Mesh-free approximations via the error reproducing kernel method and applications to nonlinear systems developing shocks

Error estimates for the error reproducing kernel method (ERKM) are provided. The ERKM is a mesh-free functional approximation scheme [A. Shaw, D. Roy, A NURBS-based error reproducing kernel method with applications in solid mechanics, Computational Mechanics (2006), to appear (available online)], wherein a targeted function and its derivatives are first approximated via non-uniform rational B-splines (NURBS) basis function. Errors in the NURBS approximation are then reproduced via a family of non-NURBS basis functions, constructed using a polynomial reproduction condition, and added to the NURBS approximation of the function obtained in the first step. In addition to the derivation of error estimates, convergence studies are undertaken for a couple of test boundary value problems with known exact solutions. The ERKM is next applied to a one-dimensional Burgers equation where, time evolution leads to a breakdown of the continuous solution and the appearance of a shock. Many available mesh-free schemes appear to be unable to capture this shock without numerical instability. However, given that any desired order of continuity is achievable through NURBS approximations, the ERKM can even accurately approximate functions with discontinuous derivatives. Moreover, due to the variation diminishing property of NURBS, it has advantages in representing sharp changes in gradients. This paper is focused on demonstrating this ability of ERKM via some numerical examples. Comparisons of some of the results with those via the standard form of the reproducing kernel particle method (RKPM) demonstrate the relative numerical advantages and accuracy of the ERKM.

On an asymptotic method of Krylov-Bogoliubov for overdamped nonlinear systems

A general asymptotic method based on the work of Krylov-Bogoliubov is developed to obtain the response of nonlinear over damped systems. A second-order system with both roots real is treated first and the method is then extended to higher-order systems. Two illustrative examples show good agreement with results obtained by numerical integration.

Nonaxisymmetric unsteady motion over a rotating disk in the presence of free convection and magnetic field

The nonaxisymmetric unsteady motion produced by a buoyancy-induced cross-flow of an electrically conducting fluid over an infinite rotating disk in a vertical plane and in the presence of an applied magnetic field normal to the disk has been studied. Both constant wall and constant heat flux conditions have been considered. It has been found that if the angular velocity of the disk and the applied magnetic field squared vary inversely as a linear function of time (i.e. as (1??t*)?1, the governing Navier-Stokes equation and the energy equation admit a locally self-similar solution. The resulting set of ordinary differential equations has been solved using a shooting method with a generalized Newton's correction procedure for guessed boundary conditions. It is observed that in a certain region near the disk the buoyancy induced cross-flow dominates the primary von Karman flow. The shear stresses induced by the cross-flow are found to be more than these of the primary flow and they increase with magnetic parameter or the parameter ? characterizing the unsteadiness. The velocity profiles in the x- and y-directions for the primary flow at any two values of the unsteady parameter ? cross each other towards the edge of the boundary layer. The heat transfer increases with the Prandtl number but reduces with the magnetic parameter.

Unsteady free convection flow under the influence of a magnetic field

The unsteady free convection boundary layer at the stagnation point of a two-dimensional body and an axisymmetric body with prescribed surface heat flux or temperature has been studied. The magnetic field is applied parallel to the surface and the effect of induced magnetic field has been considered. It is found that for certain powerlaw distribution of surface heat flux or temperature and magnetic field with time, the governing boundary layer equations admit a self-similar solution locally. The resulting nonlinear ordinary differential equations have been solved using a finite element method and a shooting method with Newton's corrections for missing initial conditions. The results show that the skin friction and heat transfer coefficients, and x-component of the induced magnetic field on the surface increase with the applied magnetic field. In general, the skin friction, heat transfer and x-component of the induced magnetic field for axisymmetric case are more than those of the two-dimensional case. Also they change more when the surface heat flux or temperature decreases with time than when it increases with time. The skin friction, heat transfer and x-component of the induced magnetic field are significantly affected by the magnetic Prandtl number and they increase as the magnetic Prandtl number decreases. The skin friction and x-component of the magnetic field increase with the dissipation parameter, but heat transfer decreases.

WLS method for parameter estimation in water distribution networks

The weighted-least-squares method based on the Gauss-Newton minimization technique is used for parameter estimation in water distribution networks. The parameters considered are: element resistances (single and/or group resistances, Hazen-Williams coefficients, pump specifications) and consumptions (for single or multiple loading conditions). The measurements considered are: nodal pressure heads, pipe flows, head loss in pipes, and consumptions/inflows. An important feature of the study is a detailed consideration of the influence of different choice of weights on parameter estimation, for error-free data, noisy data, and noisy data which include bad data. The method is applied to three different networks including a real-life problem.

A fast quasi-Newton method for semi-supervised SVM

Due to its wide applicability, semi-supervised learning is an attractive method for using unlabeled data in classification. In this work, we present a semi-supervised support vector classifier that is designed using quasi-Newton method for nonsmooth convex functions. The proposed algorithm is suitable in dealing with very large number of examples and features. Numerical experiments on various benchmark datasets showed that the proposed algorithm is fast and gives improved generalization performance over the existing methods. Further, a non-linear semi-supervised SVM has been proposed based on a multiple label switching scheme. This non-linear semi-supervised SVM is found to converge faster and it is found to improve generalization performance on several benchmark datasets. (C) 2010 Elsevier Ltd. All rights reserved.

Simulation of Free-Surface Flow in a Tank Using the Navier-Stokes Model and Unstructured Finite Volume Method

Modelling free-surface flow has very important applications in many engineering areas such as oil transportation and offshore structures. Current research focuses on the modelling of free surface flow in a tank by solving the Navier-Stokes equation. An unstructured finite volume method is used to discretize the governing equations. The free surface is tracked by dynamically adapting the mesh and making it always surface conforming. A mesh-smoothing scheme based on the spring analogy is also implemented to ensure mesh quality throughout the computaiton. Studies are performed on the sloshing response of a liquid in an elastic container subjected to various excitation frequencies. Further investigations are also carried out on the critical frequency that leads to large deformation of the tank walls. Another numerical simulation involves the free-surface flow past as submerged obstacle placed in the tank to show the flow separation and vortices. All these cases demonstrate the capability of this numerical method in modelling complicated practical problems.