基于空间基函数优化组合的微波加热温度分布数值求解

针对微波加热是一个多物理场各自演变及相互耦合的过程,无法直接求得媒质温度分布偏微分方程(PDE)解析解的问题,本文提出一种快速及准确求解微波加热温度分布的新方法.首先,本文在无限维PDE降维到有限维ODE温度模型的基础上,分析了ODE温度模型阶次选择与温度分布求解精度量化关系.其次,通过自适应变异粒子群算法(AMPSO)优化误差函数近似得到最优的空间基函数转换矩阵,利用该矩阵将空间基函数进行线性优化组合,进一步降低ODE模型阶数,进而使得在一定误差范围内可以更快速的求解微波加热过程中媒质的温度分布.再次,通过数值仿真实验证明,温度分布求解误差主要产生于模型阶次的选择,且优化后的低维ODE模型的温度分布精度相对误差控制在1.64%以内,求解速度提升72.2%.最后,使用多物理场耦合有限元方法求解微波加热PDE温度模型,进一步验证了优化后的低维ODE温度模型的准确性,充分验证了本文方法的有效性.

Numerical solution of microwave heating temperature distribution based on optimal combination of spatial basis functions

A new method is proposed for fast and accurately solving temperature distribution in this paper, aiming at theproblem that microwave heating is a process of the evolution and mutual coupling of multiple physical fields and cannot todirectly obtain the analytical solution of the PDE of medium temperature distribution. Firstly, we analyzed the quantitativerelationship between the order selection of ODE temperature model and the solution accuracy of temperature distributionbased on the dimensionality reduction of infinite dimensional PDE to finite dimensional ODE model. Secondly, the approximateoptimal spatial basis function conversion matrix is obtained using optimizing the error function by AMPSO. Thespatial basis functions are linearly optimized and combined by using this matrix to further reduce the order of ODE model.Furthermore, the temperature distribution of materials during microwave heating process can be solved more quickly withina certain error range. Thirdly, the simulation experiment shows that the solution error of temperature distribution mainlycomes from the selection of model order, and the relative error of the temperature distribution accuracy of the optimizedlow dimensional ODE model is controlled within 1.64%, and the solution speed can be improved by 72.2%. Finally, thefinite element method is used to solve the microwave heating PDE temperature model, which further verifies the accuracyof the optimized low dimensional ODE temperature model and fully verifies the effectiveness of this method.