Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I come from a Computer Science background and I've been around searching for a concise tutorial that tells me how Finite Element Method is used to solve a certain partial differential equation. Unfortunately I was depressed with the large number of advanced studies and papers that assumed prior knowledge.

Is it possible to understand the entire process of implementing FEM easily?

share|improve this question
2  
It might be expeditious for you to say a bit more about the "certain partial differential equation" you want to solve. The FEM is fairly simply applied to elliptic linear PDE's in a modest number of dimensions, but if your case is not of this form, then it's probably best to explain that now. BTW a "Computer Science background" suggests you are familiar with discrete mathematics and linear algebra in particular. If that is not the case, also worth some clarificiation. –  hardmath Nov 30 '11 at 16:13
    
Thank you. Well, the equation I want to solve is the [dynamic version of Euler-Bernoulli beam equation](en.wikipedia.org/wiki/Euler-Bernoulli_beam_equation#Dynamic beam equation) which is $\cfrac{\partial^2 }{\partial x^2}\left(EI\cfrac{\partial^2 w}{\partial x^2}\right) = - \mu\cfrac{\partial^2 w}{\partial t^2} + q(x)$. (sorry for the long reply due to problems in my country) –  al-Amjad Tawfiq Isstaif Jan 3 '12 at 10:41
1  
Just glancing at it, this doesn't seem to be of elliptic type, though it is a linear PDE. I'm assuming constants $E,I,\mu$ are positive. You should also have some boundary/initial conditions specified. If it is of hyperbolic type (as I suspect), then you may still be able to give a FEM approximation, though typically one might apply a finite element approach in the "space" variable and a finite difference stepping approach in the "time" dimension. –  hardmath Jan 3 '12 at 16:09
add comment

2 Answers

up vote 3 down vote accepted

After a lot of search, I found this excellent crash course:

A Finite Element Crash Course

share|improve this answer
add comment

here is the solution to your problem Euler-bernoulli cantilever beam deflection equation Solution by using finite difference method

D4y/dx4 =- p/ EI In the finite difference method, the derivatives can be approximated by the following expressions : dy/dx|i ~ (yi+1 - yi )/ 2(Δx) d2y/dx2|i ~ (yi+1 - 2yi + yi-1)/( Δx)2 d3y/dx3|i ~ (yi+2 - 2yi+1 +2 yi-1 - yi-2)/ 2(Δx)3 d4y/dx4|i ~ (yi+2 - 4yi+1 +6 yi - 4 yi-1 + yi-2)/ (Δx)4

Where i is the node sequence. xi = i/(n-1) where n = number of nodes Applying the boundary conditions: y(x=0) =0 , gives y0 = 0. dy(x=0)/dx = 0, dy/dx|x=0 ~ (yi+1 - yi )/ 2(Δx) ; gives y1 = y-1 d2y(x=L)/dx2 = 0, d3y(x=L)/dx3 = 0, at end node

Solution: for n=7, at node 0, dy/dx|0 = 0 at node 1 d4y/dx4|1 ~ (y1+2 - 4y1+1 +6 y1 - 4 y1-1 + y1-2)/ (Δx)4 ~ (y3 - 4y2 +6 y1 - 4 y0 + y-1)/ (Δx)4 ~ (y3 - 4y2 +7 y1)/ (Δx)4 at node 2 d4y/dx4|2 ~ (y2+2 - 4y2+1 +6 y2 - 4 y2-1 + y2-2)/ (Δx)4 ~ (y4 - 4y3 +6 y2 - 4 y1 + y0)/ (Δx)4 ~ (y4 - 4y3 +6 y2 - 4 y1 )/ (Δx)4 at node 3 d4y/dx4|3 ~ (y3+2 - 4y3+1 + 6 y3 - 4 y3-1 + y3-2)/ (Δx)4 ~ (y5 - 4y4 +6 y3 - 4 y2 + y1)/ (Δx)4 at node 4 d4y/dx4|4 ~ (y4+2 - 4y4+1 + 6 y4 - 4 y4-1 + y4-2)/ (Δx)4 ~ (y6 - 4y5 + 6 y4 - 4 y3 + y2)/ (Δx)4 at node 5 d4y/dx4|5 ~ (y5+2 - 4y5+1 + 6 y5 - 4 y5-1 + y5-2)/ (Δx)4 ~ (y7 - 4y6 + 6 y5 - 4 y4 + y3)/ (Δx)4 From the Boundary conditions: d2y/dx2|i@x=L ~ (yi+1 - 2yi + yi-1)/( Δx)2 ~(yi+1 - 2yi + yi-1) = 0 ~(y7 - 2y6 + y5) = 0 ≡ y7 = 2y6 - y5

d4y/dx4|5 ~ (2y6 - y5 - 4y6 + 6 y5 - 4 y4 + y3)/ (Δx)4 ~ (– 2y6 + 5 y5 - 4 y4 + y3)/ (Δx)4

at node 6 d4y/dx4|6 ~ (y6+2 - 4y6+1 + 6 y6 - 4 y6-1 + y6-2)/ (Δx)4 ~ (y8 - 4y7 + 6 y6 - 4 y5 + y4)/ (Δx)4

Use boundary conditions at end note (x=L) d3y/dx3|i@x=L ~ (yi+2 - 2yi+1 +2 yi-1 - yi-2)/ 2(Δx)3 ~ (y6+2 - 2y6+1 +2 y6-1 - y6-2)/ 2(Δx)3 ~ (y8 - 2y7 +2 y5 - y4)/ 2(Δx)3 = 0 ≡ y8 - 2y7 = -2 y5 + y4

d4y/dx4|6 ~ (y8 - 2y7 - 2y7 + 6 y6 - 4 y5 + y4)/ (Δx)4
~ (-2 y5 + y4 - 2y7 +6 y6 - 4 y5 + y4)/ (Δx)4 ~ ( - 2y7 + 6 y6 - 6 y5 + 2 y4)/ (Δx)4 y7 = 2y6 - y5 d4y/dx4|6 ~ ( - 2(2y6 - y5 ) + 6 y6 - 6 y5 + 2 y4)/ (Δx)4 ~( ( - 4y6 + 2 y5 ) + 6 y6 - 6 y5 + 2 y4)/ (Δx)4 ~ ( 2 y6 - 4 y5 + 2 y4)/ (Δx)4

Arrange equations :

Solving the above linear equations gives the values of yi which is the deflection of the beam at each node. yi = 1.0e-4
y0 y1 y2 y3 y4 y5 y6 0 -0.479 -1.623 -3.193 -5.002 -6.918 -8.860

increase number of nodes for more accurate solution No of nodes RMS % 7 5.79 21 0.97 51 0.24 101 0.08

with my complements .... sumaia sumaia@onetel.com

share|improve this answer
2  
This answer is so badly formatted that it is unreadable. Please read the description of the markup language and fix the formatting in your answer. Thanks. –  MJD Aug 5 '12 at 3:02
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.