MATH 212 Advanced Calculus 2 for Electrical Engineering
Advanced Calculus 2 for Nanotechnology Engineering
NE 217
1-D Finite-Element Methods
with Tent Test Functions
Douglas Wilhelm Harder
Department of Electrical and Computer Engineering
University of Waterloo
Waterloo, Ontario, Canada
Copyright © 2011 by Douglas Wilhelm Harder. All rights reserved.
1-D Finite-element Methods with Tent Test Functions
Outline
This topic discusses an introduction to finite-element methods
– Background
– Define tent-shaped test functions
– Perform integration-by-parts to get the linear equation
• Approximate the solution by a linear solution
• Find the system of equations
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1-D Finite-element Methods with Tent Test Functions
Outcomes Based Learning Objectives
By the end of this laboratory, you will:
– Understand the use of test functions in one dimension
– Understand how to create a system of linear equations using these test
functions in one dimenion
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1-D Finite-element Methods with Tent Test Functions
The Target Equation
At this point, we have generalized the finite-difference method to
allow arbitrarily spaced points in one dimension
The ultimate goal is to do this in 2 and 3 dimensions
– In order to achieve this, we need one further idea
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1-D Finite-element Methods with Tent Test Functions
Test Functions
Up to this point, we have used uniform test functions:
1 x1  x  x3
2  x   
0 otherwise
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1-D Finite-element Methods with Tent Test Functions
Test Functions
Two problems:
– The impulse function appears in the integrals, and
– This will not generalize to two and three dimensions!
1 x1  x  x3
2  x   
0 otherwise
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1-D Finite-element Methods with Tent Test Functions
Test Functions
Solution:
– Use a tent function
2  x 
3  x 
4  x 
5  x 
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1-D Finite-element Methods with Tent Test Functions
Test Functions
Defining such functions is straight-forward:
 x  x1
 x  x x1  x  x2
 2 1
 x  x3
2  x   
x2  x  x3
 x2  x3
 0
otherwise


x2  x1 x2  x3

1 ,
If x = x2, both
x2  x1 x2  x3
x3  x3
x1  x1
0

0
if x = x1,
, and if x = x3,
x2  x3
x2  x1
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1-D Finite-element Methods with Tent Test Functions
Integration by Parts
Now, substitute this test function into the integral:
b
b
  x
 d2


x
u
x
dx


x
a 2    d 2 x    a 2    0 dx
which yields
3
3
  x
d

d
 d

dx
 2  x1  u  x     2  x   u  x   dx   2  x 
dx
dx
0

 dx

x1 
x1
x1
x3
x
x
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1-D Finite-element Methods with Tent Test Functions
Integration by Parts
The tent function disappears at x1 at x3:
0x
3
3
  x
d

d
 d


x
u
x


x
u
x
dx


x
dx
    2   
 
2 
 2  1

dx
dx
0

 dx

x1 
x1
x1
3
x
x
thus, this simplifies to:
x3
x3
  x
d
 d

   2  x   u  x   dx   2  x 
dx
dx
0
 dx

x1 
x1
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1-D Finite-element Methods with Tent Test Functions
Integration by Parts
The tent function disappears at x1 at x3:
0x
3
3
  x
d

d
 d


x
u
x


x
u
x
dx


x
dx
    2   
 
2 
 2  1

dx
dx
0

 dx

x1 
x1
x1
3
x
x
thus, this simplifies to:
x3
x3
  x
d
 d

   2  x   u  x   dx   2  x 
dx
dx
0
 dx

x1 
x1
A similar property will be used in 2 and 3 dimensions
– remember this fact!
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1-D Finite-element Methods with Tent Test Functions
Integration by Parts
Recall that both the test function 2(x) and u  x  are defined on
x1  x  x3 :
x  x1
 x  x2
u

u
x1  x  x2
2
 1 x x
x2  x1

1
2
u  x  
u x  x3  u x  x2 x  x  x
3
2
3
 2 x2  x3
x3  x2
 x  x1
x x

2  x    2 1
 x  x3
 x2  x3
x1  x  x2
x2  x  x3
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1-D Finite-element Methods with Tent Test Functions
Integration by Parts
On the intervals, the derivatives are defined:
 u2  u1
x x
d
 2 1
u  x  
dx
 u3  u2
 x3  x2
 1
x x
d

2  x    2 1
dx
 1
 x2  x3
x1  x  x2
x2  x  x3
x1  x  x2
x2  x  x3
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1-D Finite-element Methods with Tent Test Functions
Integration by Parts
Thus:
x3
d
 d


x
u
x




 dx 
x  dx 2 
 dx

1
x2

x1
3
1 u2  u1
1 u3  u2
dx  
dx
x2  x1 x2  x1
x  x3 x3  x2
x2 2
x

1 u2  u1
1 u3  u2
x

x

 2 1
 x3  x2 
x2  x1 x2  x1
x2  x3 x3  x2

u2  u1 u3  u2

x2  x1 x3  x2
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1-D Finite-element Methods with Tent Test Functions
Integration by Parts
In general:
k
uk  uk 1
1
d
 d


