I can't figure out weither the problem resides in your model or in the computations, but when I had (many years ago) to use discrete convolution to approximate continuous convolution, I noticed that the best way to obtain coherent results is to use the composed midpoint rule to approximate the integral

h(t)=\int_0^t f(\tau) g(t-\tau) d\tau. For two causal signals t->f(t)], t-> g(t) with respective support [0,N*δ] and [0,M*δ], sampled in the middle of each [i*δ,(i+1)*δ] interval, define the discrete sequences f_i = f((i-1/2)*δ), i=1...N g_i = g((i-1/2)*δ), i=1..M then define h_i = h((i*δ) for i=0...N+M by h_0 = 0 h_i = δ*\sum_{k=1}^i \tilde f_{i-k+1}*g_k, i = 1...N+M-1 h_{N+M} = 0 h_i for i = 1...N+M-1 is the approximation of the integral by the composed midpoint rule and is directly given in Scilab by δ*conv(f,g) where f and g are defined above. Here is an example: function y=f(t) y=sin(2*%pi*t); endfunction function y=g(t) y=sin(%pi*t); endfunction δ=0.01; N=50; // t->f(t) has support [0,0.5] M=100; // t->g(t) has support [0,1] h=[0 δ*conv(f((1/2:N)*δ), g((1/2:M)*δ)) 0]; th=(0:N+M)*δ; // t->h(t) has support [0,1.5] tf=(0:N)*δ; tg=(0:M)*δ; clf plot(tf,f(tf),tg,g(tg),th,h) legend f g f*g S. Le 03/02/2023 à 11:52, Heinz Nabielek a écrit : This is my latest code version: the 'convoluted secondary failure fraction' is nicely below primary failure, but seems too low. Heinz m=2; // Weibull modulus in mechanism #1 k=1E-7; // corrosion rate(s-1) in mechanism #1 n=500; // number of time steps in hourly intervals t=(1:n)'; // timesteps in hours ts= 3600*t; // timesteps in seconds PHI= 1 - exp(-((k*ts)^m)); // failure fraction in mechanism #1 deriv=m*k*((k*ts)^(m-1)) .* exp(-((k*ts)^m)); //derivative of PHI phi=3600*deriv; //derivative of PHI integrated over an hour f= 1 - exp(-ts/3E5); // "pure" failure fraction in mechanism #2 F=(convol(phi,f)/n)(1:n); //"convoluted" failure fraction mechanism #2 plot("nl",t, [PHI phi f F'],'-');xgrid; legend('PHI primary failure','phi primary failure rate','pure secondary failure fraction','convoluted secondary failure fraction',4); xlabel('time (hours)'); ylabel('failure fraction/ failure rate'); On 03.02.2023, at 11:24, Heinz Nabielek <heinznabie...@gmail.com><mailto:heinznabie...@gmail.com> wrote: On 03.02.2023, at 11:13, Stéphane Mottelet <stephane.motte...@utc.fr><mailto: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'); -- Stéphane Mottelet Ingénieur de recherche EA 4297 Transformations Intégrées de la Matière Renouvelable Département Génie des Procédés Industriels Sorbonne Universités - Université de Technologie de Compiègne CS 60319, 60203 Compiègne cedex Tel : +33(0)344234688 http://www.utc.fr/~mottelet This email and any attachments are intended solely for the use of the individual or entity to whom it is addressed and may be confidential and/or privileged. 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