Tutorial : How To Implement Finite Element Method

In the previous post I discussed some of the concepts in finite element method. It is important to note that although the case that we have taken up is that of longitudinal deformation of a bar, the findings and discussions are equally applicable to any problem which can be solved by finite element method. Examples of application will be presented later, this post, however deals with the general steps in implementing the method.

Tutorial : Finite Element Method Fundamentals Cont...

In the last post we had gotten as far as the weak formulation of the problem. We had also posed a few questions. Their answers will be given in this post. The discussion will be general though and applicable to all the problems amenable to finite element approximation.

Weighted Residual!!

What is meant by weighted residual? Remember from the previous post, the residual of the system were all the terms moved to the left hand side of the differential equation. The right hand side then had no terms left and it was obviously equal to zero. In other words, the residual was equal to zero. If we multiply this residual by some weighing function, the resultant is still zero (no surprises!). Now the idea of a weighted residual formulation is to multiply the residual by some weighing function and then to integrate this product over the entire domain of the problem. For a one dimensional problem, this is equivalent to integrating over the length of the domain. The integral is then put equal to zero. Of course, because the residual is zero.

$$ \int_{\Omega}r(x)w(x)dx=0$$

Tutorial : Finite Element Method Fundamentals

hello again,

In the previous post I left you guys with a simple code which gives a one dimensional finite element approximation to longitudinal deformation of a bar. Let us learn about the problem in more details.

Problem Definition and its Differential Equation

The problem is explained with the help of the following diagram

Fig. 1

As shown in Fig. 1, a bar fixed at one end is subjected to uniform axial force $f_{o}$ and an end load $P$. At the fixed end the deformation $u$ is zero. The objective is to obtain an expression for the deformation $u$ as a function of $x$. To elaborate the implementation of FEM on this simple problem, let us consider its differential equation.

This equation in terms of the unknown function $u(x)$ is valid in the domain $0 < x< l$ .