In finite element analysis, nonlinear time-history analysis is a realistic and accurate analysis type for dynamic or seismic analysis due to its solutions contain wealthy data and complete response time-history. The most commonly used method, probably the only practical procedure, in nonlinear time-history analysis is the direct time integration method. In general, every direct time integration method could be classified as either an implicit method or an explicit method. Understanding the differences between the two categories in both theoretical and practical aspects is very important for engineers to make the best analysis strategy for a specific dynamic or seismic analysis. In this treatise, the most popular method in each category, i.e., implicit Newmark method and explicit central difference method, will be introduced and used in transient analyses and results comparisons. In total, five cases studies are included in this thesis, including three cases with linear elastic materials and two cases with quasi-brittle masonry material. These five cases are studied to answer the main research questions of this research: What differences can be observed in comparisons of solutions obtained from implicit and explicit methods for linear elastic material in transient analysis and for quasi-brittle material under seismic load? Also, how are the performances of both methods with respect to the stability and accuracy aspects? The results comparisons show that for linear elastic material, both methods could generate accurate solutions with proper time steps. However, the explicit method shows a better representation of the high-frequency response of the structure. For quasi-brittle masonry material, both methods are possible to generate good results. However, the implicit method may have convergence problems during the iterations, which could lead to inaccurate predictions of nonlinear behavior of the structure. The explicit method shows very good predictions for nonlinear behavior, but the computation time is quite long and the critical time step is highly influenced by many factors in the finite element model. Moreover, the mass scaling technique, which is generally used in the explicit method, is also investigated in this thesis.

Throughout the years, incremental iterative approaches have been shown to be excellent tools in describing the complex behaviour of structures under a wide range of circumstances. However, robustness issues arise for quasi-brittle structures due to the potential lost of convergence during the development of abrupt fracture mechanisms. In order to overcome these robustness issues, the framework of sequentially linear analysis (SLA) has been proposed: an event-by-event strategy in which a sequence of scaled linear analyses with decreasing secant stiffness is performed, representing local damage increments. The current SLA-framework is based on a fixed crack approach, potentially causing the development of severe spurious stresses and inaccuracies due to the misalignment of the crack with the principal stress directions. To this end, Hendriks and Rots proposed a model consisting of several parallel fractions or layers, from now on called the sublayer model. Each of the layers is elastic-perfectly brittle, but has different properties, chosen such to represent the overall constitutive softening behaviour as accurate as possible. The layers fail independent of each other and have their own specific crack direction. The main idea is to mimick a rotating crack by a superposition of sublayers with a fixed crack direction. The main goal of this thesis is to further elaborate, generalize and verify the sublayer model for quasi-brittle materials and capture the influence of rotating cracks on the structural response within the framework of existing regular sequentially linear analysis. In this thesis, the frameworks of regular SLA and the sublayer model were connected by a general transition from any saw-tooth law to sublayer material properties. An externalized procedure was created to automatically generate an input file for DIANA FEA and thereby facilitate verification of the sublayer model. Furthermore, the 2-dimensional framework of the sublayer model has been extended towards 3-dimensional structures to broaden the range of application. On top of that, concepts were proposed to improve the sublayer model: the tapered ripple band, reducing the required number of sublayers to reach a specific state by adding more saw-teeth near the end of the softening curve, and an improved algorithm, making use of the fact that the order of brittle fracture of the sublayers is known in advance, such that only those integration points that can actually become critical are monitored, thereby reducing computational efforts significantly. In this thesis, the sublayer model is proved to mimick a rotating crack within the framework of existing regular sequentially linear analysis based on a set of structural case studies (notched beam, shear notched beam, DEN-beam, full scale facade and full scale concrete dam). Compared to regular SLA, effects of stress locking are reduced, less wide localization bands are found and a more realistic collapse pattern is observed, thereby making the sublayer model more interesting for application in engineering practice. Furthermore, it has been shown that for the 3-dimensional framework of the sublayer model the same conclusions as for 2D can be made. In the authors opinion, the contributions of this thesis are a step towards a robust generally applicable computational method to simulate the complex structural behaviour of quasibrittle materials.

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.