3.1 Uncertainty in Buckling Load

The behavior of structures prone to buckling may be affected by several factors, which cannot be quantified deterministically. For example, there may be imperfections in the shells with respect to their nominal dimensions due to uncertainty associated with the manufacturing processes. Also, parameters such as Young's modulus or Poisson's ratio may be affected by uncertainties due to variability associated with batches of raw material. Therefore, these sources of uncertainty must be explicitly accounted for such that their effect can be properly quantified. In the following, it is assumed that the material and geometrical properties of a structure are collected in the vector $$$ \boldsymbol{\xi} $$$. The uncertainty associated with $$$ \boldsymbol{\xi} $$$ is characterized by a random variable vector $$$ \boldsymbol{\Xi} $$$ with probability density function $$$ {p}_{\boldsymbol{\Xi}}\left(\boldsymbol{\xi} \right) $$$ [27]. As the buckling load of a structure $$$ {P}_{\mathrm{cr}} $$$ depends on the properties of the structure $$$ \boldsymbol{\xi} $$$, which are assumed to be uncertain, it is clear that the buckling load becomes uncertain as well. In other words, the buckling load $$$ {P}_{\mathrm{cr}} $$$ is a random variable, with its own probability density function. Calculating the probability density function associated with the buckling load may be challenging for problems of engineering interest, particularly in the tails of the distribution. However, the calculation of second-order statistics (i.e., mean and variance) may be more tractable while still providing valuable insights on the uncertainty associated with the buckling load.

3.2 Estimation of Second-Order Statistics of Buckling Loads by Means of Monte Carlo Simulation

In the last equation, $$$ \hat{\mu_4}\left({P}_{\mathrm{cr}},{\boldsymbol{\Xi}}_n\right) $$$ denotes the estimator of the fourth-order central moment of the non-linear buckling load. It is estimated using Equation (A2) in Appendix A.

Equations (15) and (16) indicate that the variances of the estimators for the second-order statistics depend on the number of samples $$$ n $$$. It is desirable that these variances are as small as possible, as this increases the confidence that the quantities being estimated are good approximations of the exact second-order statistics. In other words, it is desirable to obtain estimates of these statistics with sufficient precision. However, small variances (or equivalently, high precision) may entail a large number of simulations $$$ n $$$, which can be quite costly from a numerical viewpoint, as it implies performing $$$ n $$$ non-linear buckling analyses. Therefore, in practical applications, it is expected that $$$ n $$$ is actually small and therefore, the estimators will possess large variability. Such a concept is illustrated schematically in Figure 2, where the probability density associated with an estimator $$$ \hat{\mu} $$$ (which represents either mean or variance, see red curve) is relatively flat, reflecting high uncertainty on the true value of the sought statistic.

4.1 Control Variates

The summation of the two terms described above leads to the Control Variates estimator of Equation (17). In essence, such an estimator allows one to aggregate the results stemming from linear and non-linear buckling analyses but still leads to conclusions about the non-linear buckling load. In fact, the effect of the linear buckling load cancels out from Equation (17), as the subtraction between the second and third terms of the right-hand side of that equation is equal to zero. But in that process, the presence of those second and third terms helps in decreasing the variance of the statistics of the buckling load. Indeed, this estimator usually possesses a relatively small variance, as illustrated schematically in Figure 2 with the probability density function in blue color.

As noted from the above description, the application of Control Variates is completely non-intrusive. That is, it is not necessary to access to system's matrices. It just suffices to conduct linear and non-linear buckling analyses for different sets of samples. Then, the Control Variates estimate in Equation (17) merges the information contained in these samples to produce an estimate of the sought statistic involving non-linear buckling analysis. Indeed, the role of the samples of the linear buckling load in the estimator is simply to exploit correlations to reduce the variance of the estimator of the sought statistic. In that sense, the optimal control parameter $$$ {\gamma}^{\ast } $$$ in Equation (19) plays a pivotal role. To understand its role better, consider the case where there is a high covariance between buckling loads calculated using linear and non-linear buckling analysis. In such a situation, performing linear buckling analysis is almost as good as performing non-linear buckling analysis, meaning that both types of analyses are capable of uncovering the effects of uncertainty in the buckling load. In such a situation of high covariance, the control parameter $$$ {\gamma}^{\star } $$$ will naturally approach to 1 and thus, the Control Variates estimator provides more importance to the information carried by the samples of the linear buckling load, which translates into an estimator of the sought statistic with reduced variance. In the (unlikely) event that the covariance $$$ \hat{\delta} $$$ between the linear and non-linear buckling load is zero, the control parameter itself becomes zero and the estimator of Equation (17) reduces to its plain Monte Carlo counterpart. In such a case, as there is no covariance $$$ \hat{\delta} $$$, the Control Variates estimate cannot leverage on the information carried by the samples of the linear buckling load and therefore, the sought estimate is produced based on the samples of the non-linear buckling analysis only.

