Radial point interpolation based finite difference method for mechanics problems

A radial point interpolation based finite difference method (RFDM) is proposed in this paper. In this novel method, radial point interpolation using local irregular nodes is used together with the conventional finite difference procedure to achieve both the adaptivity to irregular domain and the stability in the solution that is often encountered in the collocation methods. A least‐square technique is adopted, which leads to a system matrix with good properties such as symmetry and positive definiteness. Several numerical examples are presented to demonstrate the accuracy and stability of the RFDM for problems with complex shapes and regular and extremely irregular nodes. The results are examined in detail in comparison with other numerical approaches such as the radial point collocation method that uses local nodes, conventional finite difference and finite element methods. Copyright © 2006 John Wiley & Sons, Ltd.     

High-order finite difference modal method for diffraction gratings

A new high-order finite difference modal method (FDMM) is developed for analyzing diffraction gratings in conical and classical mountings. The difference scheme is constructed by enforcing the internal interface conditions in each grating layer to high-order derivatives, and it gives a high order of accuracy for computing the eigenmodes of the grating layer. Between different layers, the interface conditions are implemented using a Fourier matching scheme and a point matching scheme. Compared with the standard Fourier modal method, the high-order FDMM is more efficient since the matrices in the discretized eigenvalue problems are sparse. Numerical examples are used to illustrate the performance of the method.     

A method for calculating meshless finite difference weights

An exact method for calculating the finite difference weights for arbitrary distributed points in multiple dimensions is presented. The method avoids the numerical ill conditioning associated with a small‐scale factor (ε) by delaying the limiting of ε → 0. The Gaussian Radial Basis function is approximated by a Taylor's series expansion in the variable ε, and the resultant matrix equation is solved using the Fraction Free LU decomposition method. Copyright © 2007 John Wiley & Sons, Ltd.     

Solving the nonlinear Poisson-type problems with F-Trefftz hybrid finite element model

A hybrid finite element model based on F-Trefftz kernels (fundamental solutions) is formulated for analyzing Dirichlet problems associated with two-dimensional nonlinear Poisson-type equations including nonlinear Poisson-Boltzmann equation and diffusion-reaction equation. The nonlinear force term in the Poisson-type equation is frozen by introducing the imaginary terms at each Picard iteration step, and then the induced Poisson problem is solved by the present hybrid finite element model involving element boundary integrals only, coupling with the particular solution method with radial basis function interpolation. The numerical accuracy of the present method is investigated by numerical experiments for problems with complex geometry and various nonlinear force functions.     

Bottomhole assembly analysis by finite difference differential method

This paper presents a three‐dimensional finite difference differential method for bottomhole assembly (BHA) analysis under static loads. The analysis is required to optimize the BHA configurations for drilling directional boreholes in the petroleum industry. The optimization of BHA configurations ensures the controlled cruising of the drill bit to drill the hole along a planned trajectory. The model incorporates the contact response between drillstring and wellbore wall, the upper tangent point problem, stabilizer configurations, bent sub model and other considerations for numerical solutions. The overall formulations are presented in a matrix format within convenient coordinate systems developing a transformation matrix. Key solution algorithms for computer programming are then described in detail. One analytical solution for beams under weight and torque is used to verify the method. Also the analytical solutions of three other simple BHA configurations are used for verification. The verified method is then applied in a series of parametric investigations to characterize the responses of typical BHAs. The BHA responses studied include the side force at bit and lateral deformation of the assembly along its length with varying weights on bit and hole inclination angles for typical building, dropping and holding assemblies. BHAs with bent sub are analyzed with various tool face angles and bent angles. The effect of wellbore curvature on side forces is also demonstrated. Copyright © 2007 John Wiley & Sons, Ltd.     

Generalized finite difference method for solving two-dimensional non-linear obstacle problems

In this study, the obstacle problems, also known as the non-linear free boundary problems, are analyzed by the generalized finite difference method (GFDM) and the fictitious time integration method (FTIM). The GFDM, one of the newly-developed domain-type meshless methods, is adopted in this study for spatial discretization. Using GFDM can avoid the tasks of mesh generation and numerical integration and also retain the high accuracy of numerical results. The obstacle problem is extremely difficult to be solved by any numerical scheme, since two different types of governing equations are imposed on the computational domain and the interfaces between these two regions are unknown. The obstacle problem will be mathematically formulated as the non-linear complementarity problems (NCPs) and then a system of non-linear algebraic equations (NAEs) will be formed by using the GFDM and the Fischer–Burmeister NCP-function. Then, the FTIM, a simple and powerful solver for NAEs, is used solve the system of NAEs. The FTIM is free from calculating the inverse of Jacobian matrix. Three numerical examples are provided to validate the simplicity and accuracy of the proposed meshless numerical scheme for dealing with two-dimensional obstacle problems.     

