Abstract

In mathematical models of microwave heating with infinite-dimensional characteristics, it is difficult to use traditional numerical methods to improve computational efficiency. In this work, we propose a fast and accurate method to calculate the temperature distribution of materials under microwave heating. First, we analysed the relationship between the choice of model order and the solution accuracy by downscaling the infinite-dimensional heat conduction partial differential equation (PDE) model into a finite-dimensional ordinary differential equation (ODE) model. Additionally, the effect of different boundary conditions on the global temperature distribution was analysed. Second, the equilibrium conversion matrix was calculated using the singular value decomposition (SVD) truncation method under homogeneous boundary conditions. Using this matrix, a lower-dimensional microwave heating ODE model was further obtained. Finally, the numerical simulation results showed that the root mean square error (RMSE) was only 0.07 and the maximum relative error was only −0.85%. The computation time of the equilibrium conversion matrix was 2.12 ∼ 3.00 ms, and the model calculation time was reduced by 97.78%. We compared the calculated temperature rise curves with those obtained using the conventional COMSOL model. The SVD truncation method achieved an efficient and accurate solution for the microwave heating model.

摘要

在具有无限维特征的微波加热数学模型中，传统的数值方法难以提高计算效率。在这项工作中，我们提出了一种快速准确的方法来计算微波加热下材料的温度分布。首先，我们通过将无限维热传导偏微分方程 (PDE) 模型降尺度为有限维常微分方程 (ODE) 模型，分析了模型阶数的选择与求解精度之间的关系。此外，还分析了不同边界条件对全球温度分布的影响。其次，在齐次边界条件下使用奇异值分解 (SVD) 截断法计算平衡转换矩阵。使用该矩阵，进一步获得了低维微波加热 ODE 模型。最后，数值模拟结果表明，均方根误差（RMSE )仅为0.07，最大相对误差仅为-0.85%。平衡转换矩阵计算时间为2.12∼3.00 ms，模型计算时间减少97.78%。我们将计算出的温升曲线与使用传统 COMSOL 模型获得的温升曲线进行了比较。SVD 截断方法实现了微波加热模型的高效准确求解。