Iason Papaioannou

Bayesian Updating with Structural Reliability Methods

Bayesian updating is a powerful method to learn and calibrate models with data and observations. Because of the difficulties involved in computing the high-dimensional integrals necessary for Bayesian updating, Markov chain Monte Carlo (MCMC) sampling methods have been developed and successfully applied for this task. The disadvantage of MCMC methods is the difficulty of ensuring the stationarity of the Markov chain. We present an alternative to MCMC that is particularly effective for updating mechanical and other computational models, termed Bayesian updating with structural reliability methods (BUS). With BUS, structural reliability methods are applied to compute the posterior distribution of uncertain model parameters and model outputs in general. An algorithm for the implementation of BUS is proposed, which can be interpreted as an enhancement of the classic rejection sampling algorithm for Bayesian updating. This algorithm is based on the subset simulation, and its efficiency is not dependent on the number of random variables in the model. The method is demonstrated by application to parameter identification in a dynamic system, Bayesian updating of the material parameters of a structural system, and Bayesian updating of a random field–based finite-element model of a geotechnical site.

Bayesian analysis of rare events

In many areas of engineering and science there is an interest in predicting the probability of rare events, in particular in applications related to safety and security. Increasingly, such predictions are made through computer models of physical systems in an uncertainty quantification framework. Additionally, with advances in IT, monitoring and sensor technology, an increasing amount of data on the performance of the systems is collected. This data can be used to reduce uncertainty, improve the probability estimates and consequently enhance the management of rare events and associated risks. Bayesian analysis is the ideal method to include the data into the probabilistic model. It ensures a consistent probabilistic treatment of uncertainty, which is central in the prediction of rare events, where extrapolation from the domain of observation is common. We present a framework for performing Bayesian updating of rare event probabilities, termed BUS. It is based on a reinterpretation of the classical rejection-sampling approach to Bayesian analysis, which enables the use of established methods for estimating probabilities of rare events. By drawing upon these methods, the framework makes use of their computational efficiency. These methods include the First-Order Reliability Method (FORM), tailored importance sampling (IS) methods and Subset Simulation (SuS). In this contribution, we briefly review these methods in the context of the BUS framework and investigate their applicability to Bayesian analysis of rare events in different settings. We find that, for some applications, FORM can be highly efficient and is surprisingly accurate, enabling Bayesian analysis of rare events with just a few model evaluations. In a general setting, BUS implemented through IS and SuS is more robust and flexible.

Learning soil parameters and updating geotechnical reliability estimates under spatial variability – theory and application to shallow foundations

ABSTRACT Field data is commonly used to determine soil parameters for geotechnical analysis. Bayesian analysis allows combining field data with other information on soil parameters in a consistent manner. We show that the spatial variability of the soil properties and the associated measurements can be captured through two different modelling approaches. In the first approach, a single random variable (RV) represents the soil property within the area of interest, while the second approach models the spatial variability explicitly with a random field (RF). We apply the Bayesian concept exemplarily to the reliability assessment of a shallow foundation in a silty soil with spatially variable data. We show that the simpler RV approach is applicable in cases where the measurements do not influence the correlation structure of the soil property at the vicinity of the foundation. In other cases, it is expected to underestimate the reliability, and a RF model is required to obtain accurate results.

Multilevel Estimation of Rare Events

This work is motivated by the need to estimate the probability of rare events in engineering systems with random inputs. We introduce a multilevel estimator which is based on and generalizes the idea of subset simulation. The novel estimator employs a hierarchy of approximations to the system response computed with different resolutions. This leads to reduced computational costs compared to subset simulation. We study the statistical properties and implementation details of the proposed estimator. Markov chain Monte Carlo (MCMC) runs are required within the estimator, and we demonstrate that the nestedness of the associated multilevel failure domains enables a perfect MCMC simulation without burn-in. We show that nestedness follows from certain simple one-dimensional failure domains. In high dimensions we propose a modification of the multilevel estimator which uses level-dependent stochastic input dimensions. We report on numerical experiments in one- and two-dimensional physical space; in particular, we...

Hydroelastic Analysis of Pontoon-type Very Large Floating Structures in Random Seas

The hydroelastic response of very large floating structures (VLFS) is obtained by resolving the interaction between the surface waves and the floating elastic body. We carry out the analysis in the frequency domain, assuming that the surface waves can be described by a directional wave spectrum. Applying the modal expansion method, we obtain a discrete representation of the required transfer matrices for a finite number of frequencies, while the influence of the wave direction is obtained by numerical integration of the directional components of the spectrum. The boundary element method is used to solve the Laplace equation together with the fluid boundary conditions for the velocity potential, whereas the finite element method is adopted for solving the deflection of the floating plate. Moreover, we compute the variance of the response for two different cases of mean wave angles.