A hybrid finite difference and moving least square method for elasticity problems

In this paper, a novel hybrid finite difference and moving least square (MLS) technique is presented for the two-dimensional elasticity problems. A new approach for an indirect evaluation of second order and higher order derivatives of the MLS shape functions at field points is developed. As derivatives are obtained from a local approximation, the proposed method is computationally economical and efficient. The classical central finite difference formulas are used at domain collocation points with finite difference grids for regular boundaries and boundary conditions are represented using a moving least square approximation. For irregular shape problems, a point collocation method (PCM) is applied at points that are close to irregular boundaries. Neither the connectivity of mesh in the domain/boundary or integrations with fundamental/particular solutions is required in this approach. The application of the hybrid method to two-dimensional elastostatic and elastodynamic problems is presented and comparisons are made with the boundary element method and analytical solutions.     

Solution of Fokker-Planck equation by finite element and finite difference methods for nonlinear systems

The response of a structural system to white noise excitation (deltacorrelated) constitutes a Markov vector process whose transitional probability density function (TPDF) is governed by both the forward Fokker-Planck and backward Kolmogorov equations. Numerical solution of these equations by finite element and finite difference methods for dynamical systems of engineering interest has been hindered by the problem of dimensionality. In this paper numerical solution of the stationary and transient form of the Fokker-Planck (FP) equation corresponding to two state nonlinear systems is obtained by standard sequential finite element method (FEM) using C⁰ shape function and Crank-Nicholson time integration scheme. The method is applied to Van-der-Pol and Duffing oscillators providing good agreement between results obtained by it and exact results. An extension of the finite difference discretization scheme developed by Spencer, Bergman and Wojtkiewicz is also presented. This paper presents an extension of the finite difference method for the solution of FP equation up to four dimensions. The difficulties associated in extending these methods to higher dimensional systems are discussed. This paper is dedicated to Prof R N Iyengar of the Indian Institute of Science on the occasion of his formal retirement.     

A mesh-free minimum length method for 2-D problems

A mesh-free minimum length method (MLM) has been proposed for 2-D solids and heat conduction problems. In this method, both polynomials as well as modified radial basis functions (RBFs) are used to construct shape functions for arbitrarily distributed nodes based on minimum length procedure, which possess Kronecker delta property. The shape functions are then used to formulate a mesh-free method based on weak-form formulation. Both Gauss integration (GI) and stabilized nodal integration (NI) are employed to numerically evaluate Galerkin weak form. The numerical examples show that the MLM achieves better accuracy than the 4-node finite elements especially for problems with steep gradients. The method is easy to implement and works well for irregularly distributed nodes. Some numerical implementation issues for MLM are also discussed in detail.     

A collocation mesh-free method based on multiple basis functions

Two dominant shape functions are used to approximate scattered points in mesh-free methods, e.g. the interpolating radial basis function (RBF) and the approximating moving least squares (MLS). In the present paper, a new shape function is developed as a linear interpolating function of both MLS and RBF. This function inherits the properties of both MLS and RBF and is regularized by a control parameter μ, which takes different values in the domain [0,1]. Based on the proposed shape function, the collocation method is applied to solve initial and boundary value problems in one and two dimensions. The present method gives good results and achieves good convergence trends for different values of μ, compared with MLS and RBF individually, for a large number of nodes.     

Axisymmetric nonlinear analysis of shallow spherical shells by a combined boundary element and finite difference method

This paper is concerned with the development of the mixed boundary element method and finite element method for the analysis of spherical annular shells under axisymmetric loads. The boundary element techniques are used to solve the equilibrium equation of shells and the central difference operator is adopted to deal with the compatibility equations. Iterative techniques are used throughout the analysis procedure. A number of numerical examples are given in the paper to illustrate the validity of the present approach.     

Improved numerical solver for Kansa''s method based on affine space decomposition

Radial Basis functions (RBFs) have been successfully developed as a truly mesh-free method to find the numerical solutions of partial differential equations (PDEs). In particular, the asymmetric RBF collocation method (Kansa's method) is one of the most frequently used methods due to its ease of implementation. To achieve high accuracy, the resultant system of RBF–PDE problem usually becomes badly conditioned. We propose in this paper an improved solution method based on an affine space decomposition that decouples the influence between the interior and boundary collocations. Numerical examples are given to compare the proposed method with several direct methods.     

