Analysis of the element-free Galerkin method for Signorini problems

Li, Xin; Dong, Haiyun

doi:10.1016/j.amc.2018.10.053

Cited by 21 publications

(8 citation statements)

References 41 publications

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“…An increasing number of the meshless methods with different approximation methods and discretization schemes have been proposed and demonstrated during the past decades. Among them, the mature meshless methods include the smoothed particle hydrodynamics (Liu et al, 2019), element-free Galerkin method (Li and Dong, 2019), hp-cloud approach (Jafari and Azhari, 2017), reproducing kernel particle approach (Wang and Li, 2020), natural element approach (Somireddy and Rajagopal, 2015), local Petrov-Galerkin method (Abbaszadeh and Dehghan, 2020), finite point approach (Shojaei et al, 2017), element-free manifold approach (Gao and Wei, 2017), radial point interpolation approach (Zhou et al, 2020c), etc.…”

Section: Introductionmentioning

confidence: 99%

A Hermite interpolation element-free Galerkin method for piezoelectric materials

Zhou

Xue

2022

Journal of Intelligent Material Systems and Structures

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Piezoelectric materials have played an important role in industry due to a number of beneficial properties. However, most numerical methods for the piezoelectric materials need mesh, in which the mesh generation and remeshing are prominent difficulties. This paper proposes a Hermite interpolation element-free Galerkin method (HIEFGM) for piezoelectric materials, where the Hermite approximate approach and interpolation element-free Galerkin method (IEFGM) are combined. Based on the constitutive equation, geometric equation, and Galerkin integral weak form, the HIEFGM formulation for piezoelectric materials is established. In the proposed method, the problem domain is discretized by many nodes rather than the meshes, so the pre-processing of numerical computation is simplified. Furthermore, a new approximation technique based on the moving least squares method and Hermite approximate approach is used to derive the approximation function of field quantities. The derived approximation function has the Kronecker delta property and considers the field quantity normal derivatives of boundary nodes, which avoids the problem of imposing the essential boundary conditions and improves the accuracy of meshless approximation. The effects of the scaling factor, node density, and node arrangement on the accuracy of the proposed method are investigated. Numerical examples are given for assessing the proposed method and the results uniformly demonstrate the proposed method has excellent performance in analyzing piezoelectric materials.

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Section: Introductionmentioning

confidence: 99%

A Hermite interpolation element-free Galerkin method for piezoelectric materials

Zhou

Xue

2022

Journal of Intelligent Material Systems and Structures

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“…Author of Reference developed and analyzed the complex variable element‐free Galerkin (CVEFG) method for solving 3D elliptic problems and 3D wave equations. The main aim of Reference is to extend the EFG method to solve Signorini boundary value problems. Authors of Reference developed a complex variable boundary element‐free method based on the combined field integral equation (CFIE) and the complex variable moving least squares (CVMLS) approximation for solving the exterior Neumann problem for the Helmholtz equation.…”

Section: Introductionmentioning

confidence: 99%

A proper orthogonal decomposition variational multiscale meshless interpolating element‐free Galerkin method for incompressible magnetohydrodynamics flow

Abbaszadeh

Dehghan

Navon

2020

Numerical Methods in Fluids

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Summary In the recent decade, the meshless methods have been handled for solving most of PDEs due to easiness of the meshless methods. One of the popular meshless methods is the element‐free Galerkin (EFG) method that was first proposed for solving some problems in the solid mechanics. The test and trial functions of the EFG are based on the special basis. Recently, some modifications have been developed to improve the EFG method. One of these improvements is the variational multiscale EFG procedure. In the current article, the shape functions of interpolation moving least squares approximation have been applied to the variational multiscale EFG technique for solving the incompressible magnetohydrodynamics flow. In order to reduce the elapsed CPU time of simulation, we employ a reduced‐order model based on the proper orthogonal decomposition technique. The current combination can be referred to as the reduced‐order variational multiscale EFG technique. To illustrate the reduction in CPU time used as well as the efficiency of the proposed method, we applied it for the two‐dimensional cases.

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“…This method is still an important method in the current application of MLS method. For example, the EFG method was employed in solving the Signorini boundary value problems by Li et al [ 14 ]. The difficulty of nonlinear inequality restrictions is solved by a projection iterative scheme.…”

Section: Introductionmentioning

confidence: 99%

A Novel Reconstruction Method for Measurement Data Based on MTLS Algorithm

Tang

et al. 2020

Sensors

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Reconstruction methods for discrete data, such as the Moving Least Squares (MLS) and Moving Total Least Squares (MTLS), have made a great many achievements with the progress of modern industrial technology. Although the MLS and MTLS have good approximation accuracy, neither of these two approaches are robust model reconstruction methods and the outliers in the data cannot be processed effectively as the construction principle results in distorted local approximation. This paper proposes an improved method that is called the Moving Total Least Trimmed Squares (MTLTS) to achieve more accurate and robust estimations. By applying the Total Least Trimmed Squares (TLTS) method to the orthogonal construction way in the proposed MTLTS, the outliers as well as the random errors of all variables that exist in the measurement data can be effectively suppressed. The results of the numerical simulation and measurement experiment show that the proposed algorithm is superior to the MTLS and MLS method from the perspective of robustness and accuracy.

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Analysis of the element-free Galerkin method for Signorini problems

Cited by 21 publications

References 41 publications

A Hermite interpolation element-free Galerkin method for piezoelectric materials

A Hermite interpolation element-free Galerkin method for piezoelectric materials

A proper orthogonal decomposition variational multiscale meshless interpolating element‐free Galerkin method for incompressible magnetohydrodynamics flow

A Novel Reconstruction Method for Measurement Data Based on MTLS Algorithm

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