Analytical Solutions of a Conduction Problem with Two Different Kinds of Boundary Conditions

Abstract

Conduction problems are widely encountered in science and engineering, such as heat transfer through conductive materials [1] and water flow through porous media [2]. Such physical processes are governed by similar differential equations and are studied traditionally by seeking the mathematical solutions. However, analytical solutions can only be derived under very limited simple boundary and initial conditions [3]. Therefore, much efforts have been spent in developing numerical tools since the popularization of computers [4], and the finite element method (FEM) is, probably, the most successful and widely used one [5]. Use of the FEM usually needs the discretization of both the space and the time, and a more refined mesh and a smaller time step generally yield more exact numerical results. However, this is not the case for conduction problems as unreasonable oscillatory results are often obtained for a given mesh if the time step is smaller than a threshold value [6-9]. For instance, use of four-noded rectangular elements for conduction problems without causing oscillatory results needs the time increment (Δt) larger than L2c/(2k), while for eight-noded elements the threshold is reduced to L2c/(20k). Herein, L denotes the characteristic length of elements while c and k are volumetric capacity and conductivity coefficient of the concerned material [7]. The dilemma, on the other hand, is that a small enough time step is required for numerical convergence for problems where nonlinear material behavior presents [8-10]. It was found that the traditional mass-distributing scheme, which yields the so-called consistent mass matrix, may generate an incorrect neighboring nodal response even though the physical laws are correctly applied at the elementary level. Pan et al. [9] therefore suggested two new mass-distributing schemes which were free of numerical oscillation. Alternatively, mass lumping techniques yielding a diagonal mass matrix were employed by many other authors to remove the possible numerical oscillation [11-13]. Some special techniques were also proposed in the framework of finite element method (known as the stabilized finite element methods) to repress the unphysical oscillations, as recently reviewed and compared by Sendur [14].