Probabilistic failure envelopes of strip foundations on soils with non-stationary characteristics of undrained shear strength

This paper investigates probabilistic failure envelopes of strip foundations on spatially variable soils with profiles of undrained shear strength su linearly increasing with depth using the lower bound random finite element limit analysis. The spatially variable su is characterised by a non-stationary random field with linearly increasing mean and constant coefficient of variation (COV) with depth. The deterministic uniaxial capacities and failure envelopes are firstly derived to validate numerical models and provide a reference for the subsequent probabilistic analysis. Results indicate that the random field parameters COVsu (COV of su) and Δ (dimensionless autocorrelation distance) have a considerable effect on the probabilistic normalised uniaxial capacities which alters the size of probabilistic failure envelopes. However, COVsu and Δ have an insignificant effect on the shape of probabilistic failure envelopes is observed in the V-H, V-M and H-M loading spaces, such that failure envelopes for different soil variabilities can be simply scaled by the uniaxial capacities. In contrast to COVsu and Δ, the soil strength heterogeneity index κ = μkB/μsu0 has the lowest effect on the probabilistic normalised uniaxial capacity factors but the highest effect on the shape of the probabilistic failure envelopes. A series of expressions are proposed to describe the shape of deterministic and probabilistic failure envelopes for strip foundations under combined vertical, horizontal and moment (V-H-M) loading.

Stochastic Nonlinear Ground Response Analysis Considering Existing Boreholes Locations by the Geostatistical Method

The field and laboratory evidence of nonlinear soil behavior, even at small strains, emphasizes the ‎importance of employing nonlinear methods in seismic ground response analysis. Additionally, ‎determination of dynamic characteristics of soil layers always includes some degree of uncertainty. Most of ‎previous stochastic studies of ground response analysis have focused only on uncertainties of soil ‎parameters, and the effect of soil sample location has been mostly ignored. This study attempts to couple ‎nonlinear time-domain ground response analysis with uncertainty of soil parameters considering existing ‎boreholes’ ‎location through a geostatistical method using a program written in MATLAB. To evaluate ‎the efficiency of the proposed method, stochastic seismic ground responses at construction location were compared with those of the non-stationary random ‎field method‎ through real site data. The ‎results demonstrate that applying the boreholes’ ‎location significantly affects not only the ground ‎responses but also their Coefficient Of Variation (COV). Furthermore, the mean value of the seismic ‎responses is affected more considerably by the values of soil parameters at the vicinity of the construction location. It is also inferred that considering boreholes’ location may reduce the COV of the seismic ‎responses. Among the surface responses in the studied site, the values of Peak Ground Displacement (PGD) ‎and Peak Ground Acceleration (PGA) reflect the highest and ‎lowest dispersion due to uncertainties of soil ‎properties through both non-stationary random field and geostatistical methods.

A Dependent Lindeberg Central Limit Theorem for Cluster Functionals on Stationary Random Fields

In this paper, we provide a central limit theorem for the finite-dimensional marginal distributions of empirical processes (Zn(f))f∈F whose index set F is a family of cluster functionals valued on blocks of values of a stationary random field. The practicality and applicability of the result depend mainly on the usual Lindeberg condition and on a sequence Tn which summarizes the dependence between the blocks of the random field values. Finally, in application, we use the previous result in order to show the Gaussian asymptotic behavior of the proposed iso-extremogram estimator.

On maxima of stationary fields

Let $\{X_{\textbf{n}} \colon \textbf{n}\in{\mathbb Z}^d\}$ be a weakly dependent stationary random field with maxima $M_{A} :=, \sup\{X_{\textbf{i}} \colon \textbf{i}\in A\}$ for finite $A\subset{\mathbb Z}^d$ and $M_{\textbf{n}} := \sup\{X_{\textbf{i}} \colon \mathbf{1} \leq \textbf{i} \leq \textbf{n} \}$ for $\textbf{n}\in{\mathbb N}^d$ . In a general setting we prove that ${\mathbb{P}}(M_{(N_1(n),N_2(n),\ldots, N_d(n))} \leq v_n)$ $= \exp(\!- n^d {\mathbb{P}}(X_{\mathbf{0}} > v_n , M_{A_n} \leq v_n)) + {\text{o}}(1)$ for some increasing sequence of sets $A_n$ of size $ {\text{o}}(n^d)$ , where $(N_1(n),N_2(n), \ldots,N_d(n))\to(\infty,\infty, \ldots, \infty)$ and $N_1(n)N_2(n)\cdots N_d(n)\sim n^d$ . The sets $A_n$ are determined by a translation-invariant total order $\preccurlyeq$ on ${\mathbb Z}^d$ . For a class of fields satisfying a local mixing condition, including m-dependent ones, the main theorem holds with a constant finite A replacing $A_n$ . The above results lead to new formulas for the extremal index for random fields. The new method for calculating limiting probabilities for maxima is compared with some known results and applied to the moving maximum field.

Scaling limits of solutions of linear evolution equations with random initial conditions

We consider a linear equation [Formula: see text], where [Formula: see text] is a generator of a semigroup of linear operators on a certain Hilbert space related to an initial condition [Formula: see text] being a generalised stationary random field on [Formula: see text]. We show the existence and uniqueness of generalised solutions to such initial value problems. Then we investigate their scaling limits.

Spectral diagonal ensemble Kalman filters

A new type of ensemble Kalman filter is developed, which is based on replacing the sample covariance in the analysis step by its diagonal in a spectral basis. It is proved that this technique improves the approximation of the covariance when the covariance itself is diagonal in the spectral basis, as is the case, e.g., for a second-order stationary random field and the Fourier basis. The method is extended by wavelets to the case when the state variables are random fields which are not spatially homogeneous. Efficient implementations by the fast Fourier transform (FFT) and discrete wavelet transform (DWT) are presented for several types of observations, including high-dimensional data given on a part of the domain, such as radar and satellite images. Computational experiments confirm that the method performs well on the Lorenz 96 problem and the shallow water equations with very small ensembles and over multiple analysis cycles.