Mixed Meshless Local Petrov-Galerkin Methods for Solving Linear Fourth-Order Differential Equations

Jarak, Tomsilav; Jalušić, Boris; Sorić, Jurica

doi:10.21278/tof.44101

Cited by 5 publications

(3 citation statements)

References 19 publications

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“…The finite-strain-incompressible-elasticity problem was analyzed by Gültekin et al [17] using a variational technique based on a finite element method with a solution for volumetric constraints. The mixed meshless local Petrov-Galerkin approach was applied by Jarac et al [18] to gradient elasticity, and they found high accuracy with a low order in the meshless approximation functions. The neural networks (ANN) technique was applied by Saikia et al [19] to recuperate the stress errors in finite element solutions.…”

Section: Introductionmentioning

confidence: 99%

Mesh-Free MLS-Based Error-Recovery Technique for Finite Element Incompressible Elastic Computations

2023

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The finite element error and adaptive analysis are implemented in finite element procedures to increase the reliability of numerical analyses. In this paper, the mesh-free error-recovery technique based on moving least squares (MLS) interpolation is applied to recover the errors in the stresses and displacements of incompressible elastic finite element solutions and errors are estimated in energy norms. The effects of element types (triangular and quadrilateral elements) and the formation of patches (mesh-free patch, mesh-dependent element-based patch, and mesh-dependent node-based patch) for error recovery in MLS and conventional least-square interpolation-error quantification are also assessed in this study. Numerical examples of incompressible elasticity, including a problem with singularity, are studied to display the effectiveness and applicability of the mesh-free MLS interpolation-error recovery technique. The mixed formulation (displacement and pressure) is adopted for a finite element analysis of the incompressible elastic problem. The rate of convergence, the effectivity of the error estimation, and modified meshes for desired accuracy are used to assess the effectiveness of the error estimators. The error-convergence rates are computed in the original FEM solution, in the post-processed solution using mesh-free MLS-based displacement, stress recovery, mesh-dependent patch-based least-square-based displacement, and stress recovery (ZZ) as (0.9777, 2.2501, 2.0012, 1.6710 and 1.5436), and (0.9736, 2.0869, 1.6931, 1.8806 and 1.4973), respectively, for four-node quadrilateral, and six-node triangular meshes. It is concluded that displacement-based recovery was more effective in the finite element incompressible elastic analysis than stress-based recovery using mesh-free and mesh-dependent patches.

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Section: Introductionmentioning

confidence: 99%

Mesh-Free MLS-Based Error-Recovery Technique for Finite Element Incompressible Elastic Computations

2023

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“…The smoothed strains and smoothed deformation gradients were evaluated on subdomains selected by either edge information (edge-based S-FEM, ES-FEM) or nodal information (node-based S-FEM, NS-FEM). Jarak et al [9] presented meshless methods based on the mixed Meshless Local Petrov Galerkin approach used for solving linear fourthorder differential equations. The method was successfully adapted for an application in gradient elasticity obtaining accurate results even with a low order of meshless approximation functions.…”

Section: Introductionmentioning

confidence: 99%

Techniques for Mesh Independent Displacement Recovery in Elastic Finite Element Solutions

Ahmed

2021

TFAMENA

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In this study, techniques for mesh dependent and independent displacement recovery for an a posteriori error estimation are presented. The error recovery of the field variable is made by fitting a higher order polynomial to the displacement over a mesh independent patch (support domain) using the moving least square (MLS) interpolation procedure. The mesh dependent recovery procedure is based on the recovery of the displacement over an element patch that consists of all elements surrounding the element under consideration using the least square (LS) interpolation procedure. The two-dimensional benchmark examples are analysed using linear and quadratic triangular elements to demonstrate the effectiveness and reliability of error estimations. Global and elemental errors of a finite element solution in the energy and L 2 norms are calculated directly from the post-processed displacement. The quality of error estimation obtained using the mesh independent displacement recovery technique in terms of convergence properties, effectivity and adaptive meshes under different error norms has been compared with that of the mesh dependent displacement recovery using the MLS interpolation and least square (LS) interpolation procedures. The performance of an adaptive scheme based on a mesh independent error estimator is compared with the adaptive scheme based on a mesh dependent error estimator. The numerical results show that the finite element analysis based on mesh independent recovery is very effective in converging to a predefined accuracy in a solution with a significantly smaller number of degrees of freedom.

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“…In order to avoid that problem, a mixed MLPG method, or Meshless Finite Volume Method (FVM) is proposed by Atluri et al [32], with MLS trial functions and Heaviside test functions. Several other kinds of mixed MLPG methods were also developed for solving fourth order ODEs or PDEs arising in formulations involving strain gradient effects [33,34]. In these mixed MLPG methods, even though the mechanical strain and / or strain gradients are used as independent variables, these additional variables are eliminated at the node level and only the displacement DoFs are retained at the nodes in the final formulations.…”

Section: Introductionmentioning

confidence: 99%

A New Meshless Fragile Points Method (FPM) With Minimum Unknowns at Each Point, For Flexoelectric Analysis Under Two Theories with Crack Propagation. Part I: Theory and Implementation

Guan,

Dong,

Atluri

2020

Preprint

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Flexoelectricity" refers to a phenomenon which involves a coupling of the mechanical strain gradient and electric polarization. In this study, a meshless Fragile Points Method (FPM), is presented for analyzing flexoelectric effects in dielectric solids. Local, simple, polynomial and discontinuous trial and test functions are generated with the help of a local meshless Differential Quadrature approximation of derivatives. Both primal and mixed FPM are developed, based on two alternate flexoelectric theories, with or without the electric gradient effect and Maxwell stress. The first theory is fully nonlinear and is recommended at the nano-scale, while the second theory is linear and is sufficient at the micro-scale. In the present primal as well as mixed FPM, only the displacements and electric potential are retained as explicit unknown variables at each internal Fragile Point in the final algebraic equations. Thus the number of unknowns in the final system of algebraic equations is kept to be absolutely minimal. An algorithm for simulating crack initiation and propagation using the present FPM is presented, with classic stress-based criterion as well as a Bonding-Energy-Rate(BER)-based criterion for crack development. The present primal and mixed FPM approaches represent clear advantages as compared to the current methods for computational flexoelectric analyses, using primal as well as mixed Finite Element Methods, Element Free Galerkin (EFG) Methods, Meshless Local Petrov Galerkin (MLPG) Methods, and Isogeometric Analysis (IGA) Methods, because of the following new features: they are simpler Galerkin meshless methods using polynomial trial and test functions; minimal DoFs per Point make it very user-friendly; arbitrary polygonal subdomains make it flexible for modeling complex geometries; the numerical integration of the primal

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Mixed Meshless Local Petrov-Galerkin Methods for Solving Linear Fourth-Order Differential Equations

Cited by 5 publications

References 19 publications

Mesh-Free MLS-Based Error-Recovery Technique for Finite Element Incompressible Elastic Computations

Mesh-Free MLS-Based Error-Recovery Technique for Finite Element Incompressible Elastic Computations

Techniques for Mesh Independent Displacement Recovery in Elastic Finite Element Solutions

A New Meshless Fragile Points Method (FPM) With Minimum Unknowns at Each Point, For Flexoelectric Analysis Under Two Theories with Crack Propagation. Part I: Theory and Implementation

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