tensile tests in matlab - matlab

Question

The problem says:
Three tensile tests were carried out on an aluminum bar. In each test the strain was measured at the same values of stress. The results were
where the units of strain are mm/m.Use linear regression to estimate the modulus of elasticity of the bar (modulus of elasticity = stress/strain).
I used this program for this problem:
function coeff = polynFit(xData,yData,m)
% Returns the coefficients of the polynomial
% a(1)*x^(m-1) + a(2)*x^(m-2) + ... + a(m)
% that fits the data points in the least squares sense.
% USAGE: coeff = polynFit(xData,yData,m)
% xData = x-coordinates of data points.
% yData = y-coordinates of data points.
A = zeros(m); b = zeros(m,1); s = zeros(2*m-1,1);
for i = 1:length(xData)
temp = yData(i);
for j = 1:m
b(j) = b(j) + temp;
temp = temp*xData(i);
end
temp = 1;
for j = 1:2*m-1
s(j) = s(j) + temp;
temp = temp*xData(i);
end
end
for i = 1:m
for j = 1:m
A(i,j) = s(i+j-1);
end
end
% Rearrange coefficients so that coefficient
% of x^(m-1) is first
coeff = flipdim(gaussPiv(A,b),1);
The problem is solved without a program as follows
MY ATTEMPT
T=[34.5,69,103.5,138];
D1=[.46,.95,1.48,1.93];
D2=[.34,1.02,1.51,2.09];
D3=[.73,1.1,1.62,2.12];
Mod1=T./D1;
Mod2=T./D2;
Mod3=T./D3;
xData=T;
yData1=Mod1;
yData2=Mod2;
yData3=Mod3;
coeff1 = polynFit(xData,yData1,2);
coeff2 = polynFit(xData,yData2,2);
coeff3 = polynFit(xData,yData3,2);
x1=(0:.5:190);
y1=coeff1(2)+coeff1(1)*x1;
subplot(1,3,1);
plot(x1,y1,xData,yData1,'o');
y2=coeff2(2)+coeff2(1)*x1;
subplot(1,3,2);
plot(x1,y2,xData,yData2,'o');
y3=coeff3(2)+coeff3(1)*x1;
subplot(1,3,3);
plot(x1,y3,xData,yData3,'o');
What do I have to do to get this result?

Answer 1

As a general advice:
avoid for loops wherever possible.
avoid using i and j as variable names, as they are Matlab built-in names for the imaginary unit (I really hope that disappears in a future release...)
Due to m being an interpreted language, for-loops can be very slow compared to their compiled alternatives. Matlab is named MATtrix LABoratory, meaning it is highly optimized for matrix/array operations. Usually, when there is an operation that cannot be done without a loop, Matlab has a built-in function for it that runs way way faster than a for-loop in Matlab ever will. For example: computing the mean of elements in an array: mean(x). The sum of all elements in an array: sum(x). The standard deviation of elements in an array: std(x). etc. Matlab's power comes from these built-in functions.
So, your problem. You have a linear regression problem. The easiest way in Matlab to solve this problem is this:
%# your data
stress = [ %# in Pa
34.5 69 103.5 138] * 1e6;
strain = [ %# in m/m
0.46 0.95 1.48 1.93
0.34 1.02 1.51 2.09
0.73 1.10 1.62 2.12]' * 1e-3;
%# make linear array for the data
yy = strain(:);
xx = repmat(stress(:), size(strain,2),1);
%# re-formulate the problem into linear system Ax = b
A = [xx ones(size(xx))];
b = yy;
%# solve the linear system
x = A\b;
%# modulus of elasticity is coefficient
%# NOTE: y-offset is relatively small and can be ignored)
E = 1/x(1)
What you did in the function polynFit is done by A\b, but the \-operator is capable of doing it way faster, way more robust and way more flexible than what you tried to do yourself. I'm not saying you shouldn't try to make these thing yourself (please keep on doing that, you learn a lot from it!), I'm saying that for the "real" results, always use the \-operator (and check your own results against it as well).
The backslash operator (type help \ on the command prompt) is extremely useful in many situations, and I advise you learn it and learn it well.
I leave you with this: here's how I would write your polynFit function:
function coeff = polynFit(X,Y,m)
if numel(X) ~= numel(X)
error('polynFit:size_mismathc',...
'number of elements in matrices X and Y must be equal.');
end
%# bad condition number, rank errors, etc. taken care of by \
coeff = bsxfun(#power, X(:), m:-1:0) \ Y(:);
end
I leave it up to you to figure out how this works.