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.

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.

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.

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.

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.

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.

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.

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.

Three-dimensional inversion of borehole-surface electrical data based on quasi-analytical approximation

3D inversion of borehole-surface electrical data for complex geo-electrical models is still a challenging problem in geophysical exploration. We have developed a program for 3D inversion to borehole-surface electrical data based on the quasi-analytical approximation （QA） and re-weighted regularized conjugate gradient method （RRCG） algorithms using Visual Fortran 6.5. Application of the QA approximation to forward modeling and Frechet derivative computations speeds up the calculation dramatically. The trial calculation for synthetic data of theoretical model showed that the program is fast and highly precise.

Dynamic interaction of twin vertically overlapping lined tunnels in an elastic half space subjected to incident plane waves

The scattering of plane harmonic P and SV waves by a pair of vertically overlapping lined tunnels buried in an elastic half space is solved using a semi-analytic indirect boundary integration equation method. Then the effect of the distance between the two tunnels, the stiffness and density of the lining material, and the incident frequency on the seismic response of the tunnels is investigated. Numerical results demonstrate that the dynamic interaction between the twin tunnels cannot be ignored and the lower tunnel has a significant shielding effect on the upper tunnel for high-frequency incident waves, resulting in great decrease of the dynamic hoop stress in the upper tunnel; for the low-frequency incident waves, in contrast, the lower tunnel can lead to amplification effect on the upper tunnel. It also reveals that the frequency-spectrum characteristics of dynamic stress of the lower tunnel are significantly different from those of the upper tunnel. In addition, for incident P waves in low-frequency region, the soft lining tunnels have significant amplification effect on the surface displacement amplitude, which is slightly larger than that of the corresponding single tunnel.

Diffraction of plane P waves around an alluvial valley in poroelastic half-space

Diffraction of plane P waves around an alluvial valley of arbitrary shape in poroelastic half-space is investigated by using an indirect boundary integral equation method. Based on the Green＇s fimctions of line source in poroelastic half-space, the scattered waves are constructed using the fictitious wave sources close to the interface of the valley and the density of ficti- tious wave sources are determined by boundary conditions. The precision of the method is verified by the satisfaction extent of boundary conditions, and the comparison between the degenerated solutions and available results in single-phase case. Finally, the nature of diffraction of plane P waves around an alluvial valley in poroelastic half-space is investigated in detail through nu- merical examples.

Application of A Fast Multipole BIEM for Flow Diffraction from A 3D Body

A Fast Multipole Method (FMM) is developed as a numerical approach to the reduction of the computational cost and requirement memory capacity for a large in solving large-scale problems. In this paper it is applied to the boundary integral equation method (BIEM) for current diffraction from arbitrary 3D bodies. The boundary integral equation is discretized by higher order elements, the FMM is applied to avoid the matrix/vector product, and the resulting algebraic equation is solved by the Generalized Conjugate Residual method (GCR). Numerical examination shows that the FMM is more efficient than the direct evaluation method in computational cost and storage of computers.

Focusing phenomenon and near-trapped modes of SH waves

In this study, the null-field boundary integral equation method （BIEM） and the image method are used to solve the SH wave scattering problem containing semi-circular canyons and circular tunnels. To fully utilize the analytical property of Circular geometry, the polar coordinates are used to expand the closed-form fundamental solution to the degenerate kernel, and the Fourier series is also introduced to represent the boundary density. By collocating boundary points to match boundary condition on the boundary, a linear algebraic system is constructed. The unknown coefficients in the algebraic system can be easily determined. In this way, a semi-analytical approach is developed. Following the experience of near-trapped modes in water wave problems of the full plane, the focusing phenomenon and near-trapped modes for the SH wave problem of the half-plane are solved, since the two problems obey the same mathematical model. In this study, it is found that the SH wave problem containing two semi-circular canyons and a circular tunnel has the near-trapped mode and the focusing phenomenon for a special incident angle and wavenumber. In this situation, the amplification factor for the amplitude of displacement is over 300.

Dynamic rupture process of the 1999 Chi-Chi,Taiwan,earthquake

In this study, we preliminarily investigated the dynamic rupture process of the 1999 Chi-Chi, Taiwan, earthquake by using an extended boundary integral equation method, in which the effect of ground surface can be exactly included. Parameters for numerical modeling were carefully assigned based on previous studies. Numerical results indicated that, although many simplifications are assumed, such as the fault plane is planar and all heterogeneities are neglected, distribution of slip is still consistent roughly with the results of kinematic inversion, implying that for earthquakes in which ruptures run up directly to the ground surface, the dynamic processes are controlled by geometry of the fault to a great extent. By taking the common feature inferred by various kinematic inversion studies as a restriction, we found that the critical slip-weakening distance Dc should locate in a narrow region [60 cm, 70 cm], and supershear rupture might occur during this earthquake, if the initial shear stress before the mainshock is close to the local shear strength.

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.

Nonlinear Capillary-Gravity Waves Produced by a Vertically Oscillating Plate

The nonlinear capillary-gravity wave produced by a vertically oscillating plate, in which the contact-angle model is considered, is studied by use of the Boundary Integral Equation Method (BIEM). The present numerical experiment shows that the code is robust and efficient for modeling the generation and propagation of capillary-gravity waves. It is found that the wave heights of stationary periodic nonlinear waves radiated away from the plate are dependent on the parameters involved in the contact-angle model. The effect of the contact-angle hysteresis and the nonlinearity of capillary-gravity waves on the wave profile is discussed in the paper.