AI 中文总结
本文针对基于FFT的求解器无法处理混合边界条件等问题，提出改进的约束极小化求解器，扩展至含不等式约束的接触力学模拟，验证了其有效性并拓展了应用场景。
AI 中文摘要
本文聚焦于克服基于快速傅里叶变换（FFT）的求解器局限性这一新兴活跃研究方向，旨在使其能应用各类边界条件（BCs），而非仅周期边界条件。近期已提出一种基于各类离散三角变换（DTTs）的完整框架，可结合离散傅里叶变换，以考虑单胞各面及位移或牵引力向量各分量定义的任意类型边界条件。Amouzou-Adoun等人近期的并行实现已证明其鲁棒性与通用性，但该方法无法处理同一面上混合边界条件（如同一面部分区域为狄利克雷边界条件、另一部分为诺伊曼边界条件），也无法对域内点施加狄利克雷边界条件，或定义不同节点位移间的运动学关系。本文从基于位移的方法及DTTs应用出发，提出简单修改方案以解决上述问题。该改进求解器先从离散局部方程视角引入，再从带等式约束的约束极小化及拉格朗日乘子引入的角度讨论。对紧凑拉伸类试样进行模拟并验证；为进一步拓展，将算法从等式约束扩展至考虑不等式约束，以模拟接触力学。为验证，将无摩擦接触的刚性球形压头所得结果与小压入深度/球径比下的赫兹理论对比，小应变与有限应变的对比显示该比值增大时差异显著增大。本文认为，所提方法将显著拓展基于FFT的求解器的应用领域。
英文摘要
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.
