBEM and Muskhelishvili’s solutions are obtained in the framework of infinite domain,no special treatment is needed for solving the problem of edge cracks.Different mixed boundary conditions edge crack problems with varies of computational parameters are given to certify the high precision,efficiency and applicability of the proposed method compared with other alternating methods and extend finite element method.

Improving Behavior of Computer Game Bots Using Fictitious Play

In modern computer games, ＂bots＂ - intelligent realistic agents play a prominent role in the popularity of a game in the market. Typically, bots are modeled using finite-state machine and then programmed via simple conditional statements which are hard-coded in bots logic. Since these bots have become quite predictable to an experienced games＇ player, a player might lose interest in the game. We propose the use of a game theoretic based learning rule called fictitious play for improving behavior of these computer game bots which will make them less predictable and hence, more a enjoyable game.

INTERACTIONS BETWEEN TWO SEDIMENTING PARTICLES WITH DIFFERENT SIZES

An improved implementation of Distributed Lagrange multiplier/fictitious domain method was presented and used to simulate the interactions between two circular particles sedimenting in a two_dimensional channel. The simulation results were verified by comparison with experiments. The results show that the interactions between two particles with different sizes can be described as drafting, kissing, tumbling and separating. Only for small diameter ratio, the two particles will interact undergoing repeated DKT (Drafting, Kissing and Tumbling) process. Otherwise, the two particles will separate after their tumbling. The results also show that, during the interaction process, the motion of the small particle is strongly affected while the large particle is affected slightly.

ON FICTITIOUS DOMAIN METHOD FOR THE NUMERICAL SOLUTION TO HEAT CONDUCTION EQUATION WITH DERIVATIVE BOUNDARY CONDITIONS

This paper investigates the stability and convergence of some knowndifference schemes for the numerical solution to heat conduction equation withderivative boundary conditions by the fictitious domain method.The discrete vari-ables at the false mesh points are firstly eliminated from the difference schemes andthe local truncation errors are then analyzed in detail.The stability and convergenceof the schemes are proved by energy method.An improvement is proposed to obtainbetter schemes over the original ones.Several numerical examples and comparisonswith other schemes are presented.

FICTITIOUS DOMAIN METHODS FOR NAVIER-STOKES EQUATIONS BASED ON PENALTY FACTOR ε

Abstract In this article we briefly studied the fictitious domain methods for steady and nonsteady Navier-Stokes equations based on Penalty factorεon the extended domain. The convergence u?→u in H01（Ω）d and L2（Ω）d（d=2,3） is given as well as p?→p in L02（Ω）.

On L∞Stability and Convergence of Fictitious Domain Method for the Numerical Solutionto Parabolic Di

This paper investigates some known difference schemes for the numerical solution to parabolic differential equation with derivative boundary conditions by the fictitious domain method.The stability and convergence in L ∞ are proven.

Application of the Fictitious Domain Method for Navier-Stokes Equations

To apply the fictitious domain method and conduct numericalexperiments, a boundary value problem for an ordinary differential equation is considered. The results of numerical calculations for different valuesof the iterative parameter τ and the small parameter ε are presented. Astudy of the auxiliary problem of the fictitious domain method for NavierStokes equations with continuation into a fictitious subdomain by highercoefficients with a small parameter is carried out. A generalized solutionof the auxiliary problem of the fictitious domain method with continuationby higher coefficients with a small parameter is determined. After all theabove mathematical studies, a computational algorithm has been developedfor the numerical solution of the problem. Two methods were used to solvethe problem numerically. The first variant is the fictitious domain methodassociated with the modification of nonlinear terms in a fictitious subdomain.The model problem shows the effectiveness of using such a modification. Theproposed version of the method is used to solve two problems at once that arisewhile numerically solving systems of Navier-Stokes equations: the problem ofa curved boundary of an arbitrary domain and the problem of absence of aboundary condition for pressure in physical formulation of the internal flowproblem. The main advantage of this method is its universality in developmentof computer programs. The second method used calculation on a uniform gridinside the area. When numerically implementing the solution on a uniformgrid inside the domain, using this method it’s possible to accurately take intoaccount the boundaries of the curved domain and ensure the accuracy of thevalue of the function at the boundaries of the domain. Methodical calculationswere carried out, the results of numerical calculations were obtained. Whenconducting numerical experiments in both cases, quantitative and qualitativeindicators of numerical results coincide.