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To circumvent the constraint in application of the conventional differential quadrature (DQ) method that the solution domain has to be a regular region, an interpolation-based local differential quadrature (LDQ) method is proposed in this paper. Instead of using regular nodes placed on mesh lines in the DQ method (DQM), irregularly distributed nodes are employed in the LDQ method. That is, any spatial derivative at a nodal point is approximated by a linear weighted sum of the functional values of irregularly distributed nodes in the local physical domain. The feature of the new approach lies in the fact that the weighting coefficients are determined by the quadrature rule over the irregularly distributed local supporting nodes with the aid of nodal interpolation techniques developed in the paper. Because of this distinctive feature, the LDQ method can be consistently applied to linear and nonlinear problems and is really a mesh-free method without the limitation in the solution domain of the conventional DQM. The effectiveness and efficiency of the method are validated by two simple numerical examples by solving boundary-value problems of a linear and a nonlinear partial differential equation. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

By using the natural boundary reduction an overlapping domain decomposition method is designed to solve some exterior two-dimensional time-dependent parabolic problems. The governing equation is first discretized in time, leading to a sequence of boundary value problems with respect to time step in an unbounded domain. Then artificial boundaries are introduced. For each time level, an overlapping domain decomposition method, which is based on the natural boundary reduction, is constructed to solve the exterior elliptic boundary value problem on a two-dimensional domain. It is shown that the algorithm is equivalent to Schwarz alternating method. The convergence of this algorithm is given. The contraction factor for the exterior circular domain is also discussed. In the end of this paper, some numerical examples are presented, which illustrate the feasibility and the effectiveness of the proposed methods in this paper. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Imposition of essential boundary conditions in meshfree methods is made difficult because the shape functions used do not possess the ‘delta’ property. Various procedures have been proposed including penalty, Lagrange multipliers and collocation. It is shown in this paper that the success of a procedure depends on the arrangement of the nodal points. Certain methods of imposing boundary conditions are shown not to work for unstructured nodal arrangements. Much previous work has been demonstrated using structured grids thus hiding these drawbacks. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We study the boundary-value problems for singularly perturbed differential-difference equations with small shifts. Similar boundary-value problems are associated with expected first-exit time problems of the membrane potential in models for activity of neurons (SIAM J. Appl. Math. 1994; 54: 249–283; 1982; 42: 502–531; 1985; 45: 687–734) and in variational problems in control theory. In this paper, we present a numerical method to solve boundary-value problems for a singularly perturbed differential-difference equation of mixed type, i.e. which contains both type of terms having negative shifts as well as positive shifts, and consider the case in which the solution of the problem exhibits rapid oscillations. The stability and convergence analysis of the method is given. The effect of small shift on the oscillatory solution is shown by considering the numerical experiments. The numerical results for several test examples demonstrate the efficiency of the method. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new non-iterative numerical method for solving inverse boundary value problems for linear elliptic equations of second order. Here we assume that the Dirichlet and Neumann boundary conditions are given only on part of the domain, we have to reconstruct the solution and its normal derivative on the unaccessible part of the domain, which is the well-known ill-posed Cauchy problem. We propose to solve such inverse problems directly by using the recently developed radial basis meshless method. Several numerical experiments are given to demonstrate the effectiveness and efficiency of the proposed method. Copyright © 2003 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, to solve the problems of the interaction between the unilateral interface cracks and elastic wave, a new numerical method based on boundary element method (BEM) has been proposed. In this method, an iterating–correcting process of the solution in frequency domain and the correction in time domain has been designed to eliminate the mutual interference between the time and frequency domain. As an illustration to the algorithm, an interface crack between two half-planes under the attack of harmonic plane elastic wave has been analysed in detail. The contact configurations and stress field at the crack surface, and dynamic stress intensity factor (DSIF) at the crack tip have been given. Copyright © 2001 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

One of major difficulties in the implementation of meshless methods is the imposition of essential boundary conditions as the approximations do not pass through the nodal parameter values. As a consequence, the imposition of essential boundary conditions in meshless methods is quite awkward. In this paper, a displacement constraint equations method (DCEM) is proposed for the imposition of the essential boundary conditions, in which the essential boundary conditions is treated as a constraint to the discrete equations obtained from the Galerkin methods. Instead of using the methods of Lagrange multipliers and the penalty method, a procedure is proposed in which unknowns are partitioned into two subvectors, one consisting of unknowns on boundary Γu, and one consisting of the remaining unknowns. A simplified displacement constraint equations method (SDCEM) is also proposed, which results in a efficient scheme with sufficient accuracy for the imposition of the essential boundary conditions in meshless methods. The present method results in a symmetric, positive and banded stiffness matrix. Numerical results show that the accuracy of the present method is higher than that of the modified variational principles. The present method is a exact method for imposing essential boundary conditions in meshless methods, and can be used in Galerkin-based meshless method, such as element-free Galerkin methods, reproducing kernel particle method, meshless local Petrov–Galerkin method. Copyright © 2001 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Helically symmetric structures such as helical springs, machine screws and wire rope strands are commonly used structural items. When they are subjected to axial loads (tensile and torsional), these structures may still exhibit the helically symmetric characteristic after loading. If this is the case this feature can be used to substantially simplify the analysis of these structures in numerical simulations. In this paper, the formulation of helically symmetric boundary conditions for finite element (FE) modelling is presented. The helically symmetric relationship is ensured by using constraint equations which relate the displacements of the corresponding nodes on the corresponding artificial boundaries of the model. The application of these helically symmetric equations renders it possible to reduce the finite element model size greatly and improve the accuracy of the results. Examples of a simple circular cross-section bar and wire rope strand are presented which demonstrate the validity of the formulation. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, an adaptive mesh method is employed to solve a class of singular boundary value problems. The approach is based on an area-preserving map and some mesh shape control in two-dimensional space. Two benchmark problems, which both involve singularities in physical domains, are tested. © 1998 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper is concerned with several refinement techniques of finite element meshes for treating elliptic boundary value problems in domains with re-entrant edges and corners. A priori mesh grading is explained, and it is combined with the well-known adaptive finite element method. For two representative examples the numerically determined error norms are recorded, and the different strategies are compared. [ABSTRACT FROM AUTHOR]

With respect to a given boundary value problem, the corresponding conventional boundary integral equation is shown to yield non-equivalent solutions, which are dependent upon Poisson's ratio and geometry. In the paper a systematic method for establishing a necessary and sufficient boundary integral formulation has been proposed for two-dimensional elastostatic problems. Numerical analyses show that the conventional boundary integral equation yields incorrect results when the scale in the fundamental solution approaches a degenerate scale value. However, the results of the necessary and sufficient boundary integral equation are in good agreement with analytical solutions of the boundary value problem. [ABSTRACT FROM AUTHOR]

We present in this paper a nine-point, fourth-order difference scheme for the numerical solution of the quasilinear Poisson equation in polar co-ordinates, [This equation can not be represented in ASCII.TXT] with appropriate boundary conditions. A separate difference scheme of order four valid at r = 0 has also been obtained. The method is based on five evaluations of the function f. The numerical results of two problems obtained using this scheme are listed. The numerical results demonstrate the fourth-order accuracy of the scheme. [ABSTRACT FROM AUTHOR]