Parallel methods to solve large scale stochastic linear and nonlinear mechanics problems in a domain decomposition framework

Abstract

Parallel methods to solve large-scale stochastic linear and nonlinear mechanics problems in a domain decomposition framework Mechanics problems with inherent uncertainties are mathematically modeled using stochastic partial differential equations (sPDE). Numerical solution of these sPDE-s becomes prohibitive as the dimension of the problem --- characterized by mesh resolution and number of random variables --- grows. In this work, a domain decomposition based methodology is proposed to solve such large scale sPDE-s for both linear and nonlinear mechanics problems. The methods are built around stochastic collocation, thereby allowing reuse of existing codes. They are demonstrated to be accurate, faster than the state-of-the-art, and scalable on a parallel computer. Recently, domain decomposition (DD) methods have been successful in reducing the computational complexity and achieving parallelization for linear elliptic sPDE-s. This improvement is due to faster convergence of Karhunen-Loeve expansion in smaller domains and inherent parallelizability of DD methods. However, in order to make this approach more suitable for widespread applications in high performance computing framework, and to re-use existing finite element solvers, departure from the stochastic Galerkin method is necessary. To address this issue, in this thesis a stochastic collocation based formulation is proposed in the finite element tearing and interconnecting - dual primal (FETI-DP) framework. Using this formulation a set of methods are proposed for linear and nonlinear sPDE-s. For linear problems, the non-intrusive formulation uses collocation at both subdomain and interface levels. However, for nonlinear problems a deterministic nonlinear problem is solved using the Newton-Raphson method at each collocation point. The FETI-DP method is then invoked at the Jacobian level for the solution of the linearized system. Finally, at the post processing stage, realizations of subdomain solutions are computed by sampling from the true distribution for...