A Composite State Convergence Scheme for Bilateral Teleoperation Systems

State convergence is a novel control algorithm for bilateral teleoperation of robotic systems. First, it models the teleoperation system on state space and considers all the possible interactions between the master and slave systems. Second, it presents an elegant design procedure which requires a set of equations to be solved in order to compute the control gains of the bilateral loop. These design conditions are obtained by turning the master-slave error into an autonomous system and imposing the desired dynamic behavior of the teleoperation system. Resultantly, the convergence of master and slave states is achieved in a well-defined manner. The present study aims at achieving a similar convergence behavior offered by state convergence controller while reducing the number of variables sent across the communication channel. The proposal suggests transmitting composite master and slave variables instead of full master and slave states while keeping the operator’s force channel intact. We show that, with these composite and force variables; it is indeed possible to achieve the convergence of states in a desired way by strictly following the method of state convergence. The proposal leads to a reduced complexity state convergence algorithm which is termed as composite state convergence controller. In order to validate the proposed scheme in the absence and presence of communication time delays, MATLAB simulations and semi-real time experiments are performed on a single degree-of-freedom teleoperation system.      

Numerical analysis of confined turbulent flow

This work outlines a second order accurate, coupled, conservative new numerical scheme for solving a two dimensional incompressible turbulent flow filed. Mean vorticity, ω, and mean stream function, ψ, are used as the mean flow dependent variables. The turbulent kinetic energy $\text{k}$ and the turbulent energy decay rate, ?, are used to define the turbulent state. In the present computational scheme two systems of equations and variables are considered: the mean flow system, ψ-ω, and the turbulent state system, $\text{k-?}$. Every system is solved implicity in a coupled double loop manner, and all the flow equations are solved iteratively in the global sense. Since the turbulence boundary conditions have a non-regular variation near a solid wall, they are coupled to the equations implicitly in both systems. In this way the numerical instabilities due to the irregular form of the equations near the solid walls are suppressed. The rate of convergence of the new numerical scheme of the coupled systems ψ-ω and $\text{k-?}$ is twice that realized when solving these equations separately. The necessary conditions for convergence of the numerical equations are investigated as well as the rate of convergence features. The detailed stability conditions are derived. As an example, the axisymmetric mixing of two confined jets with an internal heat source is considered with this numerical scheme.     

Optimal periodic control problem: a modified trigonometric approximation and quasi-stationarity conditions

The constrained optimal periodic control problem for a system described by differential equations and endowed with inertial controllers is considered, A sequence of discretized problems using trigonometric polynomials is proposed to approximate the problem. Instantaneous constraints for the state and control are handled by a new and more precise approach that imposes only a small number of non-linear but easily computable constraints. The convergence conditions for a sequence of optimal solutions of discretized problems are derived. The inclusion in the approximating scheme of various quasi-stationarity conditions for the control and state variables is analysed. Extension of a new approximating approach for inertialess and smooth problems is also discussed.     

Steady-State Computations Using Summation-by-Parts Operators

This paper concerns energy stability on curvilinear grids and its impact on steady-state calulations. We have done computations for the Euler equations using fifth order summation-by-parts block and diagonal norm schemes. By imposing the boundary conditions weakly we obtain a fifth order energy-stable scheme. The calculations indicate the significance of energy stability in order to obtain convergence to steady state. Furthermore, the difference operators are improved such that faster convergence to steady state are obtained.     

Chattering phenomena in discontinuous control systems

The reduction of chatter in variable structure control (VSC) systems is considered. In this scheme the time derivative of the control law is discontinuous on a suitable manifold of an augmented order state space. For the case of differential equations with uncertain RHS, the proposed scheme maintains the robustness properties of classical VSC if an estimator of inaccessible state components is introduced. A separation property, together with the convergence of the proposed estimator are proved. Some simulation results relating to an induction motor control problem are finally presented.     

Development of a multigrid code for 3-D Navier-Stokes equations and its application to a grid-refinement study

A multigrid acceleration technique has been developed to solve the three-dimensional Navier-Stokes equations efficiently. An explicit multistage Runge-Kutta type-of time-stepping scheme is used as the basic algorithm in conjunction with the multigrid scheme. A grid-refinement study has been conducted to obtain grid converged solutions for transonic flow over a finite wing. Present solutions indicate that the number of multigrid cycles required to achieve a given level of convergence does not increase with the number of mesh points employed, making it a very attractive scheme for fine meshes.     

An iterative scheme for numerical solution of steady incompressible viscous flows

An iterative solution scheme is proposed for application to steady incompressible viscous flows in simple and complex geometries. The iterative scheme solves the vorticity-stream function form of the Navier-Stokes equations in generalized curvilinear coordinates. The flow system of equations are cast into a Newton's iterative form which are solved using the modified strongly implicit procedure. The solution scheme is benchmarked using two test cases, namely: a shear-driven steady laminar flow in a square cavity; and a simple laminar flow in a complex expanding channel. The iterative process to steady-state convergence in both test cases is highly stable and the convergence rate is without spurious oscillations. At convergence, the flow solutions are second-order accurate.     