To summarize the concepts described above, the application of Control Variates for estimating buckling loads can be visualized as follows. First, a relatively small number of samples of the buckling load are generated considering non-linear buckling analysis. As the number of samples drawn is small, the estimators drawn out of those samples (such as mean and variance) will be most likely highly uncertain (implying that their variances are high). But then, if one performs additional sampling using linear buckling analyses, it is possible to exploit covariance between linear and non-linear buckling analyses to reduce the variance of the estimators associated with the non-linear buckling load.

4.2 Control Variates With Splitting

The previous section has presented the application of the Control Variates framework. For its practical implementation, note that the same samples of the linear and non-linear buckling load are used for both evaluating the optimal control parameter (see Equation (19)) and the sought statistic (see Equation (17)). However, such a strategy induces bias in the estimator of the statistic, as documented, for example, in Reference [30]. The effect of bias can be particularly notorious in the case that the sample set $$$ {\boldsymbol{\Xi}}_n $$$ possesses a small number $$$ n $$$ of samples, which is expected to be precisely the case in practical applications, because performing $$$ n $$$ non-linear buckling analyses is numerically demanding. A remedy to eliminate bias is to resort to a Splitting approach, as proposed in Reference [19]. This is quite a convenient scheme, as it does not demand performing additional buckling analyses (neither linear nor non-linear). The splitting approach consists of dividing the set of available samples into subsets. Then, the associated estimators (for example, mean, variance, optimal control parameter) are estimated for each of these subsets. Finally, the estimators for the subsets are aggregated in such a way that bias is effectively eliminated. As discussed in Reference [19], the number of subsets to be considered should be equal to or larger than 3. However, a large number of subsets may increase the variance. Therefore, in this paper, a minimum of 3 subsets is considered to implement the splitting approach.

How does the splitting approach work in practice? First, each of the sample sets $$$ {\boldsymbol{\Xi}}_n $$$ and $$$ {\boldsymbol{\Xi}}_m $$$ is partitioned into three subsets $$$ {\boldsymbol{\Xi}}_{m^{\ast },k} $$$ and $$$ {\boldsymbol{\Xi}}_{n^{\ast },k} $$$, where $$$ k=1,2,3 $$$ and $$$ {n}^{\ast }=n/3 $$$ and $$$ {m}^{\ast }=m/3 $$$. Here, it is implicitly assumed that $$$ n $$$ and $$$ m $$$ are selected such that they are multiples of 3. For each subset $$$ k $$$, the subset controller $$$ \tau (k) $$$ is defined [19], as shown in Table 1.

4.3 Estimation of the Mean

4.4 Estimation of the Variance

5.1 Folded Plate

The first example of a folded plate, as depicted in Figure 3, is to show that the Control Variates method works successfully in the presence of two types of stability points: bifurcation points and a limit point (snap-through).

A distributed load $$$ p $$$ is applied to the plate at the hinged sleeve along the line $$$ x={L}_x $$$. The folded plate is simply supported at the edges $$$ x=0 $$$ and $$$ x=k\cdotp {L}_x $$$, with boundary conditions $$$ u=v=w=0 $$$. The lengths $$$ {L}_x $$$ and $$$ {L}_y $$$ are set to 100 cm and the plate thickness is defined as $$$ t=5 $$$ cm. A length factor $$$ k $$$ allows for control of the symmetry of the system and thus also the buckling behavior. For $$$ k=1.75 $$$, the system is non-symmetric, whereas a symmetric system occurs when $$$ k=2.00 $$$. For an initial analysis, the Young's modulus is set to $$$ E=1,000 $$$ kN/cm$$$ {}^2 $$$ and the height to $$$ h=10 $$$ cm. Each subarea to the left and right of the hinged sleeve is discretized with $$$ 10\times 10 $$$ elements. The vertical displacements of all nodes of the hinged sleeve are linked. The resulting load $$$ P=p\;{L}_y $$$ is computed for an increasing vertical displacement $$$ {w}_c $$$ at $$$ x={L}_x $$$, $$$ y={L}_y/2 $$$ using the arc-length method with a displacement control of $$$ \Delta w=0.02 $$$ cm. The resulting load–displacement curve for a non-symmetric system ($$$ k=1.75 $$$) is depicted in Figure 4.

