The numerical simulation of brittle failure with nonlinear finite element analysis (NLFEA) remains a challenge due to robustness issues. These problems are attributed to the softening material behaviour and the iterative nature of the Newton-Raphson type methods used in NLFEA. However, robust numerical simulations become increasingly important, for example due to recent developments in Groningen.  To address these issues, sequentially linear analysis (SLA) was developed which exploits the fact that a linear analysis is inherently stable. By assuming a stepwise material degradation the nonlinear response of a structure can be approximated with a sequence of linear analyses. Although this approach has been proven to be effective for several case studies, the numerical performance is still a problem that has to be solved. After every linear analysis, a single element is damaged resulting in incremental damage. As a result, the system of equations only changes locally between these linear analyses. Traditional solution techniques do not exploit this property and calculate a matrix factorisation every linear analysis, resulting in high computational times per analysis step. Since SLA typically requires many linear analyses to obtain the desired structural response, this leads to unacceptable analysis times. The aim of this thesis is to improve the computational performance of SLA by developing numerical solution techniques which exploit the incremental approach of SLA. To this extend, the following methods have been developed. A direct solution technique has been developed which is based on the Woodbury matrix identity. This identity allows for the numerically cheap computation of the inverse of a low-rank corrected matrix. In this approach, the expensive matrix factorisation does not have to be calculated every linear analysis step. Instead, the old factorisation can be reused along with some additional matrix- and vector multiplications and solving a significantly smaller linear system of equations. An optimal strategy is derived to determine at which point a new factorisation should be calculated. An improved preconditioner for the conjugate gradient (CG) method has been developed. Instead of an incomplete factorisation, the complete factorisation is used as a preconditioner which reduces the number of required CG iterations significantly. The point at which too many CG iterations are required and a new factorisation is necessary, is determined using the same strategy as the first method. From numerical experiments it follows that both methods perform significantly better than the direct solution method, especially for large 3-dimensional problems. The best performance is achieved using the Woodbury matrix identity resulting in the solver no longer being the dominant factor in SLA. Furthermore, significantly larger problems are not solvable in time frames in which previously only smaller problems were solved.

Sequentially linear analysis (SLA), an event-by-event procedure for finite element (FE) simulation of quasi-brittle materials, is based on sequentially identifying a critical integration point in the FE model, to reduce its strength and stiffness, and the corresponding critical load multiplier (λ crit), to scale the linear analysis results. In this article, two strategies are proposed to efficiently reuse previous stiffness matrix factorisations and their corresponding solutions in subsequent linear analyses, since the global system of linear equations representing the FE model changes only locally. The first is based on a direct solution method in combination with the Woodbury matrix identity, to compute the inverse of a low-rank corrected stiffness matrix relatively cheaply. The second is a variation of the traditional incomplete LU preconditioned conjugate gradient method, wherein the preconditioner is the complete factorisation of a previous analysis step's stiffness matrix. For both the approaches, optimal points at which the factorisation is recomputed are determined such that the total analysis time is minimised. Comparison and validation against a traditional parallel direct sparse solver, with regard to a two-dimensional (2D) and three-dimensional (3D) benchmark study, illustrates the improved performance of the Woodbury-based direct solver over its counterparts, especially for large 3D problems. @en