 Hi, hello, so my name is Elizabeth Bismuth and I have started my PhD four months ago and the research I'm presenting today is obviously not mine but it's from Professor Daniel Straub and Rezu Slouk who is finishing his PhD now. And so I'm doing the research on the supervision and this is directly related to my work so I will be able to answer your question at the end. Just as a note, my background is I have been a structural engineer for five years before that and I came into academia just now and I'm particularly interested in the optimal inspection strategies in structural system because it's something that the industry wish to tackle. I've seen firsthand but sometimes the dialogue is difficult with the tools available for these complex questions. So for any deteriorating system, deterioration has uncertainty and needs to be quantified and this uncertainty can be reduced through inspection of the systems and the question that is quite important to an operator is what is the optimal inspection plan, inspection and maintenance plan that I can apply for the minimum inspection for the minimum cost during the lifecycle of the structure. So the objective of the inspection planning of the optimal inspection plan is to minimize the total expected lifetime costs and the risk to the system and there are four questions that are asked generally in the inspection planning, when, where, what and how. What and how are questions that have been answered during this session, they concern what to inspect, so strain, vibration and how to inspect it, what are the methods, but I will be focusing on when and where. So when to inspect and where to inspect and a multi-component structure. So here we're looking at planning based on detail models, so for instance we have an offshore structure or a wind turbine and these are subject to various loads during the lifetime and we'll have the structure expanse and in consequence we'll have cry growth due to fatigue, corrosion, etc. And these evolution of crack, for instance, have been described in literature and they've been described with some formulas and here for this example, we will look at the Paris law for the propagation of the fatigue crack growth. So the optimal inspection planning is described as a sequential decision problem, but obviously that means that we have to look at if we want to an optimal solution, we have to look at everything that is possibly happening and calculate the associated costs and how likely they are to happen. So the generic decision tree looks like this. Now if you have multiple components, you increase exponentially the size of the system and then if you want to be, if you want to increase the number of time steps, possible times where you can inspect your system, maintain it, then again the system complexity increases. So there are a few answers to this question, one of which is for instance the partial observable Markov decision process, which will be tackled in the next presentation and these approaches for the moment have been limited to simple models, so single component because they are quite computationally demanding as they require to observe every possible state and determine the optimal policy for each possible state of the system during its lifetime. The heuristic approach has been proved to be quite efficient here. We are looking for instance at a threshold approach or a constant inspection intervals and actually the simple heuristics have shown that they provide a quite comparable expected cost to the other more complex methods and they lead to quite simple problems as in if you decide how many times regular inspection intervals and quite simple decision between repair and no repair. However when you go to a system level, multiple components, this again the computationally effort increases a lot and the heuristic system level are much more difficult to define. How can you calculate the probability of failure of the system when you have a quite complex system with multiple components? So these challenges, computational challenges can be tackled when we propose a solution which is based on simple heuristics again for defining a wide range of inspection repair strategies at the system level and then we tackle the computational problem for the calculation probability of failure with Bayesian networks and we combine this with a Monte Carlo approach to integrate over future inspection results. So this is explained in this way. Basically as soon as you know for a given inspection result and you know how many inspections you've done during a lifetime component, when you know this you know how much your total life cycle system costs. The inspection implementation costs and the repair costs. So as soon as you have the path traced this is quite easy to calculate the costs. The issue of course is for each strategy that you can choose you have different inspection outcomes which depend on the deterioration state of the model and here you have to do a Monte Carlo approach but once you have basically calculated this for all the strategies you've defined with the heuristic you can define what is the best strategy by choosing the one that minimizes the costs in relation to the inspection obtained through the Monte Carlo approach. So the probability of failure is calculated with a Bayesian network approach then the Monte Carlo approach for the inspection, integrating the inspection and you'll see with a few parameters. So the Bayesian network briefly is composed of nodes and the characteristic of this network is that it's quite easy to calculate the probability of the system if we know the conditional probability of each node in relation to its parents and the deterioration can be modeled this way by modeling the deterioration sequentially as a Markov process in this case where we have the current state depends only on the state of the proof step. So at the component level this approach is shown to be quite effective. The observations are at the component level here and we extend this Bayesian network to a hierarchical damage Bayesian network by replicating the single component for each of the components and then correlating the components by a hyperparameters. So these hyperparameters correlate the initial state of the component whether it's the initial deterioration state or the time dependent or time invariant parameters. And the advantage of the dynamic Bayesian network is that it enables us to discretize each of the variables with a discretization intervals, calculate the conditional probabilities of conditional probability tables for each of the variables. So here we are looking at matrices and not integrals anymore and propagating information from one component to the other through the hyperparameters and of course get the total