Two bifurcation points (A) and (B) occur before the snap-through point (C). In addition, associated eigenvectors $$$ {\boldsymbol{\varphi}}_{cr} $$$ are depicted in Figure 4. The first eigenvectors in points (A) and (B) represent a local buckling of the left or right part of the folded plate, whereas the eigenvectors at point (C) indicate the snap-through as a global buckling failure mode. By applying the corresponding eigenvectors as small imperfections at points (A) and (B), the resulting secondary equilibrium paths, illustrated by the red dashed lines, can be traced. A non-linear buckling analysis yields a buckling load of $$$ {P}_{cr}=9.92 $$$ kN at bifurcation point (A), $$$ {P}_{cr}=14.07 $$$ kN at point (B) and $$$ {P}_{cr}=14.67 $$$ kN at the snap-through point (C). The critical load from the linear buckling analysis is $$$ {\tilde{P}}_{cr}=10.83 $$$ kN.

In the following, the length factor $$$ k $$$, height $$$ h $$$, and Young's modulus $$$ E $$$ are modeled as truncated Gaussian random variables. Their mean values $$$ \mu $$$ and standard deviations $$$ \sigma $$$ are listed in Table 2.

The stochastic non-linear buckling response is analyzed via a Monte Carlo simulation with 5000 samples, resulting in a mean value of the critical buckling load of $$$ {\mu}_{P_{\mathrm{cr}}}=9.94 $$$ kN and a coefficient of variation of $$$ {\delta}_{P_{\mathrm{cr}}}=18.82\% $$$. All buckling loads correspond to the load at which the first diagonal element becomes negative. Despite the significant influence of the three random input parameters defined in Table 2 on the buckling behavior, a strong correlation of $$$ \rho =0.99 $$$ between the linear and non-linear buckling analysis is evident, see Figure 5. These conditions are well-suited for applying the Control Variates method. The results are given in Table 3.

Finally, it should be noted that even if a bifurcation path exists in the pre-buckling stage, the method still performs successfully. The only essential requirement is a strong correlation between the linear and non-linear buckling analyses, while the type of stability point plays a subordinate role. If a sufficiently strong correlation exists, the Control Variates method yields reliable results.

5.2 Composite Shell Panel

This example demonstrates the use of the Control Variates approach to predict buckling loads in a composite shell panel subjected to a single load and random geometric imperfections. For details on the shell panel model, see [35]. Figure 6 illustrates the system modeled with a $$$ 30\times 30 $$$ FE mesh. The panel is simply supported along the two lateral edges. System symmetry cannot be exploited due to applied random geometrical imperfections.

The transversely isotropic material properties of the laminate are provided in Table 4. The panel consists of a three-layer laminate with a total thickness of $$$ h=12.7 $$$ mm. The corresponding stacking sequence is given in Figure 6.

First of all, the buckling behavior is analyzed without any imperfections. The load–displacement curves, evaluating the load $$$ P $$$ vs. the vertical displacement $$$ {w}_c $$$ at the center of the panel, are depicted in Figure 7. These curves are obtained using the arc-length method with a displacement control $$$ \Delta w=0.5 $$$ mm.

A pronounced non-linear pre-buckling behavior is observed. Consequently, the non-linear buckling analysis using the criterion in Equation (12) results in a critical load of $$$ {P}_{\mathrm{cr}}=1.73 $$$ kN, while the linear buckling analysis according to Equation (9) yields a higher critical load of $$$ {\tilde{P}}_{\mathrm{cr}}=2.02 $$$ kN.

The correlation length influences the imperfection shape, thereby it also influences the buckling behavior and the second-order statistics of the buckling load. Figure 8 illustrates the mean $$$ {\mu}_{P_{\mathrm{cr}}} $$$ and the coefficient of variation $$$ {\delta}_{P_{\mathrm{cr}}} $$$ of the buckling load as functions of the correlation length.

