  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.
