Constrained minimization problems and FFT-based solvers: application to local Dirichlet boundary conditions and contact mechanics

Abstract

The present paper focuses on a recent and active research axis to overcome the limitations of FFT-based solvers in order to apply various types of boundary conditions (BCs) and not only periodic BCs. A complete framework based on discrete trigonometric transforms (DTTs) of various types, possibly combined with discrete Fourier transform, has been proposed recently to account for any type of BCs defined per face of the unit-cell and per component of the displacement or traction vector. A parallel implementation by Amouzou-Adoun et al. recently proved its robustness and versatility. The main novelty consists of few modifications introduced in the displacement-based FFT/DTT iterations in order to account for Dirichlet BC on a selected part of a boundary face (and Neumann BC on the other part), or on interior points, and also to apply kinematic relations between displacements on different nodes. Introduced from the viewpoint of local equations, these modifications are then discussed from the perspective of constrained minimization and saddle point systems. The current proposition is identified as a 'null-space' method in the category of 'segregated' methods (see Benzi et al., 2005): the reaction forces (acting locally to enforce the Dirichlet BC) are evaluated a posteriori once the displacement field is found. This approach is original in the context of FFT-based solver's extension to Dirichlet BCs. Starting from this modified FFT/DTT-based algorithm to account for equality constraints, the second important novelty consists in embedding this algorithm in a conventional active-set strategy to account for inequality constraints, with an application in the context of contact mechanics. Two implementations have been proposed, in the context of small strain and finite strains. It is believed that these propositions significantly expand the field of applications of FFT-based solvers.