For the small correlation length $$$ {\ell}_c=150 $$$ mm, a minimal mean value of $$$ {\mu}_{P_{\mathrm{cr}}}=1.68 $$$ kN and a maximum coefficient of variation of $$$ {\delta}_{P_{\mathrm{cr}}}={\sigma}_{P_{\mathrm{cr}}}/{\mu}_{P_{\mathrm{cr}}}=10\% $$$ can be observed. From this point, the mean value converges to the imperfection-free buckling load $$$ {P}_{\mathrm{cr}}=1.73 $$$ kN, and the coefficient of variation decreases. This is because, for large correlation lengths, the effect of geometric imperfections vanishes. Examining the vertical axis of Figure 8 (left), the mean value shows small variation. However, the correlation length has a significant influence on the coefficient of variations, see Figure 8 (right). Therefore, two different correlation lengths are selected to investigate the effectiveness of Control Variates: The smaller length of $$$ {\ell}_c=150 $$$ mm, which results in wavy realizations, and a larger length of $$$ {\ell}_c=500 $$$ mm, which produces more uniform imperfection shapes as depicted in Figure 9.

In addition, the load–displacement curves for two realizations of the selected correlation lengths are depicted in Figure 7. For these realizations, a small difference in the buckling load can be observed, consistent with the slight deviations in the mean value shown in Figure 8 (left). In Figure 10, the eigenvectors $$$ {\boldsymbol{\varphi}}_{cr} $$$ at the stability point are depicted for the panel without and with the imperfection from Figure 9 (left).

The eigenvectors are quite similar. However, the eigenvector with imperfections is non-symmetric, leading to different post-buckling behavior. Table 5 summarizes the second-order statistics for both correlation lengths, as evaluated using a Monte Carlo simulation with 5000 realizations.

For instance, Figure 11 shows the Monte Carlo convergence test for generated random imperfections with the correlation length $$$ {\ell}_c=150 $$$ mm. For 1000 samples, the relative error of $$$ {\delta}_{P_{\mathrm{cr}}} $$$ compared to the reference solution provided in Table 5 is smaller than $$$ 5\% $$$.

Figure 12 shows the correlations between buckling loads calculated using non-linear ($$$ {P}_{\mathrm{cr}} $$$) and linear analysis ($$$ {\tilde{P}}_{\mathrm{cr}} $$$) for the small and large correlation length. For each correlation length, 5000 random fields are generated, and the buckling loads with the linear and non-linear buckling analysis are calculated.

It is noteworthy that a strong correlation of $$$ \rho =0.97 $$$ is observed even for the small correlation length of $$$ {\ell}_c=150 $$$ mm. Compared to the larger length $$$ {\ell}_c=500 $$$ mm, the correlation between linear and non-linear buckling analyses decreases only slightly. This is a good condition for using Control Variates. Calculating the linear solutions is 4.8 times faster than performing the non-linear buckling analyses. However, for the small correlation length, the subspace eigenvalue solver sometimes requires more iteration steps to compute the correct eigenvalue. This means that the chosen eigenvalue solver and its properties significantly influence the effectiveness of the Control Variates approach.

The results of the estimates of second-order statistics using the Control Variates approach are provided for both correlation lengths in Table 6. The estimates produced using Monte Carlo and Control Variates with Splitting are practically identical in terms of both accuracy and precision, measured in terms of the coefficient of variation $$$ \delta $$$ in Table 6.

For a smaller correlation length $$$ {\ell}_c=150 $$$ mm, the time factor between linear and non-linear buckling analysis is $$$ {f}_s=5.58 $$$. According to Equation (33), it follows an equivalent number of $$$ {n}_e=285 $$$ simulations using the Control Variates approach. Compared to an equivalent number of $$$ {n}_e=600 $$$ Monte Carlo simulations, the proposed approach is approximately $$$ 600/285\approx 2.1 $$$ times faster. For the larger correlation length, the factor between linear and non-linear buckling analysis increases to $$$ {f}_s=5.75 $$$, as the subspace eigenvalue solver converges faster for smoother imperfection shapes. In addition, the equivalent number decreases to $$$ {n}_e=227 $$$ compared to $$$ {n}_e=285 $$$ for the smaller correlation length. This can be explained by analyzing Figure 12, as the correlation coefficient between linear and non-linear buckling loads is $$$ \rho =0.99 $$$, which is larger than the correlation coefficient associated with a shorter correlation length. In other words, for the case of a longer correlation length, linear buckling analysis provides a better approximation for calculating the exact buckling load. This results in better performance of the Control Variates method, with a speed-up factor of approximately $$$ 600/227\approx 2.6 $$$.