A transient multipole method for short-term simulations of ground heat exchangers (GHEs) is presented. The two-dimensional unsteady heat equation over a GHE cross section is separated into two problems: (i) a transient heat equation with homogeneous boundary conditions, and (ii) a steady-state heat equation with nonhomogeneous boundary conditions. An eigenfunction expansion is proposed for the solution of the transient heat equation, where the treatment of boundary conditions is considered by a multipole expansion of the eigenfunctions. A singular value decomposition is applied to extract the eigenvalues of the problem. The solution of the steady-state heat equation is obtained from a multipole expansion. The proposed method is validated against reference results for the evaluation of the eigenvalues and for the steady-state temperature field. The complete transient solution is validated against finite-element analysis simulations. The proposed method is meshless, and its accuracy is only dependent on the evaluation of eigenvalues and on the number of terms in each of the multipole expansions.

提出了一种用于地面换热器（GHE）短期仿真的瞬态多极方法。GHE横截面上的二维非稳态热方程分为两个问题：（i）具有均匀边界条件的瞬态热方程，和（ii）具有非均匀边界条件的稳态热方程。针对瞬态热方程的解提出了本征函数展开式，其中边界条件的处理通过本征函数的多极展开来考虑。应用奇异值分解来提取问题的特征值。稳态热方程的解是从多极膨胀得到的。对照参考结果验证了该方法的有效性，以评估特征值和稳态温度场。完整的瞬态解决方案已通过有限元分析仿真进行了验证。所提出的方法是无网格的，其准确性仅取决于特征值的估计以及每个多极展开中项的数量。