On 03.02.2023, at 11:13, Stéphane Mottelet <stephane.motte...@utc.fr> wrote:
>
> Thanks for the code.
>
> Just a remark on the notations, you should write :
>
> F(T) = Int_{0}^{T} PHI(t) . f(T-t) . dt
>
> i.e. not F(t) since t is mute.
>
> However, you should pay attention to the delay notion associated with
> convolution and the relationships between discrete convolution and continuous
> convolution. I am not sure that the output of conv matches with a given
> discretization of the integral above. Maybe rectangle method, but I am not
> sure at all. Anyway, you should have F(0)=0 which does not seem to be the
> case in your graph.
F(t), of course.
But the formula is fundamentally wrong and one should have seen it already from
a dimensional consideration.
Correct should be F(t) = Int_{0}^{T} (d PHI(t)/dt) . f(T-t) . dt
Hope you can help between conv and convol and possible something else.
Heinz
> Dear SciLab Friends:
>
> I have an object consisting of many (~10,000) small components that can fail
> in a statistical way during long-term operation at extreme conditions.
>
> My primary failure model is described by PHI(t) going monotonically from zero
> to one at times from t=0 to T. In the computer, this is realized in n
> timesteps.
>
> Mechanism 2: There is a secondary failure mechanism that starts only when a
> component has previously failed by mechanism 1 and occurs after some delay.
> The delay function is f(t) and final expression for the evolution of the
> mechanism 2 failure is given by:
>
> F(t) = Int_{0}^{T} PHI(t) . f(T-t) . dt
>
> a classical convolution Faltungsintegral.
>
> In Scilab, I write
>
> F= conv(PHI, f, "same")/ n;
>
> and the resulting function F looks reasonable for most of the time. Under
> some conditions, however, I can have
>
> F>PHI
>
> at early stages, but this is physically impossible. Because it comes later, F
> must always be below PHI.
> What do I do wrong?
> Heinz
>
> _________________________
> PS: massively simplified code below......
>
> k=1E-7; // corrosion rate(s-1) in mechanism #1
> n=100; // number of timesteps
> t=(1:n)'; // time in days
> ts= 3600*t; // time in seconds
> PHI= 1 - exp(-((k*ts)^2)); // failure fraction in mechanism #1
> plot("nl", t, PHI, 'r--');
> f= 1 - exp(-ts/3E5); // "pure" failure fraction in mechanism #2
> F=conv(PHI,f,"same")/length(t); //"convoluted" failure fraction mechanism #2
> plot(t,F,'b--');
> legend('failure fraction #1','subsequent failure fraction #2',4);
> xlabel('time (hours)');
> ylabel('failure fraction');
> title('Component failure evolution #1 is followed by #2');
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