To calculate the bounds of the failure probability, a global search in the epistemic domainsΘh andΘx is required. A naive approach to the problem is prohibitive in the
majority of cases due to the high numerical effort. In fact, a double loop approach has to be adopted, where the inner loop estimates the failure probability and the outer loop searches for the bounds of the probability. The ALS method is utilized to speed up the computation of (precise) failure probabilities, and to ease the search procedure for lower and upper failure probabilities.
4.3.1 The global search for lower and upper failure probabilities The objective function for the global search in Θh and Θx is the failure probability
estimate pˆF. The search can be seen as an iterative procedure that, after some steps,
converges towards the sought lower and upper failure probability bounds.
4.3.1.1 The search in the bounded domain of hyper-parameters Θh
Let the search for the failure probability bounds be restricted to the domain Θh only.
Hence, Θh, defines the set of all probability distributions functions to be considered in
the analysis. Although any element ofΘh is associated with a different value of failure
probability, the limit state does not change as we search in Θh. This is because the
limit state depends upon the structural system and not upon the uncertainty model (Credal set),C, that defines the probability distributions over the state variables.
Since the important direction is defined as any direction pointing towards the failure domain, during the search in Θh, an approximatea can be set for the entire analysis,
ing the distribution functions modifies the location of the most probable point on the limit state surface. Hence, the directiona, set at beginning of the analysis, might not be the optimal one for all the distributions analysed. This motivates the implementation of an adaptive algorithm capable of searching and updating new optimal directions, as shown in Section 3.1.
Each step of the search procedure requires the estimation of a failure probability. Using conventional sampling approaches a new (independent) simulation must be per- formed to estimate each of these failure probabilities. However, the proposed strategy allows it to combine results from different simulations to increase the efficiency of the procedure. This is possible noting that points in the SNS,u, depend onΘh through the
transformation T only, and therefore the corresponding points x in the original state space are independent ofΘh.
Since the limit state does not change as we search in Θh, any point u˜ on the limit
state can be transformed back onto the original space, and then re-mapped to the SNS for the next simulation. When a new simulation is deployed, the standard points on the limit state u˜, previously found, can be used (i) to determine an accurate initial
important direction and (ii) to compute an approximate estimation of the failure prob- ability bounds. Making use of the state points collected during previous simulations, an accurate initial direction can be obtained orientingatowards the region of the space with the highest probability density. On the other side, an approximation of the failure probability bounds can be obtained processing the collected state points. Letai denote
the direction of the current simulation andai´1the direction of the previous simulation.
LetTi andTi´1 be the transformation functions of simulation iand i´1 respectively.
Standard points from simulationican be obtained mapping standard points from sim- ulation i´1 onto the original space, as x“Ti´1puq, and then re-mapping them back
onto the standard space of the next simulation as uremap “Ti´1pTi´1puqq, where T´1
denotes the inverse transformation,T. At the current simulation, the failure probability pFpiq“ ż Rd´1 wpuK aiqfUpu K αiq du d´1 (4.8)
can be computed using the limit state pointsu˜remapobtained from previous simulations.
However, these points are no longer drawn from a probability distribution. Therefore, in order to be able to compute the failure probability, an importance probability density function hU is built to model the re-mapped state points. The density function hU
has a multi-modal distribution with density peaks centred on the re-mapped points and weighed using the metric properties of the SNS. An approximation of the failure probability can then be obtained as
pFpiq“ ż Rd´1 wpuKairemapq hUpuKai´1q fUpuKairemapq hUpuKai´1q dud´1. (4.9)
By means of the importance sampling ratioq“fUpuKairemapq{hUpu K
ai´1q, the probability pFpiq can now be computed using the points from (previous) simulation i´1 as
˜ pFpiq“ 1 NL NL ÿ j qtjuwpuKairemaptjuq “ 1 NL NL ÿ j qtju wpTi´1pTi´1puKtαi´ju1qqq, (4.10) where, qtju denotes the weighing ratio obtained on line j, and N
L is the number of
simulated lines.
4.3.1.2 The search in the bounded domain of structural parametersΘx
Imprecision of structural parameters, characterized by Θx, requires an extension of
the procedure developed so far. In fact, the bounded variables, x P Θx, change the
shape of the limit state boundary, which needs to be addressed with a simultaneous second search, tied to the search in Θh. The proposed strategy takes advantage of an
augmented probability space, where the interval variables are transformed into dummy normal random variables having an interval mean value and a fixed arbitrary standard deviation. In simple terms, this permits a combined consideration of the bounded domain Θx together withΘh. Each dummy imprecise random variable is defined with
an interval mean value µx “ x, and a real-valued standard deviation σx to be fixed
with some convenient value. The only requirement for the value of σx is that it should
neither be too large nor be too small, to avoid numerical issues in computing the failure probability. The standard deviation σx can be set, for example, as a fraction of the
interval radiusσx“ 1k px´xq{2, where,kPNcan be any positive integer. By defining
these dummy imprecise random variables, a thorough search can be performed in both domains Θh and Θx simultaneously. The failure probability bounds are computed on
the found argument optima. Note that during the global search, sampling outside the intervals may occur. However, points outside the intervals are solely used to drive the search process. In cases where the physical model restricts the evaluation to the range of the intervals, truncated normal random variables can be used, where lower and upper limits are equal to the endpoints of the intervals. Two additional reliability analyses at the end of the search, run on the argument optima, are needed to estimate the failure probability bounds.
When the limit state surface is only slightly non-linear the search procedure can be far more efficient. In fact, in this case the important directions in the original space are all oriented towards the same region of the state space. This implies that, as we search inΘh andΘx, the coordinates of the important directions do not significantly change.
Therefore, the important direction in the original space can be used to identify those (conjugate) states in the SNS that are the nearest and furthest from the limit state surface. Here, the state that is the nearest to the limit state surface (also called upper
conjugate state), is also where the failure probability has its maximum; whereas the furthest state from the limit state surface (also called lower conjugate state), is where the failure probability has its minimum.
4.3.2 Applicability of the strategy
The proposed strategy is generally applicable. It enables computation of the failure probability in high dimensional spaces, i.e. when a large, albeit finite, number of random variables are present. Moreover, the efficiency of the strategy is independent of the magnitude of the failure probability, as shown in Figure 3.3a, which makes it particularly suitable for the reliability assessment of large safety-critical systems. Although the approach is applicable to estimate the failure probability of any system, it is particularly efficient when the limit state surface is moderately non-linear (i.e. the performance functions do not show repetitive narrow spikes). In engineering practice, this applies to the majority of problems, as shown in [24], hence the proposed approach seems particularly promising to solve real-case examples.
In case of systems characterized by a large number of failure modes, the algorithm can be adopted to efficiently estimate the failure probability associated to each failure mode. Then, the individual failure probabilities can be combined to calculate the proba- bility of failure of the entire system (i.e. calculating the conditional failure probability). The approach is particularly efficient to support the estimation of the probability bounds associated to the representation of the uncertainty as intervals or fuzzy variables with no restrictions in terms of dimension. In fact, the method is able to deal with any finite number of intervals and fuzzy variables, as also shown in the example "Large scale finite element model" presented in Section 4.4.2.