In this paper, an efficient and accurate Chebyshev expansion method is presented for solving large-scale transient heat conduction problems. Based on the Chebyshev expansion method, the matrix exponential is approximated with a series of Chebyshev matrix polynomials. Furthermore, according to the characteristics of practical thermal loads, an efficient method is developed to decrease the computational cost of temperature response induced by heat sources and nonhomogeneous boundary conditions. A theoretical method is developed to investigate the relationship of the computational cost of the proposed method and the time step, and the results indicate that under the given truncation criterion, the computational cost decreases with the increasing of the time step. Since the computational cost is sparse matrix–vector multiplications and only a few of vectors are stored in the computer memory, the proposed method has great advantages both in computational cost and storage requirement for large-scale transient heat conduction problems. In addition, a stability analysis is developed and the results show that the proposed method is unconditionally stable. Numerical examples exhibit that the proposed method has excellent efficiency and accuracy.

本文提出了一种有效且准确的切比雪夫膨胀方法来解决大规模瞬态热传导问题。基于Chebyshev展开方法，使用一系列Chebyshev矩阵多项式近似矩阵指数。此外，根据实际热负荷的特点，开发了一种有效的方法来减少由热源和非均匀边界条件引起的温度响应的计算成本。提出了一种理论方法研究该方法的计算量与时间步长的关系，结果表明，在给定的截断准则下，计算量随时间步长的增加而减小。由于计算量是矩阵-向量的稀疏乘积，并且只有很少的向量被存储在计算机存储器中，因此该方法在计算量和大规模瞬态热传导问题的存储需求上都具有很大的优势。此外，进行了稳定性分析，结果表明该方法是无条件稳定的。数值算例表明，该方法具有良好的效率和准确性。