02.05. Basis functions - I

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From the course by University of Michigan

The Finite Element Method for Problems in Physics

215 ratings

University of Michigan

The Finite Element Method for Problems in Physics

215 ratings

This course is an introduction to the finite element method as applicable to a range of problems in physics and engineering sciences. The treatment is mathematical, but only for the purpose of clarifying the formulation. The emphasis is on coding up the formulations in a modern, open-source environment that can be expanded to other applications, subsequently. The course includes about 45 hours of lectures covering the material I normally teach in an introductory graduate class at University of Michigan. The treatment is mathematical, which is natural for a topic whose roots lie deep in functional analysis and variational calculus. It is not formal, however, because the main goal of these lectures is to turn the viewer into a competent developer of finite element code. We do spend time in rudimentary functional analysis, and variational calculus, but this is only to highlight the mathematical basis for the methods, which in turn explains why they work so well. Much of the success of the Finite Element Method as a computational framework lies in the rigor of its mathematical foundation, and this needs to be appreciated, even if only in the elementary manner presented here. A background in PDEs and, more importantly, linear algebra, is assumed, although the viewer will find that we develop all the relevant ideas that are needed. The development itself focuses on the classical forms of partial differential equations (PDEs): elliptic, parabolic and hyperbolic. At each stage, however, we make numerous connections to the physical phenomena represented by the PDEs. For clarity we begin with elliptic PDEs in one dimension (linearized elasticity, steady state heat conduction and mass diffusion). We then move on to three dimensional elliptic PDEs in scalar unknowns (heat conduction and mass diffusion), before ending the treatment of elliptic PDEs with three dimensional problems in vector unknowns (linearized elasticity). Parabolic PDEs in three dimensions come next (unsteady heat conduction and mass diffusion), and the lectures end with hyperbolic PDEs in three dimensions (linear elastodynamics). Interspersed among the lectures are responses to questions that arose from a small group of graduate students and post-doctoral scholars who followed the lectures live. At suitable points in the lectures, we interrupt the mathematical development to lay out the code framework, which is entirely open source, and C++ based. Books: There are many books on finite element methods. This class does not have a required textbook. However, we do recommend the following books for more detailed and broader treatments than can be provided in any form of class: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, T.J.R. Hughes, Dover Publications, 2000. The Finite Element Method: Its Basis and Fundamentals, O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu, Butterworth-Heinemann, 2005. A First Course in Finite Elements, J. Fish and T. Belytschko, Wiley, 2007. Resources: You can download the deal.ii library at dealii.org. The lectures include coding tutorials where we list other resources that you can use if you are unable to install deal.ii on your own computer. You will need cmake to run deal.ii. It is available at cmake.org.

From the lesson

2

In this unit you will be introduced to the approximate, or finite-dimensional, weak form for the one-dimensional problem.

02.01. The Galerkin, or finite-dimensional weak form 23:14

02.01q. Response to a question 7:28

02.02. Basic Hilbert spaces - I 15:51

02.03. Basic Hilbert spaces - II 9:28

02.04. The finite element method for the one-dimensional, linear, elliptic partial differential equation 22:53

02.04q. Response to a question 6:21

02.05. Basis functions - I 14:55

02.06. Basis functions - II 14:43

02.07. The bi-unit domain - I 11:43

02.08. The bi-unit domain - II 16:19

02.09. The finite dimensional weak form as a sum over element subdomains - I 16:08

02.10. The finite dimensional weak form as a sum over element subdomains - II 12:24

02.10ct.1. Intro to C++ (functions) 13:27

02.10ct.2. Intro to C++ (C++ classes) 16:43

Meet the Instructors

Krishna Garikipati, Ph.D.
Professor of Mechanical Engineering, College of Engineering - Professor of Mathematics, College of Literature, Science and the Arts

So we are going to continue with our development of the Galerkin weak form,

for this one dimensional linear elliptic PDE.

Let's begin by recalling where we were.

So we've got as far as here;

if this is our domain of interest,

or this is the physical problem we're looking at,

we have this bar,

we have our x axis, positions L0, okay?

And this is our domain omega, without those endpoints.

What we did already is to partition this by the use of points that we're calling nodes.

We have partitioned it into subdomains,

each of which is denoted as omega_sub_omega_e, indicating an element.

The nodes that we have used are labeled x1,

x2 and so on;

in general, this would be the point X_e and that would be the point X_e+1.

The point with x coordinate x=L is node,

n - number of nodes,

n nodes - and that indicating number of nodes and we also observe that the number of

nodes is equal to our number of elements plus one in one dimension.

Okay. In one dimension also,

for the very specific manner in which we are

partitioning omega for this simplest of finite element examples.

Simplest of finite element cases.

