Meshless element free Galerkin method for unsteady nonlinear heat transfer problems

Singh, Akhilendra; Singh, I.V.; Prakash, R.

doi:10.1016/j.ijheatmasstransfer.2006.08.039

Cited by 102 publications

(32 citation statements)

References 20 publications

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“…(22) and (23) into Eqs. (18) and (19) leads to the residuals of di erential equation (R ) and Dirichlet boundary condition (R ) de ned as:…”

Section: Proposed Mixed Discrete Least Squares Meshless (Mdlsm) Methodsmentioning

confidence: 99%

“…The Galerkin procedure was utilized to discretize the governing PDEs, leading to symmetric coe cient matrices. This method was successfully used to investigate various engineering problems such as static and dynamic analyses of shell structures [16], temperature eld problems [17], 2D fracture problems [18], and unsteady non-linear heat transfer [19]. The rate of convergence of the EFG method was shown to be higher than that of FEM [19].…”

Section: Introductionmentioning

confidence: 99%

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Mixed discrete least squares meshless method for solving the linear and non-linear propagation problems

2017

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Abstract. A Mixed formulation of Discrete Least Squares Meshless (MDLSM) as atruly mesh-free method is presented in this paper for solving both linear and non-linear propagation problems. In DLSM method, the irreducible formulation is deployed, which needs to calculate the costly second derivatives of the MLS shape functions. In the proposed MDLSM method, the complex and costly second derivatives of shape functions are not required. Furthermore, using the mixed formulation, both unknown parameters and their gradients are simultaneously obtained circumventing the need for post-processing procedure performed in irreducible formulation to calculate the gradients. Therefore, the accuracy of gradients of unknown parameters is increased. In MDLSM method, the set of simultaneous algebraic equations is built by minimizing a least squares functional with respect to the nodal parameters. The least squares functional is de ned as the sum of squared residuals of the di erential equation and its boundary condition. The proposed method automatically leads to symmetric and positive-de nite system of equations and, therefore, is not subject to the Ladyzenskaja-Babuska-Brezzi (LBB) condition. The proposed MDLSM method is validated and veri ed by a set of benchmark problems. The results indicate the ability of the proposed method to e ciently and e ectively solve the linear and non-linear propagation problems.

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“…(22) and (23) into Eqs. (18) and (19) leads to the residuals of di erential equation (R ) and Dirichlet boundary condition (R ) de ned as:…”

Section: Proposed Mixed Discrete Least Squares Meshless (Mdlsm) Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mixed discrete least squares meshless method for solving the linear and non-linear propagation problems

2017

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“…Via incorporating fracture mechanics, many numerical simulations focus on evaluating the thermal stress intensity factors of originally existing fractures, which is essential for predicting the thermal induced fracture propagation. During these numerical investigations, the continuum-based numerical methods, i.e., the finite element method (FEM) [8], the extended finite element method (XFEM) [9], the meshless methods (MMs) [10] and the boundary element methods (BEMs) [11], are generally adopted. However, due to the limited capacities of the continuum-based numerical methods in simulating complex discontinuous geometries as well as the fracture mechanics on treating complex fracture patterns, only simple macroscale thermal fracture problems are studied in these numerical simulations.…”

Section: Basic Formulasmentioning

confidence: 99%

Geothermal-Related Thermo-Elastic Fracture Analysis by Numerical Manifold Method

Liu

et al. 2018

Energies

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One significant factor influencing geothermal energy exploitation is the variation of the mechanical properties of rock in high temperature environments. Since rock is typically a heterogeneous granular material, thermal fracturing frequently occurs in the rock when the ambient temperature changes, which can greatly influence the geothermal energy exploitation. A numerical method based on the numerical manifold method (NMM) is developed in this study to simulate the thermo-elastic fracturing of rocklike granular materials. The Voronoi tessellation is incorporated into the pre-processor of NMM to represent the grain structure. A contact-based heat transfer model is developed to reflect heat interaction among grains. Based on the model, the transient thermal conduction algorithm for granular materials is established. To simulate the cohesion effects among grains and the fracturing process between grains, a damage-based contact fracture model is developed to improve the contact algorithm of NMM. In the developed numerical method, the heat interaction among grains as well as the heat transfer inside each solid grain are both simulated. Additionally, as damage evolution and fracturing at grain interfaces are also considered, the developed numerical method is applicable to simulate the geothermal-related thermal fracturing process.

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“…The EFGM has been used to model a variety of physics, e.g. 2D linear elasticity [8][9][10], static and dynamic fracture mechanics [11,12], plate and shell analysis [13][14][15], vibration [7,16,17], electromagnetics [18], heat transfer [8,[19][20][21], metal forming [22,23], biomechanics [24,25] and geomechanics [26]. While the EFGM is superior to the FEM in terms of accuracy and convergence, and there are no issues of volumetric locking [27], MLS shape functions are computationally more expensive and complicate the imposition of essential boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Finite deformation elasto-plastic modelling using an adaptive meshless method

Ullah

Augarde

2013

Computers & Structures

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'Finite deformation elasto-plastic modelling using an adaptive meshless method. ', Computers and structures., Further information on publisher's website:http://dx.doi.org/10. 1016/j.compstruc.2012.04.001 Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Computers and structures. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reected in this document. Changes may have been made to this work since it was submitted for publication. A denitive version will be subsequently published in Computers and structures, 2012Computers and structures, , 10.1016Computers and structures, /j.compstruc.2012 Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Problems including both material and geometric nonlinearities are of practical engineering importance in many applications. Efficient computational modelling of these problems with minimum computer resources remains challenging. Often these problems are modelled with the finite element method (FEM) with adaptive analysis, in which an error estimation procedure automatically determines regions for coarse and fine discretization. Meshless methods offer the attractive possibility of simpler adaptive procedures involving no remeshing, simply insertion or deletion of nodes. However, as meshless methods are computationally more expensive than the FEM, the use of the minimum possible number and proper distribution of nodes are important issues. In this study an adaptive meshless approach for nonlinear solid mechanics is developed based on the element free Galerkin method. An existing error estimation procedure for linear elasto-static problems, is extended here for nonlinear problems including finite deformation and elasto-plasticity, and a new adaptive procedure is described and demonstrated.

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Meshless element free Galerkin method for unsteady nonlinear heat transfer problems

Cited by 102 publications

References 20 publications

Mixed discrete least squares meshless method for solving the linear and non-linear propagation problems

Mixed discrete least squares meshless method for solving the linear and non-linear propagation problems

Geothermal-Related Thermo-Elastic Fracture Analysis by Numerical Manifold Method

Finite deformation elasto-plastic modelling using an adaptive meshless method

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