The goal of the thesis is to investigate iterative solvers and develop methods that can accelerate sampling-based sensitivity studies, e.g. Monte Carlo simulations. The direct solvers that are currently frequently used do not easily allow for a reuse of information between the different but similar realizations in a sample. The presented research has mainly been carried out in the framework of nonlinear finite element analysis with shell elements. In order to evaluate the proposed methods, PETSc was used. The methods have been implemented in an in-house finite element code developed together with an industrial partner using PETSc. This enables the users of this finite element code to evaluate the methods on industrial finite element models in the future.
The usage of a preconditioner is crucial for the fast convergence of an iterative solver. Therefore, the H-Cholesky factorization and Balancing Domain Decomposition by Constraints (BDDC) preconditioners are investigated. The efficiency of both preconditioners is analyzed on a number of finite element models in elasticity, including plate and shell problems. In order to accelerate the convergence of consecutive linear systems in the Newton-Raphson iteration, the augmented preconditioned conjugate gradient method is tested and improved numerically. Both the numerical stability and efficiency of the augmented preconditioned conjugate gradient method are investigated.
Finally, the reuse of the Cholesky factorization to accelerate sampling-based sensitivity studies is investigated. The Newton-Raphson iteration is accelerated by using the solution of a previous realization as an initial guess in the Newton-Raphson iteration of a new realization in the sample. Improving the initial guess in the Newton-Raphson iteration is crucial for good performance in a Monte Carlo simulation. It is also investigated how the iterative solver behaves close to structural instabilities such as buckling.