A STUDY ON THE WEIGHT FUNCTION OF THE MOVING LEAST SQUARE APPROXIMATION IN THE LOCAL BOUNDARY INTEGRAL EQUATION METHOD

The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the basic function and of the weight function,and is mainly determined by that of the weight function.Therefore,the weight function greatly affects the accuracy of results obtained.Different kinds of weight functions,such as the spline function, the Gauss function and so on,are proposed recently by many researchers.In the present work,the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method.The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed.Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and α in Gauss and exponential weight functions are in the range of reasonable values,respectively,and the higher the smoothness of the weight function,the better the features of the solutions.

RESEARCH ON THE COMPANION SOLUTION FOR A THIN PLATE IN THE MESHLESS LOCAL BOUNDARY INTEGRAL EQUATION METHOD

The meshless local boundary integral equation method is a currently developed numerical method, which combines the advantageous features of Galerkin finite element method(GFEM), boundary element method(BEM) and element free Galerkin method(EFGM), and is a truly meshless method possessing wide prospects in engineering applications. The companion solution and all the other formulas required in the meshless local boundary integral equation for a thin plate were presented, in order to make this method apply to solve the thin plate problem.

Improved non-singular local boundary integral equation method

When the source nodes are on the global boundary in the implementation of local boundary integral equation method （LBIEM）, singularities in the local boundary integrals need to be treated specially. In the current paper, local integral equations are adopted for the nodes inside the domain and moving least square approximation （MLSA） for the nodes on the global boundary, thus singularities will not occur in the new al- gorithm. At the same time, approximation errors of boundary integrals are reduced significantly. As applications and numerical tests, Laplace equation and Helmholtz equation problems are considered and excellent numerical results are obtained. Furthermore, when solving the Helmholtz problems, the modified basis functions with wave solutions are adapted to replace the usually-used monomial basis functions. Numerical results show that this treatment is simple and effective and its application is promising in solutions for the wave propagation problem with high wave number.

ISOTROPICALIZED SPLINE INTEGRAL EQUATION METHOD FOR THE ANALYSIS OF ANISOTROPIC PLATES

In the view of Reissner's and Kirchhoff's theories, respectively, we formulate the isotropicalized governing equations for the anisotropic plates, and give the proof of the equivalence relation between these two plate-models for the simply-supported rectangular orthotropic plates. The well-known fundamental solutions of the isotrqpic plates are utlized for the spline integral equation analysis of anisotropic plates.Even with sparse meshes the satisfactory results can be obtained. The analysis of plates on two-parameter elastic foundation is so simple as the common case that only a few terms should be added to the formulas of fictitious loads.

Resolving double-sided inverse heat conduction problem using calibration integral equation method

In this paper, a novel calibration integral equation is derived for resolving double-sided, two-probe inverse heat conduction problem of surface heat flux estimation. In contrast to the conventional inverse heat conduction techniques, this calibration approach does not require explicit input of the probe locations, thermophysical properties of the host material and temperature sensor parameters related to thermal contact resistance, sensor capacitance and conductive lead losses. All those parameters and properties are inherently contained in the calibration framework in terms of Volterra integral equation of the first kind. The Laplace transform technique is applied and the frequency domain manipulations of the heat equation are performed for deriving the calibration integral equation. Due to the ill-posed nature, regularization is required for the inverse heat conduction problem, a future-time method or singular value decomposition (SVD) can be used for stabilizing the ill-posed Volterra integral equation of the first kind.

Roll Flattening Analytical Model in Flat Rolling by Boundary Integral Equation Method

In order to improve rolled strip quality, precise plate shape control theory should be established. Roll flat- tening theory is an important part of the plate shape theory. To improve the accuracy of roll flattening calculation based on semi infinite body model, especially near the two roll barrel edges, a new and more accurate roll flattening model is proposed. Based on boundary integral equation method, an analytical model for solving a finite length semi infinite body is established. The lateral surface displacement field of the finite length semi-infinite body is simulated by finite element method （FEM） and lateral surface displacement decay functions are established. Based on the boundary integral equation method, the numerical solution of the finite length semi-infinite body under the distribu ted force is obtained and an accurate roll flattening model is established. Different from the traditional semi-infinite body model, the matrix form of the new roll flattening model is established through the mathematical derivation. The result from the new model is more consistent with that by FEM especially near the edges.

