S-PIM for Heat Transfer and Thermoelasticity Problems

Abstract

Heat transfer refers to the transition of thermal energy in media that can be solids or fluids. It is a very important physical phenomenon in most of the systems in engineering and science [1,2]. As one of the typical field problems with temperature as the field variable, the heat transfer problem is described by the second law of thermodynamics leading to a set of PDEs known as Helmholtz equations. It is one of the most fundamental topics in the analysis and design of mechanical, electrical, civil, biological and chemical systems. The Helmholtz equation is a rather a quite general PDEs that governs a number of other physical problems, including torsional deformation of bars, irrotational flow and acoustic problems. Therefore, the formulations and procedures presented in this chapter shall, in principle, applicable also these problems. In fact the S-FEM has been found particularly effective for acoustic problems [3][4][5][6][7][8][9].Compared to solid mechanics problems of vector fields of primary variables (displacements), the temperature field in heat transfer problems is scalar. In this sense, heat transfer problems are easier to deal with numerically. A large variety of numerical methods have been used to analyze the problems of heat transfer, such as the finite element method (FEM) [10-13], the finite difference method (FDM) [14,15] and types of meshfree methods [16][17][18][19].In this chapter, we provide a detailed formulation for the recent S-PIM models for general heat transfer and thermoelastic problems. First, we will give a brief description about the setting of general heat transfer problems. We will then present the field variable approximation, the gradient smoothing operation and the corresponding weakened weak (W 2 ) forms for S-PIM models. The Smoothed Point Interpolation Methods Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 09/30/15. For personal use only.