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Polynomial Toolbox: polyBox Version 2.1

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Polynomial Toolbox: polyBox Version 2.1



06 May 2013 (Updated )

Polanomial approximation, polynomials as admissible functions, a comprehensive tutorial.

Demonstrate 2: Constrained Polynomials and Beam Measurements

Demonstrate 2: Constrained Polynomials and Beam Measurements

This data set represents uncertain measurements of the deflection of a doubly clamped beam. The aim is to fit a curve through the data which fulfille the theoritical known constraints existing at both ends of the beam.

(c) 2013 Paul O'Leary and Matthew Harker


The constraints on a doubly clamped beam are

$$ y(0) = 0$$
$$ \frac{d y(0)}{d x} = 0 $$
$$ y(1) = 0$$
$$ \frac{d y(1)}{d x} = 0 $$

Prepare MATLAB for The Computation

close all;
clear all;
% Set some defaults
FontSize = 12;
set(0,'DefaultTextInterpreter', 'latex');

Load a Dataset Containing Perturbed Measurements and Constraints

The data points are contained in x and y.

load fitBeamData;

Define the Required Constraints

c1 = [0,x(1),0];
c2 = [1,x(1),0];
c3 = [0,x(end),0];
c4 = [1,x(end),0];
Cs = [c1; c2; c3; c4];

Perform the Constrained Approximation

Call polyfitconstrained to compute the constrained fit.

d = 5;
[pc, Lp, S] = polyfitconstrained( x, y, d, Cs );
% Evaluate at a lagre number of points
nrPts = 500;
xc = linspace( min(x), max(x), nrPts)';
yc = polyval( pc, xc );

Plot the Results of the Approximation

fig1 = figure;
grid on;
hold on;
plot( xc, yc, 'b', 'LineWidth', 1.5);
title( 'Constrained Polynomial Aproximation' );
% Add the positions of the constraints to the plot
[n,m] = size(cs);
csText = [];
for k = 1:n
    xk = cs(k,2);
    vk = polyvander( xk, d );
    yk = vk * pc;
    plot( xk, yk, 'ko', 'markerFaceColor', 'r');

Zoom in on one End of the Beam

Zooming in on the end of the beam show that the fit fulfils both the zero value and zero derivative constraints.


Present the Covariance of the Aolution as an Image

This is the covariance matrix of the approximation for the constrained y. Note: the zero covariance at the points where the value constraint is zero.

fig2 = figure;
imagesc( x, x, S.Ly );
axis image;
title('Covariance of the Approximation');
hold on;

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