Purpose

The study aims to present a new meshless method based on the Taylor series expansion. The compact supported radial basis functions (CSRBFs) are very attractive, can be considered as a numerical tool for the engineering problems and used to obtain the trial solution and its derivatives without differentiating the basis functions for a meshless method. A meshless based on the CSRBF and Taylor series method has been developed for the solutions of engineering problems.

Design/methodology/approach

This paper is devoted to present a truly meshless method which is called a radial basis Taylor series method (RBTSM) based on the CSRBFs and Taylor series expansion (TSE). The basis function and its derivatives are obtained without differentiating CSRBFs.

Findings

The RBTSM does not involve differentiation of the approximated function. This property allows us to use a wide range of CSRBF and weight functions including the constant one. By using a different number of terms in the TSE, the global convergence properties of the RBTSM can be improved. The global convergence properties are satisfied by the RBTSM. The computed results based on the RBTSM shows excellent agreement with results given in the open literature. The RBTSM can provide satisfactory results even with the problem domains which have curved boundaries and irregularly distributed nodes.

Originality/value

The CSRBFs have been widely used for the construction of the basic function in the meshless methods. However, the derivative of the basis function is obtained with the differentiation of the CSRBF. In the RBTSM, the derivatives of the basis function are obtained by using the TSE without differentiating the CSRBF.

中文翻译：

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径向基泰勒级数法及其应用

目的

该研究旨在提出一种基于泰勒级数展开的新无网格方法。紧凑支持径向基函数（CSRBFs）非常有吸引力，可以被认为是工程问题的数值工具，用于在不区分无网格方法的基函数的情况下获得试解及其导数。一种基于 CSRBF 和泰勒级数方法的无网格已经被开发用于解决工程问题。

设计/方法/方法

本文致力于提出一种基于 CSRBF 和泰勒级数展开 (TSE) 的真正无网格方法，称为径向基泰勒级数方法 (RBTSM)。基函数及其导数是在不区分 CSRBF 的情况下获得的。

发现

RBTSM 不涉及近似函数的微分。这个属性允许我们使用广泛的 CSRBF 和权重函数，包括常数函数。通过在 TSE 中使用不同数量的项，可以改进 RBTSM 的全局收敛特性。RBTSM 满足全局收敛特性。基于 RBTSM 的计算结果与公开文献中给出的结果非常吻合。即使对于具有弯曲边界和不规则分布节点的问题域，RBTSM 也能提供令人满意的结果。

原创性/价值

CSRBFs 已被广泛用于构建无网格方法中的基本函数。然而，基函数的导数是通过CSRBF的微分获得的。在 RBTSM 中，基函数的导数是通过使用 TSE 获得的，而不对 CSRBF 进行微分。