Purpose

This meshless collocation method is applicable not only to the Helmholtz equation with Dirichlet boundary condition but also mixed boundary conditions. It can calculate not only the high wavenumber problems, but also the variable wave number problems.

Design/methodology/approach

In this paper, the authors developed a meshless collocation method by using barycentric Lagrange interpolation basis function based on the Chebyshev nodes to deduce the scheme for solving the three-dimensional Helmholtz equation. First, the spatial variables and their partial derivatives are treated by interpolation basis functions, and the collocation method is established for solving second order differential equations. Then the differential matrix is employed to simplify the differential equations which is on a given test node. Finally, numerical experiments show the accuracy and effectiveness of the proposed method.

Findings

The numerical experiments show the advantages of the present method, such as less number of collocation nodes needed, shorter calculation time, higher precision, smaller error and higher efficiency. What is more, the numerical solutions agree well with the exact solutions.

Research limitations/implications

Compared with finite element method, finite difference method and other traditional numerical methods based on grid solution, meshless method can reduce or eliminate the dependence on grid and make the numerical implementation more flexible.

Practical implications

The Helmholtz equation has a wide application background in many fields, such as physics, mechanics, engineering and so on.

Originality/value

This meshless method is first time applied for solving the 3D Helmholtz equation. What is more the present work not only gives the relationship of interpolation nodes but also the test nodes.

中文翻译：

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使用重心拉格朗日插值搭配法求解3D亥姆霍兹方程

目的

这种无网格配置方法不仅适用于具有 Dirichlet 边界条件的 Helmholtz 方程，也适用于混合边界条件。它不仅可以计算高波数问题，还可以计算变波数问题。

设计/方法/方法

在本文中，作者开发了一种基于切比雪夫节点的重心拉格朗日插值基函数的无网格搭配方法，推导出了求解三维亥姆霍兹方程的方案。首先用插值基函数处理空间变量及其偏导数，建立求解二阶微分方程的搭配方法。然后使用微分矩阵来简化给定测试节点上的微分方程。最后，数值实验证明了所提出方法的准确性和有效性。

发现

数值实验表明，该方法具有所需的配置节点数少、计算时间短、精度更高、误差更小、效率更高等优点。更重要的是，数值解与精确解吻合得很好。

研究限制/影响

与有限元法、有限差分法等传统基于网格求解的数值方法相比，无网格法可以减少或消除对网格的依赖，使数值实现更加灵活。

实际影响

亥姆霍兹方程在物理学、力学、工程等诸多领域有着广泛的应用背景。

原创性/价值

这种无网格方法首次应用于求解 3D Helmholtz 方程。更重要的是，目前的工作不仅给出了插值节点的关系，还给出了测试节点的关系。