A hybrid radial boundary node method based on radial basis point interpolation

The hybrid boundary node method (HBNM) retains the meshless attribute of the moving least squares (MLS) approximation and the reduced dimensionality advantages of the boundary element method. However, the HBNM inherits the deficiency of the MLS approximation, in which shape functions lack the delta function property. Thus in the HBNM, boundary conditions are implemented after they are transformed into their approximations on the boundary nodes with the MLS scheme.This paper combines the hybrid displacement variational formulation and the radial basis point interpolation to develop a direct boundary-type meshless method, the hybrid radial boundary node method (HRBNM) for two-dimensional potential problems. The HRBNM is truly meshless, i.e. absolutely no elements are required either for interpolation or for integration. The radial basis point interpolation is used to construct shape functions with delta function property. So unlike the HBNM, the HRBNM is a direct numerical method in which the basic unknown quantity is the real solution of nodal variables, and boundary conditions can be applied directly and easily, which leads to greater computational precision. Some selected numerical tests illustrate the efficiency of the method proposed.     

脊形波导有限差分法分析

采用有限差分法,求波动方程的本征值和本征矢量,得到脊形波导的基模和高阶模场分布情况,得到的场分布与其他方法所得结果基本一致,同时分析了脊形波导中脊高与模式分布之间的关系。     

A particle method for two‐phase flows with large density difference

A new numerical approach for solving incompressible two‐phase flows is presented in the framework of the recently developed Consistent Particle Method (CPM). In the context of the Lagrangian particle formulation, the CPM computes spatial derivatives based on the generalized finite difference scheme and produces good results for single‐phase flow problems. Nevertheless, for two‐phase flows, the method cannot be directly applied near the fluid interface because of the abrupt discontinuity of fluid density resulting in large change in pressure gradient. This problem is resolved by dealing with the pressure gradient normalized by density, leading to a two‐phase CPM of which the original singlephase CPM is a special case. In addition, a new adaptive particle selection scheme is proposed to overcome the problem of ill‐conditioned coefficient matrix of pressure Poisson equation when particles are sparse and non‐uniformly spaced. Numerical examples of Rayleigh–Taylor instability, gravity current flow, water‐air sloshing and dam break are presented to demonstrate the accuracy of the proposed method in wave profile and pressure solution. Copyright © 2015 John Wiley & Sons, Ltd.     

A combined scheme of generalized finite difference method and Krylov deferred correction technique for highly accurate solution of transient heat conduction problems

This paper presents a numerical framework for the highly accurate solutions of transient heat conduction problems. The numerical framework discretizes the temporal direction of the problems by introducing the Krylov deferred correction (KDC) approach, which is arbitrarily high order of accuracy while remaining the computational complexity same as in the time-marching of first-order methods. The discretization by employing the KDC method yields a boundary value problem of the inhomogeneous modified Helmholtz equation at each time step. The meshless generalized finite difference method (GFDM) or meshless finite difference method (MFDM), a meshless method, is then applied to the solution of resulting boundary value problems at each time step. Six numerical experiments in one-, two-, and three-dimensional cases show that the proposed hybrid KDC-GFDM scheme allows big time step size for a long-time dynamic simulation and has a great potential for the problems with complex boundaries. In addition, some comparisons are also presented between the present method, the COMSOL software, and the GFDM with implicit Euler method.     

A novel domain decomposition method for highly oscillating partial differential equations

This paper is devoted to designing a novel domain decomposition method (DDM) for highly oscillating partial differential equations (PDE), especially, where the asymmetric meshless collocation method using radial basis functions (RBF), also Kansa's method is applied for a numerical solutions. It is found that the numerical error become worse when the original solution become more oscillating. To conquer this defect, we use a novel domain decomposition method which is motivated by time parallel algorithm. This DDM is based on a decomposition of computational domain by a coarse centers and a finer distribution of distinct centers. A corrector is designed to obtain better numerical solution after several iteration. Theoretical analysis and numerical examples are given to demonstrate the accuracy and efficiency of the proposed algorithm.     

A nonstandard finite difference scheme for a strain-gradient theory

Strain-gradient theories have proved very useful in describing computational aspects of phase transforming materials such as shape memory alloys. A significant feature implicit to these theories is a relation between the driving force acting on a phase boundary and its velocity. Numerical calculation of the kinetic relation using standard finite difference methods shows significant quantitative and qualitative departures from the analytical kinetic relation. In this paper we derive a nonstandard finite difference scheme and show that the kinetic relation evaluated using this scheme displays the correct qualitative behaviour and matches the analytical solution significantly better quantitatively. In particular, the nonstandard finite difference scheme eliminates spurious lattice trapping in the kinetic relation.     

A new fast finite element method for dislocations based on interior discontinuities

A new technique for the modelling of multiple dislocations based on introducing interior discontinuities is presented. In contrast to existing methods, the superposition of infinite domain solutions is avoided; interior discontinuities are specified on the dislocation slip surfaces and the resulting boundary value problem is solved by a finite element method. The accuracy of the proposed method is verified and its efficiency for multi‐dislocation problems is illustrated. Bounded core energies are incorporated into the method through regularization of the discontinuities at their edges. Though the method is applied to edge dislocations here, its extension to other types of dislocations is straightforward. Copyright © 2006 John Wiley & Sons, Ltd.