Predicting effective thermal conductivity by solving a Partial Differential Equation (PDE) defined on a high-resolution Representative Volume Element (RVE) is a computationally intensive task. In this paper, we tackle the task by proposing an efficient and implementation-friendly computational method that can fully leverage the computing power offered by hardware accelerators, namely, graphical processing units (GPUs). We first employ the Two-Point Flux-Approximation scheme to discretize the PDE and then utilize the preconditioned conjugate gradient method to solve the resulting algebraic linear system. The construction of the preconditioner originates from FFT-based homogenization methods, and an engineered linear programming technique is utilized to determine the homogeneous reference parameters. The fundamental observation presented in this paper is that the preconditioner system can be effectively solved using multiple Fast Cosine Transformations (FCT) and parallel tridiagonal matrix solvers. Regarding the fact that default multiple FCTs are unavailable on the CUDA platform, we detail how to derive FCTs from FFTs with nearly optimal memory usage. Numerical experiments including the stability comparison with standard preconditioners are conducted for 3D RVEs. Our performance reports indicate that the proposed method can achieve a 5-fold acceleration on the GPU platform over the pure CPU platform and solve the problems with degrees of freedom and reasonable contrast ratios in less than 30 s.

中文翻译：

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预测有效导热系数的快速余弦变换加速方法


通过求解在高分辨率代表性体积元 (RVE) 上定义的偏微分方程 (PDE) 来预测有效导热率是一项计算密集型任务。在本文中，我们通过提出一种高效且易于实施的计算方法来解决该任务，该方法可以充分利用硬件加速器（即图形处理单元（GPU））提供的计算能力。我们首先采用两点通量近似方案对偏微分方程进行离散化，然后利用预条件共轭梯度法来求解所得代数线性系统。预处理器的构造源自基于 FFT 的均质化方法，并利用工程线性编程技术来确定齐次参考参数。本文提出的基本观察是，可以使用多个快速余弦变换（FCT）和并行三对角矩阵求解器有效地求解预处理器系统。鉴于 CUDA 平台上默认多个 FCT 不可用的事实，我们详细介绍了如何从具有接近最佳内存使用率的 FFT 导出 FCT。针对 3D RVE 进行了数值实验，包括与标准预处理器的稳定性比较。我们的性能报告表明，所提出的方法可以在 GPU 平台上实现比纯 CPU 平台 5 倍的加速，并在不到 30 秒的时间内解决自由度和合理对比度的问题。