Polynomial chaos expansions and stochastic finite-element methods - Risk and Reliability in Geotechnical Engineering - page 289

2B

(a)

A

E , n

t

(b)

Figure 6.8 Example #3: Foundation on a soil mass with spatially varying Young's modulus. (a) Scheme of the
foundation, (b) finite element mesh.

∫

ρ

log (, )()

xx xx x ′ φ ′ d ′ = λ φ ( ).

(6.85)

E i ii

As no analytical solution to the Fredholm equation exists for this type of autocorrelation
function, the latter is solved by expanding the eigenmodes onto an orthogonal polynomial
basis, see details in Blatman (2009, Appendix B). Note that other numerical methods have
been proposed in the literature (Phoon et al., 2002a,b, 2005; Li et al., 2007). Eventually, 38
modes are retained in the truncated expansion:

38

∑ λ
1

log(,) E x ωµ σ ≈ + ξ ωφ ()(), x

(6.86)

log E log E i i i

i =

where μ log E = 3.8689 and σ log E = 0.2936 in this application. The first nine eigenmodes are
plotted in Figure 6.9 for the sake of illustration. Note that these 38 modes allow one to
account for 99% of the variance of the Gaussian field.
The average settlement under the foundation is computed by finite-element analysis. It
may be considered as a random variable Y = M(ξ), where ξ is the standard normal vector of
dimension 38 that enters the truncated KL expansion. Of interest is the sensitivity of this aver-
age settlement to the various modes appearing in the KL expansion. To address this problem, a
sparse PC expansion is built using an LHS ED of size n = 200. It allows one to get a LOO error
<5%. From the obtained expansion, the total Sobol' indices related to each input variable ξ i
(i.e., each mode in the KL expansion) are computed and plotted in Figure 6.10 .
It appears that only seven modes contribute to the variability of the settlement. This may
be explained by the fact that the model response is an averaged quantity over the domain of
the application of the load, which is therefore rather insensitive to small-scale fluctuations