Narrow Results

An enhancement of the FEM using Kriging interpolation (K-FEM) was recently proposed. This method offers advantages over the conventional FEM and mesh-free methods. With Kriging interpolation, the approximated field over an element is influenced not only by its own element nodes but also by a set of satellite nodes outside the element. This results in incompatibility along interelement boundaries. Consequently, the convergence of the solutions is questionable. In this paper, the convergence is investigated through several numerical tests. It is found that the solutions of the K-FEM with an appropriate range of parameters converge to the corresponding exact solutions.

This review paper presents a methodological study on possible and existing meshfree methods for solving the partial differential equations (PDEs) governing solid mechanics problems, based mainly on the research work in the past two decades at the authors group. We start with a discussion on the general steps in a meshfree method based on nodes, with the displacements as the primary variables. We then examine the major techniques used in each of these steps: (1) techniques for displacement function approximations using nodes, (2) approximation of the gradient of the displacements or strains based on nodes and a background T-cells that can be automatically generated and refined, and (3) formulation techniques for producing algebraic equations. The function approximation techniques include node-based interpolation methods, cell-based interpolation methods, function smoothing techniques, and moving least squares approximation techniques. The gradient approximation includes direct differentiation, gradient smoothing, and special strain construction. Formulation techniques include strong-form, weakform, local weakform, weak-strong-form, and weakened weakform (W2). In theory, a meshfree method can be developed using a combination of function approximation, gradient approximation, and formulation techniques, which can lead to matrix of a large number of possible methods. This review attempts to provide an overall methodological review, rather than a usual review of comparing different methods. We hope to show readers the differences between the forests, and just between the trees.

This paper represents some basic mathematic theories for G${}^s$ spaces of functions that can be used for weakened weak (W²) formulations, upon which the smoothed finite element methods (S-FEMs) and the smoothed point interpolation methods (S-PIMs) are based for solving mechanics problems. We first introduce and prove properties of G${}^s$ spaces, such as the lower boundedness and convergence of the norms, which are in contrast with H¹ spaces. We then prove the equivalence of the G^s norms and its corresponding semi-norms. These mathematic theories are important and essential for the establishment of theoretical frame and the development of relevant numerical approaches. Finally, numerical examples are presented by using typical S-FEM models known as the NS-FEM and $\alpha$S-FEM to examine the properties of a smoothed method based on G^s spaces, in comparison with the standard FEM with weak formulation.

It is well known that a high-order point interpolation method (PIM) based on the standard Galerkin formations is not conforming, and thus the solution may not always be convergent. This paper proposes a new interesting technique called quasi-conforming point interpolation method (QC-PIM) for solving elasticity problems, by devising a novel scheme that smears the discontinuity. In the QC-PIM, the problem domain is first discretized by a set of background cells (typically triangles that can be automatically generated), and the average displacements on the interfaces of the two neighboring cells are assumed to be equal. We prove that when the size of background cells approaches to zero, all the additional potential energy coming from the discontinuous displacement field becomes zero, which ensures the pass of the standard patch test and hence the convergence. Numerical experiments verify that QC-PIM can produce the convergent solutions with higher accuracy and convergent rate that is in between fully conforming linear and quadratic models.

In this paper, coupled with preconditioning technique, a preconditioned accelerated over relaxation (PAOR) iterative method for solving the absolute value equations (AVEs) is presented. Some comparison theorems are given when the matrix of the linear term is an irreducible $L$-matrix. Comparison results show that the convergence rate of the PAOR iterative method is better than that of the accelerated over relaxation (AOR) iterative method whenever both are convergent. Numerical experiments are provided in order to confirm the theoretical results studied in this paper.

In this paper, based on the basic principle of the SPH method’s kernel approximation, a new kernel approximation was constructed to compute first-order derivative through Taylor series expansion. Derivative in Newton’s method was replaced to propose a new SPH iterative method for solving nonlinear equations. The advantage of this method is that it does not require any evaluation of derivatives, which overcame the shortcoming of Newton’s method. Quadratic convergence of new method was proved and a variety of numerical examples were given to illustrate that the method has the same computational efficiency as Newton’s method.

A multiple variational iteration method (VIM) is proposed to effectively solve the second-order nonlinear two-point boundary value problems. For problems where convergence speed of the original VIM is slow or the original method is divergent, the multiple VIM method (MVIM) presented in this paper can readily improve the rate of convergence.

The periodic behaviors of a linear fourth-order difference solution to the Benjamin–Bona–Mahony (BBM)-type equation with time-periodic boundaries are analyzed in this paper. Firstly, we employ a variable transformation to change the original BBM-type equation with time-periodic boundaries into a new BBM-type equation with zero boundaries. We then construct a fourth-order linear finite difference method to discrete the new BBM-type equation. The solvability, convergence, stability and accuracy of the approximating solution are discussed. The computation procedure of the present method is given in detail. Numerical results show that the proposed difference method is reliable and efficient for time-periodic simulation.

This work proposes a hybrid block numerical method of tenth order for the direct solution of fifth-order initial value problems. The formulas that constitute the block method are derived from a continuous approximation obtained through interpolation and collocation techniques. In order to obtain better accuracy, sixth-order derivatives are incorporated to develop the formulas. The main characteristics of the method are analyzed, namely, the order, local truncation errors, zero-stability, consistency and convergence. The proposed strategy performs well, as shown by some numerical examples and the corresponding efficiency curves. Compared to existing numerical methods in the literature, the proposed method is competitive and the numerical approximations it provides are significantly close to the precise solutions.

This paper presents a new numerical iteration method for solving the absolute value equations. The proposed method uses the generalized Newton method as a predictor step, and the five-point open Newton–Cotes formula is considered the corrector step. The convergence of the proposed method is studied in detail. The proposed method solves large systems effectively due to its simplicity and effectiveness. In this paper, we have solved the beam equation, using the finite difference method to transform it into a system of absolute value equations, and then solved it using the proposed method. Several numerical examples were provided to demonstrate the accuracy and effectiveness of the new approach. In addition, the novel approach solves absolute value equations with greater accuracy and precision than other existing methods.