Polynomial chaos expansions and stochastic finite-element methods - Risk and Reliability in Geotechnical Engineering - page 291

0.5

0.4

0.3

0.2

0.1

0

2 4 6 8 10 12 14

KL eigenmodes

Figure 6.10 Example #3: Foundation on a soil mass with spatially varying Young's modulus—total Sobol'
indices. (After Blatman, G. and B. Sudret. 2011a. J. Comput. Phys. 230, 2345-2367.)

6.6 SuMMarY anD outlook

Accounting for uncertainties has become a crucial issue in modern geotechnical engineering
due to the large variability of soil properties as well as the associated limited information.
The uncertainty analysis framework that is nowadays widely used in many fields applies
equally to geotechnics. Starting from a computational model used for assessing the sys-
tem performance, the input parameters are represented by random variables or fields. The
effect of the input uncertainty onto the model response (i.e., the system performance) can be
assessed by a number of numerical methods.
MCS offers a sound framework for uncertainty propagation; however, its low efficiency
precludes its use for analyses involving finite-element models. Classical structural reliability
methods such as FORM and SORM may be used, at the price of some linearizations and
approximations. In contrast, the so-called PCEs allow for an accurate, intrinsic representa-
tion of the model output. The series expansion coefficients may be computed using nonin-
trusive schemes that are similar in essence to MCS: a sample set of input vectors is simulated
and the corresponding model output is evaluated. From these data, algorithms such as least-
square minimization or LAR may be used.
The resulting series expansion can be post-processed to compute the statistical moments
of the model response (mean value, standard deviation, etc.), quantiles, and confidence
intervals, or even the PDF. In the latter case, the PC expansion is used as a surrogate to the
original computational model. Owing to its polynomial expression, it can be used together
with MCS and kernel-smoothing techniques.
PC expansions also appear extremely efficient in the context of sensitivity analysis. The
Sobol' indices, which are considered as the most accurate sensitivity measures, may be
analytically computed from PC expansions coefficients by simple algebra. The PC expan-
sion itself may be reordered so as to exhibit the so-called Sobol' decomposition, which
allows for detecting the linear and nonlinear effects of the input variables as well as their
interaction.