🤖 AI Summary
This work addresses the limitations of conventional FFT-based solvers in handling non-periodic boundary conditions, particularly their inability to simultaneously impose different types of boundary conditions on the same surface, prescribe displacements at interior points, or enforce kinematic constraints between nodes. By integrating discrete trigonometric transforms (DTTs) with the displacement method and incorporating both equality and inequality constraints via Lagrange multipliers, the authors develop an extended FFT framework capable of accommodating localized Dirichlet boundary conditions and contact problems. The proposed approach uniquely enables a unified treatment of mixed boundary conditions, internal displacement constraints, and inter-node kinematic relationships, and is further generalized to frictional contact scenarios. Numerical experiments demonstrate excellent agreement with Hertzian contact theory in simulating the interaction between a compact tension specimen and a rigid spherical indenter under small strains, thereby validating the method’s accuracy and versatility.
📝 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. However, this approach is not able to account for a mix of BCs on a same face (for example Dirichlet BC on a part of a face and Neumann BC on the other part of the same face), neither to prescribe Dirichlet BC to points defined inside the domain, nor to define kinematic relations between displacement on different nodes. Starting from the displacement-based approach together with the use of DTTs, simple modifications are proposed to account for all these questions. The modified solver, first introduced from the viewpoint of discrete local equations, is then discussed from the perspective of constrained minimization with equality constraints and the introduction of Lagrange multipliers. Simulations of a compact tension-like specimen are performed and validated. To go further, the algorithm is then extended from equality constraints to account for inequality constraints to simulate contact mechanics. For the sake of validation, results obtained with a rigid spherical indenter with frictionless contact are compared to the Hertz theory for small ratios (indentation depth/sphere radius). Comparison between small and finite strains demonstrates increasing discrepancies when increasing this ratio. It is believed that the proposed methodology will significantly expand the field of applications of FFT-based solvers.