ANALYSIS OF SLOPING ELASTIC PILE UNDER ARBITRARY LOADS BY LINE-LOADED INTEGRAL EQUATION METHOD

For analysis of displacement and stress, an elastic sloping pile embedded in a homogeneous isotropic elastic half space under arbitrary loads at the top can be decomposed into two plane systems, i.e., the inclined plane xOz and its normal plane yOz . Let Mindlin's forces be the fundamental loads with unknown intensity function X(t),Y(t),Z(t) ,parallel to x,y,z_axis respectively, be distributed along the t axis of the pile in and concentrated forces Q x,Q y,Z ,couples M y,M x at top of the pile. Then, according to the boundary conditions of elastic pile, the problem is reduced to a set of Fredholm_Volterra type equations. Numerical solution is given and the accuracy of calculation can be checked by the reciprocal theorem of work.

Fast Multipole Accelerated Boundary Integral Equation Method for Evaluating the Stress Field Associated with Dislocations in a Finite Medium

In this paper,we develop an efficient numericalmethod based on the boundary integral equation formulation and new version of fast multipole method to solve the boundary value problem for the stress field associated with dislocations in a finite medium.Numerical examples are presented to examine the influence from material boundaries on dislocations.

On a Fast Integral Equation Method for Diffraction Gratings

The integral equation method for the simulation of the diffraction by optical gratings is an efficient numerical tool if profile gratings determined by simple crosssection curves are considered.This method in its recent version is capable to tackle profile curves with corners,gratings with thin coated layers,and diffraction scenarios with unfavorably large ratio period over wavelength.We discuss special implementational issues including the efficient evaluation of the quasi-periodic Green kernels,the quadrature algorithm,and the iterative solution of the arising systems of linear equations.Finally,as an example we present the simulation of echelle gratings which demonstrates the efficency of our approach.

EXTRAPOLATION FOR COLLOCATION METHOD OF THE FIRST KIND VOLTERRA INTEGRAL EQUATIONS

1. Introduction It is known that the following Cauchy problem for a parabolic partial differential equation (where the values at the right boundary, u.(1, t)=v(t) are unknown and sought for) is ill-posed: the solution (v) does not depend continuously on the data (g). In order to treat the ill-posedness and develop the numerical method, one reformulates the problem as a Volterra integral equation of the first kind wish a convolution type kernel (see Sneddon [1], Carslaw and Jaeger [2])

THE FUNDAMENTAL SOLUTION METHOD FOR INCOMPRESSIBLE NAVIER STOKES EQUATIONS

A complete boundary integral formulation for incompressible Navier Stokes equations with time discretization by operator splitting is developed by using the fundamental solutions of the Helmhotz operator equation with different orders. The numerical results for the lift and the drag hysteresis associated with a NACA0012 aerofoil oscillating in pitch are good in comparison with available experimental data.

THE INTEGRATION METHODS OF VACCO DYNAMICS EQUATION OFNONLINEAR NONHOLONOMIC SYSTEM

This paper presents the integration methods for vacco dynmmies equations of nonlinear nonholononic system,First.vacco dynamies equations are written in the canonical form and the field form.second the gradient methods the single-componentmethods and the field method are used to integrate the dynamics equations of the corresponding holonomic system respectively.And considering the restriction of nonholonomic construint to the initial conditions the solutions of Vacco dynamics cquations of nonlinear nonholonomic system are obtained.

UNIQUENESS OF SOLUTION OF FIELD POINT OF SINGULAR SOURCE OUTSIDE-REGION-DISTRIBUTION METHOD

The uniqueness of solution of field point, inside a convex region due to singular source(s) with kernel function decreasing with distance increasing, outside-region-distribution(s) such that the boundary condition expressed by the response of the source(s) is satisfied, is proved by using the condition of kernel function decreasing with distance increasing anal an integral inequality. Examples of part of these singular sources such as Kelvin's point force, Point-Ring-Couple (PRC) etc. are given. The proof of uniqueness of solution of field point in a twisted shaft of revolution due to PRC distribution is given as an example of application.

