An adaptive interpolation element free Galerkin method based on a posteriori error estimation of FEM for Poisson equation

Zhang, Xiaohua; Hu, Zhicheng; Wang, Min

doi:10.1016/j.enganabound.2021.05.020

Cited by 4 publications

(5 citation statements)

References 27 publications

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“…For the static elastic mechanics problem, the equilibrium equation is mainly expressed by the displacement, which has the similar properties of the two-dimensional elliptic equation. In theory, when the mesh is infinitely subdivided, the displacement error will converge to 0, which can also verify the correctness of the numerical method, and is usually called the boundedness of the error [49,50]. That is, when the element size is h, the error order of the finite element solution u h and the analytical solution u is called the convergence rate, which can be expressed as:…”

Section: Error Estimationmentioning

confidence: 99%

“…Therefore, The numerical comparison results show that ARIMA predicts the variable force relatively well. The prediction result was F2 by the ARIMA algorithm, and then according to the energy equation and variational principle, it is easy to obtain the equilibrium equation [48][49][50]. In Example 2, the corresponding stress, strain and displacement cloud are calculated under t = 10.82 s and predictive force F2 = 2194.1522 N. The solution process is the same as Section 2.…”

Section: Examplementioning

confidence: 99%

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ARIMA-FEM Method with Prediction Function to Solve the Stress–Strain of Perforated Elastic Metal Plates

Chen

Dai

Zheng

2022

Metals

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Stress analysis and deformation prediction have always been the focuses of the field of mechanics. The accurate force prediction in plate deformation plays important role in the production, processing and performance analysis of materials. In this paper, we propose an ARIMA-FEM method, which can be used to solve some mechanical problems of 2D porous elastic plate. We have given a detailed theory and solving steps of ARIMA-FEM. In addition, three numerical examples are given to predict the stress–strain of thin porous elastic metal plates. This article uses CST, LST and Q4 elements to discrete the rectangular plates, square plates and circle plates with holes. As for variable force prediction, this paper compared with linear regression, nonlinear regression and neural network prediction, and the results show that the ARIMA method has a higher prediction accuracy. Furthermore, we calculate the numerical solution at four mesh scales, and the numerical convergence is consistent with the theoretical convergence, which also shows the effectiveness of our method. The image smoothing algorithm is applied to keep edge information with high resolution, which can more concisely describe the plate internal changes. Finally, the application scope of ARIMA-FEM, model expansion, superconvergence analysis and other issues have been given enlightening views in the discussion section. In fact, this algorithm combined statistics and mechanics. It also reflects the knowledge integration of interdisciplinary and uses it better to serve practical applications.

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Section: Error Estimationmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

ARIMA-FEM Method with Prediction Function to Solve the Stress–Strain of Perforated Elastic Metal Plates

Chen

Dai

Zheng

2022

Metals

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“…The local and global approximation errors can be measured by a posteriori error estimation, while the node refinement process decides whether a refinement is needed or not according to the error data. The majority of error estimators in the FEM framework are classified as either recovery-based error estimators or residual-based error estimators [33]. The employment of recovery methods in the computation of a posteriori error estimators that are used in the present study is one of the most important applications.…”

Section: Adaptive Mesh Refinementmentioning

confidence: 99%

Adaptive Finite Element Model for Simulating Crack Growth in the Presence of Holes

Alshoaibi

Fageehi

2021

Materials

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This study presents a developed finite element code written by Visual Fortran to computationally model fatigue crack growth (FCG) in arbitrary 2D structures with constant amplitude loading, using the linear elastic fracture mechanics (LEFM) concept. Accordingly, optimizing an FCG analysis, it is necessary to describe all the characteristics of the 2D model of the cracked component, including loads, support conditions, and material characteristics. The advancing front method has been used to generate the finite element mesh. The equivalent stress intensity factor was used as the onset criteria of crack propagation, since it is the main significant parameter that must be precisely predicted. As such, a criterion premised on direction (maximum circumferential stress theory) was implemented. After pre-processing, the analysis continues with incremental analysis of the crack growth, which is discretized into short straight segments. The adaptive mesh finite element method was used to perform the stress analysis for each increment. The displacement extrapolation technique was employed at each crack extension increment to compute the SIFs, which are then assessed by the maximum circumferential stress theory to determine the direction of the crack growth and predict the fatigue life as a function of crack length using a modified form of Paris’ law. The application examples demonstrate the developed program’s capability and performance.

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“…The IEFG method not only has the advantage of directly applying boundary conditions but also has the advantage of having a smaller radius of influence compared to EFG under the same basic functions. The IEFG method has been applied to potential problems [34,35], elastoplasticity problems [28], crack problems [36], structural dynamic analysis [37], prevention of groundwater contamination [38], elastoplasticity problems [39], Poisson equation [40], elastic large deformation problems [41], Oldroyd equation [42], etc.…”

Section: Introductionmentioning

confidence: 99%

A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

Wang¹,

Wu²,

Xu³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

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By introducing the dimensional splitting (DS) method into the multiscale interpolating element-free Galerkin (VMIEFG) method, a dimension-splitting multiscale interpolating element-free Galerkin (DS-VMIEFG) method is proposed for three-dimensional (3D) singular perturbed convection-diffusion (SPCD) problems. In the DS-VMIEFG method, the 3D problem is decomposed into a series of 2D problems by the DS method, and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method. The improved interpolation-type moving least squares (IIMLS) method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems. The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems. The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes. For extremely small singular diffusion coefficients, the numerical solution will avoid numerical oscillation and has high computational stability. KEYWORDSDimension-splitting multiscale interpolating element-free Galerkin (DS-VMIEFG) method; interpolating variational multiscale element-free Galerkin (VMIEFG) method; dimension splitting method; singularly perturbed convection-diffusion problems

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An adaptive interpolation element free Galerkin method based on a posteriori error estimation of FEM for Poisson equation

Cited by 4 publications

References 27 publications

ARIMA-FEM Method with Prediction Function to Solve the Stress–Strain of Perforated Elastic Metal Plates

ARIMA-FEM Method with Prediction Function to Solve the Stress–Strain of Perforated Elastic Metal Plates

Adaptive Finite Element Model for Simulating Crack Growth in the Presence of Holes

A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

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