Heat Conduction Analysis of 3-D Axisymmetric and Anisotropic FGM Bodies by Meshless Local Petrov–Galerkin Method

Abstract

The meshless local Petrov-Galerkin method is used to analyze transient heat conduction in 3-D axisymmetric solids with continuously inhomogeneous and anisotropic material properties. A 3-D axisymmetric body is created by rotation of a cross section around an axis of symmetry. Axial symmetry of geometry and boundary conditions reduces the original 3-D boundary value problem into a 2-D problem. The cross section is covered by small circular subdomains surrounding nodes randomly spread over the analyzed domain. A unit step function is chosen as test function, in order to derive local integral equations on the boundaries of the chosen subdomains, called local boundary integral equations. These integral formulations are either based on the Laplace transform technique or the time difference approach. The local integral equations are nonsingular and take a very simple form, despite of inhomogeneous and anisotropic material behavior across the analyzed structure. Spatial variation of the temperature and heat flux (or of their Laplace transforms) at discrete time instants are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares method. The Stehfest algorithm is applied for the numerical Laplace inversion, in order to retrieve the time-dependent solutions.

Fracture analysis of cracks in magneto-electro-elastic solids by the MLPG

“…With increasing gradient parameter γ the SIF is decreasing. A similar phenomenon is observed for an edge crack in an elastic FGM strip under a mechanical loading [10] and for a cracked piezoelectric FGM specimen [35]. For a crack in a homogeneous magnet-electric-elastic solid analyzed in the previous example the SIF, EDIF, MIIF are uncoupled.…”

“…(35) and selecting Heaviside unit step functions as test functions u * (x), v * (x), m * (x) and β * (x) in each subdomain, one can recast these equations into the following forms…”

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

“…Previous applications of the MLPG method to heat conduction studies [12][13][14][15] have targeted linearly-variable problems with the application of the Laplace transform for the elimination of the time dependence of the differential equation using ordinary circular-type integration sites around uniformly distributed nodes. The present MLPG approach successfully addressed non-linear problems of elevated complexity, i.e.…”

“…The transient heat conduction behavior of functionally graded anisotropic materials with continuously variable material properties have been investigated with the application of meshless local boundary integral equations with Moving Least-Squares approximations using Laplace transformation for the time integration of 2D problems [12,13], whereas anisotropic linearlygraded transient three-dimensional problems, either axisymmetric [14], or fully 3D [15], have been efficiently tackled in a similar fashion using Meshless Local Petrov-Galerkin methods.…”

Analysis of thick functionally graded plates by local integral equation method

“…The price which is to be paid in that approach is the loss of a pure boundary integral character of the formulation. Recently, meshless methods are becoming popular, and they have been successfully applied to 2-D and 3-D axisymmetric transient heat conduction analyses for isotropic and anisotropic FGMs [4][5][6][7][8] and in elasticity [9][10][11].…”