Three-dimensional numerical modeling of surface-to-borehole electromagnetic method for monitoring reservoir

Dynamic exploration for oil and gas requires careful monitoring of reservoir contents for safety and efficiency of oil extraction. This paper proposes a multi-source and multi-azimuth walk-around vertical electromagnetic profiling （MM-VEP） technique for surface-to-borehole electromagnetic surveying. Based on the difference in conductivities between reservoirs with different concentrations of oil and water, MM-VEP can be used to monitor reservoirs as they are injected with water. The MM-VEP response in five azimuth planes is modeled with three-dimensional （3D） integral equation calculations. The progress of waterflooding in four stages for enhanced oil recovery is shown to be indicated by field anomalies MM-VEP caused by variations in the reservoir resistivity. Numerical modeling demonstrates that MM-VEP measurements provides enough quantitative information from an underground reservoir to accurately detect oil deposits and monitor the progress of waterflooding.

Interaction of Linear Waves with Infinitely Long Horizontal Cylinders Studied by Boundary Element Method

The two-dimensional problems concerning the interaction of linear water waves with cylinders of arbitrary shape in two-layer deep water are investigated by use of the Boundary Integral Equation Method （BIEM）. Simpler new expressions for the Green functions are derived, and verified by comparison of results obtained by BIEM with those by an analytical method. Examined are the radiation and scattering of linear waves by two typical configurations of cylinders in two- layer deep water. Hydrodynamic behaviors including hydrodynamic coefficients, wave forces, reflection and transmission coef- ficients and energies are analyzed in detail, and some imeresting physical phenomena are observed.

METHOD TO CALCULATE BENDING CENTER AND STRESS INTENSITY FACTORS OF CRACKED CYLINDER UNDER SAINT-VENANT BENDING

Using the single crack solution and the regular solution elf plane harmonic function, the problem of Saint-Venant bending of a cracked cylinder by a transverse force was reduced to solving two sets of integral equations and its general solution was then obtained. Based on the obtained solution, a method to calculate the bending center and the stress intensity factors of the cracked cylinger whose cross-section is not thin-walled, but of small torsion rigidity is proposed. Some numerical examples are given.

A MESHLESS METHOD OF A THIN PLATE

A meshless approach to analysis of arbitrary Kirechhoff plates bythe local boundary integral equation (LBIF) method is presented. Themethod combines the advantageous features of all the three meth- ods:the Galerkin finite element method (GFEM), the boundary elementmethod (BEM) and the element- free Galerkin method (EFGM). It is atruly meshless method, which means that the discretization is inde-pendent of geometric subdivision into elements or cells, but is onlyboundary integration, however over a local boundary cen- tered) overa domain in question.

A moving Kriging interpolation-based boundary node method for two-dimensional potential problems

In this paper, a meshfree boundary integral equation （BIE） method, called the moving Kriging interpolation- based boundary node method （MKIBNM）, is developed for solving two-dimensional potential problems. This study combines the DIE method with the moving Kriging interpolation to present a boundary-type meshfree method, and the corresponding formulae of the MKIBNM are derived. In the present method, the moving Kriging interpolation is applied instead of the traditional moving least-square approximation to overcome Kronecker＇s delta property, then the boundary conditions can be imposed directly and easily. To verify the accuracy and stability of the present formulation, three selected numerical examples are presented to demonstrate the efficiency of MKIBNM numerically.

A hybrid method for inverse cavity scattering problem for shape

In this paper, we give a hybrid method to numerically solve the inverse open cavity scattering problem for cavity shape, given the scattered solution on the opening of the cavity. This method is a hybrid between an iterative method and an integral equations method for solving the Cauchy problem. The idea of this hybrid method is simple, the operation is easy, and the computation cost is small. Numerical experiments show the feasibility of this method, even for cases with noise.

PRECONDITIONED CONJUGATE GRADIENT METHODS FOR INTEGRAL EQUATIONS OF THE SECOND KIND DEFINED ON THE HALF-LINE

We consider solving integral equations of the second kind defined on the half-line [0, infinity) by the preconditioned conjugate gradient method. Convergence is known to be slow due to the non-compactness of the associated integral operator. In this paper, we construct two different circulant integral operators to be used as preconditioners for the method to speed up its convergence rate. We prove that if the given integral operator is close to a convolution-type integral operator, then the preconditioned systems will have spectrum clustered around 1 and hence the preconditioned conjugate gradient method will converge superlinearly. Numerical examples are given to illustrate the fast convergence.