Convergence Acceleration for Hyperbolic Systems Using Semicirculant Approximations

The iterative solution of systems of equations arising from systems of hyperbolic, time-independent partial differential equations (PDEs) is studied. The PDEs are discretized using a finite volume or finite difference approximation on a structured grid. A convergence acceleration technique where a semicirculant approximation of the spatial difference operator is employed as preconditioner is considered. The spectrum of the preconditioned coefficient matrix is analyzed for a model problem. It is shown that, asymptotically, the time step for the forward Euler method could be chosen as a constant, which is independent of the number of grid points and the artificial viscosity parameter. By linearizing the Euler equations around an approximate solution, a system of linear PDEs with variable coefficients is formed. When utilizing the semicirculant (SC) preconditioner for this problem, which has properties very similar to the full nonlinear equations, numerical experiments show that the favorable convergence properties hold also here. We compare the results for the SC method to those of a multigrid (MG) scheme. The number of iterations and the arithmetic complexities are considered, and it is clear that the SC method is more efficient for the problems studied. Also, the MG scheme is sensitive to the amount of artificial dissipation added, while the SC method is not.     

An extension of MacCormack's method for flows with higher-order equations and in different configurations

The numerical scheme for the computation of a shock discontinuity developed by MacCormack has been extended to solve a number of differential equations, including cases explicitly containing higher-order derivatives: (1) Korteweg-de Vries equation with a term of third-order derivative, (2) a system of nonlinear equations governing nonsteady one-dimensional plasma flow in cylindrical coordinate, (3) equations of solar wind. Comparisons with previous results are made, if available, to illustrate the advantages of the present method. The question of convergence of the numerical calculation is discussed.     

On the application of the minimum polynomial extrapolation method to incompressible flows with heat transfer

In this paper, the Minimum Polynomial Extrapolation method (MPE) is used to accelerate the convergence of the Characteristic–Based–Split (CBS) scheme for the numerical solution of steady state incompressible flows with heat transfer. The CBS scheme is a fractional step method for the solution of the Navier–Stokes equations while the MPE method is a vector extrapolation method which transforms the original sequence into another sequence converging to the same limit faster then the original one without the explicit knowledge of the sequence generator. The developed algorithm is tested on a two-dimensional benchmark problem (buoyancy–driven convection problem) where the Navier–Stokes equations are coupled with the temperature equation. The obtained results show the feature of the extrapolation procedure to the CBS scheme and the reduction of the computational time of the simulation.     

Neural network optimized with evolutionary computing technique for solving the 2-dimensional Bratu problem

In this paper, a stochastic technique is developed to solve 2-dimensional Bratu equations using feed-forward artificial neural networks, optimized with genetic and interior-point algorithms. The 2-dimensional equations are first transformed into a 1-dimensional boundary value problem, and a mathematical model of the transformed equation is then formulated with neural networks using an unsupervised error. Network weights are optimized to minimize the error. Evolutionary computing based on genetic algorithms is used as a tool for global search, integrated with an interior-point method for rapid local convergence. The methodology is applied to solve three cases of boundary value problems for the Bratu equations. The accuracy, convergence and effectiveness of the scheme is validated for a large number of simulations. Comparison of results is made with the exact solution derived using MATHEMATICA, and is found to be in good agreement.     

Iterative numerical solutions and boundary conditions for the parabolized Navier-Stokes equations

A new numerical iteration scheme for solving the parabolized Navier-Stokes (PNS) equations is presented. This scheme has all the features and advantages of the successive line over relaxation (SLOR) technique, and thus it can be easily accelerated to get much higher rate of convergence of the global iteration scheme than previously suggested schemes. The choice of appropriate downstream boundary conditions for the PNS and Navier-Stokes equations is discussed in the context of boundary layer simulation. A critical comparison of accuracy and rate of convergence is performed for the flow over a flat plate.     

Iterative methods for stationary solutions to the steady-state compressible Navier-Stokes equations

Numerical experiments are presented for the solution of the steady-state compressible Navier-Stokes equations. One test problem is fixed supersonic flow past a double ellipse, and the various solution methods studied. The problem is discretized using Osher's scheme, first- and second-order accurate. The fastest convergence to steady state is obtained using Newton's method. Simplifications of Newton's method based on domain decomposition are shown to perform well, whereas line relaxation methods meet with difficulties.     

Two-level solution method for Markov chain modelling transfer lines with unreliable servers and finite buffers

Transfer lines with inter-stage buffers and unreliable servers are often modelled by means of Markov chains. Because of the large number of states, solving the steady-state equations of the chain is not a trivial task. This paper proposes a two-level iterative scheme for computing the steady-state probability distribution. In the framework of non-negative matrix theory some general results are proved which guarantee the convergence of the proposed procedure. Moreover, numerous numerical experiments are given, which show that the two-level iterative scheme enjoys a very good rate of convergence. The method also works suitably for solving the steady-state probability equation of chains modelling systems with more than three stages.     

