In the current work, a number of algorithms are developed and compared for the numerical solution of periodic (quasi-static) linear elastic mechanical boundary-value problems (BVPs) based on two different discretizations of Fourier series. The first is standard and based on the trapezoidal approximation of the Fourier mode integral, resulting in trapezoidal discretization (TD) of the truncated Fourier series. Less standard is the second discretization based on piecewise-constant approximation of the Fourier mode integrand, yielding a piecewise-constant discretization (PCD) of this series. Employing these, fixed-point algorithms are formulated via Green-function preconditioning (GFP) and finite-difference discretization (of differential operators; FDD). In particular, in the context of PCD, this includes an algorithm based on the so-called "discrete Green operator" (DGO) recently introduced by Eloh et al. (2019), which employs GFP, but not FDD. For computational comparisons, the (classic) benchmark case of a cubic inclusion embedded in a matrix (e.g., Suquet, 1997; Willot, 2015) is employed. Both discontinuous and smooth transitions in elastic stiffness at the matrix-inclusion (MI) interface are considered. In the context of both TD and PCD, a number of GFP- and FDD-based algorithms are developed. Among these, one based on so-called averaged-forward-backward-differencing (AFB) is shown to result in the greatest improvement in convergence rate. As it turns out, AFB is equivalent to the "rotated scheme" (R) of Willot (2015) in the context of TD. In the context of PCD, comparison of the performance and convergence behavior of AFB/R- and DGO-based algorithms shows that the former is more efficient than the latter for larger phase contrasts.

中文翻译：

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用于非均质材料准静态力学行为数值模拟的光谱算法的开发和比较

在当前的工作中，基于傅立叶级数的两种不同离散化，针对周期性（准静态）线性弹性机械边界值问题 (BVP) 的数值求解开发了多种算法并进行了比较。第一个是标准的，基于傅立叶模式积分的梯形近似，导致截断傅立叶级数的梯形离散化 (TD)。较不标准的是基于傅立叶模式被积函数的分段常数近似的第二个离散化，产生该系列的分段常数离散化 (PCD)。采用这些，定点算法是通过格林函数预处理 (GFP) 和有限差分离散化（微分算子；FDD）来制定的。特别是，在 PCD 的上下文中，这包括基于所谓的“