Algebraic Multigrid Based Preconditioning Approaches for Generalized Continuum Models and Indirect Displacement Control Techniques

The contribution deals with algebraic multigrid (AMG) based preconditioning methods for the iterative solution of a coupled linear system of equations arising in numerical simulations of failure of quasi‐brittle materials using generalized continuum approaches. In particular, the focus is on the solution of large and sparse linear systems originating from a gradient‐enhanced micropolar formulation with coupled fields of displacements, microrotations, and non‐local damage. Moreover, due to the possible presence of snap‐back behavior in quasi‐brittle materials, indirect displacement control techniques are discussed in the context of iterative linear solvers. The respective linear systems exhibit a distinct block structure representing the coupled fields of unknowns, which requires specialized preconditioners to iteratively solve the fully coupled linear system. Firstly, the present paper describes and investigates monolithic multigrid strategies for such problems which treat the block structure within the AMG hierarchy. An evaluation of the performance of the monolithic multigrid strategy for displacement controlled problems is carried out. This is done by a comparison to the performance of a previously published AMG based preconditioning strategy that applies the AMG to each field separately in a block preconditioning fashion. The results obtained for 2D plane strain and 3D triaxial compression indicate that this monolithic multigrid strategy performs similarly to the reference approach in terms of iteration counts in the majority of the investigated test cases. However, for distinct choices of material parameters, the respective strategy is shown to outperform the reference approach prior to localization and damage initiation. Secondly, for problems relying on indirect displacement control, a novel monolithic solution scheme is proposed that extends the existing block preconditioner to accommodate the additional constraint equation. We present a block preconditioner for the augmented system, for which we show in a 2D simulation of borehole failure that it significantly enhances computational efficiency by combining indirect displacement control and iterative linear solution techniques.