We developed an implicit stress return algorithm for nonlocal models involving implicit evolution rules of nonlocal quantities (e.g. damage energy). This involves solving two sets of linear equations: equilibrium equation at the structural scale at every increment, and nonlocal interaction equation at the material scales at every iteration. The sparsity of the 2nd set of equations over the course of failure process is exploited. Efficiency vs. a semi-implicit algorithm is demonstrated.
Load-Deformation curves with points corresponding to the sparsity of the 2nd set of equations
Sparsity and size at different stages of failure
Efficiency of the algorithm, against a semi-implicit one