Okay. This is as far as we got and

what this allows us to do is to write the Galerkin weak form

as follows. We can now write it as a sum over e going from one to number of

elements of an integral over omega e of wh comma x

Sigma_h e d x equals sum over e going from

one to number of elements integral over omega e_w h f a dx

plus wh at l,

which of course is our last node times t which is the traction times a_real.

So this is our Galerkin weak form.

Now observe that each integral really is

redefined in this expression as an integral over the relevant element subdomain.

What we need to do to proceed is to talk of how we are going to write out

a representation for our trial solution,

right?

Our trial solution, uh of x and the weighting function.

Okay? This is what we need to do.

Now in doing this - so observe that because of the fact that we're able

to use this decomposition of our integrals over these element subdomains,

we really only need to worry about how to carry out

this representation over an element subdomain. Okay?

So let me state that.

Need to represent uh and wh over omega e, right?

Where, of course, e goes from one to number of elements.

Right? So we have the advantage of needing to

focus only upon what often gets called a local representation, right?

Local being local to the element.

Okay? All right.

So let's look at how we can go ahead and do this now.

So let's then focus upon a single element;

that is xe that is xe+1 and omega e. Okay?

All right.

So it's over this subdomain that we need to write

out our trial solution and weighting function.

The way we can do this now is to define what we will call local basis functions.

We defined local basis functions on omega e. Okay?

All right. And what you're going to observe is as you may expect,

there is going to be a finite number of these local basis functions.

Okay? So we define local basis functions on

omega e and let me also state that we will define

a finite number of basis functions.

Because we have the advantage of working just locally over the elements of domain,

we're going to focus upon this element subdomain and define

a finite number of basis functions over that subdomain,

but then if there is a finite number of

basis functions over a single subdomain and there is a finite number of subdomains,

what we're going to observe, of course,

is that there is a finite number of basis functions over the domain omega.

Okay? So we will have a finite number of basis functions over omega e

and therefore, over omega itself.

So we have a finite number of basis functions.

Naturally, of course, our weak form is going to be finite dimensional.

Right? So that's the connection.

So when it comes down to defining a finite number of basis functions,

we are assured of having a finite dimensional weak form together with a weak form.

Okay? That's how we want to think about it. All right.

So let's talk about what these basis functions are going to be.

We are going to take,

in this initial entry level,

very first presentation of a finite element method - we

are going to take the simplest possible basis functions.

We have an element and it is by choice,

by the way, that I have chosen an element to have only two nodes on it.

We are going to use those two nodes to write out - to expand

fields of interest over the element in terms of just two basic functions. Okay?

So what are we going to do is the following;

we will define two basis functions over omega

e. These basis functions

are going to be polynomials.

And since we are in one dimension,

if we have two basis functions and the basis is complete,

it follows that we are talking of representing functions as what type of polynomials.

Think about that. If you have a complete basis where you are going to use

just two basic functions in one dimension with polynomials,

what kind of polynomials are we able to represent?

Let me just add on that bit about complete basis.

Well, you probably got the answer.

What this implies is that we are essentially going to be representing functions here

as linear polynomials

on Omega e. Okay?

This is the simplest sort of

polynomial basis function we can

adopt in the most standard type of finite element method.

There are finite element methods in which we can even use constant polynomials,

they need a little more work than we need to do - than we should be

doing at this early stage of developing the finite element method.

So we go with linear polynomials.

Okay? All right.

Let's actually try to write these down now.

So the way this works is the follows;

graphically, this is what we will do.

We have omega e and I'm going to write the nodal coordinates here.

Our basis functions are the following; they're linears, right?

That one and that one.

This is n1 for obvious reasons that we're going to call n2.

Okay. So n1 is a function of position,

and n2 is also a function of position.

Okay. With these, what we will be able to do now is to write out

our finite dimensional trial solution and for the very - just for this case,

I'm going to specify that this is the finite dimensional trial solution

over element e - restricted to e and obviously restricted to

element e because it's over element e that we've defined our basis functions.

So the way we write it out in this case is sum_e going from

one to number of nodes in

the element N_sub_n_sub_e is for a number of - the capital N is then for number,

the little n is for nodes and e is element.

Okay? So we have two nodes and the elements,

so Nne is equal to two.

So the way we write this is then N_A - function of

position times some degree of freedom that is being interpolated.

And we will write that degree of freedom as da_sub_e,

which in detail is simply n1,

functional position times a degree of freedom - sorry,

that degree of freedom should be one - degree of

freedom one corresponding to element e plus basis

function 2 function of position times degree of

freedom 2 corresponding to element e. Okay.

So in this, what we have is that na of X is

the basis function.

All right? And da of sub e is the degree of freedom.

Also this is any - is

basis function A where A equals one comma two and da_sub_e is degree of

freedom - A equals one comma two on element e. Okay.

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