Generalized finite element method with time-independent enrichment functions for 3D transient heat diffusion problems

Iqbal, M.; Masood, Kashif; Aljuhni, Aiman; Ahmad, Afaq

doi:10.1016/j.ijheatmasstransfer.2019.118969

Cited by 11 publications

(4 citation statements)

References 39 publications

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“…The parameters R c and C are constants which control the shape of Ψ q ( x ). Similar global enrichments have been shown to lead to efficient approximations in two‐dimensional (2D) and 3D heat transfer problems . The resulting numerical scheme with global enrichments has been shown in these works to efficiently approximate the solution for localized heat sources to engineering accuracy.…”

Section: Boundary Value Problem and Variational Formulationmentioning

confidence: 83%

“…One can tune the enrichment functions to give a very smooth approximation to the second derivative as well, but this would be very specific to the problem under consideration, unlike the general nature of the enrichment function. In the current study, they are kept in the general form, as used in References and .…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…The use of multiple enrichment functions greatly reduces the dependency of the PUFEM solution on FEM meshes. As suggested in Reference , It is always practical and more beneficial to increase the number of enrichment functions rather than refine the mesh grids. To start with, we use a very coarse mesh of 64 elements in the 3D cube and enrich the solution space with different numbers of enrichment functions, Q = 2,3,…6.…”

Section: Numerical Experimentsmentioning

confidence: 99%

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A residual a posteriori error estimate for partition of unity finite elements for three‐dimensional transient heat diffusion problems using multiple global enrichment functions

Iqbal

Gimperlein

Laghrouche

et al. 2020

Numerical Meth Engineering

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In this article, a study of residual based a posteriori error estimation is presented for the partition of unity finite element method (PUFEM) for three-dimensional (3D) transient heat diffusion problems. The proposed error estimate is independent of the heuristically selected enrichment functions and provides a useful and reliable upper bound for the discretization errors of the PUFEM solutions. Numerical results show that the presented error estimate efficiently captures the effect of h-refinement and q-refinement on the performance of PUFEM solutions. It also efficiently reflects the effect of ill-conditioning of the stiffness matrix that is typically experienced in the partition of unity based finite element methods. For a problem with a known exact solution, the error estimate is shown to capture the same solution trends as obtained by the classical L 2 norm error. For problems with no known analytical solutions, the proposed estimate is shown to be used as a reliable and efficient tool to predict the numerical errors in the PUFEM solutions of 3D transient heat diffusion problems. K E Y W O R D Sdiffusion problems, enrichment functions, error estimate, GFEM, PUFEM INTRODUCTIONDifferent numerical methods evolved over the years to reduce the computational cost and memory requirements for complex engineering problems. One of the features of all these numerical methods is that they are subjected to various sources of numerical errors, 1 which can question the reliability of their results. Babuška and Strouboulis 2 emphasized on the reliability of the finite element method (FEM) computations and mentioned that unreliable finite element results can lead to very serious consequences. Over the years, numerous researchers devised different methods for the error estimation of h, p, and hp versions of FEM. [3][4][5] The basic mathematical theory of FEM and its error estimation can be found in the book of Babuška et al. 6 A comprehensive explanation of the error estimation procedures and their theory is also presented by Ainsworth and Oden. 7Int J Numer Methods Eng. 2020;121:2727-2746. wileyonlinelibrary.com/journal/nme

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Section: Boundary Value Problem and Variational Formulationmentioning

confidence: 83%

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

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A residual a posteriori error estimate for partition of unity finite elements for three‐dimensional transient heat diffusion problems using multiple global enrichment functions

Iqbal

Gimperlein

Laghrouche

et al. 2020

Numerical Meth Engineering

Self Cite

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“…e finite element method (FEM) is a reliable computational tool to solve partial differential equations (PDE) in science and engineering [1][2][3][4]. A domain of complex geometry is partitioned into a finite number of nonoverlapping subdomains of simplex shapes by introducing the concept of discretization [5].…”

Section: Introductionmentioning

confidence: 99%

PSBFEM-Abaqus: Development of User Element Subroutine (UEL) for Polygonal Scaled Boundary Finite Element Method in Abaqus

Yang

2021

Mathematical Problems in Engineering

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The polygonal scaled boundary finite element method (PSBFEM) is a novel method integrating the standard scaled boundary finite element method (SBFEM) and the polygonal mesh technique. This work discusses developing a PSBFEM framework within the commercial finite element software Abaqus. The PSBFEM is implemented by the User Element Subroutine (UEL) feature of the software. The details on the main procedures to interact with Abaqus, defining the UEL element, and solving the stiffness matrix by the eigenvalue decomposition are present. Moreover, we also develop the preprocessing module and the postprocessing module using the Python script to generate meshes automatically and visualize results. Several benchmark problems from two-dimensional linear elastostatics are solved to validate the proposed implementation. The results show that PSBFEM-UEL has significantly better than FEM convergence and accuracy rate with mesh refinement. The implementation of PSBFEM-UEL can conveniently use arbitrary polygon elements by the polygon/quadtree discretizations in the Abaqus. The developed UEL and the associated input files can be downloaded from https://github.com/hhupde/PSBFEM-Abaqus.

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An enriched finite element method for efficient solutions of transient heat diffusion problems with multiple heat sources

Iqbal¹,

Alam

Ahmad

et al. 2021

Engineering with Computers

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Generalized finite element method with time-independent enrichment functions for 3D transient heat diffusion problems

Cited by 11 publications

References 39 publications

A residual a posteriori error estimate for partition of unity finite elements for three‐dimensional transient heat diffusion problems using multiple global enrichment functions

A residual a posteriori error estimate for partition of unity finite elements for three‐dimensional transient heat diffusion problems using multiple global enrichment functions

PSBFEM-Abaqus: Development of User Element Subroutine (UEL) for Polygonal Scaled Boundary Finite Element Method in Abaqus

An enriched finite element method for efficient solutions of transient heat diffusion problems with multiple heat sources

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