Latex for Bloggers: Problems & Solutions

A few days back a visitor commented that the my equations on this blog are not showing in Google Chrome. It turned out that it was not working in any browser, let alone Google Chrome. I checked out the java script and everything seemed OK. Then I surmised that one of the servers that the script refers to might be down. I checked out and found the following to be not working in the mathtex3.js:

http://www.cyberroadie.org/cgi-bin/mathtex.cgi?.

Video Tutorial : How To Implement Finite Element Method 4

In the previous post I discussed the procedure for finding out the global stiffness matrix through local elements. In this post we are going to take this process further, develop the entire linear system, elaborate on finding out the force vector and finally solve the system by applying boundary conditions.

Video Tutorial : How To Implement Finite Element Method 3

From this post we saw how to formulate one row of the stiffness matrix. But this is not how you want the computer to calculate it. In this post we are going to explore, through some animations, how to calculate the entire global stiffness matrix by assembling local or elementary matrices. Please make sure to read the instructions at the end to view the animations full screen and to have play and pause control.

Latex for Bloggers: Watchmath not Working

Alright a little diversion. For those of you who have seen previous posts know that I use a lot of latex equations. You can too it is simple.

Writing Latex Equations

Here are a few simple steps which you could find on a number of blogs. I am sorry if I do not remember those tens of blogs / sites to give them due credits.

1. Go to the design tab of your blogger dashboard.

2. Then click the Page Elements.

3. By default you should have your blog post on the left and some free space on the right (with some default gadgets already included). This should be the layout of your blog page.

4. On the right hand side of your page, click add a gadget. Another window will pop up. Scroll down in this window until you fine HTML/Java Script. Click on the plus sign and it will be added.

5. In the configure HTML/Java Script window copy and paste the following code:

Video Tutorial : How To Implement Finite Element Method 2

In the previous post I discussed some fundamental concepts in implementing the finite element method. In today's post we will move further. The post ends with a small animation on the calculation of the stiffness matrix.

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 $0 < x< l$ .