一种用于预测有效热导率的快速余弦变换加速方法
A fast cosine transformation accelerated method for predicting effective thermal conductivity
AI总结：
本文提出一种基于快速余弦变换与GPU加速的高效数值方法，用于快速预测高分辨率RVE的有效热导率，相比纯CPU实现可获得5倍加速。
摘要：
通过求解定义在高分辨率代表性体积单元（RVE）上的偏微分方程（PDE）来预测有效热导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现的计算方法来应对这一任务，该方法能够充分利用硬件加速器（即图形处理单元（GPU））所提供的计算能力。我们首先采用两点通量近似格式对PDE进行离散化，然后利用预处理共轭梯度法求解所得的代数线性系统。预处理子的构造源于基于FFT的均匀化方法，并利用一种工程化的线性规划技术来确定均匀参考参数。本文提出的基本观察是，预处理系统可以通过多次快速余弦变换（FCT）和并行三对角矩阵求解器来有效求解。考虑到CUDA平台上默认不提供多次FCT，我们详细说明了如何从FFT推导出FCT，并实现近乎最优的内存使用。针对三维RVE进行了数值实验，包括与标准预处理子的稳定性比较。我们的性能报告表明，所提出的方法在GPU平台上相比纯CPU平台可以实现5倍的加速，并且在不到30秒的时间内求解具有512^3自由度和合理对比度的问题。
英文摘要：
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 $512^3$ degrees of freedom and reasonable contrast ratios in less than $30$ seconds.
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