Transient thermal conduction with variable conductivity using the Meshless Local Petrov–Galerkin method

Abstract

a b s t r a c tA numerical solution of the transient heat conduction problem with spatiotemporally variable conductivity in 2D space is obtained using the Meshless Local Petrov-Galerkin (MLPG) method. The approximation of the field variables is performed using Moving Least Squares (MLS) interpolation. The accuracy and the efficiency of the MLPG schemes are investigated through variation of (i) the domain resolution, (ii) the order of the basis functions, (iii) the shape of the integration site around each node, (iv) the conductivity range, and (v) the volumetric heat capacity range. Steady-state boundary conditions of the essential type are assumed. The results are compared with those calculated by a typical Finite Element method. Specific rectangular-type integration sites are introduced during both steady-state and transient MLPG integration, in order to provide complete surface coverage of the domain without overlapping, and the accuracy of the method is demonstrated in all cases studied. Computational efficiency is also investigated with this MLPG method and found to be slower than FE methods during construction stage, but it clearly surpasses that of FEM approaches during the solution stage on a wide parameter range.

Modelling Thermal Conduction in Nanoparticle Aggregates in the Presence of Surfactants

“…The approach to variables and derivatives is achieved with the DC PSE method [ 34 ]. All integrals are calculated in cubic sectors around each node, as this has been shown to increase the stability of the method [ 32 , 35 ], and square grids are used with increased resolution in the area around the interfaces. A complete analysis of the mesh construction and the approach to variables and integrals is given in [ 35 ].…”

“…In addition, a method for the modelling the effects of the nanolayer and the surfactants is presented. The Meshless Local Petrov–Galerkin (MLPG) method [ 31 , 32 , 33 ] is used for the solution of the heat transfer equation throughout the working domain and for the calculation of the effective conductivity. Approaches to field functions and their derivatives are made using the Discretisation-Corrected Particle Strength Exchange (DC PSE) method [ 34 ], which has been shown to provide stable and fast solutions to such problems, and the integration is performed in cubic sectors around each node.…”

An Efficient Meshless Numerical Method for Heat Conduction Studies in Particle Aggregates

“…A solution to this problem could be the gradual refinement of the grid, so that the transition to the higher discretization region becomes sufficiently smooth [12]. However, defining the nodes and the vectors normal to the interfaces at the limits of the refinement in complex three-dimensional geometries with multiple interconnections has proven computationally difficult and time-consuming.…”

“…Creating this well-coherent mesh can be particularly difficult and usually requires timely user involvement, especially in complex geometries with steep gradients of the field variables, where substantial mesh refinement is usually needed. Because of this difficulty, new numerical methods have appeared recently, commonly called meshfree methods [10][11][12][13]. These methods overcome many of the problems associated with well-connected mesh creation through its complete elimination in the solution process.…”

Development of the Meshless Local Petrov-Galerkin Method to Analyze Three-Dimensional Transient Incompressible Laminar Fluid Flow

“…The MLPG method does not need any mesh for either field interpolation or background integration. So far, considerable efforts have been made to use the MLPG method to solve various problems in the solid and fluid mechanics fields such as elasticity (Atluri and Zhu 2000; Baradaran et al 2011;Bagheri et al 2011;Mahmoodabadi et al 2014), elastodynamics (Batra and Ching 2002;Soares et al 2012;Heaney et al 2010), fracture (Feng et al 2009), crack analysis (Sladek et al 2010;Sladek et al 2012), heat transfer (Sladek et al 2004; Baradaran and Mahmoodabadi 2009; Baradaran and Mahmoodabadi 2010;Mahmoodabadi et al 2011;Karagiannakis et al 2016;Li et al 2018), vibrations (Andreaus et al 2005;Gu and Liu 2001a;Rashidi Moghaddam and Baradaran 2017), plate and shell (Sladek et al 2007;Li et al 2008;Sladek et al 2013;Vaghefi et al 2016), convection-diffusion (Chen et al 2018), Klein-Gordon equation (Darani 2017), electric-field integral equation (Honarbakhsh 2017), metal removal in laser drilling (Abidouab et al 2018) and fluid flow (Arefmanesh et al 2010;Najafi et al 2012).…”