PSA, SFEM and reliability methods

Random variables

There is little doubt in the historical prevalence of perturbation methods to construct the statistical relationships linking the input and output variables of a system. In these approaches, a Taylor series expansion is applied to the mathematical operator describ- ing this relation. Typically truncation to first or second order terms is employed. Perturbation theory is efficient for the evaluation of the first two statistical moments - it requires only the knowledge of the mean and variance of the input parameters. However, it becomes inaccurate when major parameters have a sufficiently large coef- ficient of variation (COV), normally around 0.15, according to [33]. This would not be unexpected with space structures [38]. Note that COV is defined as the ratio between the standard deviation and mean, i.e. COV = σ/µ. The degree of nonlinearity present also negatively affects the applicability of perturbation schemes [38].

Within a more general setting, permitting model parameters of significant scatter, a Monte Carlo-based approach is the traditionally assumed path [39]. Realisations of the stochastic structure are explicitly generated, each one resulting in a deterministic solution case. Upon collection of a sufficiently large sample of the population, the response variability is estimated by direct statistical manipulation. This procedure requires some knowledge of the distributions of the random model properties.

The conventional probabilistic structural analysis dictates that they are modelled as independent random variables. Explicitly, the probability P (a ≤ X ≤ b) of a random quantity X in an interval [a, b], and its mean E(X), are given by its PDF fX as

P (a ≤ X ≤ b) = Z b a fX(x)dx (2.21) E[g(X)] = Z ∞ −∞ g(x)fX(x)dx (2.22)

where E is formally the mathematical expectation operator. The nth _{central moment}

is given by

mn=

Z ∞

−∞

with the variance being m2 or var(X). The covariance cov(X, Y ) is

cov(X, Y ) = E[(X − E(X))(Y − E(Y ))] (2.24) and must be zero if X and Y are independent.

The standard PSA is advantageous primarily due to its straightforward implementa- tion. Upon performing a direct Monte Carlo simulation (MCS), the procedure tends to be quite representative of the real uncertain structure and is thus the de facto ver- ification technique for more advanced schemes. Such use is plentiful in literature, for instance in [40]–[42] in the framework of spacecraft design and analysis. Prior to these contributions, Esnault and Klein [38] compiled a guideline for the COV of various loads, materials and basic geometries encountered in space applications.

Among the intensely debated topics in PSA central place is taken by methods targeting reduction of the required number of samples for MCS. Normally, for each random variable, such as Young’s modulus or shell thickness, an appropriate PDF is assigned. Sampling of the global problem is done by taking uncorrelated realisations of all random variables. For this unoptimised procedure, required population sample sizes commonly encountered in literature are in the vicinity of 500 to 2000 realisations ([34], [40], [42], [43]). Sampling techniques obviating the use of such na¨ıve approaches are summarised in [32] and include

• Importance: more samples are generated in critical regions defined a priori. A widely adopted approach.

• Adaptive: after each simulation, the sampling distribution is modified based on the preceding results.

• Directional: uniformly distributed sample direction vectors are defined. Failure probability is the mean of the conditional probabilities obtained along each vector. Importance and adaptive sampling can be applied to each direction.

• Latin hypercube: the sampling space is divided into equi-probable subsets. In other words, the range of a random variable is split into a fixed number of non- overlapping, equally probable intervals.

Random fields

Many researchers have recognised modelling of the parameter space as a set of uncorre- lated random variables as an oversimplification. It is unlikely that a physical quantity governed by some form of uncertainty would deviate from its mean value uniformly over its domain. For example, geometrical variation of a component due to limitations of manufacturing processes would likely be distributed, rather than fixed for the whole part. Modelling spatial variability of parameters gives rise to random fields, and re- spectively stochastic processes, if temporal effects are also included. This idea is the foundation of the stochastic finite element method.

Comprehensive overviews of the SFEM and its underlying principles are given in [44]– [46]. In essence, random fields are expressed in terms of field variables of the form

v(x, θ), with the arguments representing spatial variation and probabilistic behaviour, respectively. Generally, E(v), var(v), along with the corresponding spatial dependency of the field variable

cov(x1, x2) = E{(v(x1, θ) − E[v(x1, θ)])(v(x2, θ) − E[v(x2, θ)])} (2.25)

fully define the stochastic field. Special treatment is usually required for the repre- sentation of (2.25). However, a more challenging aspect of SFEM is that a discrete computational domain has to be specified for the random fields over the application geometry. This translates into the necessity to manage an additional mesh in the FE model, which is one of the method’s drawbacks. Note that both perturbation and MCS approaches can be employed with the stochastic FEM in likeness to PSA.

In spite of SFEM’s academic popularity, deployment in commercial software remains currently limited, although work is being done in this direction [47]. Specialised SFEM programs also exist, and are discussed by Stefanou [34], whereas a MATLAB imple- mentation was laid out in [31] and is freely accessible. Some practical applications have been published, such as design optimisation of shell structures [48].

Reliability methods

A resurgence of approaching problems in stochastic mechanics from the point of view of structural reliability has been observed in the past two decades. In brief, the goal is to estimate the failure probability of the system, rather than the response confidence bands. It can be shown that direct Monte Carlo simulation demands a sample of the population of order 1/pF, where pF is the failure probability. Clearly, even the

computational capabilities of today would be insufficient to cope with solving circa 106 ∼ 108 _{realisations of a complex problem in reasonable time.}

The so-called variance reduction methods prioritise the estimation of failure probability COV, which is much smaller than that of the system response quantities and hence the direct MCS. Two recent advancements have enabled that. Line sampling, introduced in [49], incorporates the somewhat abstract idea of identifying important directions towards high failure probability regions in the input parameter space. Realisations are then evaluated along this direction from randomly selected starting points, and the intersection with the failure regions are found. In [50] the procedure was shown to require only a few hundred samples for certain problems, contrasting with ∼ 107 for conventional MCS. Application to spacecraft ensued shortly after, proving the robust- ness of the method [40], which even under the most unfavourable conditions performs comparably to the direct MCS.

The second strategy is called subset simulation [51]. Calculation of pF is achieved by ex-

pressing it as the product of a set of conditional failure probabilities, facilitated through intermediate failure events. The original problem is then reduced to the evaluation of the conditional probabilities, which can be made sufficiently large upon appropriate modelling. To that end, a Markov chain MCS based on the Metropolis-Hastings al- gorithm is employed. The method was found to have a performance similar to line sampling for a satellite structure [41], with both schemes requiring only a few hundred realisations for pF of order 10