Dimitris G. Giovanis

Stochastic Finite Element Method

Chapter 3 presents the fundamentals of the Stochastic Finite Element Method in the framework of the stochastic formulation of the virtual work principle. The resulting stochastic partial differential equations are solved with either non-intrusive Monte Carlo simulation methods, or intrusive approaches such as the versatile spectral stochastic finite element method. Additional approximate methodologies such as the Neumann and Taylor series expansion methods are also presented together with some exact analytic solutions that are available for statically determinate stochastic structures. The concept of the variability response function is then developed and generalized for general stochastic finite element systems.

Representation of a Stochastic Process

Chapter 2 describes various methods used for the simulation of a stochastic process such as point discretization methods as well as the most popular Karhunen-Loeve and spectral representation series expansion methods. Methods for the simulation of non-Gaussian fields are then presented followed by solved numerical examples.

Epistemic uncertainty assessment using Incremental Dynamic Analysis and Neural Networks

Incremental dynamic analysis (IDA) is a powerful method for the seismic performance assessment of structures. IDA is also very efficient for handling uncertainty due to the mechanical properties of the structure. In the latter case, IDA should be performed within a Monte Carlo framework requiring the execution of a vast number of nonlinear response history analyses. The increased computing effort renders the calculation of performance statistics time-consuming and hence the method is not always practical. We propose a scheme based on artificial neural networks (NN) in order to reduce the computational effort. Within a Monte Carlo approach, trained NN can rapidly generate a large sample of IDA curves and therefore allow us to easily calculate useful response statistics and fragility curves. The implementation of the proposed approach is quick, straightforward and quite accurate.

A Variability Response-Based Adaptive SSFEM

The present work sets up a methodology that allows the estimation of the spatial distribution of the second-order error of the response, as a function of the number of terms used in the truncated Karhunen-Loeve (KL) series representation of the random field involved in the problem. For this purpose, the concept of the variability response function (VRF) is adopted, as it is well recognized that VRF depends only on deterministic parameters of the problem as well as on the standard deviation of the random parameter. The criterion for selecting the number of KL terms at different parts of the structure is the uniformity of the spatial distribution of the second-order error. This way a significantly reduced number of polynomial chaos (PC) coefficients, with respect to classical PC expansion, is required in order to reach a target second-order error.

Monte Carlo Simulation vs. Polynomial Chaos in Structural Analysis: A Numerical Performance Study

The present work revisits the computational performance of non-intrusive Monte Carlo versus intrusive Galerkin methods for large-scale stochastic systems in the framework of high performance computing environments. The purpose of this work is to perform an assessment of the range of the relative superiority of these approaches with regard to a variety of stochastic parameters. In both approaches, the solution of the resulting algebraic equations is performed with a combination of primal and dual domain decomposition methods implementing specifically tailored preconditioners.