Development of a Finite Element Program for Temperature-Displacement Coupled Problems

When projecting an apartment complex, bridge or any other construction, it is important to take into consideration which loads will affect the building. For many constructions it is the self-load, or load of carry that first comes to mind. However, the forces caused by heat can be of significant importance as well. E.g. during a fire, knowing the distribution of temperature throughout the load bearing structure could potentially save lives. With the complex geometries of structures, the analytic solutions can however, be difficult, edging impossible to solve. Therefore another approach is need. This is where the finite element method comes in to play. This method makes it possible to solve these problems using numerical integration.

This thesis seeks to explain how the Finite Element Method of heat transfer works, and to develop a program that can describe the flow of heat through various geometries using this theory. The first subjects covered are the equations of the Finite Element Method in the heat transfer problem. Starting at the simplest possible equations of one-dimensional steadystate and expanding to cover both steady-state and transient, nonlinear and linear, through both one and two dimensions. After the theory of the equations has been covered, they will be implemented in the programming language Python using the interface "Spyder". To validate the results, the programs will be compared to the analytic solutions. The final programs compared to the analytic solutions shows the heat distribution can be modelled very accurately. By assuming the right boundary conditions the developed program can model both simple and more complex problems. The program does not include every aspect of heat transfer and if one were to use the program to model real life situations, it would require some additional work.