基于局部基本解法的静电场仿真分析

doi:10.19596/j.cnki.1001-246x.8308

摘要/Abstract

摘要：

将局部基本解方法应用于静电场问题的模拟与分析。局部基本解方法是利用控制方程的基本解，基于局部理论和移动最小二乘原理提出的一种无网格算法。相比于有限元和有限差分等传统网格类方法，该方法仅需离散节点，避免了复杂的网格剖分难题。作为一种半解析数值技术，物理问题的基本解被作为插值基函数建立数值离散模型，从而保证了算法的较高精度。此外，与具有全局离散格式的无网格方法相比，局部基本解法更适用于高维复杂几何和大尺度模拟。二维和三维数值试验表明，该方法具有实施方便灵活，计算精度高和计算速度快等优势。为静电场仿真研究开辟新的途径，拓展了局部基本解方法的应用领域。

关键词: 局部基本解方法, 无网格法, 静电场, 基本解

Abstract:

Localized method of fundamental solutions is applied to the simulation and analysis of electrostatic field problems. The localized method of fundamental solutions is a meshless algorithm based on local theory and moving least square approximation, which uses fundamental solution of the governing equation. Compared with traditional mesh-type methods such as finite element method and finite difference method, this method needs discrete nodes only, and avoids troublesome mesh generation. As a semi-analytical numerical technique, fundamental solutions of physical problems are used as interpolation basis functions to establish a numerical discrete model, thus ensuring high accuracy of the algorithm. In addition, compared with meshless methods with global discretization scheme, the local fundamental method is more suitable for high-dimensional complex geometry and large-scale simulation. Two- and three-dimensional numerical tests show that this method is convenient, flexible, accurate and fast. It is a new way for electrostatic field simulation. It expands application of localized method of fundamental solutions.

Key words: localized method of fundamental solutions, meshless method, electrostatic field, fundamental solution

中图分类号:

TM151

王超, 王发杰, 谷岩, 王晓. 基于局部基本解法的静电场仿真分析[J]. 计算物理, 2021, 38(5): 612-622.

Chao WANG, Fajie WANG, Yan GU, Xiao WANG. Simulation Analysis of Electrostatic Field Based on Localized Method of Fundamental Solutions[J]. Chinese Journal of Computational Physics, 2021, 38(5): 612-622.

图/表 16

图1 静电场边值问题的LMFS示意图(a) 节点分布；(b) 局部子域

Fig.1 Schematic of the LMFS for boundary value problem of electrostatic field (a) distribution of nodes; (b) local subdomain

图2 矩形接地金属槽模型

Fig.2 Model of rectangular grounding metal slot

图3 LMFS的节点分布示意图(a) 节点均匀分布；(b) 节点不规则分布

Fig.3 Nodal distribution of the LMFS (a) regular nodal distribution; (b) irregular nodal distribution

图4 规则和不规则节点分布下LMFS随支撑点数目增加的误差变化曲线

Fig.4 Error curves of the LMFS with respect to the number of supporting nodes, under regular and irregular nodal distributions

图5 规则和不规则节点分布下LMFS随总节点数目增加的误差变化曲线

Fig.5 Error curves of the LMFS with respect to the number of total nodes, under regular and irregular nodal distributions

图6 计算域内电位的精确分布及数值误差(a) 精确解；(b) 绝对误差

Fig.6 Exact solution and numerical error of electric potential in the computational domain (a) exact solution; (b) absolute error

图7 LMFS、GFDM和MFS在不同节点数目下的计算时间

Fig.7 Calculation times of LMFS, GFDM and MFS with respect to the total number of nodes

图8 LMFS、GFDM和MFS在不同节点数目下的全局误差

Fig.8 Global errors of LMFS, GFDM and MFS with respect to the total number of nodes

表1 加扰动时LMFS、GFDM和MFS的计算误差

Table 1 Computational errors of MFS, LMFS and GFDM with distrubance

s/%	MFS	LMFS	GFDM
0	2.881×10^-13	3.057×10^-9	9.355×10^-6
1	7.431×10^-3	1.386×10^-3	7.576×10^-3
3	1.966×10^-2	7.725×10^-3	1.593×10^-2
5	4.117×10^-2	8.575×10^-3	4.670×10^-2

图9 FEM的网格分布

Fig.9 Mesh distribution of the FEM

图10 LMFS的节点分布

Fig.10 Nodal distribution of the LMFS

图11 方形槽电位分布(a) FEM计算结果；(b) LMFS计算结果；(c) 相对偏差分布

Fig.11 Electric potential distributions (a) results of FEM; (b) results of FEM; (c) distribution of relative deviation

图12 S型管道模型

Fig.12 An S-shape pipe model

图13 FEM网格分布

Fig.13 Distribution of FEM mesh

图14 电位仿真结果(a) FEM；(b) N=5 443；(c) N=7 649；(d) N=13 375

Fig.14 Electric potential simulation results (a) FEM; (b) N=5 443; (c) N=7 649; (d) N=13 375

表2 LMFS和GFDM的计算结果及其与FEM计算结果的比较

Table 2 Numerical results obtained with FEM, LMFS and GFDM

坐标	FEM	LMFS	LMFS偏差/10^-3	GFDM	GFDM偏差/10^-2
(0.07, 0, 0)	1.982 01	1.976 05	3.01	1.934 52	2.40
(0.2, 0, 0)	3.871 22	3.863 86	1.90	3.769 14	2.64
(0.3, 0.1, 0)	5.339 76	5.355 57	2.96	5.314 81	0.467
(0.4, 0.2, 0)	7.359 71	7.389 45	4.04	7.885 56	7.14
(0.5, 0.2, 0)	8.563 52	8.578 86	1.79	8.893 10	3.85

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