In this paper, we consider the problem of solving a large sparse linear system; after a brief introduction on existing direct, semi-iterative and iterative methods, the paper is focused on Krylov projection methods. It presents a unified framework of these methods and derives some properties from the projection condition. Then it expands these properties for the most well-known methods. Some numerical experiments illustrate convergence behavior. Finally, some practical issues such as numerical errors, preconditioning, and parallel processing are discussed.

This paper considers modelling strategies for the multiscale analysis of heterogeneous materials. Initial focus is on computational homogenization (e.g. [1-4]) which represents a powerful, albeit computationally intensive, technique for capturing the influence of an evolving microstrucuture on the overall macroscopic behaviour. First-order homogenization assumes that the representative volume element (RVE) is infinitely small compared to the macroscopic characteristic length scale and that there is clear separation of scales. Second-order schemes [2,4,5] are subsequently discussed as a possible solution. This approach introduces a material length scale into the macroscopic constitutive model and potentially enables geometric size effects to be captured. However, the size of the RVE now becomes an important parameter for the macroscopic response and there is no clear way of determining this length scale. Second-order computational homogenization is demonstrated in the context of an indentation problem. In situations where the RVE cannot be defined and there is no clear separation of scales, computational homogenization is not applicable and full resolution of the microstructure may be necessary. This paper summarises an efficient solution strategy for such situations following the work of Miehe and Bayreuther [6], with particular focus on fracturing heterogeneous materials, which utilises a geometric multi-grid preconditioner and the scale transition methodology from computational homogenization. This solution strategy is coupled with the use of the hybrid-Trefftz stress elements for modelling cohesive cracking [7]. The resulting modelling approach is demonstrated with the analysis of concrete dog-bone specimens where the different phases of matrix, aggregate and interfacial transition zone are explicitly modelled. Such an approach is still computationally intensive and so this paper also presents a novel hybrid multiscale approach that avoids the need for the finite element mesh to fully resolve all heterogeneities [8]. Classical micromechanics is utilised to enhance the approximation space in order to capture the effect of microstructural inclusions. In comparison to where the finite element mesh fully resolves the microstructure, this novel approach significantly reduces the number of degrees of freedom required while still accurately capturing the influence of the inclusions.