1-d finite-element methods with poisson’s equation
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MATH 212NE 217
Douglas Wilhelm Harder
Department of Electrical and Computer Engineering
University of Waterloo
Waterloo, Ontario, Canada
Copyright © 2011 by Douglas Wilhelm Harder. All rights reserved.
Advanced Calculus 2 for Electrical EngineeringAdvanced Calculus 2 for Nanotechnology Engineering
1-D Finite-Element Methodswith Poisson’s Equation
1-D Finite-element Methods with Poisson’s Equation
2
Outline
This topic discusses an introduction to finite-element methods– Review of Poisson’s equation
– Defining a new kernel V(x)
– Approximate solutions using uniform test functions
1-D Finite-element Methods with Poisson’s Equation
3
Outcomes Based Learning Objectives
By the end of this laboratory, you will:– Understand how to approximate the heat-conduction/diffusion and wave
equations in two and three dimensions
– You will understand the differences between insulated and Dirichlet boundary conditions
1-D Finite-element Methods with Poisson’s Equation
4
The Target Equation
Recall the first of Maxwell’s equations (Gauss’s equation):
If we are attempting to solve for the underlying potential function, under the assumption that it is a conservative field, we have
This is in the form of Poisson’s equation
0
E
2
0
u
1-D Finite-element Methods with Poisson’s Equation
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The Target Equation
In one dimension, this simplifies to:
Define:
and thus we are solving for
2
20
xdu x
dx
2
20
def xdV x u x
dx
0V x
1-D Finite-element Methods with Poisson’s Equation
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The Integral
If , it follows that
for any test function (x) and therefore
Substituting the alterative definition of V(x) into this equation, we get
0x V x
0V x
0b
a
x V x dx
2
20
0b
a
xdx u x dxdx
1-D Finite-element Methods with Poisson’s Equation
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The Integral
Consider the first test function 1(x) :
3
1
2 2
2 210
0xb
a x
xd dx u x dx u x dxdx dx
1-D Finite-element Methods with Poisson’s Equation
8
Integration by Parts
Again, take
but before we apply integration by parts, expand the integral:
Everything in the second integral is known: bring it to the right:
2
20
0b
a
xdx u x dxdx
2
20
0b b
a a
xdx u x dx x dxdx
2
20
b b
a a
xdx u x dx x dxdx
1-D Finite-element Methods with Poisson’s Equation
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Integration by Parts
The left-hand integral is no different from before,
and performing integration by parts, we have
0
0
b b b
a aa
b b
a ax b x a
xd d dx u x x u x dx x dxdx dx dx
xd d d db u x a u x x u x dx x dx
dx dx dx dx
2
20
b b
a a
xdx u x dx x dxdx
1-D Finite-element Methods with Poisson’s Equation
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Integration by Parts
First, substituting in the first test function:
which yields
3 3
1 13 1
2 3 2 1 2 20
x x
x xx x x x
xd d d dx u x x u x x u x dx x dxdx dx dx dx
2
2 220
b b
a a
xdx u x dx x dxd x
1 1 1
1 3x x x x
1-D Finite-element Methods with Poisson’s Equation
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The System of Linear Equations
Again, recall that we approximated the solution by unknown piecewise linear functions:
where we define on
2 11 2 1 2
1 2 2 1
3 22 3 2 3
2 3 3 2
11 1
1 1
n nn n n n
n n n n
x x x xu u x x xx x x x
x x x xu u x x xx x x xu x
x x x xu u x x x
x x x x
11
1 1
defk k
k k kk k k k
x x x xu x u u
x x x x
1k kx x x
1-D Finite-element Methods with Poisson’s Equation
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Unequally Spaced Points
Recall that we approximated the left-hand integral by substituting the piecewise linear functions to get:
1 1
1 1
1
1
1
2
20
2
20
1 1
1 1 0
1 1 11 1 1 1 0
2 2
1 1 1 12 2 2
k k
k k
k
k
k
b b
k k
a a
x x
x x
x
k k k k
k k k k x
k k kk k k k k k k k x
xdx u x dx x dxd x
xdu x dx dx
d x
xu u u udx
x x x x
xu u u dx
x x x x x x x x
1kx
1-D Finite-element Methods with Poisson’s Equation
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Integration by Parts
That is, our linear equations are:
This time, let’s take an actual example
1
1
1 1 11 1 1 1 0
1 1 1 1 1
2
k
k
x
k k kk k k k k k k k x
xu u u dx
x x x x x x x x
