A constrained LArge Time INcrement algorithm for quasi-brittle fracture: Implementation and optimisation aspects

Abstract

The use of a robust and efficient solution algorithm is essential when modelling the structural response of quasi-brittle materials such as concrete and masonry. Due to the softening behaviour of these materials, the algorithm must cope with highly non-linear phenomena such as snap-backs and bifurcations. Three main algorithm classes can be identified in literature: incremental-iterative solution algorithms (e.g. Newton-Raphson methods), non-iterative solution algorithms (e.g. sequentially-linear approaches) and non-incremental solution algorithms (e.g. LArge Time INcrement or LATIN algorithms). In this contribution, a novel non-incremental LATIN-based solution algorithm is presented. A key feature of LATIN algorithms is that the whole (pseudo-) time domain is iteratively calculated in one single time increment. The algorithm starts with the linear elastic solution of the entire loading regime and evolves towards a solution which satisfies, on the one hand, internal and external equilibrium conditions and, on the other hand, the non-linear constitutive laws. Depending on the problem, the algorithm may give an indication of the load-carrying capacity of a structure already in the very first iterations.

Each iteration of the algorithm consists of a local and a global solution stage, in which the strain and stress fields are calculated for the complete space and time domains. In the local solution stage, strains and stresses verify the non-linear constitutive laws as well as a local search equation which searches for a new solution using the fields of the previous iteration. In the global solution stage, stress-strain couples are calculated which satisfy both structural equilibrium and a global search equation, which searches for an improved solution using the stresses and strains of the local solution stage. In our algorithm, a constraint function is added to the global stage. In contrast with classical LATIN algorithms, our algorithm is therefore capable of tracing snap-back responses in quasi-brittle materials. In the present contribution, special attention will be given to the choice of this constraint equation and implementational aspects of the algorithm. The aforementioned search equations are characterised by so-called search directions, which influence the robustness and efficiency of the algorithm. An optimal choice of these search directions result in a higher convergence rate. On the other hand, some search directions may lead to non-convergence of the algorithm. In our paper, we will therefore investigate the impact of these search directions on the algorithm efficiency and robustness. Several numerical examples will be considered, ranging from a simple one-dimensional truss to a complex mesoscopic masonry model. These examples include non-proportional loading, bifurcations and snap-back behaviour in their loading responses.