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The interpolating dimension splitting element-free Galerkin method for 3D potential problems

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In this paper, based on the improved interpolating moving least-squares (IMLS) method and the dimension splitting method, the interpolating dimension splitting element-free Galerkin (IDSEFG) method for three-dimensional (3D) potential problems is proposed. The key of the IDSEFG method is to split a 3D problem domain into many related two-dimensional (2D) subdomains. The shape function is constructed by the improved IMLS method on the 2D subdomains, and the Galerkin weak form based on the dimension splitting method is used to obtain the discretized equations. The discrete equations on these 2D subdomains are coupled by the finite difference method. Take the improved element-free Galerkin (IEFG) method as a comparison, the advantage of the IDSEFG method is that the essential boundary conditions can be enforced directly. The effects of the number of nodes, the direction of dimension splitting, and the parameters of the influence domain on the calculation accuracy are studied through four numerical examples, the numerical solutions of the IDSEFG method are compared with the numerical solutions of the IEFG method and the analytical solutions. It is verified that the numerical solutions of the IDSEFG method are highly consistent with the analytical solution, and the calculation efficiency of this method is significantly higher than that of the IEFG method.
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Vol.:(0123456789)
1 3
Engineering with Computers (2022) 38 (Suppl 4):S2703–S2717
https://doi.org/10.1007/s00366-021-01408-5
ORIGINAL ARTICLE
The interpolating dimension splitting element‑free Galerkin method
for3D potential problems
QianWu1· MiaojuanPeng1· YuminCheng2
Received: 17 December 2020 / Accepted: 20 April 2021 / Published online: 27 May 2021
© The Author(s), under exclusive licence to Springer-Verlag London Ltd., part of Springer Nature 2021
Abstract
In this paper, based on the improved interpolating moving least-squares (IMLS) method and the dimension splitting method,
the interpolating dimension splitting element-free Galerkin (IDSEFG) method for three-dimensional (3D) potential prob-
lems is proposed. The key of the IDSEFG method is to split a 3D problem domain into many related two-dimensional (2D)
subdomains. The shape function is constructed by the improved IMLS method on the 2D subdomains, and the Galerkin
weak form based on the dimension splitting method is used to obtain the discretized equations. The discrete equations on
these 2D subdomains are coupled by the finite difference method. Take the improved element-free Galerkin (IEFG) method
as a comparison, the advantage of the IDSEFG method is that the essential boundary conditions can be enforced directly.
The effects of the number of nodes, the direction of dimension splitting, and the parameters of the influence domain on
the calculation accuracy are studied through four numerical examples, the numerical solutions of the IDSEFG method are
compared with the numerical solutions of the IEFG method and the analytical solutions. It is verified that the numerical
solutions of the IDSEFG method are highly consistent with the analytical solution, and the calculation efficiency of this
method is significantly higher than that of the IEFG method.
Keywords Meshless method· Dimension splitting method· Improved interpolating moving least-squares method· Finite
difference method· Interpolating dimension splitting element-free Galerkin method· Potential problem
1 Introduction
There are many complicated problems, such as crack propa-
gation and large deformation, in science and engineering
fields must be solved with numerical methods. The boundary
element method and the finite element method are the major
numerical methods which based on meshes or elements.
In recent years, the meshless method has made great
progress [1, 2]. When meshless method is used to solve a
problem, only discrete nodes are distributed on the problem
domain and its boundary without meshing. Without domain
discretization, this method uses a node-based approxima-
tion to construct the approximation function or interpolation
function. This fully ensures the calculation accuracy and
efficiency. Moreover, the approach leads to a flexible choice
of nodes in the domain for the specific characteristics. It is
shown that the method has good adaptability and calcula-
tion accuracy. Therefore, as a new and efficient method in
scientific and engineering computing, the meshless method
has gradually become a research hotspot.
Currently, the element-free Galerkin method (EFG) [3]
and the reproducing kernel particle method (RKPM) [4, 5]
are major meshless methods. Based on the moving least-
squares (MLS) approximation [6], the EFG method has
high calculation accuracy, because the MLS approximation
is obtained from the ordinary least squares method with
the best approximation [710]. Because of the complicated
shape function of the MLS approximation, the computa-
tional efficiency of the EFG method is low, especially for
solving three-dimensional (3D) problems. In addition, the
* Miaojuan Peng
mjpeng@shu.edu.cn
* Yumin Cheng
ymcheng@shu.edu.cn
1 Department ofCivil Engineering, School ofMechanics
andEngineering Science, Shanghai University,
Shanghai200444, China
2 Shanghai Key Laboratory ofMechanics inEnergy
Engineering, Shanghai Institute ofApplied Mathematics
andMechanics, School ofMechanics andEngineering
Science, Shanghai University, Shanghai200072, China
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
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