Academic literature on the topic 'Spectral stochastic finite element'

The ability to predict the wave climate has a great impact on a wide range of sectors, including coastal and offshore engineering, marine renewable energy and shipping. The state of the art in wave prediction is called spectral wave modelling and is based on a phase-averaged, spectral description of the sea-surface elevation. The governing equation, called the action balance equation, is five-dimensional and describes the generation, propagation and evolution of action density in geographic space, spectral space and time. Due to the multidimensional nature of the equation the feasible resolutions are restricted by the computational costs. The aim of this work is to propose schemes which can increase the range of possible resolutions in spectral wave modelling, with the use of adaptivity in space and angle. Thus, this work focuses on the development of an unstructured, adaptive finite element spectral wave model (Fluidity-SW). A sub-grid scale model for the spatial discretisation is used, which retains the stability of discontinuous systems, with continuous degrees of freedom. Then, a new framework for angular adaptivity is developed, with results in dynamic angular and spatial anisotropy of the angular mesh. Finally a spatially h−adaptive scheme is implemented, which can dynamically treat the spatial gradients of the solution fields. The resulting framework is thoroughly verified and validated in a wide range of test cases and realistic scenarios, against analytical solutions, wave measurements and the results obtained with the widely used SWAN model. Thus, the overall ability of the code to simulate surface gravity wind-waves in fixed and adaptive spatial and angular meshes is demonstrated.

In this thesis, the failures that occurred during the construction of the Jamuna Bridge Abutment in Bangladesh have been investigated. In particular, the influence of heterogeneity on slope stability has been studied using statistical methods, random field theory and the finite element method. The research is divided into three main parts: the statistical characterization of the Jamuna River Sand, based on an extensive in-situ and laboratory database available for the site; calibration of the laboratory data against a double-hardening elastoplastic soil model; and stochastic finite element slope stability analyses, using a Monte Carlo simulation, to analyse the slope failures accounting for heterogeneity. The sand state has been characterised in terms of state parameter, a meaningful quantity which can fully represent the mechanical behaviour of the soil. It was found that the site consists of predominantly loose to mildly dilative material and is very variable. Also, a Normal distribution was found to best represent the state parameter and a Lognormal distribution was found to best represent the tip resistance.The calibration of the constitutive model parameters was found to be challenging, as alternative approaches had to be adopted due to lack of appropriate test results available for the site. Single-variate random fields of state parameter were then linked to the constitutive model parameters based on the relationships found between them, and a parametric study of the abutment was then carried out by linking finite elements and random field theory within a Monte Carlo framework.It was found that, as the degree of anisotropy of the heterogeneity increases, the range of structural responses increases as well. For the isotropic cases, the range of responses was relatively smaller and tended to result in more localised failures. For the anisotropic cases, it was found that there are two different types of deformation mechanism. It was also found that, as the vertical scale of fluctuation becomes bigger, the range of possible structural responses increases and failure is more likely. Finally, it was found that the failed zones observed during the excavation of the West Guide Bund of the Jamuna Bridge Abutment could be closely predicted if heterogeneity was considered in the finite element analyses. In particular, it was found that, for such a natural deposit, a large degree of anisotropy (in the range of 20) could account for the deformation mechanisms observed on site.

The discretization of the stationary diffusion equation with random parameters by the Stochastic Finite Element Method requires the solution of a highly structured but very large linear system of equations. Depending on the stochastic properties of the diffusion coefficient together with the stochastic discretization we consider three solver cases. If the diffusion coefficient is given by a stochastically linear expansion, e.g. a truncated Karhunen-Loeve expansion, and tensor product polynomial stochastic shape functions are employed, the Galerkin matrix can be transformed to a block-diagonal matrix. For the solution of the resulting sequence of linear systems we study Krylov subspace recycling methods whose success depends on the ordering and grouping of the linear systems as well as the preconditioner. If we use complete polynomials for the stochastic discretization instead, we show that decoupling of the Galerkin matrix with respect to the stochastic degrees of freedom is impossible. For a stochastically nonlinear diffusion coefficient, e.g. a lognormal random field, together with complete polynomials serving as stochastic shape functions, we introduce and test the performance of a new Kronecker product preconditioner, which is not exclusively based on the mean value of the diffusion coefficient.

We propose a variant of the stochastic finite element method, where the random elements occuring in the problem formulation are approximated by simple random elements, i.e. random elements with only a finite number of possible values.