A numerical study on preconditioning and partitioning schemes for reactive transport in a PEMFC catalyst layer

A numerical method used to simulate PEMFC catalyst layer transport and electrochemistry is described. The set of nonlinear equations is discretized using the finite volume method and solved using an inexact Newton method. A block “ILU porous” preconditioner is used to precondition the linear system. A “porous partitioning” scheme is used to partition the domain for parallel processors. Geometries with low porosities are found to require a much smaller number of iterations for convergence compared to geometries with high porosities. The porous partitioning scheme is shown to outperform the standard partitioning scheme for cases run with more than 4 processors. The block “ILU porous” preconditioner was not found to be more effective than the block ILU(1) preconditioner. Finally, the importance of using rigorous convergence criteria in these simulations is demonstrated by comparing the computed total consumption values of different species at different nonlinear rms tolerance values.     

Unstructured Grid-Based Discontinuous Galerkin Method for Broadband Electromagnetic Simulations

A parallel, unstructured, high-order discontinuous Galerkin method is developed for the time-dependent Maxwell's equations, using simple monomial polynomials for spatial discretization and a fourth-order Runge–Kutta scheme for time marching. Scattering results for a number of validation cases are computed employing polynomials of up to third order. Accurate solutions are obtained on coarse meshes and grid convergence is achieved, demonstrating the capabilities of the scheme for time-domain electromagnetic wave scattering simulations.     

Mixed finite-difference scheme for analysis of simply supported thick plates

A mixed finite-difference scheme is presented for the stress and free vibration analysis of simply supported nonhomogeneous and layered orthotropic thick plates. The analytical formulation is based on the linear, three-dimensional theory of orthotropic elasticity and a Fourier approach is used to reduce the governing equations to six first-order ordinary differential equations in the thickness coordinate. The governing equations possess a symmetric coefficient matrix and are free of derivatives of the elastic characteristics of the plate. In the finite difference discretization two interlacing grids are used for the different fundamental unknowns in such a way as to reduce both the local discretization error and the bandwidth of the resulting finite-difference field equations. Numerical studies are presented for the effects of reducing the interior and boundary discretization errors and of mesh refinement on the accuracy and convergence of solutions. It is shown that the proposed scheme, in addition to a number of other advantages, leads to highly accurate results, even when a small number of finite difference intervals is used.     

Mathematical Modelling Of Wreck Events Originated By Dam Collapse

Based on Saint-Venant (shallow water) equations, in this paper the mathematical model of wreck events produced by dam collapse is constructed. A two-layer difference scheme with non-linear regularisation is used for the numerical solution of the aforementioned model. The convergence of this difference scheme in Eulerian variables with non-linear regularisation to the smooth solutions of one-dimensional Saint-Venant equations are considered for a Cauchy problem with periodic (in spatial variables) solutions. The proof of difference scheme convergence is conducted using the energetic method. The existence and uniqueness of the difference scheme solution is proved. That the difference scheme converges in mesh norm $L_2$ with speed $O\lpar h^2\rpar$ in the class of sufficiently smooth solutions of the difference scheme is also proved.     

An unfactored implicit scheme with multigrid acceleration for the solution of the Navier-Stokes equations

An unfactored implicit difference scheme for the steady state solution of the multidimensional Navier-Stokes equations of a compressible fluid is presented. The hyperbolic part is approximated by a high resolution scheme based on flux-vector splitting and upwind-biased differences to avoid the necessity of artificial dissipation terms and to construct a diagonal dominant solution matrix. Consequently, an iterative inversion of the solution matrix can be performed without any time step restriction. The rate of convergence is improved by using the indirect multigrid concept in form of the FAS scheme. The method is formulated for a body-fitted, curvilinear coordinate system. The computational results for laminar subsonic, transonic and supersonic steady-state flows which are compared with analytical and other numerical results as well as with experimental data illustrate the efficiency and the accuracy of the algorithm.     

Solution domain decomposition method for viscous incompressible flow

A solution domain decomposition method is developed for steady state solution of the biharmonic-based Navier-Stokes equations. It consists of a domain decomposition in conjunction with Chebyshev collocation for spatial discretization. The interactions between subdomains are effectively decoupled by means of a superposition of auxiliary solutions to yield a set of independent elementary problems which can be solved concurrently on multiprocessor computers. Assessments are carried out to a number of test problems including the two-dimensional steady flow in a driven square cavity. Illustrative examples indicate a good performance of the proposed methodology which does not affect the convergence and stability of the discretization scheme. Spectral accuracy is retained with absolute error decaying in an exponential fashion. The numerical solutions for the driven cavity compare favorably against previously published numerical results except for a slight overprediction in the vertical velocity component at Reynolds number of 400. TheC ³ continuity is speculated to be its cause.