probability of failure of the entire system. So this is an example of a relativity of the system with the different inspection results for different components depending on where the inspection has been made and you can see that the peaks signify the inspection occurrences. The accuracy of the DBN is quite good it's been compared to the MCMC method and it's shown to have quite accuracy to the other method but the MCMC is very computational demanding when you have more and more observations. So the DBN has the advantage of being computationally efficient and the computation does not increase with the number of observations you have during your lifetime of the system. So for inspecting components this is your reliability curve. So here we have the situation where the component has no defect observed so the reliability of the component doesn't go back up every time you inspect it. And for a non-inspecting component through the hierarchical dynamic Bayesian network you have the inspection on another on a non-inspected component of an inspection on another component is reflected in the reliability of that component itself. So for the heuristics we've defined five parameters the first one is the inspection campaigns are performed at fixed interval delta t then the number of inspected components the fixed number of inspected components for each campaign is fixed as a certain number then the components that will be inspected are selected following their value of information but actually in this case we are looking at a proxy of value of information so for instance the probability of failure of the component or the criticality of the component of the system and then we use the thresholds in order to add an inspection campaign if needed and the repairs are carried out if the damages exceed a repair criterion in the case that I'm going to present now we will assume that we will repair a component as soon as the crack is detected so here present the case study of the Daniel system so the Daniel system is made of n number of components all are equivalent and they are interchangeable so the observation one can replace the other so there is no prioritization of components for that idealized system which would not be the case for instance for a real life structure but this is something that will be developed further in the research we constructed the algorithm around the Danek Bayesian network model so we determine the probabilistic model so here we have all the parameters and you will note that you will note you have the correlations here for the deterioration and for the hyperparameters that will be reflected in the correlation we reflected in the modeling of the hyperparameters and here we are going to look at two cost models so you can associate the different costs to repairs and system failures depending on what your structure is like and so just to explain briefly how the probability of failure updating for the system works so here we have the example we have an inspection campaign every 10 years so the first 10 years we did the inspection on the three or however many components with fixed number of components we defined the calculated probability of failure didn't exceed the threshold so we carried on then when we arrived at the probability of failure is calculated let's say every year at year 19 we noticed that at year 20 the probability of failure would be exceeded so we carried out another inspection at year 19 in order to reduce the probability of failure of the system and at this you will see that there are four points so B was without inspection if we didn't do anything C is when we do an additional inspection but again at year 20 at this point it would have been again higher than the threshold so we did another one so in total we added two additional inspections at this year and we went back to a level that is acceptable and then at year 20 we did again the inspection as planned etc so this is what's the for an inspection results what the probability of failure of the system looks like these are the components conditional components probability they are corresponding and this is just the expression of so the cost associated to an inspection history you have the failure risk component repair component inspection inspection campaign and system probability of failure so for each inspection campaign you have a cost associated and then basically the optimization works that we generate a random number a sample of deterioration states from there we draw a sample of observations and we integrate the costs over that these observations and this gives us these histograms graphs which will give us the optimal inspection strategies so you can see that for an inspection interval of five years fixed inspection intervals the optimal number of components to be inspected is in Daniel's system of ten components is three this is with the thresholds of two times ten times five if the interval inspection is ten years then the ideal inspection strategy is about six components and then we can basically vary the threshold probability the inspection fixed time inspection times between each inspection and we can vary the number of inspected components with the defined heuristic at the beginning and this gives us the optimal for the case one optimal strategy which is five components inspected every ten years with a probability threshold of three times ten minus five and for case two we have a different result because the costs of the different risks and repairs and inspection are different and here we have that one component inspected every ten years with a probability threshold of three ten minus five is sufficient so this was just Daniel's system the probability of the failure of the system depending on what components have failed is quite easy to write but actually if we want to apply this to a more complex system with components that are more critical than others then we have to do another much more analysis and currently we're extending this optimization to a simple Zias frame so with cross bracing and we're looking at hotspots for fatigue and we will look at the value of information based not only on the probability of failure of the members but also the criticality, the redundancy factor etc so these are the conclusions I have shown you a pragmatic solution that can be applicable to a number of problems and it shows promising results for an extension to more complex systems of course we need to look at the heuristics on which this is based and we can propose to an operator the operator can propose any kind of heuristics or plans that would fit in best so in order to eliminate those that wouldn't make any sense I thank you very much for your attention and thank you for making this week today I was really not planned