x
u
x
dx

x  dx k     dx    x xk  xk 1 xk  xk 1 dx 
k 1
k 1
xk 1
x

xk 1

xk
uk 1  uk
1
dx
xk 1  xk xk 1  xk
uk  uk 1
uk 1  uk
1
1
 xk  xk 1  
 xk 1  xk 
xk  xk 1 xk  xk 1
xk 1  xk xk 1  xk
uk  uk 1 uk 1  uk


xk  xk 1 xk 1  xk
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1-D Finite-element Methods with Tent Test Functions
Integration by Parts
Substituting this back into our equation, we get:
  x
uk 1  uk uk  uk 1 k 1

  k  x 
dx
xk 1  xk xk  xk 1 xk 1
0
x
or, expanding the left-hand side:
k 1
  x


uk 1
1
1
1
uk 1  

uk 1   k  x 
dx
 uk 
xk  xk 1
xk 1  xk
0
 xk 1  xk xk  xk 1 
xk 1
x
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1-D Finite-element Methods with Tent Test Functions
Integration by Parts
  x
In the special case that
is a constant, this simplifies further:
0


uk 1
1
1
1
 xk 1  xk 1
uk 1  

u

u

 k
k 1
xk  xk 1
x

x
x

x
x

x
0
2
k
k 1 
k 1
k
 k 1 k
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1-D Finite-element Methods with Tent Test Functions
The Test Functions
The function to find the approximations is straight-forward:
function [ v ] = uniform1d( x, uab, rho )
n = length( x ) - 2;
idx = 1./diff(x);
M = diag( -(idx( 1:end - 1 ) + idx( 2: end )) ) + ...
diag( idx( 2:end - 1 ), -1 ) + ...
diag( idx( 2:end - 1 ), +1 );
xk 2
b = zeros ( n, 1 );
k 1
int( rho, a, b ) approximates

for k = 1:n
b(k) = int( rho, x(k), x(k + 1), x(k + 2) );
end
 x    x dx
xk
b(1)
= b(1)
- idx(1)*uab(1);
b(end) = b(end) - idx(end)*uab(end);
v = [uab(1); M \ b; uab(2)];
end
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1-D Finite-element Methods with Tent Test Functions
The Test Functions
In Matlab, recall that we may evaluate functions at a vector of points:
– Let a, b, c be the points
x = linspace( a, c, n );
rhox = rho( x ).*((x < b).*(x – a)/(b – a) + (x >= b).*(x – c)/(b – c));
xa
 b  a
  x  
xc
 b  c
a xb
b xc
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1-D Finite-element Methods with Tent Test Functions
The Test Functions
For example, we get the following graph with this Matlab code:
>>
>>
>>
>>
>>
>>
>>
>>
a = 1; b = 2; c = 5;
x = linspace( a, c, 100 );
tentx = (x < b).*(x - a)/(b - a) + (x >= b).*(x - c)/(b – c);
cosx = cos(x).*tentx;
hold on
plot( x, cosx, 'b' );
plot( x, cos(x), 'r' );
plot( x, tentx, 'm' );
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1-D Finite-element Methods with Tent Test Functions
The Test Functions
d2
 u  x   2 u  x   sin 4  x 
dx
2
Using the exact same example from
the last uniform case:
>>
>>
>>
>>
>>
>>
x_uneq = [0 0.16 0.25 0.33 0.41 0.5 0.59 0.67 0.75 0.84 1]';
u_uniform = uniform1d( x_uneq, [0, 0], @rho );
u_tent =
tent1d( x_uneq, [0, 0], @rho );
plot( x_uneq, u_uniform, 'b+' );
hold on
plot( x_uneq, u_tent, 'rx' );
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1-D Finite-element Methods with Tent Test Functions
The Test Functions
To demonstrate that this is better than the previous approximation:
>> plot( x_uneq, u_uniform - u(x_uneq), 'b+' );
>> plot( x_uneq, u_tent - u(x_uneq), 'rx' );
>> norm( u_uniform - u(x_uneq) )
ans =
0.0012
>> norm( u_tent - u(x_uneq) )
ans =
4.7368e-011
Recall that the function
u(x) is the exact solution
8 × 10–4
–1.8 × 10–11
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1-D Finite-element Methods with Tent Test Functions
Summary
In this topic, we have used tent-shaped test functions as opposed to
uniform test functions
– The integration-by-parts was simpler
• We avoided the end points as they were zero
• No impulse functions…
– More work is required for the code
– Better solution for our example
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1-D Finite-element Methods with Tent Test Functions
What’s Next?
We will next look at how we can extend this to two dimensions
– Use triangles to divide up the region instead of intervals
• Tessellations
– Approximate the solution using interpolating polynomials
Intervals
Tessellation
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1-D Finite-element Methods with Tent Test Functions
References
[1] Glyn James, Advanced Modern Engineering Mathematics, 4th Ed.,
Prentice Hall, 2011, §§9.2-3.
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1-D Finite-element Methods with Tent Test Functions
Usage Notes
•
•
These slides are made publicly available on the web for anyone to use
If you choose to use them, or a part thereof, for a course at another
institution, I ask only three things:
– that you inform me that you are using the slides,
– that you acknowledge my work, and
– that you alert me of any mistakes which I made or changes which you make, and
allow me the option of incorporating such changes (with an acknowledgment) in
my set of slides
Sincerely,
Douglas Wilhelm Harder, MMath
dwharder@alumni.uwaterloo.ca
26