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 both linear and nonlinear problems. From implementation viewpoint, the proposed methods have two advantages: (i) existing mechanics solvers can be re-used with minimal modification, and (ii) it is independent of the probability distribution of the input random variables. Numerical studies suggest a significant improvement in speed-up compared to the current literature and good parallel performance for the linear case. For the nonlinear case, the proposed method is numerically tested for p-Laplace and plain-strain plasticity problems, where it is found to be computationally efficient, accurate, and exhibit good scalability. It is commonly observed that nonlinear phenomena in large-scale structures often occur in localized zones while the remaining parts remain linear. The traditional approach to solve this problem is to utilize a variant of Newton's method, and update the Jacobian at every Newton iteration. Therefore, the computational implementation does not utilize the localized spread of nonlinearity resulting in enormous cost. In this thesis, two closely related methods to solve large-scale problems with localized nonlinearity are introduced. In both of these methods, the FETI-DP domain decomposition is employed at the Jacobian level to solve the linearized problem. In the first method, the Jacobian matrix is computed only for certain elements around the nonlinearity whereas in the second method the initial Jacobian is re-used for all Newton iterations. The new methods can significantly reduce the computational time in parallel platforms when compared with the traditional method. Detailed numerical studies are conducted for two elasto-plastic problems under plane strain conditions --- a plate with a hole and a v-notch, respectively. Finally, the proposed method is extended to solve localized nonlinear problems in the presence of inherent uncertainties. The proposed method is found to be accurate and scalable when implemented on a stochastic elasto-plastic problem.