Boundary Element Methods for Heat Transfer with Phase Change Problems: Theory and Application

The present chapter mainly focuses on the definition of phase change problems and is the entrance for both free and moving boundary problems and their practical applications. In this chapter, we will start from the fundamentals, definitions, and formulations; an overview of the phase change problem in both one, two, and three dimensions is developed with physical and mathematical formulations. 

The present chapter focuses on the elementary development and features of one of the most important numerical techniques that were started by Brebbia CA in 1970. The chapter contains the basic strategy and development of these methods and techniques. The chapter focuses mainly on the boundary element methods and its potential problems.

In the present chapter, the author focused on the direct and indirect application of the boundary element methods to solve phase change problems. The chapter starts with the basic principles of the boundary element methods with the complete derivation of the final system of the algebraic equations. 

In the present chapter, the principles of the boundary element methods for phase change problems over domains with fixed boundaries are developed briefly. The details are developed starting from the 1-D diffusion problems over domains with fixed boundaries subjected to different types of boundary and initial conditions. In these problems, a new criterion has emerged: the existence of a moving boundary, which requires special treatment when solving overall problems. 

In the present chapter, the principles of the boundary element methods for phase change problems over domains with moving boundaries are developed in brief details. The details are developed starting from the 1-D diffusion problems over domains with fixed boundaries subjected to different types of boundary and initial conditions. In these problems, a new criterion has appeared: the existence of the moving boundary, which requires special treatment when solving overall problems.

In the present chapter, the author suggested a new approach for applying the direct boundary element methods; however, this approach is not published yet. It is a new idea based on Artificial quadratic linear approximation in the boundary element method. 

The present chapter is designated to cover overall discussion of different numerical examples, and as we will see, the results are divided into four types. 

In this chapter, we will discuss some of the practical and industrial applications that we have developed over the course of our 37-year research career.