1-D Finite-element Methods with Poisson’s Equation
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Integration by Parts
Consider the equation
with the boundary conditionsu(0) = 0
u(1) = 0
2
2 42
sind
u x u x xdx
1-D Finite-element Methods with Poisson’s Equation
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Integration by Parts
The solution to the boundary value problem
u(0) = 0
u(1) = 0
is the exact function
2
2 42
sind
u x u x xdx
2 42
1 13 1 4 5cos cos
16u x x x x x
1-D Finite-element Methods with Poisson’s Equation
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Integration by Parts
We know in one dimension if the right-hand side is close to zero, the solution is a straight line
Thus, choose 9 interior points with a focus on the centre:
>> x_uneq = [0 0.16 0.25 0.33 0.41 0.5 0.59 0.67 0.75 0.84 1];
2
2 42
sind
u x u x xdx
1-D Finite-element Methods with Poisson’s Equation
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Integration by Parts
We will compare this approximation with the approximation found using 9 equally spaced interior points– The finite difference approximation
>> x_eq = 0:0.1:1;
1-D Finite-element Methods with Poisson’s Equation
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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 ); b = zeros ( n, 1 );
for k = 1:n b(k) = 0.5*int( rho, x(k), x(k + 2) ); end b(1) = b(1) - idx(1)*uab(1); b(end) = b(end) - idx(end)*uab(end); v = [uab(1); M \ b; uab(2)];end
int( rho, a, b ) approximates b
a
x dx
1-D Finite-element Methods with Poisson’s Equation
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The Test Functions
The right-hand function of
is also straight-forward:
function [ u ] = rho( x ) u = sin( pi*x ).^4;end
2
2 42
sind
u x u x xdx
1-D Finite-element Methods with Poisson’s Equation
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The Equations
Thus, we can find our two approximations:>> x_eq = (0:0.1:1)';>> plot( x_eq, uniform1d( x_eq, [0, 0], @rho ), 'b+' );>> hold on>> x_uneq = [0 0.16 0.25 0.33 0.41 0.5 0.59 0.67 0.75
0.84 1]';>> plot( x_uneq, uniform1d( x_uneq, [0, 0], @rho ),
'rx' );
1-D Finite-element Methods with Poisson’s Equation
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The Equations
It is difficult to see which is the better function, therefore create a function storing the actual solution (as found in Maple):
function u = u(x) u = -
1/16*( ... (cos(pi*x).^4 - 5*cos(pi*x).^2 + 4)/pi^2
+ ... 3*x.*(x -
1) ... );
end
1-D Finite-element Methods with Poisson’s Equation
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The Equations
Instead, plotting the errors:>> x_eq = 0:0.1:1;
>> plot( x_eq, uniform1d( x_eq, [0, 0], @rho ) - u(x_eq), 'b+' );
>> hold on>> xnu = [0 0.16 0.25 0.33 0.41 0.5 0.59 0.67 0.75 0.84 1];>> plot( x_uneq, uniform1d( x_uneq, [0, 0], @rho ) - u(x_uneq),
'rx' );
1-D Finite-element Methods with Poisson’s Equation
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Integration by Parts
Understanding that the right-hand side has a greater influence in the centre,
appropriately changing the sample points yielded a significantly better approximation
2
2 42
sind
u x u x xdx
1-D Finite-element Methods with Poisson’s Equation
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Summary
In this topic, we have generalized Laplace’s equation to Poisson’s equation– Used the same uniform test functions
– We looked at a problem for which there is an exact solution• Changing the points allowed us to get better approximations
1-D Finite-element Methods with Poisson’s Equation
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What’s Next?
The impulse function (the derivative of a step function) is difficult to deal with…
We will next consider test functions that avoid this…– The test functions will be tents
– This generalizes to higher dimensions
1-D Finite-element Methods with Poisson’s Equation
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References
[1] Glyn James, Advanced Modern Engineering Mathematics, 4th Ed., Prentice Hall, 2011, §§9.2-3.
1-D Finite-element Methods with Poisson’s Equation
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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
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