Numerical difficulties often arise in the finite element simulation of quasi-brittle materials. Such problems are associated with material softening behaviour and the loss of positive definitiveness of the tangent stiffness matrix. These difficulties often manifest as breakdown of the nonlinear incremental-iterative solution process. Recently, Alnaas and Jefferson (2015) proposed an algorithm which circumvents these problems by employing a pseudo-tangent matrix that is based on a smoothed unloading-reloading function. This function has a small positive gradient at its intersection with the principal softening evolution function. A key feature of the proposed method is that it always uses a positive definite stiffness matrix. The authors have shown the approach to be numerically robust, reasonably efficient and accurate.

The present paper describes three acceleration techniques which further improve the convergence properties of this new smooth unloading-reloading (SUR) method. These techniques are designated ‘predictive-SUR', ‘fixing' and ‘slack tolerance'. In the ‘predictive-SUR' approach, a function is employed to predict a converged value of a damage evolution parameter based on an extrapolation in semi-log space. In the ‘fixing approach', a damage evolution parameter is updated from the last converged increment in Stage-1 iterations, and then fixed in Stage-2 iterations. The third approach uses of a slightly slacker convergence tolerance at key stages of a computation. The slacker tolerance (1% for the L2 norm of out of balance residual forces) is temporarily triggered when the number of iterations within an increment exceeds a certain limit. Subsequently, the convergence tolerance reverts to the standard tighter tolerance of 0.001%.

The paper presents a series of example computations based on the analysis of plain and reinforced quasi-brittle material structural elements. The results from these analyses are used to compare the performances of the new acceleration algorithms with that of the standard SUR algorithm. The principle finding of the work is that all three methods require less computer time than the standard SUR method, whilst having little effect on the accuracy of simulations.