Biomedical Engineering Reference
In-Depth Information
14
Solution of the one-dimensional
diffusion equation by means of
the Finite Element Method
In the present and following chapters extensive use will be made of a simple
finite element code
mlfem_nac
. This code, including a manual, can be freely
downloaded from the website: www.mate.tue.nl/biomechanicsbook.
The code is written in the program environment MATLAB. To be able to use
this environment a licence for MATLAB has to be obtained. For information about
MATLAB see: www.mathworks.com.
14.1
Introduction
It will be clear from the previous chapters that many problems in biomechanics
are described by (sets of) partial differential equations and for most problems it
is difficult or impossible to derive closed form (analytical) solutions. However, by
means of computers, approximate solutions can be determined for a very large
range of complex problems, which is one of the reasons why biomechanics as
a discipline has grown so fast in the last three decades. These computer-aided
solutions are called numerical solutions, as opposed to analytical or closed form
solutions of equations. The present and following chapters are devoted to the
numerical solution of partial differential equations, for which several methods
exist. The most important ones are the Finite Difference Method and the Finite
Element Method. The latter is especially suitable for partial differential equations
on domains with complicated geometries, material properties and boundary con-
ditions (which is nearly always the case in biomechanics). That is why the next
chapters focus on the Finite Element Method. The basic concepts of the method
are explained in the present chapter. The one-dimensional diffusion equation will
be used as an example to illustrate the key features of the finite element method.
Clearly, the one-dimensional diffusion problem can be solved analytically for a
wide variety of parameter choices, but the structure of the differential equation is
representative of a much larger class of problems to be discussed later. In addition,
the diffusion equation and the more extended, instationary (convection) diffusion
equation play an important role in many processes in biomechanics.
