基于特征正交分解降阶模型的瞬态热传导分析

Reduced order model analysis method via proper orthogonal decomposition for transient heat conduction

提出了一种用常数边界条件建立的瞬态热传导问题的特征正交分解(POD)降阶模态,对时变边界条件进行瞬态热传导降阶分析的方法,可对未知时刻的温度场进行内插和外插预测分析.首先,对常数边界瞬态热传导问题,在实验、经验或有限单元法等数值方法结果的基础上,取一些时刻的解构成瞬像矩阵,并对该矩阵进行特征正交分解以求得POD模态;其次,通过特征值误差分析,确定降阶分析模态,并用其对随时间变化的边界条件进行瞬态热传导计算分析.本文的特点是,当边界条件时变时,不需要重新计算POD模态,而且可以用很少的模态捕捉全阶模型高达99%的能量.实例表明,本文所述方法正确有效,同样的POD模态能准确预测和拟合求解域不变但边界条件以各种不同方式光滑时变时的瞬态传热问题,在气动热力学等需实时控制或快速计算的问题中具有很好的发展前景.

This paper presents a reduced order analysis method for transient heat conduction problems with time-varying boundary conditions by using the proper orthogonal decomposition （POD） modes obtained from the results of using constant boundary conditions. This method can perform interpolation and extrapolation analysis for temperature field at unknown times. First, POD modes are developed by calculating eigenvectors of an autocorrelation matrix composed of snapshots which are clustered by the given results obtained from experiments, finite element method （FEM） or other numerical methods for transient heat transfer problem with constant boundary conditions. Then, the reduced order modeling （ROM） for problems with time-varying boundary condition is obtained by projecting the finite element discrete format on reduced POD modes determined by eigen-value error analysis of the original POD modes. One feature is that the POD modes need not to be reformed when the boundary conditions are time-varying, and one can use a few modes to capture as much as 99% of energy of the whole order model. Examples show that the method developed in this paper is correct and effective. The same POD modes can accurately analyze transient heat conduction problems with the same geometric domain but variety of smooth and time-varying boundary conditions. The method has a good prospect in aero-thermodynamics analysis that needs real-time control or fast calculation.