#finite element methods

[lostinengineering]

Apple computers are good for people. We are not people, we are engineers

Finite Element Methods Professor

submitted by bfgiraffe

[regexkind]

Finite Elements Methods Scratchwork

bullshit scratchwork below the cut, read on at risk of eternal boredom

Ok, so suppose we have a two-point boundary value problem of the form

-d2u/dx2 = f in (0,1)

u(0) = 0

u’(1) = 0

where f is a fixed continuous function.

Turns out this is soluble by variational methods.

Throughout, assume that (f,g) refers to the standard integral-based inner product on [0,1], defined by Int(f(x)g(x),x,0,1) in pseudocode.

Define V = {v in L2(0,1): (v’,v’) < infty and v(0) = 0}.

The claim here is that u in V solves the boundary value problem above iff (u’,v’)=(f,v) for all v in V.

(=>)

If we have a solution to the boundary value problem, then in particular notice that for any v in V, we have

(f,v) = Int(f(x)v(x),x,0,1) = Int(-u’’(x)v(x),0,1)

Now as long as u’ and v are continuously differentiable (which is implicit (? I think) in the fact that u’’ shows up in the boundary-value problem, and that (v’,v’) < infty)), it must be that we can apply the Integration by Parts theorem, which states that

u’v]_0^1 = Int(u’’(x)v(x),x,0,1) + Int(u’(x)v’(x),0,1)

But since u’(1) = 0 and v(0) = 0, the right-hand side goes away.

In particular, Int(-u’’(x)v(x),x,0,1) = Int(u’(x),v’(x),0,1), which implies that the variational formulation given above holds.

(<=)

In the other direction, suppose that u is a continuously twice-differentiable function satisfying the variational formulation.

Then in particular, u’ is continuously differentiable and so is any v in V by definition. Thus the above reasoning gives (f,v) = (-u’’(x)v(x),x,0,1) + u’(1)v(1) (since v(0) = 0).

In particular, (f-(-u’’),v) = 0 for all v in V such that v(1) = 0 as well. Let w = f+u’’, which is at least continuous even if it’s not continuously twice-differentiable. If w is *not* identically zero (read: u doesn’t solve the DE), then w(x) is nonzero and either strictly positive or strictly negative on some closed interval [x_0-e,x_0+e] by continuity. Choose v(x) so that it is a twice-differentiable function that is zero off of this interval and twice differentiable and positive on it. This v(x) will in particular be such that v(1)=0 since our closed interval [x_0-e,x_0+e] could be chosen not to include this endpoint. But, for this v, we will contradict the assumption that (w,v) = 0. Thus it must have been the case that -u’’=f on the interval. 

Notice that plugging in v(x) = x to the equation (f-(-u’’),v) = u’(1)v(1) will imply in particular that 0 = (0,x) = u’(1), so u also satisfies the right-hand condition on its derivative.

Thus the variational formulation agrees with the DE formulation under fairly mild continuity assumptions.

[baughtaaf-blog]

Developments in Analysis and Design using Finite Element Methods

This volume contains a selection of papers presented at The Seventh International Conference on Civil and Structural Engineering and The Fifth International Conference on the Applications of Artificial Intelligence to Civil and Structural Engineering, Developments in Analysis and Design using Finite Element Methods held concurrently from 13-15 September 1999, at Oxford, England.

[lumfcva-blog]

MATLAB Guide to Finite Elements: An Interactive Approach

MATLAB Guide to Finite Elements: An Interactive Approach This successful textbook explores the numerical implementation of Finite Element Analysis using the computer program MATLAB, which is very popular today in engineering and engineering education. The book contains a short tutorial on MATLAB as well as a systematic strategy for the treatment of finite element methods. Useful to both students and researchers in engineering, it provides various examples and exercises from mechanical, civil and aerospace engineering, as well as from materials science. The book especially stresses the interactive use of MATLAB, with each example solved in an interactive manner. The accompanying CD-ROM includes more than fifty MATLAB functions specifically written to be used with this book in the form of a MATLAB Finite Element Toolbox. An extensive solutions manual is provided as well, which includes detailed solutions to all the problems in the book for classroom use. This second edition includes a new brick (solid) element with eight nodes and a one-dimensional fluid flow element. A review of the applications of finite elements in various fields such as fluid flow, heat transfer, structural dynamics, electro-magnetics, is added as well.