Galerkin Meshfree Methods: A Review and Mathematical Implementation Aspects

Nagevadiya, Bhaumik; Vaghasia, Bhavik; Rachchh, Nikunj V.; Bhoraniya, Ramesh

doi:10.1007/s40819-019-0665-4

Cited by 5 publications

(3 citation statements)

References 80 publications

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“…These finite element functions are chosen to be simple enough to allow for efficient computation and yet provide a good approximation to the true solution. Recently, the Galerkin finite element method was used to solve various problems, and the accuracy and convergence of the method were analyzed, see, for example, [9][10][11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Numerical approximation of the two-species Shigesada-Kawasaki-Teramoto model

Almusawi

Harfash

2023

Preprint

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In this article, the finite element approximation of the two-species Shigesada-Kawasaki-Teramoto model, with cross-diffusion for one species, is studied. In the beginning, the regularized problem of the model and the entropy proof of the regularized problem were presented. Also, an approxi- mation of the regularized problem using the finite element method was introduced. A number of bounds were found in different spaces for approximate solutions. In addition, depending on the bounds that are obtained, the convergence of the approximate problem to the weak form of the problem was studied, and thus the existence of solutions to the Shigesada-Kawasaki-Teramoto model was proved. Also, a practical algorithm for solving the approximate problem was presented and used to calculate the numerical results of the problem.

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Section: Introductionmentioning

confidence: 99%

Numerical approximation of the two-species Shigesada-Kawasaki-Teramoto model

Almusawi

Harfash

2023

Preprint

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“…Finite element method uses mesh interpolation, it leads to errors and cumbersome re-meshing, which consumes more time and money. Even after re-meshing and modifying, method is not sure to give accurate results [1,2]. Mesh-free methods have inherent properties to eliminate the difficulties in finite element method, that is dependency on mesh.…”

Section: Introductionmentioning

confidence: 99%

Influence of Weight-Functions on the Performance of Element-Free Galerkin Method

Nagevadiya,

Bhoraniya

2023

Preprint

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The influence of the weight functions on the performance of the element-free Galerkin method has been studied. The convergence and computational cost using various functions for 1D and 2D structural models are examined. Acceptable ranges, as well as the best scaling parameters for each weight function, are pointed out. Mesh-free methods are completely independent of mesh and approximate the governing partial differential equations based on the only set of nodes. We have discussed the analysis of six different weight functions using the physical nature of field problems and show that the continuity of mesh-free shape function i.e., moving least square shape function depends on weight function. It is observed that the so-called weight function which is showing a minimum error is not always true. The convergence & computational cost (computation time) are also important factors. The manuscript contains the solution of 1D and 2D elastostatic problems with ten different weight functions. Paper represents the comparative study of different weight functions to predict the best field behavior with systematic analysis of each case.

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“…In finite difference the grid generation is a rather trivial process, while the mesh generation in finite elements is a highly complicated subject and an active research arena of its own (e.g., Thompson et al., 1998; Lo, 2015, and others). It should be noted that there are numerical methods which solve the partial differential equations without relying on complex meshes such as analytic element methods (Janković et al., 2003; Strack, 2003) or even mesh‐free methods (Liu, 2002; Nagevadiya et al., 2019; Wu & Wang, 2021). Nevertheless, in this study we focus on the finite element method.…”

Section: Introductionmentioning

confidence: 99%

Simulation of Unconfined Aquifer Flow Based on Parallel Adaptive Mesh Refinement

Kourakos

Harter

2021

Water Resources Research

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The density of a numerical mesh is a key component for the quality of the numerical solution. Common methods in groundwater simulations rely on meshes generated prior to simulation, which lead to unnecessary or insufficient degrees of freedom (dofs). In this study we use Adaptive Mesh Refinement (AMR) to substitute the subjective mesh generation process with a more rigorous approach where new dofs are iteratively added to the solution of an initial coarse mesh based on a solution error estimation. The simulation is solved on a succession of meshes through which the mesh density is increased in areas of steep gradients in the solution variable. Here, we apply the approach to unconfined aquifer flow which is a non‐linear problem commonly encountered in groundwater modeling. By taking advantage of the iterative nature of AMR we arrive at an efficient solver approach. We treat the position of the unconfined boundary as an additional unknown which is estimated iteratively. The first iterations are executed rapidly as the mesh is rather coarse, while the free boundary adapts to the computed water table position as close as the mesh density allows. In this study we demonstrate the viability of the proposed unconfined aquifer simulation method to a series of 2D hypothetical examples under several conditions and stresses as well as a 3D test case. In all cases the AMR was able to automatically resolve the solution in areas with steep hydraulic head gradient, using appropriately higher density to achieve high localized refinement and accuracy.

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Galerkin Meshfree Methods: A Review and Mathematical Implementation Aspects

Cited by 5 publications

References 80 publications

Numerical approximation of the two-species Shigesada-Kawasaki-Teramoto model

Numerical approximation of the two-species Shigesada-Kawasaki-Teramoto model

Influence of Weight-Functions on the Performance of Element-Free Galerkin Method

Simulation of Unconfined Aquifer Flow Based on Parallel Adaptive Mesh Refinement

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