Development of an efficient domain decomposition algorithm for solving large stochastic mechanics problems

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.