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.

A FICTITIOUS DOMAIN METHOD FOR DIRICHLET PROBLEM AND ITS APPLICATIONS TO GENER ALIZED STOKES PROBLEM

This paper discusses a fictitious domain method for the linear Dirichlet problem and its applications to the generalized Stokes problem. This method treats Dirichlet boundary condit ion via a Lagrange multiplier technique and is well suited to the no-slip bound ary condition in viscous flow problems. In order to improve the accuracy of solu tions, meshes are refined according to the a posteriori error estimate. The mini -element discretization is applied to solve the generalized Stokes problem. Fin ally, some numerical results to validate this method are presented for partial d ifferential equations with Dirichlet boundary condition.

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.

Accuracy Analysis of Gravity Model Based on Fictitious Compress Recovery Approach

The potential field determined based on the fictitious compress recovery approach is influenced by the errors contained in the boundary （the Earth＇s surface or the surface corresponding to the satellite altitude） values. Given the boundary value with definite accuracy, the accuracy of the field determined based on the fictitious compress recovery approach is estimated, and it is theoretically shown that the determined field has the same accuracy level as the given boundary value.

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.

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.

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.

On L ∞ Stability and Convergence of Fictitious Domain Method for the Numerical Solution to Parabolic Differential Equation with Derivative Boundary Conditions

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.

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）.

Multigrid fictitious boundary method for incompressible viscous flows around moving airfoils

In this paper, an efficient multigrid fictitious boundary method （MFBM） coupled with the FEM solver package FEATFLOW was used for the detailed simulation of incompressible viscous flows around one or more moving NACA0012 airfoils. The calculations were carded on a fixed multigrid finite element mesh on which fluid equations were satisfied everywhere, and the airfoils were allowed to move freely through the mesh. The MFBM was employed to treat interactions between the fluid and the airfoils The motion of the airfoils was modeled by Newton-Euler equations. Numerical results of experiments verify that this method provides an efficient way to simulate incompressible viscous flows around moving airfoils.

Fully Coupled Fluid-Structure Interaction Model Based on Distributed Lagrange Multiplier/Fictitious Domain Method

This paper, with a finite element method, studies the interaction of a coupled incompressible fluid-rigid structure system with a free surface subjected to external wave excitations. With this fully coupled model, the rigid structure is taken as ＂fictitious＂ fluid with zero strain rate. Both fluid and structure are described by velocity and pressure. The whole domain, including fluid region and structure region, is modeled by the incompressible Navier-Stokes equations which are discretized with fixed Eulerian mesh. However, to keep the structure＇ s rigid body shape and behavior, a rigid body constraint is enforced on the ＂fictitious＂ fluid domain by use of the Distributed Lagrange Multipher/Fictitious Domain （DLM/ FD） method which is originally introduced to solve particulate flow problems by Glowinski et al. For the verification of the model presented herein, a 2D numerical wave tank is established to simulate small amplitude wave propagations, and then numerical results are compared with analytical solutions. Finally, a 2D example of fluid-structure interaction under wave dynamic forces provides convincing evidences for the method excellent solution quality and fidelity.

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.

Applications of the Fictitious Compress Recovery Approach in Physical Geodesy

The fictitious compress recovery approach is introduced, which could be applied to the establishment of the Rungerarup theorem, the determination of the Bjerhammar＇s fictitious gravity anomaly, the solution of the ＂downward con- tinuation＂ problem of the gravity field, the confirmation of the convergence of the spherical harmonic expansion series of the Earth＇s potential field, and the gravity field determination in three cases： gravitational potential case, gravitation case, and gravitational gradient case. Several tests using simulation experiments show that the fictitious compress recovery approach shows promise in physical geodesy applications.

SEMI-ANALYTICAL FINITE ELEMENT METHOD FOR FICTITIOUS CRACK MODEL IN FRACTURE MECHANICS OF CONCRETE

Based on the Hamiltonian governing equations of plane elasticity for sectorial domain, the variable separation and eigenfunction expansion techniques were employed to develop a novel analytical finite element for the fictitious crack model in fracture mechanics of concrete. The new analytical element can be implemented into FEM program systems to solve fictitious crack propagation problems for concrete cracked plates with arbitrary shapes and loads. Numerical results indicate that the method is more efficient and accurate than ordinary finite element method.

Microwave coherent manipulation of cold atoms in optically induced fictitious magnetic traps on an atom chip

We propose a novel on-chip platform for controlling and manipulating cold atoms precisely and coherently. The scheme is achieved by producing optically induced fictitious magnetic traps（OFMTs） with 790 nm（for -（87）Rb） circularly polarized laser beams and state-dependent potentials simultaneously for two internal atomic states with microwave coplanar waveguides. We carry out numerical calculations and simulations for controlled collisional interactions between OFMTs and addressable single atoms＇ manipulation on our designed hybrid atom chips. The results show that our proposed platform is feasible and flexible, which has wide applications including collisional dynamics investigation, entanglement generation,and scalable quantum gates implementation.

A Case Study on the Translation of Fictitious Elements in Science Fiction

As The Three-Body Problem translated by Ken Liu went popular in America, Chinese science fiction has got spotlightfrom all over the world. This article studies the translation of fictitious elements in The Three-Body Problem. Based on the specificmethods Ken Liu employs, general requirements for translators are proposed and strategies of translating fictitious elements areelicited. It intends to bring forth enlightening ideas in translating fictitious elements of science fiction and shed lights on this areaof study which few have set foot on.

Fictitious soil pile model for dynamic analysis of pipe piles under high-strain conditions

A fictitious soil pile(FSP)model is developed to simulate the behavior of pipe piles with soil plugs undergoing high-strain dynamic impact loading.The developed model simulates the base soil with a fictitious hollow pile fully filled with a soil plug extending at a cone angle from the pile toe to the bedrock.The friction on the outside and inside of the pile walls is distinguished using different shaft models,and the propagation of stress waves in the base soil and soil plug is considered.The motions of the pile—soil system are solved by discretizing them into spring-mass model based on the finite difference method.Comparisons of the predictions of the proposed model and conventional numerical models,as well as measurements for pipe piles in field tests subjected to impact loading,validate the accuracy of the proposed model.A parametric analysis is conducted to illustrate the influence of the model parameters on the pile dynamic response.Finally,the effective length of the FSP is proposed to approximate the affected soil zone below the pipe pile toe,and some guidance is provided for the selection of the model parameters.

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.