瞬态热传导问题的精细积分边界元法

对于三维瞬态热传导问题,在考虑内部热源的情况下,采用双重互易边界元法(DRBEM)结合精细积分法(PIM)进行求解。该方法根据含有内部热源的各向同性介质瞬态常系数热传导问题的控制方程,通过加权余量法推导出相应的边界积分方程,然后用双互易法(DRM)处理得到的边界积分方程,将热源项和温度关于时间导数项引起的域积分通过径向基函数(RBF)逼近后转化为边界积分。之后将边界积分方程离散,得到与时间相关的一阶常系数微分方程组,最后,在获得解析解的过程中,通过PIM处理其中的矩阵指数函数(MEF)。通过三个数值算例来验证该方法的准确性和稳定性。

Precise Integration BEM for Transient Heat Conduction Problems

For the three-dimensional transient heat conduction problem, considering the internal heat source. The dual reciprocity boundary element method (DRBEM) combined with the precise integration method (PIM) is used to solve the problem. According to the governing equation of transient constant coefficient heat conduction problem in isotropic medium with internal heat source, the corresponding boundary integral equation is derived by the weighted residual method. The dual reciprocity method (DRM) is applied to deal with the obtained boundary integral equation. The domain integral caused by the heat source term and the time derivative term of temperature is transformed into the boundary integral after being approximated by radial basis function (RBF). Then, the boundary integral equation is discretized to obtain the first-order constant coefficient differential equations related to time. Finally, in the process of obtaining the analytical solution, the matrix exponential function (MEF) is processed by PIM. Three numerical examples are given to verify the accuracy and stability of the method.