Bayesian model calibration using structural reliability methods: application to the hydrological abc model

Bayesian updating using structural reliability methods (BUS) is applied to calibrate the simple hydrological abc model to observations of a hypothetical real world case. The assumed hypothetical real world is chosen such that the abc model cannot represent it perfectly. The likelihood function is expressed in terms of the measurement error and the modeling error. The probability distributions of both errors are only approximate, due to a lack of knowledge about the true behavior. The correlation structure of the modeling error is regarded as partially uncertain and inferred in the updating process. It is highlighted that under the presence of modeling errors, the predictive distribution of the model output is not the same as the predictive distribution of the true discharge.

Bayesian Updating of Slope Reliability in Undrained Clay with Vane Shear Test Data

In-situ test data, monitoring data and other site-specific information are a common basis for assessing geotechnical performance. These information enable one to learn probabilistic models of uncertain geotechnical properties and update the reliability estimate of geotechnical structures. This learning process is facilitated by the application of Bayesian analysis, which makes optimal use of site-specific information. The objective of this study is to investigate the application of Bayesian analysis to update the probabilistic description of spatially varying soil properties and the reliability of slope stability with in-situ test data. For proper characterization of the prior information on the undrained shear strength s u , a non-stationary random field model is proposed to account for the depth-dependent nature of s u . Bayesian updating for learning the distribution of s u and updating the slope reliability is performed with the adaptive BUS approach with subset simulation. The approach is applied to a saturated clay slope in spatially variable soil. The spatial distribution of the s u is updated with vane shear test data. In addition, the effect of the borehole location on the updated slope reliability is investigated, to inform future optimal test program.

Reliability Analysis of Infinite Slopes under Random Rainfall Events

Landslide events often occur after rainfall events during which the pore water pressure builds up within surficial soil layers, leading to shallow slope failure. The spatial variability of the permeability parameters of the soil causes high gradients in pore water pressure when the rainwater infiltrates into the slope. In this paper, we compute the reliability of infinite slopes under random rainfall events considering the spatial variability of the soil permeability. We model the infiltration process in HYDRUS-1D, which applies a numerical solution of Richards equation, and combine this with a one-dimensional random field model of the hydraulic conductivity of the soil. The rainfall event is characterized in terms of its duration and average intensity and modeled through a self-similar random process. The reliability analysis of the infinite slope is based on the factor of safety concept for evaluating slope stability. To cope with the large number of random variables arising from the discretization of the random fields, we evaluate the slope reliability through subset simulation, which is an adaptive Monte Carlo method known to be especially efficient for such high dimensional problems. We study the influence of the duration and average intensity of the rainfall event on the slope reliability.

Computing the Reliability of Shallow Foundations with Spatially Distributed Measurements

In many geotechnical projects, field data is used to determine the soil parameters. In most instances, however, the statistical analysis is performed ad hoc and the spatial distribution of this data is not (expclitly) accounted for. A more formal statistical approach allows to make better use of the data and combine it in a consistent manner with other information on soil parameters. In particular, Bayesian analysis enables combining information from different sources to learn parameters and models of engineering systems, and facilitates a spatial modeling. In this paper, we apply the Bayesian concept to learn the spatial probability distribution of the friction angle of a silty soil using outcomes of direct shear tests at different locations; we then use the derived distribution to compute the reliability of a shallow foundation. We employ two different approaches for constructing the spatial probabilistic model of the friction angle. Both approaches account for the spatial variability of the soil parameter. In the first approach, we apply a single random variable for modelling the soil property within the area of interest. The inherent spatial variability of the parameter is described by the distribution of the random variable and we use the measurements to update the parameter of this distribution. We adopt the simplifying assumption of a highly fluctuating soil and use the distribution of the mean of the friction angle in conjunction with an analytical model for the bearing capacity to update the reliability of the shallow foundation. The second approach consists of modelling the spatial variability explicitly through a random field model and using the measurements to directly update the random field. Thereby, we employ a finite element model of the soil to assess the reliability of shallow foundation.

Probabilistic failure analysis of infinite slopes under random rainfall processes and spatially variable soil

ABSTRACT Rainfall-induced landslides occur during or immediately after rainfall events in which the pore water pressure builds up, leading to shallow slope failure. Thereby, low permeability layers result in high gradients in pore water pressure. The spatial variability of the soil permeability influences the probability such low permeability layers, and hence the probability of slope failure. In this paper, we investigate the influence of the vertical variability of soil permeability on the slope reliability, accounting for the randomness of rainfall processes. We model the saturated hydraulic conductivity of the soil with a one-dimensional random field. The random rainfall events are characterised by their duration and intensity and are modelled through self-similar random processes. The transient infiltration process is represented by Richards equation, which is evaluated numerically. The reliability analysis of the infinite slope is based on the factor of safety concept for evaluating slope stability. To cope with the large number of random variables arising from the discretization of the random field and the rainfall process, we evaluate the slope reliability through Subset Simulation, which is an adaptive Monte Carlo method known to be especially efficient for reliability analysis of such high-dimensional problems. Numerical investigations show higher probability of slope failure with increased spatial variability of the saturated hydraulic conductivity and with more uniform rainfall patterns.