| dc.description.abstract | With the growth of computational resources available, demand on the accuracy and speed of
simulation is steeply increasing. Accordingly, accounting for the variabilities in the simulations
plays a pivotal role in taking informed engineering decision. Variability or uncertainties can be
classi fied into two broad categories, epistemic uncertainty, and aleoteric uncertainty. Where,
epistemic uncertainty is because of the lack of knowledge or data, and aleoteric is the uncertainty
inherent to the system. In this dissertation epistemic uncertainty is considered.
Monte Carlo simulation is the most versatile method for the analysis of systems with uncertainties.
However, for large scale engineering applications where finite element is used for
solving the underlying physical problem, the computational cost of Monte Carlo becomes prohibitive.
On the other hand, spectral stochastic finite element (SSFEM) outperforms Monte
Carlo for problems with low stochastic dimensionality | that is, the number of basic random
variables. However, in SSFEM the computational cost becomes prohibitive for problems with
large stochastic dimensionality. Therefore, in this dissertation a hybrid method combining both
Monte Carlo and SSFEM is developed to solve large scale stochastic systems. That is, the target
systems are of large physical degrees of freedom and stochastic dimensionality.
The total computational cost of analyzing a stochastic system can be divided into two parts,
namely, uncertainty modeling and uncertainty propagation. Accordingly, first, a domain shape
independence property of Karhunen-Loeve (KL) expansion is proposed and mathematically
proved. Based on this property, an algorithm for the computation of KL expansion is pro
posed. Next, the rate of decay of the eigenvalues in the KL expansion | which determines
the stochastic dimensionality of the problem | is mathematically shown to be dependent on
the domain size. That is, it is shown that as the size of the domain reduces the eigenvalues in
the KL expansion decay faster. Based on these mathematical results, next, a domain decomposition
(DD) based hybrid stochastic finite element formulation is proposed and its superior
numerical performance for a serial implementation is demonstrated. In this algorithm MCS is
used to solve the interface problem and SSFEM is used to solve the subdomain level problem,
in this sense this proposed method is a hybrid method. Here, the Finite Element Tearing and
Interconnecting (FETI) is used as the DD solver. Further, the same DD based hybrid algorithm
is parallelized and is used to solve a large three dimensional elasticity problem, with large
stochastic dimensionality. In these parallel numerical studies it was observed that, although
using the proposed hybrid method brings down the computational cost to a great extent, the
cost of solving the subdomain level problem using SSFEM turns out to be a large fraction of
the total cost, for problems with very large stochastic dimensionality. Hence, fi nally the system
of linear equations of SSFEM is reformulated as generalized Sylvester equation to achieve
computational cost saving in solving the subdomain level problems. | en_US |
