An edge-based unstructured finite volume procedure for the numerical analysis of heat conduction applications

Abstract

In recent years, there has been a significant level of research on the application of unstructured mesh methods to the simulation of a variety of engineering and scientific problems. Great progress has been achieved in such area and one of the most successful methodologies consists on the use of the Finite Volume Method (FVM). The unstructured FV formulation is very flexible to deal with any kind of control volume and therefore any kind of unstructured meshes, which are particularly important when complex geometries or automatic mesh adaptation are required. In this article, an unstructured finite volume vertex centered formulation, which was implemented using an edge-based data structure, is deduced and detailed for the solution of heat conduction problems. The numerical formulation is initially described considering a tri-dimensional model and latter particularized for bi-dimensional applications using triangular meshes. The presented procedure is very flexible and efficient to solve potential problems. It can also be extended to deal with a broader class of applications, such as models involving convection-diffusion-reaction terms, after considering the appropriate discretization of the convection-type term. In order to demonstrate the potentiality of the method, some model problems are investigated and the results are validated using analytical or other well-established numerical solutions.